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<h1 >GAP - Reference Manual</h1 >
<p>Release 4 .15 .1 , 2025 -10 -18 </p>
</div >
<p><b>The GAP Group
</b>
<br />Email: <span class="URL" ><a href="mailto:support@gap-system.org" >support@gap-system.org</a></span >
<br />Homepage: <span class="URL" ><a href="https://www.gap-system.org " >https://www.gap-system.org</a></span >
</p>
<p><a id="X81488B807F2A1CF1" name="X81488B807F2A1CF1" ></a></p>
<h3>Copyright</h3>
<p>Copyright © (1987 -2025 ) for the core part of the <strong class="pkg" >GAP</strong > system by the <strong class="pkg" >GAP</strong > Group.</p>
<p>Most parts of this distribution, including the core part of the <strong class="pkg" >GAP</strong > system are distributed under the terms of the GNU General Public License Version 2 , see <span class="URL" ><a href="https://www.gnu.org/licenses/old-licenses/gpl-2.0.html " >https://www.gnu.org/licenses/old-licenses/gpl-2 .0 .html </a></span > or the <code class="file" >LICENSE</code > file in the root directory of the <strong class="pkg" >GAP</strong > installation.</p>
<p>More detailed information about copyright and licenses of parts of this distribution can be found in Section <a href="chap1_mj.html#X7950EFA183E3F666" ><span class="RefLink" >1 .4 </span ></a> of this manual.</p>
<p><strong class="pkg" >GAP</strong > has been developed over a long time and has many authors and contributors. More detailed information can be found in Section <a href="chap1_mj.html#X877A62A1781C2147" ><span class="RefLink" >1 .2 </span ></a> of this manual.</p>
<p><a id="X8537FEB07AF2BEC8" name="X8537FEB07AF2BEC8" ></a></p>
<div class="contents" >
<h3>Contents<a id="contents" name="contents" ></a></h3>
<div class="ContChap" ><a href="chap1_mj.html#X874E1D45845007FE" >1 <span class="Heading" >Preface</span ></a>
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap1_mj.html#X863F306C7D32F4B0" >1 .1 <span class="Heading" >The <strong class="pkg" >GAP</strong > System</span ></a>
</span >
</div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap1_mj.html#X877A62A1781C2147" >1 .2 <span class="Heading" >Authors and Maintainers</span ></a>
</span >
</div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap1_mj.html#X82A988D47DFAFCFA" >1 .3 <span class="Heading" >Acknowledgements</span ></a>
</span >
</div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap1_mj.html#X7950EFA183E3F666" >1 .4 <span class="Heading" >Copyright and License</span ></a>
</span >
</div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap1_mj.html#X7BF552C07E2F8F7C" >1 .5 <span class="Heading" >Further Information about <strong class="pkg" >GAP</strong ></span ></a>
</span >
</div >
</div >
<div class="ContChap" ><a href="chap2_mj.html#X8755A2C67B197C63" >2 <span class="Heading" >The Help System</span ></a>
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap2_mj.html#X7E2C53D2844DD8C3" >2 .1 <span class="Heading" >Invoking the Help</span ></a>
</span >
</div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap2_mj.html#X7BE8068878B7D7D1" >2 .2 <span class="Heading" >Browsing through the Sections</span ></a>
</span >
</div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap2_mj.html#X863FF9087EDA8DF9" >2 .3 <span class="Heading" >Changing the Help Viewer</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2_mj.html#X87C1BFB2826488B0" >2 .3 -1 SetHelpViewer</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap2_mj.html#X84AFFC817B282359" >2 .4 <span class="Heading" >The Pager Command </span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2_mj.html#X7ED03E41792C3840" >2 .4 -1 Pager</a></span >
</div ></div >
</div >
<div class="ContChap" ><a href="chap3_mj.html#X79CCD3A6821E5A37" >3 <span class="Heading" >Running GAP</span ></a>
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap3_mj.html#X782751D5858A6EAF" >3 .1 <span class="Heading" >Command Line Options</span ></a>
</span >
</div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap3_mj.html#X7FD66F977A3B02DF" >3 .2 <span class="Heading" >The gap.ini and gaprc files</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X87DF11C885E73583" >3 .2 -1 <span class="Heading" >The gap.ini file</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X84D4CF587D437C00" >3 .2 -2 <span class="Heading" >The gaprc file</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X7B0AD104839B6C3C" >3 .2 -3 <span class="Heading" >Configuring User preferences</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X7F1DF6757B248014" >3 .2 -4 DeclareUserPreference</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X870A11E7864F9CA7" >3 .2 -5 <span class="Heading" >User Preferences Defined by <strong class="pkg" >GAP</strong ></span ></a>
</span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap3_mj.html#X7CB282757ACB1C09" >3 .3 <span class="Heading" >Saving and Loading a Workspace</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X876544A57C73C488" >3 .3 -1 SaveWorkspace</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap3_mj.html#X83BF07587F2CC6CD" >3 .4 <span class="Heading" >Testing for the System Architecture</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X7C825AF087A27884" >3 .4 -1 ARCH_IS_UNIX</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X82A6893A7EC8FA72" >3 .4 -2 ARCH_IS_MAC_OS_X</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X7A14B659847B8627" >3 .4 -3 ARCH_IS_WINDOWS</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X87E7CC3B8395BBB3" >3 .4 -4 ARCH_IS_WSL</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap3_mj.html#X8719B2118511645F" >3 .5 <span class="Heading" >Global Values that Control the <strong class="pkg" >GAP</strong > Session</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X8354754E7935F935" >3 .5 -1 GAPInfo</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap3_mj.html#X818F2DDC863C381E" >3 .6 <span class="Heading" >Coloring the Prompt and Input </span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3_mj.html#X84F3481C8466C7FC" >3 .6 -1 ColorPrompt</a></span >
</div ></div >
</div >
<div class="ContChap" ><a href="chap4_mj.html#X7FE7C0C17E1ED118" >4 <span class="Heading" >The Programming Language</span ></a>
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap4_mj.html#X7B5FF6827DFBDF20" >4 .1 <span class="Heading" >Language Overview</span ></a>
</span >
</div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap4_mj.html#X80A85A707B6F4BE7" >4 .2 <span class="Heading" >Lexical Structure</span ></a>
</span >
</div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap4_mj.html#X7E90E6607F4E4943" >4 .3 <span class="Heading" >Symbols</span ></a>
</span >
</div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap4_mj.html#X7C53CEFC8641B919" >4 .4 <span class="Heading" >Whitespaces</span ></a>
</span >
</div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap4_mj.html#X87506BDC7D5F789E" >4 .5 <span class="Heading" >Keywords</span ></a>
</span >
</div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap4_mj.html#X860313A179A5163F" >4 .6 <span class="Heading" >Identifiers</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X85CF993B7D19F2C4" >4 .6 -1 IsValidIdentifier</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X839A7F8E84BBCA57" >4 .6 -2 <span class="Heading" >Conventions about Identifiers</span ></a>
</span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap4_mj.html#X7BAFE9C1817253C6" >4 .7 <span class="Heading" >Expressions</span ></a>
</span >
</div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap4_mj.html#X7A4C2D0E7E286B4F" >4 .8 <span class="Heading" >Variables</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X842B89D4860FD5DB" >4 .8 -1 IsBound</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X7BABB3E77F52626C" >4 .8 -2 Unbind</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap4_mj.html#X816FBEEA85782EC2" >4 .9 <span class="Heading" >More About Global Variables</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X7CD3523B84744EB2" >4 .9 -1 IsReadOnlyGlobal</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X850CE44478254F27" >4 .9 -2 MakeReadOnlyGlobal</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X832AAF13861968BE" >4 .9 -3 MakeReadWriteGlobal</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X847706237E72418F" >4 .9 -4 MakeConstantGlobal</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X84BB4B1E872849FF" >4 .9 -5 ValueGlobal</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X823D4BC378395B32" >4 .9 -6 IsBoundGlobal</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X829A5F0E811F77D3" >4 .9 -7 UnbindGlobal</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X7D39D3E17CF49F5B" >4 .9 -8 BindGlobal</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X876A6EB68745A510" >4 .9 -9 NamesGVars</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X7E604AF579A7BC92" >4 .9 -10 NamesSystemGVars</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X870169447AF490D8" >4 .9 -11 NamesUserGVars</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap4_mj.html#X7DF8774F7D542298" >4 .10 <span class="Heading" >Namespaces for <strong class="pkg" >GAP</strong > packages</span ></a>
</span >
</div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap4_mj.html#X815F71EA7BC0EB6F" >4 .11 <span class="Heading" >Function</span ></a>
</span >
</div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap4_mj.html#X78C70489791FDF43" >4 .12 <span class="Heading" >Function Calls</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X80B93A9C7E0A57F4" >4 .12 -1 <span class="Heading" >Function Call With Arguments</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X867D54987EF86D1D" >4 .12 -2 <span class="Heading" >Function Call With Options</span ></a>
</span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap4_mj.html#X7A274A1F8553B7E6" >4 .13 <span class="Heading" >Comparisons</span ></a>
</span >
</div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap4_mj.html#X7B66C8707B5DE10A" >4 .14 <span class="Heading" >Arithmetic Operators</span ></a>
</span >
</div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap4_mj.html#X8543285D87361BE6" >4 .15 <span class="Heading" >Statements</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X7E6A50307F4D3FAE" >4 .15 -1 <span class="Heading" >Assignments</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X825803DE78251DA6" >4 .15 -2 <span class="Heading" >Procedure Calls</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X875000188622700D" >4 .15 -3 <span class="Heading" >If</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X87AA46408783383F" >4 .15 -4 <span class="Heading" >While</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X8295CBF47FAA05C9" >4 .15 -5 <span class="Heading" >Repeat</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X78783E777867638A" >4 .15 -6 <span class="Heading" >For</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X7B60C6127E183021" >4 .15 -7 <span class="Heading" >Break</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X7CCBA2247AA366BD" >4 .15 -8 <span class="Heading" >Continue</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X812C6ABC7A182E9E" >4 .15 -9 <span class="Heading" >Return (With or without Value)</span ></a>
</span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap4_mj.html#X8732D9257FFCEA1B" >4 .16 <span class="Heading" >Syntax Trees</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4_mj.html#X81558D66810BEA67" >4 .16 -1 SyntaxTree</a></span >
</div ></div >
</div >
<div class="ContChap" ><a href="chap5_mj.html#X86FA580F8055B274" >5 <span class="Heading" >Functions</span ></a>
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap5_mj.html#X870553EF8605792F" >5 .1 <span class="Heading" >Information about a function</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X79C3BDC4781FA0FD" >5 .1 -1 NameFunction</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X877F03F77FD74C98" >5 .1 -2 NumberArgumentsFunction</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X818BAB817A4FB346" >5 .1 -3 NamesLocalVariablesFunction</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X80E108C57F90FAA3" >5 .1 -4 FilenameFunc</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X7FF7643781D2C194" >5 .1 -5 StartlineFunc</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X844F95767C74834F" >5 .1 -6 LocationFunc</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X845A929B83D46E01" >5 .1 -7 PageSource</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap5_mj.html#X851B58408520700D" >5 .2 <span class="Heading" >Calling a function with a list argument that is interpreted as
several arguments</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X7CF4DDB97D65AE52" >5 .2 -1 CallFuncList</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap5_mj.html#X83066E5A80B5FB71" >5 .3 <span class="Heading" >Wrapping a function, so the values produced are cached</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X817ED3B280A64601" >5 .3 -1 MemoizePosIntFunction</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap5_mj.html#X7EB0A85F7D128BE0" >5 .4 <span class="Heading" >Functions that do nothing</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X7DB422A2876CCC4D" >5 .4 -1 ReturnTrue</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X7C131FB17D7518FC" >5 .4 -2 ReturnFalse</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X7A0994DE7C258E55" >5 .4 -3 ReturnFail</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X818EA8C47B46A634" >5 .4 -4 ReturnNothing</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X8788D7D780FCE169" >5 .4 -5 ReturnFirst</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X810325697BDEF899" >5 .4 -6 IdFunc</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap5_mj.html#X80FE39D27CE3DE1B" >5 .5 <span class="Heading" >Function Types</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X85E40340806C2B8C" >5 .5 -1 IsFunction</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X87838FE885A9AAF9" >5 .5 -2 FunctionsFamily</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap5_mj.html#X81F732457F7BC851" >5 .6 <span class="Heading" >Naming Conventions</span ></a>
</span >
</div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap5_mj.html#X7A1721CD79F08E71" >5 .7 <span class="Heading" >Code annotations (pragmas)</span ></a>
</span >
</div >
</div >
<div class="ContChap" ><a href="chap6_mj.html#X7DB71A2A841CADA5" >6 <span class="Heading" >Main Loop and Break Loop</span ></a>
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap6_mj.html#X81667F568237B232" >6 .1 <span class="Heading" >Main Loop</span ></a>
</span >
</div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap6_mj.html#X866092F281910B74" >6 .2 <span class="Heading" >Special Rules for Input Lines</span ></a>
</span >
</div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap6_mj.html#X8074A8387C9DB9A8" >6 .3 <span class="Heading" >View and Print</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6_mj.html#X8082880F824292E9" >6 .3 -1 <span class="Heading" >Default delegations in the library</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6_mj.html#X87D445D37B31DADB" >6 .3 -2 <span class="Heading" >Recommendations for the implementation</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6_mj.html#X851902C583B84CDC" >6 .3 -3 View</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6_mj.html#X7AFA64D97A1F39A3" >6 .3 -4 Print</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6_mj.html#X815BF22186FD43C9" >6 .3 -5 ViewObj</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6_mj.html#X83A5C59278E13248" >6 .3 -6 Display</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6_mj.html#X87E546E27A1F1FAB" >6 .3 -7 SetNameObject</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap6_mj.html#X8593B49F8705B486" >6 .4 <span class="Heading" >Break Loops</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6_mj.html#X83033EEB81CF4F49" >6 .4 -1 <span class="Heading" >quit from a break loop</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6_mj.html#X7A388B808167FE09" >6 .4 -2 <span class="Heading" >return from a break loop</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6_mj.html#X82EBF01181C3C859" >6 .4 -3 OnBreak</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6_mj.html#X80711C807C99C220" >6 .4 -4 OnBreakMessage</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6_mj.html#X7A7FFA2B7C1EF5A3" >6 .4 -5 Where</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap6_mj.html#X7EE5CF2C8419F061" >6 .5 <span class="Heading" >Variable Access in a Break Loop</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6_mj.html#X79E66DA2875303B0" >6 .5 -1 <span class="Heading" >DownEnv and UpEnv</span ></a>
</span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap6_mj.html#X7BC8D2E37ADE9062" >6 .6 <span class="Heading" >Error and ErrorCount</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6_mj.html#X7E7AD8D87EBA1A08" >6 .6 -1 Error</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6_mj.html#X7A5C000D7E4984DD" >6 .6 -2 ErrorNoReturn</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6_mj.html#X86A11BCC7FECEEA4" >6 .6 -3 ErrorCount</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap6_mj.html#X83704B1080FD9B40" >6 .7 <span class="Heading" >Leaving GAP</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6_mj.html#X7ECC75048583853B" >6 .7 -1 QUIT</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6_mj.html#X838B50A9790DE55B" >6 .7 -2 GapExitCode</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6_mj.html#X7AB1567987922580" >6 .7 -3 QuitGap</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6_mj.html#X85A8DD6B7A20DD89" >6 .7 -4 ForceQuitGap</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6_mj.html#X7A2C380986F46FEE" >6 .7 -5 InstallAtExit</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6_mj.html#X843C07A4869EAA1D" >6 .7 -6 SaveOnExitFile</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap6_mj.html#X82234FD181899530" >6 .8 <span class="Heading" >Line Editing</span ></a>
</span >
</div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap6_mj.html#X7AD8D65F7BA1C3E0" >6 .9 <span class="Heading" >Editing using the <code class="code" >readline</code > library</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6_mj.html#X7C38F9E0783D9442" >6 .9 -1 <span class="Heading" >Readline customization</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6_mj.html#X846C3DED84AD7593" >6 .9 -2 <span class="Heading" >The command line history</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6_mj.html#X7C1F4D04861C1197" >6 .9 -3 SaveCommandLineHistory</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6_mj.html#X87D4EA197A263FB7" >6 .9 -4 <span class="Heading" >Writing your own command line editing functions</span ></a>
</span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap6_mj.html#X7D8E1CF47E97A764" >6 .10 <span class="Heading" >Editing Files</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6_mj.html#X82E5859C8113BA4D" >6 .10 -1 Edit</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap6_mj.html#X7B67FF1E87FE67D1" >6 .11 <span class="Heading" >Editor Support</span ></a>
</span >
</div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap6_mj.html#X83279E897ACCFFFA" >6 .12 <span class="Heading" >Changing the Screen Size</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6_mj.html#X8723E0A1837894F3" >6 .12 -1 SizeScreen</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap6_mj.html#X87847E5087D6F47D" >6 .13 <span class="Heading" >Teaching Mode</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6_mj.html#X7BE2515F82425404" >6 .13 -1 TeachingMode</a></span >
</div ></div >
</div >
<div class="ContChap" ><a href="chap7_mj.html#X8345F6817DFD6394" >7 <span class="Heading" >Debugging and Profiling Facilities</span ></a>
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap7_mj.html#X83C45B0A797AAF96" >7 .1 <span class="Heading" >Recovery from NoMethodFound-Errors</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X86B5FEC67A9394DC" >7 .1 -1 ShowArguments</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X834BD9928773DCC1" >7 .1 -2 ShowArgument</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X7D25D904800D5CBA" >7 .1 -3 ShowDetails</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X7F6996CA872478B8" >7 .1 -4 ShowMethods</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X7E5E2E7B85029E34" >7 .1 -5 ShowOtherMethods</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap7_mj.html#X7FDA1D4B87BD25A8" >7 .2 <span class="Heading" >Inspecting Applicable Methods</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X80848FF486BD6F9F" >7 .2 -1 ApplicableMethod</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap7_mj.html#X7D43A2D885B37739" >7 .3 <span class="Heading" >Tracing Methods</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X80B044017C9E4137" >7 .3 -1 TraceMethods</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X7D34CADB813A4AF1" >7 .3 -2 TraceAllMethods</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X7EB04D387C53E4C1" >7 .3 -3 UntraceMethods</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X7B3018AA82D55949" >7 .3 -4 UntraceAllMethods</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X81078D3387A38E31" >7 .3 -5 TraceImmediateMethods</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X81B000CF86BA1534" >7 .3 -6 TraceInternalMethods</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap7_mj.html#X7A9C902479CB6F7C" >7 .4 <span class="Heading" >Info Functions</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X7AA1A1CF79F20790" >7 .4 -1 NewInfoClass</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X7B3709C584B3DA1E" >7 .4 -2 DeclareInfoClass</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X7A43B9E68765EE9E" >7 .4 -3 SetInfoLevel</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X7B2ADC37783104B9" >7 .4 -4 InfoLevel</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X7BA636EF80A1435A" >7 .4 -5 ShowUsedInfoClasses</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X864E4B6886E2697D" >7 .4 -6 Info</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X800234B5815CAC97" >7 .4 -7 <span class="Heading" >Customizing <code class="func" >Info</code > (<a href="chap7_mj.html#X864E4B6886E2697D" ><span class="RefLink" >7 .4 -6 </span ></a>) statements</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X7A28F77C82D6A3E0" >7 .4 -8 InfoWarning</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap7_mj.html#X86425F067FC63A4C" >7 .5 <span class="Heading" >Assertions</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X7C7596418423660B" >7 .5 -1 SetAssertionLevel</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X876C83707F13A0FD" >7 .5 -2 AssertionLevel</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X830E443284780FB9" >7 .5 -3 Assert</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap7_mj.html#X792BA9A67E64CDED" >7 .6 <span class="Heading" >Timing</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X80355C9282B35673" >7 .6 -1 Runtimes</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X7E32B27F81870D24" >7 .6 -2 Runtime</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X844E1CFE80F41760" >7 .6 -3 NanosecondsSinceEpoch</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X7C0F91F982189624" >7 .6 -4 time </a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X7B543F357C7202CF" >7 .6 -5 Sleep</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap7_mj.html#X844CB04081A771FB" >7 .7 <span class="Heading" >Tracking Memory Usage</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X8077B50B844C4EFC" >7 .7 -1 TotalMemoryAllocated</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X8156D7208591460F" >7 .7 -2 memory_allocated</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap7_mj.html#X7FDF923D7D2937A1" >7 .8 <span class="Heading" >Profiling</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X7939F6F182FDA5F1" >7 .8 -1 <span class="Heading" >Function Profiling</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X79D6CB927BBEB940" >7 .8 -2 ProfileGlobalFunctions</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X7C893F68841B990B" >7 .8 -3 ProfileOperations</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X79D41E977DCA2BEE" >7 .8 -4 ProfileOperationsAndMethods</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X81E8A8627C34FD3B" >7 .8 -5 ProfileFunctions</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X79D394EC7BE8D008" >7 .8 -6 UnprofileFunctions</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X787AC3BE7F991344" >7 .8 -7 ProfileMethods</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X87A05F977F033693" >7 .8 -8 UnprofileMethods</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X80FEA6A08775A48E" >7 .8 -9 DisplayProfile</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X7DAF9AB9793AE203" >7 .8 -10 ClearProfile</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X7C5CE32579891120" >7 .8 -11 <span class="Heading" >An Example of Function Profiling</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X812F9CE0817110EA" >7 .8 -12 <span class="Heading" >Line By Line Profiling</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X7E9C65B17B8EF993" >7 .8 -13 <span class="Heading" >Line by Line profiling example</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X86557887796F66FA" >7 .8 -14 ProfileLineByLine</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X87CC48807DB4C008" >7 .8 -15 CoverageLineByLine</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X7C5DED9C7CC77504" >7 .8 -16 UnprofileLineByLine</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X7B705B2D8670A9C5" >7 .8 -17 UncoverageLineByLine</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X7823C83D79B36D3B" >7 .8 -18 IsLineByLineProfileActive</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X83D8A42B7BB92F5B" >7 .8 -19 DisplayCacheStats</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X79C58704838232CC" >7 .8 -20 ClearCacheStats</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap7_mj.html#X7EE874867C0BEEDD" >7 .9 <span class="Heading" >Information about the version used</span ></a>
</span >
</div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap7_mj.html#X801051CC86594630" >7 .10 <span class="Heading" >Test Files</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X8213757B7ACC76E6" >7 .10 -1 <span class="Heading" >Starting and stopping test</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X87712F9D8732193C" >7 .10 -2 Test</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X87AF67528799481F" >7 .10 -3 TestDirectory</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap7_mj.html#X85FF55448787CCA0" >7 .11 <span class="Heading" >Debugging Recursion</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X7D8968FC7E24A4E5" >7 .11 -1 SetRecursionTrapInterval</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap7_mj.html#X85679F17791D9B63" >7 .12 <span class="Heading" >Global Memory Information</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X7F1F741D7F0899D1" >7 .12 -1 <span class="Heading" >Garbage Collection</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X7848AB367F3A1221" >7 .12 -2 CollectGarbage</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X836977DE80416F3D" >7 .12 -3 GasmanStatistics</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X85327FA5872E0356" >7 .12 -4 GasmanMessageStatus</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7_mj.html#X80C683247E94769F" >7 .12 -5 GasmanLimits</a></span >
</div ></div >
</div >
<div class="ContChap" ><a href="chap8_mj.html#X7FD84061873F72A2" >8 <span class="Heading" >Options Stack</span ></a>
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap8_mj.html#X794C5B5A80203CF9" >8 .1 <span class="Heading" >Functions Dealing with the Options Stack</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8_mj.html#X7D4939FF7FB37FBE" >8 .1 -1 PushOptions</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8_mj.html#X7818A5278679FD43" >8 .1 -2 PopOptions</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8_mj.html#X83D1190984DA3B85" >8 .1 -3 ResetOptionsStack</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8_mj.html#X78D87D1081BF99FE" >8 .1 -4 OnQuit</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8_mj.html#X7F9373AD7DB88D1F" >8 .1 -5 ValueOption</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8_mj.html#X7EDA4EB67D43FE33" >8 .1 -6 DisplayOptionsStack</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8_mj.html#X832F41187B150C19" >8 .1 -7 InfoOptions</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap8_mj.html#X7BB781647CAAE9B4" >8 .2 <span class="Heading" >Options Stack – an Example</span ></a>
</span >
</div >
</div >
<div class="ContChap" ><a href="chap9_mj.html#X82BCD4297920C903" >9 <span class="Heading" >Files and Filenames</span ></a>
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap9_mj.html#X83D8AAA484EE95D9" >9 .1 <span class="Heading" >Portability</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9_mj.html#X87D278437A916905" >9 .1 -1 LastSystemError</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap9_mj.html#X7A4973627A5DB27D" >9 .2 <span class="Heading" >GAP Root Directories</span ></a>
</span >
</div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap9_mj.html#X8223D52E78AF4420" >9 .3 <span class="Heading" >GAP Package Directories</span ></a>
</span >
</div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap9_mj.html#X85030B35865A1080" >9 .4 <span class="Heading" >Directories</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9_mj.html#X82B3E24683942597" >9 .4 -1 IsDirectory</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9_mj.html#X86A71E927EEC7EAD" >9 .4 -2 Directory</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9_mj.html#X8222B1A886E6195E" >9 .4 -3 DirectoryTemporary</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9_mj.html#X7BAD8036849E8430" >9 .4 -4 DirectoryCurrent</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9_mj.html#X81DDD2E87F68E086" >9 .4 -5 ChangeDirectoryCurrent</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9_mj.html#X87ED469A85343A3C" >9 .4 -6 DirectoriesLibrary</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9_mj.html#X808E2C187DD984B4" >9 .4 -7 DirectoriesSystemPrograms</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9_mj.html#X7B225E5282534EDA" >9 .4 -8 DirectoryContents</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9_mj.html#X86F4A32C83B82369" >9 .4 -9 DirectoryDesktop</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9_mj.html#X7B0D818A808A3481" >9 .4 -10 DirectoryHome</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap9_mj.html#X8545E03E7D651456" >9 .5 <span class="Heading" >File Names</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9_mj.html#X7E352E1F87060602" >9 .5 -1 <span class="Heading" >Filename</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9_mj.html#X86C7683E7A2A2146" >9 .5 -2 PathSystemProgram</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap9_mj.html#X85EC7D9087C481B0" >9 .6 <span class="Heading" >Special Filenames</span ></a>
</span >
</div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap9_mj.html#X87271FEF86A6A0F9" >9 .7 <span class="Heading" >File Access</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9_mj.html#X8269697A7B927AF1" >9 .7 -1 IsExistingFile</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9_mj.html#X7E156EC886E11BBC" >9 .7 -2 IsReadableFile</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9_mj.html#X8412F485796B25F5" >9 .7 -3 IsWritableFile</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9_mj.html#X83A1AAD58435FC4C" >9 .7 -4 IsExecutableFile</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9_mj.html#X7D1BE00F83C4EEE8" >9 .7 -5 IsDirectoryPath</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap9_mj.html#X81A0A4FF842B039B" >9 .8 <span class="Heading" >File Operations</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9_mj.html#X8373AC6B7D5F9167" >9 .8 -1 Read</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9_mj.html#X7824CB7D7D4BAFBC" >9 .8 -2 ReadAsFunction</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9_mj.html#X86956C577FFEE1F9" >9 .8 -3 <span class="Heading" >PrintTo and AppendTo</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9_mj.html#X79813A6686894960" >9 .8 -4 <span class="Heading" >LogTo</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9_mj.html#X7CAB119378B075B7" >9 .8 -5 <span class="Heading" >InputLogTo</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9_mj.html#X7A5591D87EAFA6CC" >9 .8 -6 <span class="Heading" >OutputLogTo</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9_mj.html#X8241CEAD80415BB9" >9 .8 -7 CrcFile</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9_mj.html#X7E63ACA38142BE96" >9 .8 -8 RemoveFile</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9_mj.html#X83F3B0337C7EA5CC" >9 .8 -9 UserHomeExpand</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9_mj.html#X79EE267A7FAF28A6" >9 .8 -10 Reread</a></span >
</div ></div >
</div >
<div class="ContChap" ><a href="chap10_mj.html#X839725177BF8B5B4" >10 <span class="Heading" >Streams</span ></a>
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap10_mj.html#X7F89070B7CF52DE0" >10 .1 <span class="Heading" >Categories for Streams and the StreamsFamily</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10_mj.html#X7E974B96785E91A8" >10 .1 -1 IsStream</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10_mj.html#X7FE4096F8497B7F2" >10 .1 -2 IsClosedStream</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10_mj.html#X7FB4391283847C3A" >10 .1 -3 IsInputStream</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10_mj.html#X7C8956BB7FE2A89C" >10 .1 -4 IsInputTextStream</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10_mj.html#X7DCD6ADC86CF2472" >10 .1 -5 IsInputTextNone</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10_mj.html#X7D357CA07E7B1E78" >10 .1 -6 IsOutputStream</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10_mj.html#X8248B8A4844CB8AB" >10 .1 -7 IsOutputTextStream</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10_mj.html#X7C89CDD47E33E741" >10 .1 -8 IsOutputTextNone</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10_mj.html#X7F0F9DD47DE16DAB" >10 .1 -9 StreamsFamily</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap10_mj.html#X8461F4DF7FC20C4B" >10 .2 <span class="Heading" >Operations applicable to All Streams</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10_mj.html#X786E5520803FDE00" >10 .2 -1 CloseStream</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10_mj.html#X7F0459287E717456" >10 .2 -2 FileDescriptorOfStream</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10_mj.html#X87BC257A78F96828" >10 .2 -3 UNIXSelect</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap10_mj.html#X7D1D33A587BFD93D" >10 .3 <span class="Heading" >Operations for Input Streams</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10_mj.html#X7A5DC83D7E295568" >10 .3 -1 Read</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10_mj.html#X7D62F2877F0E45A7" >10 .3 -2 ReadAsFunction</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10_mj.html#X79E1E6A57AE58BB8" >10 .3 -3 ReadByte</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10_mj.html#X7D2CA44C7D110C4F" >10 .3 -4 ReadLine</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10_mj.html#X85C603D7867430D0" >10 .3 -5 ReadAll</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10_mj.html#X81D4FB097F631A79" >10 .3 -6 IsEndOfStream</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10_mj.html#X7B646FA3860521D1" >10 .3 -7 PositionStream</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10_mj.html#X7A777E1186EB330B" >10 .3 -8 RewindStream</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10_mj.html#X7A60AD8C7E0D0507" >10 .3 -9 SeekPositionStream</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap10_mj.html#X7F454EB286947C85" >10 .4 <span class="Heading" >Operations for Output Streams</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10_mj.html#X7D37C7A07E9C319C" >10 .4 -1 WriteByte</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10_mj.html#X79FA85498596CC99" >10 .4 -2 WriteLine</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10_mj.html#X78C113917936058D" >10 .4 -3 WriteAll</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10_mj.html#X7F4E090C86AACCF7" >10 .4 -4 <span class="Heading" >PrintTo and AppendTo (for streams)</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10_mj.html#X7BF4E44C7D51E085" >10 .4 -5 LogTo</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10_mj.html#X7B843516796B2A18" >10 .4 -6 InputLogTo</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10_mj.html#X834A6DD17B0E2062" >10 .4 -7 OutputLogTo</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10_mj.html#X8663FCD57E8BC390" >10 .4 -8 SetPrintFormattingStatus</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap10_mj.html#X80B5F2E4856D8980" >10 .5 <span class="Heading" >File Streams</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10_mj.html#X8343D04981128784" >10 .5 -1 InputTextFile</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10_mj.html#X83F53291822B7126" >10 .5 -2 OutputTextFile</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap10_mj.html#X808348977A05477A" >10 .6 <span class="Heading" >User Streams</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10_mj.html#X83531E4C7C53544F" >10 .6 -1 InputTextUser</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10_mj.html#X83E5FC9487766297" >10 .6 -2 OutputTextUser</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10_mj.html#X7DAF5B7085F4F893" >10 .6 -3 InputFromUser</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap10_mj.html#X8028E1D87CE2F059" >10 .7 <span class="Heading" >String Streams</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10_mj.html#X7ABABCDF7ED81F7F" >10 .7 -1 InputTextString</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10_mj.html#X7FEDA5167979B74D" >10 .7 -2 OutputTextString</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap10_mj.html#X8563EF8387236417" >10 .8 <span class="Heading" >Input -Output Streams</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10_mj.html#X82822D3D8339F635" >10 .8 -1 IsInputOutputStream</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10_mj.html#X820799A3824684AC" >10 .8 -2 InputOutputLocalProcess</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10_mj.html#X7CDF48447E823977" >10 .8 -3 ReadAllLine</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap10_mj.html#X8724699C7D67BA47" >10 .9 <span class="Heading" >Dummy Streams</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10_mj.html#X7C732324806716C6" >10 .9 -1 InputTextNone</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10_mj.html#X7CC5C1FC81715E38" >10 .9 -2 OutputTextNone</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap10_mj.html#X7CB5832F8721ADF3" >10 .10 <span class="Heading" >Handling of Streams in the Background</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10_mj.html#X81FB42517E3EA96D" >10 .10 -1 InstallCharReadHookFunc</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10_mj.html#X8492474C7A0B10AD" >10 .10 -2 UnInstallCharReadHookFunc</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap10_mj.html#X848DD7DC79363341" >10 .11 <span class="Heading" >Comma separated files</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10_mj.html#X86FDC1EF82CAD2DA" >10 .11 -1 ReadCSV</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10_mj.html#X8779DAC585E05A47" >10 .11 -2 PrintCSV</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap10_mj.html#X87396F857ADA3F97" >10 .12 <span class="Heading" >Opening files in the Operating System</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10_mj.html#X86B98E287AD42BE8" >10 .12 -1 OpenExternal</a></span >
</div ></div >
</div >
<div class="ContChap" ><a href="chap11_mj.html#X7882133B7BDD51BC" >11 <span class="Heading" >Processes</span ></a>
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap11_mj.html#X8390266186E61CCE" >11 .1 <span class="Heading" >Process and Exec</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11_mj.html#X7B09033178D1107A" >11 .1 -1 Process</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11_mj.html#X81402C91833986FC" >11 .1 -2 Exec</a></span >
</div ></div >
</div >
<div class="ContChap" ><a href="chap12_mj.html#X86710F997832ABA4" >12 <span class="Heading" >Objects and Elements</span ></a>
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap12_mj.html#X78497E777FB3E402" >12 .1 <span class="Heading" >Objects</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap12_mj.html#X7B130AC98415CAFB" >12 .1 -1 IsObject</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap12_mj.html#X780C66027A49D110" >12 .2 <span class="Heading" >Elements as equivalence classes</span ></a>
</span >
</div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap12_mj.html#X83BE0C20875DD285" >12 .3 <span class="Heading" >Sets</span ></a>
</span >
</div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap12_mj.html#X7BAF69417BB925F6" >12 .4 <span class="Heading" >Domains</span ></a>
</span >
</div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap12_mj.html#X84545F3985C60F5B" >12 .5 <span class="Heading" >Identical Objects</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap12_mj.html#X7961183378DFB902" >12 .5 -1 IsIdenticalObj</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap12_mj.html#X811976EC78EC5E29" >12 .5 -2 IsNotIdenticalObj</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap12_mj.html#X7F0C119682196D65" >12 .6 <span class="Heading" >Mutability and Copyability</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap12_mj.html#X811EFD727EBD1ADC" >12 .6 -1 IsCopyable</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap12_mj.html#X7999AD1D7A4F1F46" >12 .6 -2 IsMutable</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap12_mj.html#X7F0ABF2C870B0CBB" >12 .6 -3 Immutable</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap12_mj.html#X80CE136D804097C7" >12 .6 -4 MakeImmutable</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap12_mj.html#X7FBA5F4D7C6872BD" >12 .6 -5 <span class="Heading" >Mutability of Iterators</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap12_mj.html#X7ADB82997A16E853" >12 .6 -6 <span class="Heading" >Mutability of Results of Arithmetic Operations</span ></a>
</span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap12_mj.html#X786B942B82D684BD" >12 .7 <span class="Heading" >Duplication of Objects</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap12_mj.html#X846BC7107C352031" >12 .7 -1 ShallowCopy</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap12_mj.html#X7C1E70587EBDD2CB" >12 .7 -2 StructuralCopy</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap12_mj.html#X86E7193D848C53FC" >12 .8 <span class="Heading" >Other Operations Applicable to any Object </span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap12_mj.html#X85D6D47B83BD02A1" >12 .8 -1 SetName</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap12_mj.html#X7F14EF9D81432113" >12 .8 -2 Name</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap12_mj.html#X871562FD7F982C12" >12 .8 -3 InfoText</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap12_mj.html#X7F6C5C3287E8B816" >12 .8 -4 IsInternallyConsistent</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap12_mj.html#X7F4D216B7DF7BE9D" >12 .8 -5 MemoryUsage</a></span >
</div ></div >
</div >
<div class="ContChap" ><a href="chap13_mj.html#X7E8202627B421DB1" >13 <span class="Heading" >Types of Objects</span ></a>
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap13_mj.html#X846063757EC05986" >13 .1 <span class="Heading" >Families</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap13_mj.html#X7CF70EAC84284919" >13 .1 -1 FamilyObj</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap13_mj.html#X7FB4123E7E22137D" >13 .1 -2 NewFamily</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap13_mj.html#X84EFA4C07D4277BB" >13 .2 <span class="Heading" >Filters</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap13_mj.html#X82E62B997C05E05E" >13 .2 -1 RankFilter</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap13_mj.html#X7A78ECC67E2C9D78" >13 .2 -2 NamesFilter</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap13_mj.html#X7F6645D87DD26CF0" >13 .2 -3 FilterByName</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap13_mj.html#X7F9568A67F3840DE" >13 .2 -4 ShowImpliedFilters</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap13_mj.html#X836FAA18861BE387" >13 .2 -5 FiltersType</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap13_mj.html#X7CC6903E78F24167" >13 .3 <span class="Heading" >Categories</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap13_mj.html#X792A23BF82BDF66B" >13 .3 -1 IsCategory</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap13_mj.html#X85C6EB707A406A5A" >13 .3 -2 CategoriesOfObject</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap13_mj.html#X85D07C3E7F4D4043" >13 .3 -3 CategoryByName</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap13_mj.html#X87F68F887B44DBBD" >13 .3 -4 NewCategory</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap13_mj.html#X879DE2A17A6C6E92" >13 .3 -5 DeclareCategory</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap13_mj.html#X787BACEE7937EF01" >13 .3 -6 CategoryFamily</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap13_mj.html#X8698205F8648EB33" >13 .4 <span class="Heading" >Representation</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap13_mj.html#X805F1C3B7C730062" >13 .4 -1 <span class="Heading" >Basic Representations of Objects</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap13_mj.html#X86D42C7783ACA5F4" >13 .4 -2 IsRepresentation</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap13_mj.html#X7BBE93BE7977750F" >13 .4 -3 RepresentationsOfObject</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap13_mj.html#X7CC8106F809E15CF" >13 .4 -4 NewRepresentation</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap13_mj.html#X7C81FB2682AE54CD" >13 .4 -5 DeclareRepresentation</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap13_mj.html#X7C701DBF7BAE649A" >13 .5 <span class="Heading" >Attributes</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap13_mj.html#X7973C8F4782D15A1" >13 .5 -1 IsAttribute</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap13_mj.html#X7F7960338163AA88" >13 .5 -2 KnownAttributesOfObject</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap13_mj.html#X7B9654807858A3B0" >13 .5 -3 NewAttribute</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap13_mj.html#X7A00FC8A7A677A56" >13 .5 -4 DeclareAttribute</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap13_mj.html#X7A951C33839AF2C1" >13 .5 -5 IsAttributeStoringRep</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap13_mj.html#X79DE5208877AE42A" >13 .6 <span class="Heading" >Setter and Tester for Attributes</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap13_mj.html#X87D5B5AC7DAF932D" >13 .6 -1 Tester</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap13_mj.html#X7FD8952C841D2B1F" >13 .6 -2 Setter</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap13_mj.html#X8529F8A17884A32C" >13 .6 -3 AttributeValueNotSet</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap13_mj.html#X79120CE37BB69D11" >13 .6 -4 InfoAttributes</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap13_mj.html#X7851E2DB79656DB0" >13 .6 -5 DisableAttributeValueStoring</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap13_mj.html#X7E5DACBE7A9A9AD1" >13 .6 -6 EnableAttributeValueStoring</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap13_mj.html#X871597447BB998A1" >13 .7 <span class="Heading" >Properties</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap13_mj.html#X81F1C3EE83003FA0" >13 .7 -1 IsProperty</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap13_mj.html#X7E51C08286E03E7F" >13 .7 -2 KnownPropertiesOfObject</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap13_mj.html#X86711BC77B62EB02" >13 .7 -3 KnownTruePropertiesOfObject</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap13_mj.html#X7F2D6FD979FE23DD" >13 .7 -4 NewProperty</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap13_mj.html#X7F4602F082682A04" >13 .7 -5 DeclareProperty</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap13_mj.html#X7997705185C7E720" >13 .8 <span class="Heading" >Other Filters</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap13_mj.html#X821635DA7821ED74" >13 .8 -1 NewFilter</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap13_mj.html#X846EA18A7D36626C" >13 .8 -2 DeclareFilter</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap13_mj.html#X7C92D53E7920CE02" >13 .8 -3 SetFilterObj</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap13_mj.html#X8117FD03870FB02E" >13 .8 -4 ResetFilterObj</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap13_mj.html#X7E340B8C833BC440" >13 .9 <span class="Heading" >Types</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap13_mj.html#X7D3E6B6482BE5B16" >13 .9 -1 TypeObj</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap13_mj.html#X85A60A7F8083C1C4" >13 .9 -2 DataType</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap13_mj.html#X7CE39E9478AEC826" >13 .9 -3 NewType</a></span >
</div ></div >
</div >
<div class="ContChap" ><a href="chap14_mj.html#X853DF11B80068ED5" >14 <span class="Heading" >Integers</span ></a>
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap14_mj.html#X838230CE810107A3" >14 .1 <span class="Heading" >Integers: Global Variables</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap14_mj.html#X7E20D82B79DE5129" >14 .1 -1 Integers</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap14_mj.html#X818683B17F8C97F3" >14 .1 -2 IsIntegers</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap14_mj.html#X80CF510B8080C7CA" >14 .2 <span class="Heading" >Elementary Operations for Integers</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap14_mj.html#X87AEADF07DC8303B" >14 .2 -1 IsInt</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap14_mj.html#X82A854757DFA9C76" >14 .2 -2 IsPosInt</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap14_mj.html#X87CA734380B5F68C" >14 .2 -3 Int</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap14_mj.html#X87DD1EEE7EF18036" >14 .2 -4 IsEvenInt</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap14_mj.html#X8621BA927CD12EFB" >14 .2 -5 IsOddInt</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap14_mj.html#X782095927FB9F1DB" >14 .2 -6 AbsInt</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap14_mj.html#X842614817FE48D62" >14 .2 -7 SignInt</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap14_mj.html#X8197C4E882BAF14E" >14 .2 -8 LogInt</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap14_mj.html#X83D9B5C87EEA2A77" >14 .2 -9 RootInt</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap14_mj.html#X7F98A0CE7B9FD366" >14 .2 -10 SmallestRootInt</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap14_mj.html#X83B998E486893FED" >14 .2 -11 IsSquareInt</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap14_mj.html#X862D1BD786EFFDA9" >14 .2 -12 ListOfDigits</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap14_mj.html#X8185784B7E228DEA" >14 .2 -13 Random</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap14_mj.html#X7A9FD25D81D88D1B" >14 .3 <span class="Heading" >Quotients and Remainders</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap14_mj.html#X849D0F807F697D35" >14 .3 -1 QuoInt</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap14_mj.html#X795170A385AC8FEE" >14 .3 -2 BestQuoInt</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap14_mj.html#X805ADD5A826D844D" >14 .3 -3 RemInt</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap14_mj.html#X7A4FEFCA8128E3C3" >14 .3 -4 GcdInt</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap14_mj.html#X8775930486BD0C5B" >14 .3 -5 Gcdex</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap14_mj.html#X7B33143E78A8DDE3" >14 .3 -6 LcmInt</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap14_mj.html#X79B466E984CD52D4" >14 .3 -7 CoefficientsQadic</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap14_mj.html#X83124F86839DC7E6" >14 .3 -8 CoefficientsMultiadic</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap14_mj.html#X84A1900E82902B5F" >14 .3 -9 ChineseRem</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap14_mj.html#X7E404B1183DBC82A" >14 .3 -10 PowerModInt</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap14_mj.html#X82005E587F0CB02A" >14 .4 <span class="Heading" >Prime Integers and Factorization</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap14_mj.html#X86F5E4CD82FEB9F4" >14 .4 -1 Primes</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap14_mj.html#X78FDA4437EDCA70C" >14 .4 -2 IsPrimeInt</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap14_mj.html#X7CD977B17B4A7A4B" >14 .4 -3 PrimalityProof</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap14_mj.html#X8443125D7FD6F2A6" >14 .4 -4 IsPrimePowerInt</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap14_mj.html#X78744C367A94C69F" >14 .4 -5 NextPrimeInt</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap14_mj.html#X819060E17E83728A" >14 .4 -6 PrevPrimeInt</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap14_mj.html#X82C989DB84744B36" >14 .4 -7 FactorsInt</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap14_mj.html#X80E7A5D381C64CC9" >14 .4 -8 PrimeDivisors</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap14_mj.html#X786FF92C7C54BF97" >14 .4 -9 PartialFactorization</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap14_mj.html#X803D431087B6FF28" >14 .4 -10 PrintFactorsInt</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap14_mj.html#X82148B347E294C87" >14 .4 -11 PrimePowersInt</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap14_mj.html#X809E0E1B83AF7695" >14 .4 -12 DivisorsInt</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap14_mj.html#X864BF040862409FC" >14 .5 <span class="Heading" >Residue Class Rings</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap14_mj.html#X87B1210B8581D5B2" ><code >14 .5 -1 \mod</code ></a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap14_mj.html#X79CE76AD82B3E2B2" >14 .5 -2 ZmodnZ</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap14_mj.html#X838F36507D985EDA" >14 .5 -3 ZmodnZObj</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap14_mj.html#X7D0107DD79753901" >14 .5 -4 IsZmodnZObj</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap14_mj.html#X7904B6D681EBF091" >14 .6 <span class="Heading" >Check Digits</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap14_mj.html#X82BABA8F868BD425" >14 .6 -1 CheckDigitISBN</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap14_mj.html#X85F1A6A5870485B9" >14 .6 -2 CheckDigitTestFunction</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap14_mj.html#X85361FAE8088C006" >14 .7 <span class="Heading" >Random Sources</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap14_mj.html#X82E31A697E389F1D" >14 .7 -1 IsRandomSource</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap14_mj.html#X821004F286282D49" >14 .7 -2 Random</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap14_mj.html#X86FFFBC9790F9742" >14 .7 -3 <span class="Heading" >State and Reset for Random Sources</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap14_mj.html#X7AC96008820FAF1F" >14 .7 -4 <span class="Heading" >Kinds of Random Sources</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap14_mj.html#X7CB0B5BC82F8FD8F" >14 .7 -5 RandomSource</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap14_mj.html#X8653AE447D94C1DC" >14 .7 -6 <span class="Heading" >Implementing new kinds of random sources</span ></a>
</span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap14_mj.html#X7A0311DF78DB4FD8" >14 .8 <span class="Heading" >Bitfields</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap14_mj.html#X85C7BD9E7FCC6C10" >14 .8 -1 MakeBitfields</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap14_mj.html#X8068CE3781F4003C" >14 .8 -2 BuildBitfields</a></span >
</div ></div >
</div >
<div class="ContChap" ><a href="chap15_mj.html#X7FB995737B7ED8A2" >15 <span class="Heading" >Number Theory</span ></a>
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap15_mj.html#X7845C1F97A1742C7" >15 .1 <span class="Heading" >InfoNumtheor (Info Class)</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap15_mj.html#X796F0DFE7D5D211C" >15 .1 -1 InfoNumtheor</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap15_mj.html#X823386567DAC22E6" >15 .2 <span class="Heading" >Prime Residues</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap15_mj.html#X7FA3F5347B7004BA" >15 .2 -1 PrimeResidues</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap15_mj.html#X85A0C67982D9057A" >15 .2 -2 Phi</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap15_mj.html#X85296F3087611B03" >15 .2 -3 Lambda</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap15_mj.html#X7D191CF67E5018BE" >15 .2 -4 GeneratorsPrimeResidues</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap15_mj.html#X83103A5385821BAE" >15 .3 <span class="Heading" >Primitive Roots and Discrete Logarithms</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap15_mj.html#X82373F3D8277EE9E" >15 .3 -1 OrderMod</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap15_mj.html#X81AD9C7779A7BA89" >15 .3 -2 LogMod</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap15_mj.html#X84A138947E8C49A8" >15 .3 -3 DLog</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap15_mj.html#X82440BB9812FF148" >15 .3 -4 PrimitiveRootMod</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap15_mj.html#X790466C07BD90E20" >15 .3 -5 IsPrimitiveRootMod</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap15_mj.html#X7F9069D77AC48054" >15 .4 <span class="Heading" >Roots Modulo Integers</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap15_mj.html#X83449DBC80495971" >15 .4 -1 Jacobi</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap15_mj.html#X81464ABF7F10E544" >15 .4 -2 Legendre</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap15_mj.html#X83E3ED577B7A04ED" >15 .4 -3 RootMod</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap15_mj.html#X84D3F03B862841F8" >15 .4 -4 RootsMod</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap15_mj.html#X81F856E682A8ECBA" >15 .4 -5 RootsUnityMod</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap15_mj.html#X7B3A5A0378A32F83" >15 .5 <span class="Heading" >Multiplicative Arithmetic Functions</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap15_mj.html#X823707DF821E79A0" >15 .5 -1 Sigma</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap15_mj.html#X798C62847EE0372E" >15 .5 -2 Tau</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap15_mj.html#X79C1DA36827C2959" >15 .5 -3 MoebiusMu</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap15_mj.html#X7B2E061C835159B9" >15 .6 <span class="Heading" >Continued Fractions</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap15_mj.html#X874C161B83416092" >15 .6 -1 ContinuedFractionExpansionOfRoot</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap15_mj.html#X8059667580A039A6" >15 .6 -2 ContinuedFractionApproximationOfRoot</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap15_mj.html#X7C5563A37D566DA5" >15 .7 <span class="Heading" >Miscellaneous</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap15_mj.html#X8243EAA586D78ED4" >15 .7 -1 PValuation</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap15_mj.html#X85E1EFC484F648A4" >15 .7 -2 TwoSquares</a></span >
</div ></div >
</div >
<div class="ContChap" ><a href="chap16_mj.html#X7BDA99EE7CEADA7C" >16 <span class="Heading" >Combinatorics</span ></a>
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap16_mj.html#X800E48927D5C83F5" >16 .1 <span class="Heading" >Combinatorial Numbers</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap16_mj.html#X87665F748594BF29" >16 .1 -1 Factorial</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap16_mj.html#X7A9AF5F58682819D" >16 .1 -2 Binomial</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap16_mj.html#X7DC5667580522BDA" >16 .1 -3 Bell</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap16_mj.html#X792FF6EA786A5C2B" >16 .1 -4 Bernoulli</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap16_mj.html#X85037456785BB33C" >16 .1 -5 Stirling1</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap16_mj.html#X7C93E14D7BC360F0" >16 .1 -6 Stirling2</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap16_mj.html#X81B4696585C38147" >16 .2 <span class="Heading" >Combinations, Arrangements and Tuples</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap16_mj.html#X8770F16D794C0ADB" >16 .2 -1 Combinations</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap16_mj.html#X78DD5C0D81057540" >16 .2 -2 <span class="Heading" >Iterator and enumerator of combinations</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap16_mj.html#X82A6E98C85714FD0" >16 .2 -3 NrCombinations</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap16_mj.html#X7837B3357C7566C8" >16 .2 -4 Arrangements</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap16_mj.html#X7DE1ABD47D19F140" >16 .2 -5 NrArrangements</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap16_mj.html#X81601C6786120DDC" >16 .2 -6 UnorderedTuples</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap16_mj.html#X7959281584C42C52" >16 .2 -7 NrUnorderedTuples</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap16_mj.html#X86A3CA0F7CC8C320" >16 .2 -8 Tuples</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap16_mj.html#X7BA135297E8DA819" >16 .2 -9 EnumeratorOfTuples</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap16_mj.html#X86416A31807B0086" >16 .2 -10 IteratorOfTuples</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap16_mj.html#X85E18A9A87FD4CA2" >16 .2 -11 NrTuples</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap16_mj.html#X7B0143FB83F359B7" >16 .2 -12 PermutationsList</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap16_mj.html#X8629A2908050EB3A" >16 .2 -13 NrPermutationsList</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap16_mj.html#X79C159507B2BF1C9" >16 .2 -14 Derangements</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap16_mj.html#X7C1741B181A9AB9C" >16 .2 -15 NrDerangements</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap16_mj.html#X7A13D8DC8204525F" >16 .2 -16 PartitionsSet</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap16_mj.html#X7BCD7FC2876386F1" >16 .2 -17 NrPartitionsSet</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap16_mj.html#X84A6D15F8107008B" >16 .2 -18 Partitions</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap16_mj.html#X8793AEBD7E529E1D" >16 .2 -19 IteratorOfPartitions</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap16_mj.html#X7EBD746A8607D0B8" >16 .2 -20 IteratorOfPartitionsSet</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap16_mj.html#X86933C4F795C4EBD" >16 .2 -21 NrPartitions</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap16_mj.html#X820DF201871F2723" >16 .2 -22 OrderedPartitions</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap16_mj.html#X80BB9F4982CA1E8B" >16 .2 -23 NrOrderedPartitions</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap16_mj.html#X8009520C82942461" >16 .2 -24 PartitionsGreatestLE</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap16_mj.html#X7CB8D4FF8592A9BB" >16 .2 -25 PartitionsGreatestEQ</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap16_mj.html#X7A70D4F3809494E7" >16 .2 -26 RestrictedPartitions</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap16_mj.html#X800B43838742FBF4" >16 .2 -27 NrRestrictedPartitions</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap16_mj.html#X7F4EDCCA780B469D" >16 .2 -28 SignPartition</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap16_mj.html#X7DB9BEB6856EC03D" >16 .2 -29 AssociatedPartition</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap16_mj.html#X7A95D8A6820363A8" >16 .2 -30 PowerPartition</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap16_mj.html#X877D997B7F66A119" >16 .2 -31 PartitionTuples</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap16_mj.html#X7F44AD098561DE32" >16 .2 -32 NrPartitionTuples</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap16_mj.html#X8796C1D783ED9CB4" >16 .2 -33 BetaSet</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap16_mj.html#X83DC50B67D74E674" >16 .3 <span class="Heading" >Fibonacci and Lucas Sequences</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap16_mj.html#X85AE1D70803A886C" >16 .3 -1 Fibonacci</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap16_mj.html#X7830A03181D67192" >16 .3 -2 Lucas</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap16_mj.html#X821888E77EB43F67" >16 .4 <span class="Heading" >Permanent of a Matrix</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap16_mj.html#X7F0942DD83BBAB7A" >16 .4 -1 Permanent</a></span >
</div ></div >
</div >
<div class="ContChap" ><a href="chap17_mj.html#X87003045878E74DF" >17 <span class="Heading" >Rational Numbers</span ></a>
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap17_mj.html#X7A76497986DA921F" >17 .1 <span class="Heading" >Rationals: Global Variables</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap17_mj.html#X7B6029D18570C08A" >17 .1 -1 Rationals</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap17_mj.html#X826E2AA88679B3DF" >17 .2 <span class="Heading" >Elementary Operations for Rationals</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap17_mj.html#X7ED018F5794935F7" >17 .2 -1 IsRat</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap17_mj.html#X7BD6E170840F045D" >17 .2 -2 IsPosRat</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap17_mj.html#X81179AC87AC951A8" >17 .2 -3 IsNegRat</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap17_mj.html#X7D830E7482E7F528" >17 .2 -4 NumeratorRat</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap17_mj.html#X81F6B5877A81E727" >17 .2 -5 DenominatorRat</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap17_mj.html#X7EB4C646806A2BDE" >17 .2 -6 Rat</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap17_mj.html#X7C8F8693825C28A4" >17 .2 -7 Random</a></span >
</div ></div >
</div >
<div class="ContChap" ><a href="chap18_mj.html#X7DFC03C187DE4841" >18 <span class="Heading" >Cyclotomic Numbers</span ></a>
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap18_mj.html#X79E25C3085AA568F" >18 .1 <span class="Heading" >Operations for Cyclotomics</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap18_mj.html#X8631458886314588" >18 .1 -1 E</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap18_mj.html#X863D1E017BC9EB7F" >18 .1 -2 Cyclotomics</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap18_mj.html#X841C425281A6F775" >18 .1 -3 IsCyclotomic</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap18_mj.html#X869750DA81EA0E67" >18 .1 -4 IsIntegralCyclotomic</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap18_mj.html#X7DD6B95F79321D23" >18 .1 -5 Int</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap18_mj.html#X7CBA6CB678E2B143" >18 .1 -6 String</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap18_mj.html#X815D6EC57CBA9827" >18 .1 -7 Conductor</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap18_mj.html#X81DD58BB81FB3426" >18 .1 -8 AbsoluteValue</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap18_mj.html#X7808ECF37AA9004D" >18 .1 -9 RoundCyc</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap18_mj.html#X7AE2933985BE4C3E" >18 .1 -10 CoeffsCyc</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap18_mj.html#X803478CA7D2D830F" >18 .1 -11 DenominatorCyc</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap18_mj.html#X785F2CAB805DE1BE" >18 .1 -12 ExtRepOfObj</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap18_mj.html#X7DDD51B983D5BC44" >18 .1 -13 DescriptionOfRootOfUnity</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap18_mj.html#X8712419182ECD8DD" >18 .1 -14 IsGaussInt</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap18_mj.html#X7E6CF4947D0A56F7" >18 .1 -15 IsGaussRat</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap18_mj.html#X7FE3D5637B5485D0" >18 .1 -16 DefaultField</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap18_mj.html#X7EE5FB7181125E02" >18 .2 <span class="Heading" >Infinity and negative Infinity</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap18_mj.html#X8511B8DF83324C27" >18 .2 -1 IsInfinity</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap18_mj.html#X7F66A62384329705" >18 .3 <span class="Heading" >Comparisons of Cyclotomics</span ></a>
</span >
</div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap18_mj.html#X7B242083873DD74F" >18 .4 <span class="Heading" >ATLAS Irrationalities</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap18_mj.html#X8414ED887AF36359" >18 .4 -1 <span class="Heading" >EB, EC, <span class="SimpleMath" >\(\ldots\)</span >, EH</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap18_mj.html#X813CF4327C4B4D29" >18 .4 -2 <span class="Heading" >EI and ER</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap18_mj.html#X8672D7F986CBA116" >18 .4 -3 <span class="Heading" >EY, EX, <span class="SimpleMath" >\(\ldots\)</span >, ES</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap18_mj.html#X7E5985FC846C5201" >18 .4 -4 <span class="Heading" >EM , EL, <span class="SimpleMath" >\(\ldots\)</span >, EJ</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap18_mj.html#X844F0EBF849EDEB3" >18 .4 -5 NK</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap18_mj.html#X812E334E7A869D33" >18 .4 -6 AtlasIrrationality</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap18_mj.html#X79FE34337DF2CD10" >18 .5 <span class="Heading" >Galois Conjugacy of Cyclotomics</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap18_mj.html#X79EE9097783128C4" >18 .5 -1 GaloisCyc</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap18_mj.html#X7BE001A0811CD599" >18 .5 -2 ComplexConjugate</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap18_mj.html#X7E361C057E97CA66" >18 .5 -3 StarCyc</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap18_mj.html#X84438F867B0CC299" >18 .5 -4 Quadratic</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap18_mj.html#X7DDDEC3F80543B7D" >18 .5 -5 GaloisMat</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap18_mj.html#X7BB9F5957AA8C082" >18 .5 -6 RationalizedMat</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap18_mj.html#X8557FC2D7ACD6105" >18 .6 <span class="Heading" >Internally Represented Cyclotomics</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap18_mj.html#X7D3028777DE39709" >18 .6 -1 SetCyclotomicsLimit</a></span >
</div ></div >
</div >
<div class="ContChap" ><a href="chap19_mj.html#X81AA901181CA568F" >19 <span class="Heading" >Floats</span ></a>
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap19_mj.html#X7B4092CA7ABB93B0" >19 .1 <span class="Heading" >A sample run</span ></a>
</span >
</div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap19_mj.html#X8606FDCE878850EF" >19 .2 <span class="Heading" >Methods</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap19_mj.html#X86D5EA93813FB6C4" >19 .2 -1 <span class="Heading" >Float creators</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap19_mj.html#X7BCD34DC7B5A0521" >19 .2 -2 Rat</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap19_mj.html#X7D1EAE11844625F4" >19 .2 -3 Cyc</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap19_mj.html#X7A962B0983FA66E8" >19 .2 -4 SetFloats</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap19_mj.html#X819050BF8403806E" >19 .2 -5 FLOAT</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap19_mj.html#X7BD96E0585D5A1EE" >19 .2 -6 EqFloat</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap19_mj.html#X7B3133497DDE839B" >19 .2 -7 PrecisionFloat</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap19_mj.html#X801753137949DD78" >19 .2 -8 SignBit</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap19_mj.html#X7935C65D7B0F47C7" >19 .2 -9 SinCos</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap19_mj.html#X846E1196844B9E11" >19 .2 -10 Atan2</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap19_mj.html#X7981510D826EE3E5" >19 .2 -11 Log1p</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap19_mj.html#X86073E147FB3C0EA" >19 .2 -12 Erf</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap19_mj.html#X7E03FDEE824D1E8E" >19 .2 -13 <span class="Heading" >Infinity testers</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap19_mj.html#X8151581186F75BA3" >19 .2 -14 <span class="Heading" >Standard mathematical operations</span ></a>
</span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap19_mj.html#X845ACF3A78BD2771" >19 .3 <span class="Heading" >High-precision-specific methods</span ></a>
</span >
</div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap19_mj.html#X7E8F6EFB87A65F78" >19 .4 <span class="Heading" >Complex arithmetic</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap19_mj.html#X7B0269D983F96677" >19 .4 -1 Argument</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap19_mj.html#X7E57B09C80136484" >19 .5 <span class="Heading" >Interval-specific methods</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap19_mj.html#X7C34D1D185802F2F" >19 .5 -1 Sup </a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap19_mj.html#X78F1E457814FD1FD" >19 .5 -2 Inf</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap19_mj.html#X829581A485F55996" >19 .5 -3 Mid</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap19_mj.html#X7FE540B387B0012C" >19 .5 -4 AbsoluteDiameter</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap19_mj.html#X7CA771757F441592" >19 .5 -5 RelativeDiameter</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap19_mj.html#X86D22AE57E2D84B2" >19 .5 -6 IsDisjoint</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap19_mj.html#X7A5E0C3E79837EB8" >19 .5 -7 IsSubset</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap19_mj.html#X85191E1679936CE9" >19 .5 -8 IncreaseInterval</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap19_mj.html#X879EE14282DD1539" >19 .5 -9 BlowupInterval</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap19_mj.html#X7EC15DAE7CBBB42E" >19 .5 -10 BisectInterval</a></span >
</div ></div >
</div >
<div class="ContChap" ><a href="chap20_mj.html#X787B4AB77A2F5E14" >20 <span class="Heading" >Booleans</span ></a>
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap20_mj.html#X87F9AF65832E7AD2" >20 .1 <span class="Heading" >IsBool (Filter)</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap20_mj.html#X7D58580284CF7894" >20 .1 -1 IsBool</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap20_mj.html#X85E648AA8414F303" >20 .2 <span class="Heading" >Fail (Variable)</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap20_mj.html#X8294AAC9860E87E5" >20 .2 -1 fail</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap20_mj.html#X862F17B68465B399" >20 .3 <span class="Heading" >Comparisons of Booleans</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap20_mj.html#X79305F9780394190" >20 .3 -1 <span class="Heading" >Equality and inequality of Booleans</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap20_mj.html#X7FEF019482AF5923" >20 .3 -2 <span class="Heading" >Ordering of Booleans</span ></a>
</span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap20_mj.html#X79AD41A185FD7213" >20 .4 <span class="Heading" >Operations for Booleans</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap20_mj.html#X7DFE7E518088AA89" >20 .4 -1 <span class="Heading" >Logical disjunction</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap20_mj.html#X7A64D25F804973CD" >20 .4 -2 <span class="Heading" >Logical conjunction</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap20_mj.html#X84F5034185D7EC3C" >20 .4 -3 <span class="Heading" >Logical negation</span ></a>
</span >
</div ></div >
</div >
<div class="ContChap" ><a href="chap21_mj.html#X7B256AE5780F140A" >21 <span class="Heading" >Lists</span ></a>
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap21_mj.html#X86B28F5B781FFD31" >21 .1 <span class="Heading" >List Categories</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X7C4CC4EA8299701E" >21 .1 -1 IsList</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X870AA9D8798C93DD" >21 .1 -2 IsDenseList</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X7C71596C82B6EF35" >21 .1 -3 IsHomogeneousList</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X80872FAF80EB5DF9" >21 .1 -4 IsTable</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X79581E0387F7F7A9" >21 .1 -5 IsRectangularTable</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X7C84E16A85C99C8C" >21 .1 -6 IsConstantTimeAccessList</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap21_mj.html#X7B202D147A5C2884" >21 .2 <span class="Heading" >Basic Operations for Lists</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X8297BBCD79642BE6" ><code >21 .2 -1 <span >\</span >[<span >\</span >]</code ></a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap21_mj.html#X7921047F83F5FA28" >21 .3 <span class="Heading" >List Elements</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X78791B8B838A8BA0" ><code >21 .3 -1 \{\}</code ></a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap21_mj.html#X8611EF768210625B" >21 .4 <span class="Heading" >List Assignment</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X813FF1637F8D2B7F" ><code >21 .4 -1 \{\}\:\=</code ></a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X795EC9D67E34DAB0" >21 .4 -2 Add</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X7E98B11B79BA9167" >21 .4 -3 Remove</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X79D7E96F80A2D7C0" >21 .4 -4 CopyListEntries</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X79E31DB27C82D6E1" >21 .4 -5 Append</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap21_mj.html#X7963C8E17EFF86DB" >21 .5 <span class="Heading" >IsBound and Unbind for Lists</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X79EC565A7DCEC938" >21 .5 -1 IsBound</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X866F45D3797FDA00" >21 .5 -2 GetWithDefault</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X78B72FDF7BD63C0B" >21 .5 -3 Unbind</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap21_mj.html#X7DD65BEA7EDB0CD7" >21 .6 <span class="Heading" >Identical Lists</span ></a>
</span >
</div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap21_mj.html#X7ED7C0738495556F" >21 .7 <span class="Heading" >Duplication of Lists</span ></a>
</span >
</div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap21_mj.html#X808A207182B2F84F" >21 .8 <span class="Heading" >Membership Test for Lists</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X7B914A287F88ED0A" ><code >21 .8 -1 \in</code ></a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap21_mj.html#X84D6FC7E7E39ED33" >21 .9 <span class="Heading" >Enlarging Internally Represented Lists</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X78BF67A5802E93AD" >21 .9 -1 EmptyPlist</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap21_mj.html#X8016D50F85147A77" >21 .10 <span class="Heading" >Comparisons of Lists</span ></a>
</span >
</div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap21_mj.html#X845EEAF083D43CCE" >21 .11 <span class="Heading" >Arithmetic for Lists</span ></a>
</span >
</div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap21_mj.html#X84D642967B8546B7" >21 .12 <span class="Heading" >Filters Controlling the Arithmetic Behaviour of Lists</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X87ABCEE9809585A0" >21 .12 -1 IsGeneralizedRowVector</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X7FBCA5B58308C158" >21 .12 -2 IsMultiplicativeGeneralizedRowVector</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X7BAD12E67BFC90DE" >21 .12 -3 IsListDefault</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X8428E77B86722D52" >21 .12 -4 NestingDepthA</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X84B383B97FD986CD" >21 .12 -5 NestingDepthM</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap21_mj.html#X7E6A1F66781BE923" >21 .13 <span class="Heading" >Additive Arithmetic for Lists</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X86A85ADC85C451DC" >21 .13 -1 <span class="Heading" >Zero for lists</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X7B91CE4D814C2D08" >21 .13 -2 <span class="Heading" >AdditiveInverse for lists</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X842D123E7EE5E3DB" >21 .13 -3 <span class="Heading" >Addition of lists</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X7C3DC8BE78DEECDE" >21 .13 -4 <span class="Heading" >Subtraction of lists</span ></a>
</span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap21_mj.html#X782ED7F27D8C7FC1" >21 .14 <span class="Heading" >Multiplicative Arithmetic for Lists</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X79A8A5627FD42FA5" >21 .14 -1 <span class="Heading" >One for lists</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X78C6C1E2849D303A" >21 .14 -2 <span class="Heading" >Inverse for lists</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X84FDB95179BFE4CD" >21 .14 -3 <span class="Heading" >Multiplication of lists</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X82EA2A5B786181C7" >21 .14 -4 <span class="Heading" >Division of lists</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X7A0FD70C80B95C00" >21 .14 -5 <span class="Heading" >mod for lists</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X84BB2DFB8432A1A4" >21 .14 -6 <span class="Heading" >Left quotients of lists</span ></a>
</span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap21_mj.html#X8676EFE67972FD06" >21 .15 <span class="Heading" >Mutability Status and List Arithmetic</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X80FDB1457FF582E7" >21 .15 -1 ListWithIdenticalEntries</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap21_mj.html#X8196FD4779BCCA0C" >21 .16 <span class="Heading" >Finding Positions in Lists</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X79975EC6783B4293" >21 .16 -1 Position</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X7FA9648883AE1B88" >21 .16 -2 Positions</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X7B4B10AE81602D4E" >21 .16 -3 PositionCanonical</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X7D2B25B484591506" >21 .16 -4 PositionNthOccurrence</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X7A122E848464E534" >21 .16 -5 PositionSorted</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X820BA44D85930EBF" >21 .16 -6 PositionSortedBy</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X78BFE9D78347C0DA" >21 .16 -7 PositionSet</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X7FD9C1D37F300206" >21 .16 -8 PositionMaximum</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X7E6C763A82C6153B" >21 .16 -9 PositionProperty</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X7DA94D278304EC3D" >21 .16 -10 PositionsProperty</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X86C9E5C3863B3C03" >21 .16 -11 PositionBound</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X819F71047AABEA2F" >21 .16 -12 PositionsBound</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X865EF45D87ED1384" >21 .16 -13 PositionNot</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X7F42E5AD87EC9D5A" >21 .16 -14 PositionNonZero</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X87A8C62A867D6DA4" >21 .16 -15 PositionSublist</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap21_mj.html#X7865747A7CCF5812" >21 .17 <span class="Heading" >Properties and Attributes for Lists</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X83F8EC7C7BF27EFC" >21 .17 -1 IsMatchingSublist</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X7FA892828252BB3B" >21 .17 -2 IsDuplicateFree</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X7BAA9B0E81D4A884" >21 .17 -3 IsSortedList</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X80CDAF45782E8DCB" >21 .17 -4 IsSSortedList</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X780769238600AFD1" >21 .17 -5 Length</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X7B55FB967CDEF468" >21 .17 -6 ConstantTimeAccessList</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap21_mj.html#X83E558E37D1B44D4" >21 .18 <span class="Heading" >Sorting Lists</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X7FE4975F8166884D" >21 .18 -1 Sort</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X791F2B2C7E9B9A46" >21 .18 -2 SortParallel</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X87287FCA81E2B06A" >21 .18 -3 Sortex</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X800209E881E7CECB" >21 .18 -4 SortingPerm</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap21_mj.html#X80ABC25582343910" >21 .19 <span class="Heading" >Sorted Lists and Sets</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X7B16AD597CB12305" ><code >21 .19 -1 \in</code ></a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X7B4C0FEE7CDF6F2A" >21 .19 -2 IsEqualSet</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X79B940567A849216" >21 .19 -3 IsSubsetSet</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X832C23CC7FCD8892" >21 .19 -4 AddSet</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X7FCA282E789A4F4B" >21 .19 -5 RemoveSet</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X7B3469CD7EFC1A87" >21 .19 -6 UniteSet</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X8473AA657FEC3D4D" >21 .19 -7 IntersectSet</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X80B427537EB07D09" >21 .19 -8 SubtractSet</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap21_mj.html#X7DF510F7848CBBFD" >21 .20 <span class="Heading" >Operations for Lists</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X840C55A77D1BB2E1" >21 .20 -1 Concatenation</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X7CB0A6AF87C7FAF7" >21 .20 -2 Compacted</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X7ECE9056792F28BA" >21 .20 -3 Collected</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X8727F2928467C2F9" >21 .20 -4 DuplicateFreeList</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X7F5D4DD87E4378AC" >21 .20 -5 AsDuplicateFreeList</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X7FA272D984EF82ED" >21 .20 -6 Flat</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X7C4FDB007C3F54A1" >21 .20 -7 Reversed</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X8057372F83374193" >21 .20 -8 Shuffle</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X8075FBDE7B81B4C8" >21 .20 -9 Apply</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X7EF6E2BC81DBF6FB" >21 .20 -10 Perform</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X8763882A7D65F979" >21 .20 -11 PermListList</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X82CE0DE8828E4303" >21 .20 -12 <span class="Heading" >Maximum</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X82F133EC7F89665F" >21 .20 -13 <span class="Heading" >Minimum</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X842851EB7E0969F7" >21 .20 -14 <span class="Heading" >MaximumList and MinimumList</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X7E1593B979BDF2CD" >21 .20 -15 <span class="Heading" >Cartesian</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X7E76F5A782184823" >21 .20 -16 <span class="Heading" >IteratorOfCartesianProduct</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X7B5A19098406347A" >21 .20 -17 Permuted</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X86CB7DCE8510F977" >21 .20 -18 List</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X7C86D7F7795125F0" >21 .20 -19 Filtered</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X8179B13D80E935FC" >21 .20 -20 Number</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X82801DFA84E11272" >21 .20 -21 First</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X7E5B62E780421CE9" >21 .20 -22 Last</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X7F06961278166671" >21 .20 -23 ForAll</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X7AF82E747A8BDA75" >21 .20 -24 ForAny</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X7E5C72F27B657948" >21 .20 -25 Product</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X7A04B71C84CFCC2D" >21 .20 -26 Sum</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X834E4DF57F3A20F0" >21 .20 -27 Iterated</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X7D150C2881881139" >21 .20 -28 ListN</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap21_mj.html#X805CA0B68029B47A" >21 .21 <span class="Heading" >Advanced List Manipulations</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X8258477D7F72171B" >21 .21 -1 ListX</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X7AC321B87A2DCAF5" >21 .21 -2 SetX</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X82B1411E7FBE925F" >21 .21 -3 SumX</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X7FB318B47D8783DA" >21 .21 -4 ProductX</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap21_mj.html#X79596BDE7CAF8491" >21 .22 <span class="Heading" >Ranges</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X86DDC2FF7A50FBEE" >21 .22 -1 IsRange</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X83896BC481536B07" >21 .22 -2 IsRangeRep</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X7D22B2298167A58F" >21 .22 -3 ConvertToRangeRep</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap21_mj.html#X7EA3ACE27E43D174" >21 .23 <span class="Heading" >Enumerators</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X7BB462C17962647F" >21 .23 -1 IsQuickPositionList</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap21_mj.html#X81ECC2077D88E112" >21 .24 <span class="Heading" >Plain Lists</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X8438CB908367254C" >21 .24 -1 PlainListCopy</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap21_mj.html#X87BA4EBF80F16B72" >21 .24 -2 IsPlistRep</a></span >
</div ></div >
</div >
<div class="ContChap" ><a href="chap22_mj.html#X7AC531DD79B6938E" >22 <span class="Heading" >Boolean Lists</span ></a>
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap22_mj.html#X7E7832B0804221AE" >22 .1 <span class="Heading" >IsBlist (Filter)</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap22_mj.html#X7BE078187A08DCEA" >22 .1 -1 IsBlist</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap22_mj.html#X7CC745317FE54C14" >22 .2 <span class="Heading" >Boolean Lists Representing Subsets</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap22_mj.html#X7C597B2D87CA2E6E" >22 .2 -1 BlistList</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap22_mj.html#X874BEF63785AB439" >22 .2 -2 ListBlist</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap22_mj.html#X85AD5EF77EFD7451" >22 .2 -3 SizeBlist</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap22_mj.html#X7BA42D03796ED4B3" >22 .2 -4 IsSubsetBlist</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap22_mj.html#X8100080382AECFF9" >22 .3 <span class="Heading" >Set Operations via Boolean Lists</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap22_mj.html#X7970BD3883C42D91" >22 .3 -1 <span class="Heading" >UnionBlist</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap22_mj.html#X86E1F8DE85E1EE1E" >22 .3 -2 <span class="Heading" >IntersectionBlist</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap22_mj.html#X7D6FC2C58725708C" >22 .3 -3 DifferenceBlist</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap22_mj.html#X8634D25D7B4C6151" >22 .4 <span class="Heading" >Function that Modify Boolean Lists</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap22_mj.html#X79815EB77CC8A389" >22 .4 -1 UniteBlist</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap22_mj.html#X7C86C8D3853BE5EB" >22 .4 -2 UniteBlistList</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap22_mj.html#X84EB70D37EB275DF" >22 .4 -3 IntersectBlist</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap22_mj.html#X7AA138407D5A3BAC" >22 .4 -4 SubtractBlist</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap22_mj.html#X830645EC846B2E3C" >22 .4 -5 MeetBlist</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap22_mj.html#X7F14FF35786DAEF3" >22 .4 -6 FlipBlist</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap22_mj.html#X7E9F6C197A79098F" >22 .4 -7 SetAllBlist</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap22_mj.html#X87ED45A88688AE8E" >22 .4 -8 ClearAllBlist</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap22_mj.html#X7C71B225841DFC0F" >22 .5 <span class="Heading" >More about Boolean Lists</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap22_mj.html#X8453ADDA810B4C03" >22 .5 -1 IsBlistRep</a></span >
</div ></div >
</div >
<div class="ContChap" ><a href="chap23_mj.html#X82C7E6CF7BA03391" >23 <span class="Heading" >Row Vectors</span ></a>
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap23_mj.html#X7E383689817D2371" >23 .1 <span class="Heading" >IsRowVector (Filter)</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap23_mj.html#X7DFB22A07836A7A9" >23 .1 -1 IsRowVector</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap23_mj.html#X85516C3179C229DB" >23 .2 <span class="Heading" >Operators for Row Vectors</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap23_mj.html#X785DC60D8482695D" >23 .2 -1 NormedRowVector</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap23_mj.html#X8679F7DD7DFCBD9C" >23 .3 <span class="Heading" >Row Vectors over Finite Fields</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap23_mj.html#X810E46927F9E8F75" >23 .3 -1 <span class="Heading" >ConvertToVectorRep</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap23_mj.html#X83D8F5BB80089279" >23 .3 -2 ImmutableVector</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap23_mj.html#X872E17FF829DB50F" >23 .3 -3 NumberFFVector</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap23_mj.html#X85C68AED805E4B9C" >23 .4 <span class="Heading" >Coefficient List Arithmetic</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap23_mj.html#X800EC03F7E0A5F23" >23 .4 -1 AddVector</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap23_mj.html#X7854B2B67E3FE2CA" >23 .4 -2 AddCoeffs</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap23_mj.html#X7BEF28C981C42E16" >23 .4 -3 MultVector</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap23_mj.html#X8264B3EE7D56EEDD" >23 .4 -4 CoeffsMod</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap23_mj.html#X7D287281781E16A2" >23 .5 <span class="Heading" >Shifting and Trimming Coefficient Lists</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap23_mj.html#X80465E9B7A38C176" >23 .5 -1 LeftShiftRowVector</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap23_mj.html#X822CCA4781D5C5EC" >23 .5 -2 RightShiftRowVector</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap23_mj.html#X78951C0E86D857B5" >23 .5 -3 ShrinkRowVector</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap23_mj.html#X85796B6079581023" >23 .5 -4 RemoveOuterCoeffs</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap23_mj.html#X7B63F1EB83FA0CF6" >23 .6 <span class="Heading" >Functions for Coding Theory</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap23_mj.html#X7C9F4D657F9BA5A1" >23 .6 -1 WeightVecFFE</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap23_mj.html#X85AA5C6587559C1C" >23 .6 -2 DistanceVecFFE</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap23_mj.html#X7F2F630984A9D3D6" >23 .6 -3 DistancesDistributionVecFFEsVecFFE</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap23_mj.html#X85135CEB86E61D49" >23 .6 -4 DistancesDistributionMatFFEVecFFE</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap23_mj.html#X82E5987E81487D18" >23 .6 -5 AClosestVectorCombinationsMatFFEVecFFE</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap23_mj.html#X7C88671678A2BEB4" >23 .6 -6 CosetLeadersMatFFE</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap23_mj.html#X87FEC1927B3A63C8" >23 .7 <span class="Heading" >Vectors as coefficients of polynomials</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap23_mj.html#X84DE99D57C29D47F" >23 .7 -1 ValuePol</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap23_mj.html#X8328088C807AFFAF" >23 .7 -2 ProductCoeffs</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap23_mj.html#X87248AA27F05BDCC" >23 .7 -3 ReduceCoeffs</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap23_mj.html#X7F74B1637CB13B7B" >23 .7 -4 ReduceCoeffsMod</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap23_mj.html#X825F8F357FB1BF56" >23 .7 -5 PowerModCoeffs</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap23_mj.html#X833EF7AE80CE8B3C" >23 .7 -6 ShiftedCoeffs</a></span >
</div ></div >
</div >
<div class="ContChap" ><a href="chap24_mj.html#X812CCAB278643A59" >24 <span class="Heading" >Matrices</span ></a>
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap24_mj.html#X801E1B5D7EC8DDD3" >24 .1 <span class="Heading" >InfoMatrix (Info Class)</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap24_mj.html#X78EC82D27B4191DA" >24 .1 -1 InfoMatrix</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap24_mj.html#X866E55A58164FAED" >24 .2 <span class="Heading" >Categories of Matrices</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap24_mj.html#X7E1AE46B862B185F" >24 .2 -1 IsMatrix</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap24_mj.html#X7CF42B8A845BC6A9" >24 .2 -2 IsOrdinaryMatrix</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap24_mj.html#X86EC33E17DD12D0E" >24 .2 -3 IsLieMatrix</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap24_mj.html#X7899335779A39A95" >24 .3 <span class="Heading" >Operators for Matrices</span ></a>
</span >
</div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap24_mj.html#X7F5AD28E869B66CB" >24 .4 <span class="Heading" >Properties and Attributes of Matrices</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap24_mj.html#X83A9DC2085D3A972" >24 .4 -1 DimensionsMat</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap24_mj.html#X80AE547B8095A5CB" >24 .4 -2 DefaultFieldOfMatrix</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap24_mj.html#X784EC2777C06AFE4" >24 .4 -3 TraceMatrix</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap24_mj.html#X8488D69A7ADDB4E2" >24 .4 -4 DeterminantMatrix</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap24_mj.html#X824B5DC2875118B3" >24 .4 -5 DeterminantMatrixDestructive</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap24_mj.html#X80693FAB7D541804" >24 .4 -6 DeterminantMatrixDivFree</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap24_mj.html#X7F8D25897EC1630B" >24 .4 -7 IsEmptyMatrix</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap24_mj.html#X848B80437CE65FF3" >24 .4 -8 IsMonomialMatrix</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap24_mj.html#X7EEC8E768178696E" >24 .4 -9 IsDiagonalMatrix</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap24_mj.html#X8740E71C799C0BCC" >24 .4 -10 IsUpperTriangularMatrix</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap24_mj.html#X853A5B988306DBFE" >24 .4 -11 IsLowerTriangularMatrix</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap24_mj.html#X823FB2398697B957" >24 .5 <span class="Heading" >Matrix Constructions</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap24_mj.html#X7DB902CE848D1524" >24 .5 -1 IdentityMat</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap24_mj.html#X86D343A77D9B3D4D" >24 .5 -2 NullMat</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap24_mj.html#X8508A7EA812BA0CC" >24 .5 -3 EmptyMatrix</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap24_mj.html#X81042E7A7F247ADE" >24 .5 -4 DiagonalMat</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap24_mj.html#X87BADF217C19CBE1" >24 .5 -5 DiagonalMatrix</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap24_mj.html#X806C62A67A7D5379" >24 .5 -6 PermutationMat</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap24_mj.html#X7C52A38C79C36C35" >24 .5 -7 TransposedMatImmutable</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap24_mj.html#X7DBB40847E2B6252" >24 .5 -8 TransposedMatDestructive</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap24_mj.html#X8634C79E7DB22934" >24 .5 -9 KroneckerProduct</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap24_mj.html#X845EC4D18054D140" >24 .5 -10 ReflectionMat</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap24_mj.html#X7DEBC9967DFDFC18" >24 .5 -11 PrintArray</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap24_mj.html#X79CC5F568252D341" >24 .6 <span class="Heading" >Random Matrices</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap24_mj.html#X7F957F0280A87961" >24 .6 -1 RandomMat</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap24_mj.html#X7C939B4A7EDF015D" >24 .6 -2 RandomInvertibleMat</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap24_mj.html#X84743732846ACB44" >24 .6 -3 RandomUnimodularMat</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap24_mj.html#X85485DCE809E323A" >24 .7 <span class="Heading" >Matrices Representing Linear Equations and the Gaussian Algorithm</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap24_mj.html#X7A995A74838950E6" >24 .7 -1 RankMatrix</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap24_mj.html#X7BA26C3387AB434E" >24 .7 -2 TriangulizedMat</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap24_mj.html#X8384CA8E7B3850D3" >24 .7 -3 TriangulizeMat</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap24_mj.html#X7DA0D5887DB12DC4" >24 .7 -4 NullspaceMat</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap24_mj.html#X87684B0F7AB7B7DB" >24 .7 -5 NullspaceMatDestructive</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap24_mj.html#X838A519C7CD2969E" >24 .7 -6 SolutionMat</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap24_mj.html#X7A7880D27CE7C1FE" >24 .7 -7 SolutionMatDestructive</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap24_mj.html#X7AB5AC547809F999" >24 .7 -8 BaseFixedSpace</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap24_mj.html#X871FCAA97C60B2BA" >24 .8 <span class="Heading" >Eigenvectors and eigenvalues</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap24_mj.html#X7A2462CC7B0C9D66" >24 .8 -1 GeneralisedEigenvalues</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap24_mj.html#X845CA0457D65876D" >24 .8 -2 GeneralisedEigenspaces</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap24_mj.html#X8413C6FB7CEE9D59" >24 .8 -3 Eigenvalues</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap24_mj.html#X7A6B047281B52FD7" >24 .8 -4 Eigenspaces</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap24_mj.html#X8506584579D4EA18" >24 .8 -5 Eigenvectors</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap24_mj.html#X7E5405D085661B29" >24 .9 <span class="Heading" >Elementary Divisors</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap24_mj.html#X7AC4D74F81908109" >24 .9 -1 ElementaryDivisorsMat</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap24_mj.html#X7AA1C9047B102204" >24 .9 -2 ElementaryDivisorsTransformationsMat</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap24_mj.html#X85819D3F7A582180" >24 .9 -3 DiagonalizeMat</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap24_mj.html#X7CA6B51D7AE3172B" >24 .10 <span class="Heading" >Echelonized Matrices</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap24_mj.html#X7D5D6BD07B7E981B" >24 .10 -1 SemiEchelonMat</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap24_mj.html#X8251F6F57D346385" >24 .10 -2 SemiEchelonMatDestructive</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap24_mj.html#X7EFD1DB5861A54F0" >24 .10 -3 SemiEchelonMatTransformation</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap24_mj.html#X827D7971800DB661" >24 .10 -4 SemiEchelonMats</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap24_mj.html#X808F493B839BC7A6" >24 .10 -5 SemiEchelonMatsDestructive</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap24_mj.html#X86B0D4A886BC0C6E" >24 .11 <span class="Heading" >Matrices as Basis of a Row Space</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap24_mj.html#X7AD6B5F5794D9E46" >24 .11 -1 BaseMat</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap24_mj.html#X78B094597E382A5F" >24 .11 -2 BaseMatDestructive</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap24_mj.html#X78B94EFF87A455BE" >24 .11 -3 BaseOrthogonalSpaceMat</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap24_mj.html#X7AFF8BCF80C88B45" >24 .11 -4 SumIntersectionMat</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap24_mj.html#X8245D54F7AC532EB" >24 .11 -5 BaseSteinitzVectors</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap24_mj.html#X79D5E53685F0FBEE" >24 .12 <span class="Heading" >Triangular Matrices</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap24_mj.html#X7A9139D686ACB7D8" >24 .12 -1 DiagonalOfMatrix</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap24_mj.html#X84A78C057F9DAE5E" >24 .12 -2 UpperSubdiagonal</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap24_mj.html#X84D74DEA798A9094" >24 .12 -3 DepthOfUpperTriangularMatrix</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap24_mj.html#X85B403857F2855F7" >24 .13 <span class="Heading" >Matrices as Linear Mappings</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap24_mj.html#X87FA0A727CDB060B" >24 .13 -1 CharacteristicPolynomial</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap24_mj.html#X7B52560C792C1A0F" >24 .13 -2 RationalCanonicalFormTransform</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap24_mj.html#X83F55D4E79BA5D1B" >24 .13 -3 JordanDecomposition</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap24_mj.html#X85923C107A4569D0" >24 .13 -4 BlownUpMat</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap24_mj.html#X82AC277D84EC5749" >24 .13 -5 BlownUpVector</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap24_mj.html#X7E06762479A00DF4" >24 .13 -6 CompanionMatrix</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap24_mj.html#X873822B6830CE367" >24 .14 <span class="Heading" >Matrices over Finite Fields</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap24_mj.html#X7DED2522828B6C30" >24 .14 -1 ImmutableMatrix</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap24_mj.html#X8587A62F818AA0D6" >24 .14 -2 ConvertToMatrixRep</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap24_mj.html#X84A76F7A7B4166BC" >24 .14 -3 ProjectiveOrder</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap24_mj.html#X847ADC6779E33A1C" >24 .14 -4 SimultaneousEigenvalues</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap24_mj.html#X8593A5337D3B2C70" >24 .15 <span class="Heading" >Inverse and Nullspace of an Integer Matrix Modulo an Ideal</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap24_mj.html#X7D8D1E0E83C7F872" >24 .15 -1 InverseMatMod</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap24_mj.html#X7D7DF873826A7C20" >24 .15 -2 BasisNullspaceModN</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap24_mj.html#X86AE919983B242E2" >24 .15 -3 NullspaceModQ</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap24_mj.html#X787DF5F07DC7D86E" >24 .16 <span class="Heading" >Special Multiplication Algorithms for Matrices over GF(2 )</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap24_mj.html#X7C0C26027FAE0C83" >24 .16 -1 PROD_GF2MAT_GF2MAT_SIMPLE</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap24_mj.html#X81965B7D7F45E088" >24 .16 -2 PROD_GF2MAT_GF2MAT_ADVANCED</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap24_mj.html#X7F8A71F38201A250" >24 .17 <span class="Heading" >Block Matrices</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap24_mj.html#X7D675B3C79CF8871" >24 .17 -1 AsBlockMatrix</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap24_mj.html#X8633538685551E7A" >24 .17 -2 BlockMatrix</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap24_mj.html#X83FAF4158180041F" >24 .17 -3 MatrixByBlockMatrix</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap24_mj.html#X782F2EBF80C431D0" >24 .18 <span class="Heading" >Linear Programming</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap24_mj.html#X845D5F8D7D905CB8" >24 .18 -1 SimplexMethod</a></span >
</div ></div >
</div >
<div class="ContChap" ><a href="chap25_mj.html#X8414F20D8412DDA4" >25 <span class="Heading" >Integral matrices and lattices</span ></a>
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap25_mj.html#X786A64B983339767" >25 .1 <span class="Heading" >Linear equations over the integers and Integral Matrices</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap25_mj.html#X792315717F5B0294" >25 .1 -1 NullspaceIntMat</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap25_mj.html#X7D749F317DBD1E69" >25 .1 -2 SolutionIntMat</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap25_mj.html#X82CECB6E7D515CD2" >25 .1 -3 SolutionNullspaceIntMat</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap25_mj.html#X7F66E8EA7D1AA2C1" >25 .1 -4 BaseIntMat</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap25_mj.html#X8771349D865C9179" >25 .1 -5 BaseIntersectionIntMats</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap25_mj.html#X7848EF9F83D491C1" >25 .1 -6 ComplementIntMat</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap25_mj.html#X8143C1448069D846" >25 .2 <span class="Heading" >Normal Forms over the Integers</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap25_mj.html#X783CEC847D81F22A" >25 .2 -1 TriangulizedIntegerMat</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap25_mj.html#X7DBE174E8625AFA5" >25 .2 -2 TriangulizedIntegerMatTransform</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap25_mj.html#X78CD40A687FE2311" >25 .2 -3 TriangulizeIntegerMat</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap25_mj.html#X8535AC327932B89F" >25 .2 -4 HermiteNormalFormIntegerMat</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap25_mj.html#X7FDA78F979574ACC" >25 .2 -5 HermiteNormalFormIntegerMatTransform</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap25_mj.html#X87089FEC7FBEEA8F" >25 .2 -6 SmithNormalFormIntegerMat</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap25_mj.html#X839C1F9E87273A93" >25 .2 -7 SmithNormalFormIntegerMatTransforms</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap25_mj.html#X80EF38737F6D61DB" >25 .2 -8 DiagonalizeIntMat</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap25_mj.html#X81FB746E82BE6CDA" >25 .2 -9 NormalFormIntMat</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap25_mj.html#X8221694D7C99197A" >25 .2 -10 AbelianInvariantsOfList</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap25_mj.html#X80F6990983C979FB" >25 .3 <span class="Heading" >Determinant of an integer matrix</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap25_mj.html#X787599E087F4C0BA" >25 .3 -1 DeterminantIntMat</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap25_mj.html#X79F2EFEC7C3EA80C" >25 .4 <span class="Heading" >Decompositions</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap25_mj.html#X7911A60384C511AB" >25 .4 -1 Decomposition</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap25_mj.html#X843A976787600F13" >25 .4 -2 LinearIndependentColumns</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap25_mj.html#X8285776B7DD86925" >25 .4 -3 PadicCoefficients</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap25_mj.html#X7F5C619B7A9C3EB9" >25 .4 -4 IntegralizedMat</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap25_mj.html#X8512FB69824AE353" >25 .4 -5 DecompositionInt</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap25_mj.html#X839C6ABE829355F4" >25 .5 <span class="Heading" >Lattice Reduction</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap25_mj.html#X7D0FCEF8859E8637" >25 .5 -1 LLLReducedBasis</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap25_mj.html#X86D23EB885EDE60E" >25 .5 -2 LLLReducedGramMat</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap25_mj.html#X871DB00B803D5177" >25 .6 <span class="Heading" >Orthogonal Embeddings</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap25_mj.html#X842280C2808FF05D" >25 .6 -1 OrthogonalEmbeddings</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap25_mj.html#X79A692B6819353D4" >25 .6 -2 ShortestVectors</a></span >
</div ></div >
</div >
<div class="ContChap" ><a href="chap26_mj.html#X856C23B87E50F118" >26 <span class="Heading" >Vector and Matrix Objects</span ></a>
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap26_mj.html#X7A7275C27EC61ACE" >26 .1 <span class="Heading" >Concepts and Rules for Vector and Matrix Objects</span ></a>
</span >
</div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap26_mj.html#X7C6CDBFE7EB083A5" >26 .2 <span class="Heading" >Categories of Vector and Matrix Objects</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap26_mj.html#X7D963FCC7E849BE0" >26 .2 -1 IsVectorObj</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap26_mj.html#X7E7617A0781D1E4B" >26 .2 -2 IsMatrixObj</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap26_mj.html#X7D1ACCBE7E9CF501" >26 .2 -3 IsMatrixOrMatrixObj</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap26_mj.html#X78CD88A283330E72" >26 .2 -4 IsRowListMatrix</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap26_mj.html#X877A706186C89ADB" >26 .3 <span class="Heading" >Defining Attributes of Vector and Matrix Objects</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap26_mj.html#X8662026C7CCDB446" >26 .3 -1 <span class="Heading" >BaseDomain</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap26_mj.html#X85ABF33684865ED5" >26 .3 -2 <span class="Heading" >ConstructingFilter</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap26_mj.html#X818702FD7A2E9D90" >26 .3 -3 <span class="Heading" >CompatibleVectorFilter</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap26_mj.html#X828BA5E1849E3D06" >26 .3 -4 Length</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap26_mj.html#X820ED34380C10E19" >26 .3 -5 <span class="Heading" >NumberRows and NumberColumns</span ></a>
</span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap26_mj.html#X7BD7D2837BFDE649" >26 .4 <span class="Heading" >Constructing Vector and Matrix Objects</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap26_mj.html#X860E84397BD148E9" >26 .4 -1 <span class="Heading" >NewVector and NewZeroVector</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap26_mj.html#X79A6544D86261E82" >26 .4 -2 <span class="Heading" >Vector</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap26_mj.html#X7DBA8BF5844F3281" >26 .4 -3 <span class="Heading" >ZeroVector</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap26_mj.html#X7AD2210B8047FB01" >26 .4 -4 <span class="Heading" >NewMatrix, NewZeroMatrix, NewIdentityMatrix</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap26_mj.html#X879384D479EB1D82" >26 .4 -5 <span class="Heading" >Matrix</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap26_mj.html#X838F5B6C7C87C8E1" >26 .4 -6 <span class="Heading" >ZeroMatrix</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap26_mj.html#X7D807ABC7FCB4E77" >26 .4 -7 <span class="Heading" >IdentityMatrix</span ></a>
</span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap26_mj.html#X7C7F5250855C4371" >26 .5 <span class="Heading" >Operations for Base Domains of Vector and Matrix Objects</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap26_mj.html#X85D7A6A782B21E5C" >26 .5 -1 <span class="Heading" >OneOfBaseDomain and ZeroOfBaseDomain</span ></a>
</span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap26_mj.html#X7954E20987E0B260" >26 .6 <span class="Heading" >Operations for Vector and Matrix Objects</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap26_mj.html#X7FFC60A27FE6FA97" >26 .6 -1 <span class="Heading" >Comparison of Vector and Matrix Objects</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap26_mj.html#X7FBBE79478012648" >26 .6 -2 <span class="Heading" >Unpack</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap26_mj.html#X85E896F67CE2F925" >26 .6 -3 <span class="Heading" >ChangedBaseDomain</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap26_mj.html#X83DD8B39864A2C94" >26 .6 -4 <span class="Heading" >Randomize</span ></a>
</span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap26_mj.html#X7FE662477F36A21F" >26 .7 <span class="Heading" >List Like Operations for Vector Objects</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap26_mj.html#X7D5DF49C7ADB6986" >26 .7 -1 <span class="Heading" >Element Access and Assignment for Vector Objects</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap26_mj.html#X7A21731C83EE3BB0" >26 .7 -2 PositionNonZero</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap26_mj.html#X7ABDE1B685A78326" >26 .7 -3 PositionLastNonZero</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap26_mj.html#X790013817E314B2D" >26 .7 -4 ListOp</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap26_mj.html#X7FDF7655852AEAAE" >26 .8 <span class="Heading" >Arithmetical Operations for Vector Objects</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap26_mj.html#X7F8CE23F7A250072" >26 .8 -1 <span class="Heading" >Unary Arithmetical Operations for Vector Objects</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap26_mj.html#X85A815CA790094CC" >26 .8 -2 <span class="Heading" >Binary Arithmetical Operations for Vector Objects</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap26_mj.html#X876090A684E71C93" >26 .8 -3 AddVector</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap26_mj.html#X8039D013817317C3" >26 .8 -4 MultVector</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap26_mj.html#X7BE9D278852C13BC" >26 .9 <span class="Heading" >Operations for Vector Objects</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap26_mj.html#X7AC470557EC90714" >26 .9 -1 <span class="Heading" >ConcatenationOfVectors</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap26_mj.html#X7DBE956E7F9C700E" >26 .9 -2 ExtractSubVector</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap26_mj.html#X80EC354D78D7B5A6" >26 .9 -3 CopySubVector</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap26_mj.html#X866366E587991171" >26 .9 -4 WeightOfVector</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap26_mj.html#X81ACAE017C00F782" >26 .9 -5 DistanceOfVectors</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap26_mj.html#X81CC13CA7A1FF4AA" >26 .10 <span class="Heading" >Arithmetical Operations for Matrix Objects</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap26_mj.html#X819F87A07DA7E2DC" >26 .10 -1 <span class="Heading" >Unary Arithmetical Operations for Matrix Objects</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap26_mj.html#X7BBB70557A7A9591" >26 .10 -2 <span class="Heading" >Binary Arithmetical Operations for Matrix Objects</span ></a>
</span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap26_mj.html#X85FAB7E778A71C19" >26 .11 <span class="Heading" >Operations for Matrix Objects</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap26_mj.html#X870FBE817C884AB5" >26 .11 -1 MatElm</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap26_mj.html#X7C33059984635480" >26 .11 -2 SetMatElm</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap26_mj.html#X838B45F7790E9FDF" >26 .11 -3 ExtractSubMatrix</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap26_mj.html#X793CD4637F237915" >26 .11 -4 MutableCopyMatrix</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap26_mj.html#X7ED9E5D4809E3B50" >26 .11 -5 CopySubMatrix</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap26_mj.html#X809A6B3F7EA5E7D8" >26 .11 -6 CompatibleVector</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap26_mj.html#X7EE70D5A81E9ED72" >26 .11 -7 RowsOfMatrix</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap26_mj.html#X7E06762479A00DF4" >26 .11 -8 <span class="Heading" >CompanionMatrix</span ></a>
</span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap26_mj.html#X7D40EE2084A6C976" >26 .12 <span class="Heading" >Operations for Row List Matrix Objects</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap26_mj.html#X82C4FCFA808010F8" >26 .12 -1 <span class="Heading" >List Access for a Row List Matrix</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap26_mj.html#X7F89BB2482D28AAE" >26 .12 -2 <span class="Heading" >List Assignment for a Row List Matrix</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap26_mj.html#X807518367C96516F" >26 .12 -3 <span class="Heading" >Sublist Access for a Row List Matrix</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap26_mj.html#X8371789181FA136B" >26 .12 -4 <span class="Heading" >Sublist Assignment for a Row List Matrix</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap26_mj.html#X872E63867803ED78" ><code >26 .12 -5 IsBound<span >\</span >[<span >\</span >]</code ></a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap26_mj.html#X79328CB280C71DDB" ><code >26 .12 -6 Unbind<span >\</span >[<span >\</span >]</code ></a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap26_mj.html#X7BDD838579E4D2D6" >26 .12 -7 Add</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap26_mj.html#X86E355D07A41C025" >26 .12 -8 Remove</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap26_mj.html#X82D0359B81F8D442" >26 .12 -9 Append</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap26_mj.html#X7E234F717BE333EA" >26 .12 -10 ShallowCopy</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap26_mj.html#X7E9F095E85DED480" >26 .12 -11 ListOp</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap26_mj.html#X7B86A8487B12F9BD" >26 .13 <span class="Heading" >Basic operations for row/column reductions</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap26_mj.html#X7B3997D37CC44FCA" >26 .13 -1 MultMatrixRowLeft</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap26_mj.html#X794636447E8C5553" >26 .13 -2 MultMatrixRowRight</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap26_mj.html#X80AF7B267E6B9CE0" >26 .13 -3 MultMatrixColumnRight</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap26_mj.html#X843DAFE37F347471" >26 .13 -4 MultMatrixColumnLeft</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap26_mj.html#X8662EB748629502F" >26 .13 -5 AddMatrixRowsLeft</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap26_mj.html#X7CD05EE984614AB6" >26 .13 -6 AddMatrixRowsRight</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap26_mj.html#X7B1E1E417CA267A3" >26 .13 -7 AddMatrixColumnsRight</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap26_mj.html#X85ECB8C87DFD8F32" >26 .13 -8 AddMatrixColumnsLeft</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap26_mj.html#X87CCA3117F6B3F0D" >26 .13 -9 SwapMatrixRows</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap26_mj.html#X824C8A347EB9D499" >26 .13 -10 SwapMatrixColumns</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap26_mj.html#X7BEE647484978886" >26 .14 <span class="Heading" >Implementing New Vector and Matrix Objects Types</span ></a>
</span >
</div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap26_mj.html#X82EEE1D37A94F807" >26 .15 <span class="Heading" >Available Representations of Vector Objects</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap26_mj.html#X7C8050938691A914" >26 .15 -1 IsGF2VectorRep</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap26_mj.html#X82A643007EC6D1CA" >26 .15 -2 Is8BitVectorRep</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap26_mj.html#X83262B7085FA94E3" >26 .15 -3 IsPlistVectorRep</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap26_mj.html#X8730DB7D7E7DA883" >26 .15 -4 IsZmodnZVectorRep</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap26_mj.html#X7CFD844C7D80D541" >26 .16 <span class="Heading" >Available Representations of Matrix Objects</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap26_mj.html#X7F6078FF81E912E7" >26 .16 -1 IsGF2MatrixRep</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap26_mj.html#X81466B6C7CAC3A7B" >26 .16 -2 Is8BitMatrixRep</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap26_mj.html#X80C6031C7DB31A15" >26 .16 -3 IsPlistMatrixRep</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap26_mj.html#X84D0F3117DA86850" >26 .16 -4 IsZmodnZMatrixRep</a></span >
</div ></div >
</div >
<div class="ContChap" ><a href="chap27_mj.html#X7D28329B7EDB8F47" >27 <span class="Heading" >Strings and Characters</span ></a>
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap27_mj.html#X7A90690B78260194" >27 .1 <span class="Heading" >IsChar and IsString</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap27_mj.html#X80CFAE128560E064" >27 .1 -1 IsChar</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap27_mj.html#X78723B5D795A3B6D" >27 .1 -2 IsString</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap27_mj.html#X7B1B45C587A72F96" >27 .1 -3 <span class="Heading" >Strings As Lists</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap27_mj.html#X7EA6CA7486D7E9DD" >27 .1 -4 <span class="Heading" >Printing Strings</span ></a>
</span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap27_mj.html#X82E5F5AB818F32DB" >27 .2 <span class="Heading" >Special Characters</span ></a>
</span >
</div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap27_mj.html#X7E70384E7D0B7083" >27 .3 <span class="Heading" >Triple Quoted Strings</span ></a>
</span >
</div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap27_mj.html#X82AEC07487C45ECD" >27 .4 <span class="Heading" >Internally Represented Strings</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap27_mj.html#X7A17EDF8785C9F58" >27 .4 -1 IsStringRep</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap27_mj.html#X7CE2415F7FEC5809" >27 .4 -2 ConvertToStringRep</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap27_mj.html#X7FFC464683CC8023" >27 .4 -3 CopyToStringRep</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap27_mj.html#X7D944D507CBB24CD" >27 .4 -4 IsEmptyString</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap27_mj.html#X836078DC829A8221" >27 .4 -5 EmptyString</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap27_mj.html#X7DA671FC7F490C16" >27 .4 -6 CharsFamily</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap27_mj.html#X82F980A17FE84AA4" >27 .5 <span class="Heading" >Recognizing Characters</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap27_mj.html#X78566FD57B95ECBE" >27 .5 -1 IsDigitChar</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap27_mj.html#X854114A97BAFEAEA" >27 .5 -2 IsLowerAlphaChar</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap27_mj.html#X87B1A13D81353AD8" >27 .5 -3 IsUpperAlphaChar</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap27_mj.html#X84634DF67A431D26" >27 .5 -4 IsAlphaChar</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap27_mj.html#X8127954B79B8A0DA" >27 .6 <span class="Heading" >Comparisons of Strings</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap27_mj.html#X79538F138286739A" ><code >27 .6 -1 \=</code ></a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap27_mj.html#X8129E3A785F60093" ><code >27 .6 -2 \<</code ></a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap27_mj.html#X7E72717A82A309F5" >27 .7 <span class="Heading" >Operations to Produce or Manipulate Strings</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap27_mj.html#X792FB3A1849FD739" >27 .7 -1 DisplayString</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap27_mj.html#X8482132779EA7A23" >27 .7 -2 DEFAULTDISPLAYSTRING</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap27_mj.html#X7803FBCA79DB5529" >27 .7 -3 ViewString</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap27_mj.html#X7BBDF9D383595425" >27 .7 -4 DEFAULTVIEWSTRING</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap27_mj.html#X7B3CC87285DEC23D" >27 .7 -5 PrintString</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap27_mj.html#X81FB5BE27903EC32" >27 .7 -6 String</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap27_mj.html#X86AACCE987F74FA5" >27 .7 -7 StripLineBreakCharacters</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap27_mj.html#X865FBB7E788017DD" >27 .7 -8 HexStringInt</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap27_mj.html#X7BB1059185AB4F84" >27 .7 -9 StringPP</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap27_mj.html#X79C8280A853D8FA9" >27 .7 -10 WordAlp</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap27_mj.html#X798A0F35852ABDAD" >27 .7 -11 LowercaseString</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap27_mj.html#X87A2F2557DE7EE08" >27 .7 -12 LowercaseChar</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap27_mj.html#X7E7E5F5B7FED56A0" >27 .7 -13 UppercaseString</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap27_mj.html#X81E0AEE687200505" >27 .7 -14 UppercaseChar</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap27_mj.html#X86E897D486DCFEAB" >27 .7 -15 SplitString</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap27_mj.html#X864F0A9078D4DE0E" >27 .7 -16 ReplacedString</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap27_mj.html#X806379367A53D171" >27 .7 -17 NormalizeWhitespace</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap27_mj.html#X8685DE9386E57771" >27 .7 -18 NormalizedWhitespace</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap27_mj.html#X86EBB6EB829723E4" >27 .7 -19 RemoveCharacters</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap27_mj.html#X84624FEB825EC4B5" >27 .7 -20 JoinStringsWithSeparator</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap27_mj.html#X79F8FFC5876D854A" >27 .7 -21 Chomp</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap27_mj.html#X855820848179CC28" >27 .7 -22 StartsWith</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap27_mj.html#X8235AD797868E872" >27 .7 -23 StringFormatted</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap27_mj.html#X7848A9D878FD59BB" >27 .7 -24 NumbersString</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap27_mj.html#X787EAB117816578E" >27 .7 -25 StringNumbers</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap27_mj.html#X82975B6480932683" >27 .7 -26 StringOfMemoryAmount</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap27_mj.html#X844BDC8578A3B508" >27 .8 <span class="Heading" >Character Conversion</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap27_mj.html#X826D95D680F87D23" >27 .8 -1 IntChar</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap27_mj.html#X87B6C1AF7E4A6639" >27 .8 -2 CharInt</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap27_mj.html#X8159CE81798DDA76" >27 .8 -3 SIntChar</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap27_mj.html#X78E6611A829DDA3E" >27 .8 -4 CharSInt</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap27_mj.html#X78D9BD857F890C0A" >27 .9 <span class="Heading" >Operations to Evaluate Strings</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap27_mj.html#X7B6D118184F692A0" >27 .9 -1 Int</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap27_mj.html#X87AD395584294FF2" >27 .9 -2 Rat</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap27_mj.html#X796D366B7DDEFF67" >27 .9 -3 IntHexString</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap27_mj.html#X7C0C29C87CBA97B7" >27 .9 -4 Ordinal</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap27_mj.html#X7DE4CCD285440659" >27 .9 -5 EvalString</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap27_mj.html#X7D4B9D7A7995C55D" >27 .9 -6 CrcString</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap27_mj.html#X7873D1F28779B490" >27 .9 -7 HexSHA256</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap27_mj.html#X83FF17E782E6FFF3" >27 .9 -8 Pluralize</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap27_mj.html#X78F20AA1804D524F" >27 .10 <span class="Heading" >Calendar Arithmetic</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap27_mj.html#X87BA46787FF000E8" >27 .10 -1 DaysInYear</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap27_mj.html#X8791B0B386D59ADB" >27 .10 -2 DaysInMonth</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap27_mj.html#X7CED84C07CD5E2CF" >27 .10 -3 DMYDay</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap27_mj.html#X7A79DEE07A41B8EF" >27 .10 -4 DayDMY</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap27_mj.html#X87D03FC0809DB6EC" >27 .10 -5 WeekDay</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap27_mj.html#X7C74C33784CDED6C" >27 .10 -6 StringDate</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap27_mj.html#X84A6A2637FB35A32" >27 .10 -7 HMSMSec</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap27_mj.html#X879461D77C81100B" >27 .10 -8 SecHMSM</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap27_mj.html#X802469C47F886A59" >27 .10 -9 StringTime</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap27_mj.html#X870A71D47B0E936E" >27 .10 -10 SecondsDMYhms</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap27_mj.html#X78AF8EA887532B5B" >27 .10 -11 DMYhmsSeconds</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap27_mj.html#X78024C8087F3E07F" >27 .11 <span class="Heading" >Obtaining LaTeX Representations of Objects</span ></a>
</span >
</div >
</div >
<div class="ContChap" ><a href="chap28_mj.html#X867203C5877489A2" >28 <span class="Heading" >Dictionaries and General Hash Tables</span ></a>
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap28_mj.html#X81560C4083E27955" >28 .1 <span class="Heading" >Using Dictionaries</span ></a>
</span >
</div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap28_mj.html#X7B571EA282AF70D7" >28 .2 <span class="Heading" >Dictionaries</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap28_mj.html#X7E78E3E983A5C895" >28 .2 -1 NewDictionary</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap28_mj.html#X86BD015B7B889329" >28 .3 <span class="Heading" >Dictionaries via Binary Lists</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap28_mj.html#X865D5BE1830A448D" >28 .3 -1 DictionaryByPosition</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap28_mj.html#X87F7247E784021C2" >28 .3 -2 IsDictionary</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap28_mj.html#X7D776BC67ABDDCCE" >28 .3 -3 IsLookupDictionary</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap28_mj.html#X86C4F0507AD98B8A" >28 .3 -4 AddDictionary</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap28_mj.html#X808C885D7E267285" >28 .3 -5 KnowsDictionary</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap28_mj.html#X863706BF847A47EB" >28 .3 -6 LookupDictionary</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap28_mj.html#X8444087381BBA88A" >28 .4 <span class="Heading" >General Hash Tables</span ></a>
</span >
</div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap28_mj.html#X85CD6C9B85DE7C54" >28 .5 <span class="Heading" >Hash keys</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap28_mj.html#X79DEEB5783513838" >28 .5 -1 DenseIntKey</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap28_mj.html#X87FC10AC81E5F6BA" >28 .5 -2 SparseIntKey</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap28_mj.html#X84D1A83C8247E7FB" >28 .6 <span class="Heading" >Dense hash tables</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap28_mj.html#X874FAA447930C7DA" >28 .6 -1 DenseHashTable</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap28_mj.html#X7FDB74417A19E674" >28 .7 <span class="Heading" >Sparse hash tables</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap28_mj.html#X8757D3A785290640" >28 .7 -1 SparseHashTable</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap28_mj.html#X80FDDF957887B4FC" >28 .7 -2 DoubleHashArraySize</a></span >
</div ></div >
</div >
<div class="ContChap" ><a href="chap29_mj.html#X7AA1073C7E943DD7" >29 <span class="Heading" >Records</span ></a>
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap29_mj.html#X864F92347B5A3FF0" >29 .1 <span class="Heading" >IsRecord and RecNames</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap29_mj.html#X782A998E7D9EC406" >29 .1 -1 IsRecord</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap29_mj.html#X837F1E1F866FB1A0" >29 .1 -2 RecNames</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap29_mj.html#X7EAAE25D7A17F778" >29 .2 <span class="Heading" >Accessing Record Elements</span ></a>
</span >
</div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap29_mj.html#X806DE3BD78742CA4" >29 .3 <span class="Heading" >Record Assignment</span ></a>
</span >
</div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap29_mj.html#X86BC2672803863FB" >29 .4 <span class="Heading" >Identical Records</span ></a>
</span >
</div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap29_mj.html#X83A7E6607B1D63BC" >29 .5 <span class="Heading" >Comparisons of Records</span ></a>
</span >
</div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap29_mj.html#X79BE8D0E829E7ACE" >29 .6 <span class="Heading" >IsBound and Unbind for Records</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap29_mj.html#X7A13E8F87CAAA0AF" >29 .6 -1 IsBound</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap29_mj.html#X7CA9AEFE7DB71604" >29 .6 -2 Unbind</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap29_mj.html#X784897E180815EDA" >29 .7 <span class="Heading" >Record Access Operations</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap29_mj.html#X87BF90FA7F7A3B1B" >29 .7 -1 NameRNam</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap29_mj.html#X78199B6B84A017B9" >29 .7 -2 RNamObj</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap29_mj.html#X7821AC097821AC09" ><code >29 .7 -3 \.</code ></a></span >
</div ></div >
</div >
<div class="ContChap" ><a href="chap30_mj.html#X8050A8037984E5B6" >30 <span class="Heading" >Collections</span ></a>
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap30_mj.html#X8084F03A78ABD4F8" >30 .1 <span class="Heading" >IsCollection (Filter)</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap30_mj.html#X79C9FC7F86E2738C" >30 .1 -1 IsCollection</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap30_mj.html#X85D8D8F684B02DDF" >30 .2 <span class="Heading" >Collection Families</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap30_mj.html#X84E5A67E87D8DD66" >30 .2 -1 CollectionsFamily</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap30_mj.html#X856AC2DF7F7CBAAF" >30 .2 -2 IsCollectionFamily</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap30_mj.html#X864BB3748546F63F" >30 .2 -3 ElementsFamily</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap30_mj.html#X78C38017804B2EA7" >30 .2 -4 CategoryCollections</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap30_mj.html#X85ABC1B4829778C7" >30 .2 -5 DeclareCategoryCollections</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap30_mj.html#X7C3722DF8736FFDB" >30 .3 <span class="Heading" >Lists and Collections</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap30_mj.html#X877128A77826DD69" >30 .3 -1 IsListOrCollection</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap30_mj.html#X7EF8910F82B45EC7" >30 .3 -2 Enumerator</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap30_mj.html#X80CD7DDC7D0C60D5" >30 .3 -3 EnumeratorSorted</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap30_mj.html#X85E149177AC547C3" >30 .3 -4 EnumeratorByFunctions</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap30_mj.html#X7F12F40E87F3C3A7" >30 .3 -5 List</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap30_mj.html#X82CE157A7FAD8036" >30 .3 -6 SortedList</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap30_mj.html#X7E399AC97FD98217" >30 .3 -7 SSortedList</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap30_mj.html#X8289FCCC8274C89D" >30 .3 -8 AsList</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap30_mj.html#X7BCA5C6181391007" >30 .3 -9 AsSortedList</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap30_mj.html#X856D927378C33548" >30 .3 -10 AsSSortedList</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap30_mj.html#X79B130FC7906FB4C" >30 .3 -11 Elements</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap30_mj.html#X79AD18737E70B414" >30 .4 <span class="Heading" >Attributes and Properties for Collections</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap30_mj.html#X7969C48780C5C1BC" >30 .4 -1 IsEmpty</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap30_mj.html#X808A4061809A6E67" >30 .4 -2 IsFinite</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap30_mj.html#X7E3402D6799D3C24" >30 .4 -3 IsTrivial</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap30_mj.html#X7F192373850B85B9" >30 .4 -4 IsNonTrivial</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap30_mj.html#X78EF6A137E8F66B0" >30 .4 -5 IsWholeFamily</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap30_mj.html#X858ADA3B7A684421" >30 .4 -6 Size</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap30_mj.html#X865507568182424E" >30 .4 -7 Representative</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap30_mj.html#X8026085680270D37" >30 .4 -8 RepresentativeSmallest</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap30_mj.html#X7F8FEA3278239ADE" >30 .5 <span class="Heading" >Operations for Collections</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap30_mj.html#X79CA175481F8105F" >30 .5 -1 IsSubset</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap30_mj.html#X851069107CACF98E" >30 .5 -2 <span class="Heading" >Intersection</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap30_mj.html#X799F0E2F7A502DBA" >30 .5 -3 <span class="Heading" >Union</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap30_mj.html#X825AC0F07E010B07" >30 .5 -4 Difference</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap30_mj.html#X82D39CF980FDBFFA" >30 .6 <span class="Heading" >Membership Test for Collections</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap30_mj.html#X84B7FA8C7C94400F" ><code >30 .6 -1 \in</code ></a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap30_mj.html#X8151A51884B7EE2C" >30 .7 <span class="Heading" >Random Elements</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap30_mj.html#X7FF906E57D6936F8" >30 .7 -1 Random</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap30_mj.html#X811B5BD47DC5356B" >30 .7 -2 PseudoRandom</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap30_mj.html#X7EBA01EB83BC65A9" >30 .7 -3 RandomList</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap30_mj.html#X85A3F00985453F95" >30 .8 <span class="Heading" >Iterators</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap30_mj.html#X83ADF8287ED0668E" >30 .8 -1 Iterator</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap30_mj.html#X8688C20B828FC129" >30 .8 -2 IteratorSorted</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap30_mj.html#X87168A827E5B28E4" >30 .8 -3 IsIterator</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap30_mj.html#X8055FC557B5D899E" >30 .8 -4 IsDoneIterator</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap30_mj.html#X879F62F77D1D1179" >30 .8 -5 NextIterator</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap30_mj.html#X858A28667D137C4B" >30 .8 -6 IteratorList</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap30_mj.html#X7DB80BE68271247E" >30 .8 -7 TrivialIterator</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap30_mj.html#X82677D8F817D6701" >30 .8 -8 IteratorByFunctions</a></span >
</div ></div >
</div >
<div class="ContChap" ><a href="chap31_mj.html#X7E651AC287AFDCC1" >31 <span class="Heading" >Domains and their Elements</span ></a>
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap31_mj.html#X859C7AB97B34F55F" >31 .1 <span class="Heading" >Operational Structure of Domains</span ></a>
</span >
</div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap31_mj.html#X84FA03F87A17B059" >31 .2 <span class="Heading" >Equality and Comparison of Domains</span ></a>
</span >
</div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap31_mj.html#X82039A218274826F" >31 .3 <span class="Heading" >Constructing Domains</span ></a>
</span >
</div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap31_mj.html#X7EA77DE17DD8A231" >31 .4 <span class="Heading" >Changing the Structure</span ></a>
</span >
</div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap31_mj.html#X860FCCBE7A41412F" >31 .5 <span class="Heading" >Changing the Representation</span ></a>
</span >
</div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap31_mj.html#X7D72F11B82F4A036" >31 .6 <span class="Heading" >Domain Categories</span ></a>
</span >
</div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap31_mj.html#X7CBDD36E7B7BE286" >31 .7 <span class="Heading" >Parents</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap31_mj.html#X7BC856CC7F116BB0" >31 .7 -1 Parent</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap31_mj.html#X7B58FDEF80338DD6" >31 .8 <span class="Heading" >Constructing Subdomains</span ></a>
</span >
</div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap31_mj.html#X86D579707B112970" >31 .9 <span class="Heading" >Operations for Domains</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap31_mj.html#X86B4AC017FAF4D12" >31 .9 -1 IsGeneralizedDomain</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap31_mj.html#X7E353DD1838AB223" >31 .9 -2 GeneratorsOfDomain</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap31_mj.html#X826A21287FD3ACC0" >31 .9 -3 Domain</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap31_mj.html#X7C2B0C1280237CB0" >31 .10 <span class="Heading" >Attributes and Properties of Elements</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap31_mj.html#X81278E53800BF64D" >31 .10 -1 Characteristic</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap31_mj.html#X8046262384895B2A" >31 .10 -2 OneImmutable</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap31_mj.html#X8040AC7A79FFC442" >31 .10 -3 ZeroImmutable</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap31_mj.html#X86DEB543824C40EB" >31 .10 -4 MultiplicativeZeroOp</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap31_mj.html#X814D78347858EC13" >31 .10 -5 IsOne</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap31_mj.html#X82BDA47282F9BBA7" >31 .10 -6 IsZero</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap31_mj.html#X7CB5896082D29173" >31 .10 -7 IsIdempotent</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap31_mj.html#X78EE524E83624057" >31 .10 -8 InverseImmutable</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap31_mj.html#X84BB723C81D55D63" >31 .10 -9 AdditiveInverseImmutable</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap31_mj.html#X84F59A2687C62763" >31 .10 -10 Order</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap31_mj.html#X7B3BC7BA7BB2646D" >31 .11 <span class="Heading" >Comparison Operations for Elements</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap31_mj.html#X7EF67D047F03CA6F" >31 .11 -1 <span class="Heading" >\= and \<</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap31_mj.html#X7EFE013B8634D214" >31 .11 -2 CanEasilyCompareElements</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap31_mj.html#X7A2914307963E370" >31 .12 <span class="Heading" >Arithmetic Operations for Elements</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap31_mj.html#X8481C9B97B214C23" >31 .12 -1 <span class="Heading" >\+, \*, \/, \^, \mod</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap31_mj.html#X7A37082878DB3930" >31 .12 -2 LeftQuotient</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap31_mj.html#X80761843831B468E" >31 .12 -3 Comm</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap31_mj.html#X86A62A937A42B82E" >31 .12 -4 LieBracket</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap31_mj.html#X7E8F1FB87C229BB0" >31 .12 -5 Sqrt</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap31_mj.html#X80A2D8A7874B268B" >31 .13 <span class="Heading" >Relations Between Domains</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap31_mj.html#X7C03098C838ADE40" >31 .13 -1 UseSubsetRelation</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap31_mj.html#X78039B628262BFA8" >31 .13 -2 UseFactorRelation</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap31_mj.html#X839BE6467E8474D9" >31 .13 -3 UseIsomorphismRelation</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap31_mj.html#X863C35007C7AA914" >31 .13 -4 InstallSubsetMaintenance</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap31_mj.html#X7BB7EE5078EF6F47" >31 .13 -5 InstallFactorMaintenance</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap31_mj.html#X79F97F0F78D89186" >31 .13 -6 InstallIsomorphismMaintenance</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap31_mj.html#X7B97A0307EA161E5" >31 .14 <span class="Heading" >Useful Categories of Elements</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap31_mj.html#X7FBD4F65861C2DF2" >31 .14 -1 IsExtAElement</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap31_mj.html#X7F346AA47AEC39AB" >31 .14 -2 IsNearAdditiveElement</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap31_mj.html#X78D042B486E1D7F7" >31 .14 -3 IsAdditiveElement</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap31_mj.html#X7CE2353F836F6E0A" >31 .14 -4 IsNearAdditiveElementWithZero</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap31_mj.html#X87F3552A789C572D" >31 .14 -5 IsAdditiveElementWithZero</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap31_mj.html#X84B0929982B51CB4" >31 .14 -6 IsNearAdditiveElementWithInverse</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap31_mj.html#X7C0E4AE883947778" >31 .14 -7 IsAdditiveElementWithInverse</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap31_mj.html#X860D1E387DD5CCCF" >31 .14 -8 IsExtLElement</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap31_mj.html#X809E0C097E480AF1" >31 .14 -9 IsExtRElement</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap31_mj.html#X797D3B2A7A2B2F53" >31 .14 -10 IsMultiplicativeElement</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap31_mj.html#X82BC294F7D388AE8" >31 .14 -11 IsMultiplicativeElementWithOne</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap31_mj.html#X8703BFC2841BBD63" >31 .14 -12 IsMultiplicativeElementWithZero</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap31_mj.html#X7FDB14E57814FA3B" >31 .14 -13 IsMultiplicativeElementWithInverse</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap31_mj.html#X802F34F280B29DF4" >31 .14 -14 IsVector</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap31_mj.html#X799AEDE180C31276" >31 .14 -15 IsNearRingElement</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap31_mj.html#X84BF40CA86C07361" >31 .14 -16 IsRingElement</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap31_mj.html#X7C724689784EEF3D" >31 .14 -17 IsNearRingElementWithOne</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap31_mj.html#X875B67208017608E" >31 .14 -18 IsRingElementWithOne</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap31_mj.html#X80CD04ED85B6B2F9" >31 .14 -19 IsNearRingElementWithInverse</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap31_mj.html#X8113834E84FD0435" >31 .14 -20 IsRingElementWithInverse</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap31_mj.html#X7ABEF00C870789D2" >31 .15 <span class="Heading" >Useful Categories for all Elements of a Family</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap31_mj.html#X7979AFAA80FF795A" >31 .15 -1 IsAssociativeElement</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap31_mj.html#X78A286418205CE44" >31 .15 -2 IsAdditivelyCommutativeElement</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap31_mj.html#X8137FA8D86714AC0" >31 .15 -3 IsCommutativeElement</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap31_mj.html#X810D2E5E832594AA" >31 .15 -4 IsFiniteOrderElement</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap31_mj.html#X796957D0805A0221" >31 .15 -5 IsJacobianElement</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap31_mj.html#X7844399D7847AB24" >31 .15 -6 IsZeroSquaredElement</a></span >
</div ></div >
</div >
<div class="ContChap" ><a href="chap32_mj.html#X7C9734B880042C73" >32 <span class="Heading" >Mappings</span ></a>
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap32_mj.html#X859A13548515A5D7" >32 .1 <span class="Heading" >Direct Products and their Elements</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap32_mj.html#X87FD9FE787023FF0" >32 .1 -1 IsDirectProductElement</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap32_mj.html#X78F8A1168280E06D" >32 .1 -2 DirectProductFamily</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap32_mj.html#X7CF6FEFB8290D5CB" >32 .2 <span class="Heading" >Creating Mappings</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap32_mj.html#X79D0D2F07A14D039" >32 .2 -1 GeneralMappingByElements</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap32_mj.html#X7D55E1977ED70E01" >32 .2 -2 <span class="Heading" >MappingByFunction</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap32_mj.html#X865FC25A87D36F3D" >32 .2 -3 InverseGeneralMapping</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap32_mj.html#X7BD2D5A87CD6B213" >32 .2 -4 RestrictedInverseGeneralMapping</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap32_mj.html#X7ED1E4E27CCE2DCA" >32 .2 -5 CompositionMapping</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap32_mj.html#X86486B687B7077AC" >32 .2 -6 CompositionMapping2</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap32_mj.html#X7A926D167C3155F6" >32 .2 -7 IsCompositionMappingRep</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap32_mj.html#X87775B438008DCA5" >32 .2 -8 ConstituentsCompositionMapping</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap32_mj.html#X795FF8DC785F110A" >32 .2 -9 ZeroMapping</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap32_mj.html#X7EBAE0368470A603" >32 .2 -10 IdentityMapping</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap32_mj.html#X86452F8587CBAEA0" >32 .2 -11 <span class="Heading" >Embedding</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap32_mj.html#X8769E8DA80BC96C1" >32 .2 -12 <span class="Heading" >Projection</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap32_mj.html#X800014D683A81009" >32 .2 -13 RestrictedMapping</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap32_mj.html#X7E5A430D7F838F1C" >32 .3 <span class="Heading" >Properties and Attributes of (General) Mappings</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap32_mj.html#X83C7494E828CC9C8" >32 .3 -1 IsTotal</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap32_mj.html#X86D44C8A78BF1981" >32 .3 -2 IsSingleValued</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap32_mj.html#X7CC95EB282854385" >32 .3 -3 IsMapping</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap32_mj.html#X7F065FD7822C0A12" >32 .3 -4 IsInjective</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap32_mj.html#X784ECE847E005B8F" >32 .3 -5 IsSurjective</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap32_mj.html#X878F56AB7B342767" >32 .3 -6 IsBijective</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap32_mj.html#X7B6FD7277CDE9FCB" >32 .3 -7 Range</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap32_mj.html#X7DE8173F80E07AB1" >32 .3 -8 Source </a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap32_mj.html#X784F871383FB599B" >32 .3 -9 UnderlyingRelation</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap32_mj.html#X786581DE871A47D0" >32 .3 -10 UnderlyingGeneralMapping</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap32_mj.html#X83B4FF15847F06FC" >32 .4 <span class="Heading" >Images under Mappings</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap32_mj.html#X7D23C1CE863DACD8" >32 .4 -1 ImagesSource</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap32_mj.html#X85ADB89B7C8DD7D0" >32 .4 -2 ImagesRepresentative</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap32_mj.html#X7D51184B7EE5B2CF" >32 .4 -3 ImagesElm</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap32_mj.html#X8781348F7F5796A0" >32 .4 -4 ImagesSet</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap32_mj.html#X7CFAB0157BFB1806" >32 .4 -5 ImageElm</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap32_mj.html#X87F4D35A826599C6" >32 .4 -6 <span class="Heading" >Image</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap32_mj.html#X86114B2E7E77488C" >32 .4 -7 <span class="Heading" >Images</span ></a>
</span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap32_mj.html#X79BB1EC07C828667" >32 .5 <span class="Heading" >Preimages under Mappings</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap32_mj.html#X78EF1FE77B0973C0" >32 .5 -1 PreImagesRange</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap32_mj.html#X7FBB830C8729E995" >32 .5 -2 PreImagesElm</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap32_mj.html#X7D212F727CAE971A" >32 .5 -3 PreImageElm</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap32_mj.html#X7AE24A1586B7DE79" >32 .5 -4 PreImagesRepresentative</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap32_mj.html#X856BAFC87B2D2811" >32 .5 -5 PreImagesSet</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap32_mj.html#X836FAEAC78B55BF4" >32 .5 -6 <span class="Heading" >PreImage</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap32_mj.html#X85C8590E832002EF" >32 .5 -7 <span class="Heading" >PreImages</span ></a>
</span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap32_mj.html#X7E2E16277940FA0B" >32 .6 <span class="Heading" >Arithmetic Operations for General Mappings</span ></a>
</span >
</div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap32_mj.html#X834E02BB7D4B4AE5" >32 .7 <span class="Heading" >Mappings which are Compatible with Algebraic Structures</span ></a>
</span >
</div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap32_mj.html#X8008FCCC7F4C731F" >32 .8 <span class="Heading" >Magma Homomorphisms</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap32_mj.html#X7DC72CF28539A251" >32 .8 -1 IsMagmaHomomorphism</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap32_mj.html#X8181676787E760A2" >32 .8 -2 MagmaHomomorphismByFunctionNC</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap32_mj.html#X79D0216E871B7051" >32 .8 -3 NaturalHomomorphismByGenerators</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap32_mj.html#X806F892C862F29F9" >32 .9 <span class="Heading" >Mappings that Respect Multiplication</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap32_mj.html#X7BEFF95883EAEC78" >32 .9 -1 RespectsMultiplication</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap32_mj.html#X7EE4DA097AE9CBC1" >32 .9 -2 RespectsOne</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap32_mj.html#X7F27AE9C84A4DF90" >32 .9 -3 RespectsInverses</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap32_mj.html#X819DD174829BF3AE" >32 .9 -4 IsGroupGeneralMapping</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap32_mj.html#X81A5A5CF846E5FBF" >32 .9 -5 KernelOfMultiplicativeGeneralMapping</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap32_mj.html#X7F09B6E28080DCB4" >32 .9 -6 CoKernelOfMultiplicativeGeneralMapping</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap32_mj.html#X8455A5A67C35178B" >32 .10 <span class="Heading" >Mappings that Respect Addition</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap32_mj.html#X7A3321E878925C3A" >32 .10 -1 RespectsAddition</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap32_mj.html#X8130D8907B92F746" >32 .10 -2 RespectsAdditiveInverses</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap32_mj.html#X7D342736781EB280" >32 .10 -3 RespectsZero</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap32_mj.html#X7B99EF287A8A0BD9" >32 .10 -4 IsAdditiveGroupGeneralMapping</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap32_mj.html#X7EC0E9907D6631D6" >32 .10 -5 KernelOfAdditiveGeneralMapping</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap32_mj.html#X813C6D7980213F41" >32 .10 -6 CoKernelOfAdditiveGeneralMapping</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap32_mj.html#X7C24431C81532575" >32 .11 <span class="Heading" >Linear Mappings</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap32_mj.html#X87842ED97FA19973" >32 .11 -1 RespectsScalarMultiplication</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap32_mj.html#X780BE6307A3271A9" >32 .11 -2 IsLeftModuleGeneralMapping</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap32_mj.html#X7F6841107E59107F" >32 .11 -3 IsLinearMapping</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap32_mj.html#X7E88C32A82E942DA" >32 .12 <span class="Heading" >Ring Homomorphisms</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap32_mj.html#X7C8DA031799B79D5" >32 .12 -1 IsRingGeneralMapping</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap32_mj.html#X7988102883675606" >32 .12 -2 IsRingWithOneGeneralMapping</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap32_mj.html#X86B14F908601DEA9" >32 .12 -3 IsAlgebraGeneralMapping</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap32_mj.html#X842AD44679C5BDC2" >32 .12 -4 IsAlgebraWithOneGeneralMapping</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap32_mj.html#X8324DA78879DF4D7" >32 .12 -5 IsFieldHomomorphism</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap32_mj.html#X7E4A55567BED0F88" >32 .13 <span class="Heading" >General Mappings</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap32_mj.html#X8656AB8A7D672CAE" >32 .13 -1 IsGeneralMapping</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap32_mj.html#X791690817E23D90C" >32 .13 -2 IsConstantTimeAccessGeneralMapping</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap32_mj.html#X81CFF5F87BBEA8AD" >32 .13 -3 IsEndoGeneralMapping</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap32_mj.html#X7D6F78587C00CDD0" >32 .14 <span class="Heading" >Technical Matters Concerning General Mappings</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap32_mj.html#X7D28581F82481163" >32 .14 -1 IsSPGeneralMapping</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap32_mj.html#X80D02AD183E01F16" >32 .14 -2 IsGeneralMappingFamily</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap32_mj.html#X86CFADBA7F2FE446" >32 .14 -3 FamilyRange</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap32_mj.html#X7C3736E281A9E505" >32 .14 -4 FamilySource</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap32_mj.html#X7AE54FB67E2E6374" >32 .14 -5 FamiliesOfGeneralMappingsAndRanges</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap32_mj.html#X7E1E26E37C413F6F" >32 .14 -6 GeneralMappingsFamily</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap32_mj.html#X7CF92CC37A6BBDA5" >32 .14 -7 TypeOfDefaultGeneralMapping</a></span >
</div ></div >
</div >
<div class="ContChap" ><a href="chap33_mj.html#X838651287FCCEFD8" >33 <span class="Heading" >Relations</span ></a>
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap33_mj.html#X7DED7F1F78D31785" >33 .1 <span class="Heading" >General Binary Relations</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap33_mj.html#X788D722F82165551" >33 .1 -1 IsBinaryRelation</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap33_mj.html#X7A1D8EEF8034B0B5" >33 .1 -2 BinaryRelationByElements</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap33_mj.html#X81878EEF873B34D5" >33 .1 -3 <span class="Heading" >IdentityBinaryRelation</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap33_mj.html#X80DDCDD387BA23F2" >33 .1 -4 EmptyBinaryRelation</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap33_mj.html#X7899E59181C46EBB" >33 .2 <span class="Heading" >Properties and Attributes of Binary Relations</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap33_mj.html#X79D69B667F5FE8FE" >33 .2 -1 IsReflexiveBinaryRelation</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap33_mj.html#X785916A181555368" >33 .2 -2 IsSymmetricBinaryRelation</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap33_mj.html#X7823942478124563" >33 .2 -3 IsTransitiveBinaryRelation</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap33_mj.html#X870F72C38550A0A4" >33 .2 -4 IsAntisymmetricBinaryRelation</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap33_mj.html#X782B7C8A8136532F" >33 .2 -5 IsPreOrderBinaryRelation</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap33_mj.html#X7A1228207AB4FBA3" >33 .2 -6 IsPartialOrderBinaryRelation</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap33_mj.html#X80D3735C84D1CDC2" >33 .2 -7 IsHasseDiagram</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap33_mj.html#X82D6CB4B7CCE9E25" >33 .2 -8 IsEquivalenceRelation</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap33_mj.html#X85E2FD8B82652876" >33 .2 -9 Successors</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap33_mj.html#X7B4D22A17E752A91" >33 .2 -10 DegreeOfBinaryRelation</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap33_mj.html#X8278E4457C3C3A0D" >33 .2 -11 PartialOrderOfHasseDiagram</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap33_mj.html#X78032F927F078E19" >33 .3 <span class="Heading" >Binary Relations on Points</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap33_mj.html#X79E40E9385274F89" >33 .3 -1 BinaryRelationOnPoints</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap33_mj.html#X7D9323C283867515" >33 .3 -2 RandomBinaryRelationOnPoints</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap33_mj.html#X8315C7A47CEB6BB3" >33 .3 -3 <span class="Heading" >AsBinaryRelationOnPoints</span ></a>
</span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap33_mj.html#X7D9A14AE799142EF" >33 .4 <span class="Heading" >Closure Operations and Other Constructors</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap33_mj.html#X8252B17C864A4904" >33 .4 -1 ReflexiveClosureBinaryRelation</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap33_mj.html#X820811E9785A7274" >33 .4 -2 SymmetricClosureBinaryRelation</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap33_mj.html#X853BFAD9858DCDF7" >33 .4 -3 TransitiveClosureBinaryRelation</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap33_mj.html#X79672B3A7BCB6991" >33 .4 -4 HasseDiagramBinaryRelation</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap33_mj.html#X85C22B3D812957C0" >33 .4 -5 StronglyConnectedComponents</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap33_mj.html#X86AAE6027A3AEF72" >33 .4 -6 PartialOrderByOrderingFunction</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap33_mj.html#X7DAA67338458BB64" >33 .5 <span class="Heading" >Equivalence Relations</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap33_mj.html#X7A44D73D8150266A" >33 .5 -1 EquivalenceRelationByPartition</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap33_mj.html#X82CD1C00810F6042" >33 .5 -2 EquivalenceRelationByRelation</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap33_mj.html#X7B70215E7E3F9CA4" >33 .5 -3 EquivalenceRelationByPairs</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap33_mj.html#X7C5AA9B97EE42DA5" >33 .5 -4 EquivalenceRelationByProperty</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap33_mj.html#X85A2A8E27AF52769" >33 .6 <span class="Heading" >Attributes of and Operations on Equivalence Relations</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap33_mj.html#X877389B683DD8F1A" >33 .6 -1 EquivalenceRelationPartition</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap33_mj.html#X79DC914C82D7903B" >33 .6 -2 GeneratorsOfEquivalenceRelationPartition</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap33_mj.html#X82BE360381476D92" >33 .6 -3 JoinEquivalenceRelations</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap33_mj.html#X79EE13287DEB11B1" >33 .7 <span class="Heading" >Equivalence Classes</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap33_mj.html#X8424996186DB14EA" >33 .7 -1 IsEquivalenceClass</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap33_mj.html#X78F967E77EB16386" >33 .7 -2 EquivalenceClassRelation</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap33_mj.html#X879439897EF4D728" >33 .7 -3 EquivalenceClasses</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap33_mj.html#X7BB985BA7FD7A82E" >33 .7 -4 EquivalenceClassOfElement</a></span >
</div ></div >
</div >
<div class="ContChap" ><a href="chap34_mj.html#X7E4AAA7382D42361" >34 <span class="Heading" >Orderings</span ></a>
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap34_mj.html#X79B1262585CE5427" >34 .1 <span class="Heading" >IsOrdering (Filter)</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap34_mj.html#X7EFDF115780934AF" >34 .1 -1 IsOrdering</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap34_mj.html#X85E6445C87283BEC" >34 .1 -2 OrderingsFamily</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap34_mj.html#X85C4CAA784BD7F01" >34 .2 <span class="Heading" >Building new orderings</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap34_mj.html#X78B5D91278EFAFC9" >34 .2 -1 OrderingByLessThanFunctionNC</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap34_mj.html#X813D5BEB80506CE4" >34 .2 -2 OrderingByLessThanOrEqualFunctionNC</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap34_mj.html#X7F62235B87C20A54" >34 .3 <span class="Heading" >Properties and basic functionality</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap34_mj.html#X84FA448B7B4DDFDC" >34 .3 -1 IsWellFoundedOrdering</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap34_mj.html#X867AC932843AD921" >34 .3 -2 IsTotalOrdering</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap34_mj.html#X814E5E7D85EDCAC7" >34 .3 -3 IsIncomparableUnder</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap34_mj.html#X872497B9782B97B4" >34 .3 -4 FamilyForOrdering</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap34_mj.html#X7D08ED6882015BFB" >34 .3 -5 LessThanFunction</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap34_mj.html#X857E800583E9026D" >34 .3 -6 LessThanOrEqualFunction</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap34_mj.html#X87F51D737C695D41" >34 .3 -7 IsLessThanUnder</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap34_mj.html#X8308B7DF7AAF6D9C" >34 .3 -8 IsLessThanOrEqualUnder</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap34_mj.html#X834CD021878745BC" >34 .4 <span class="Heading" >Orderings on families of associative words</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap34_mj.html#X7C1808AE84B989AE" >34 .4 -1 IsOrderingOnFamilyOfAssocWords</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap34_mj.html#X8175B8887868F29A" >34 .4 -2 IsTranslationInvariantOrdering</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap34_mj.html#X816CD4BD82D41ED0" >34 .4 -3 IsReductionOrdering</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap34_mj.html#X7B6051C282EA88D5" >34 .4 -4 OrderingOnGenerators</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap34_mj.html#X79B2DEB786680F51" >34 .4 -5 LexicographicOrdering</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap34_mj.html#X802EB44B7E7B1F57" >34 .4 -6 ShortLexOrdering</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap34_mj.html#X7B6ED9327E0A2099" >34 .4 -7 IsShortLexOrdering</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap34_mj.html#X849DD7C6782333D5" >34 .4 -8 WeightLexOrdering</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap34_mj.html#X7C7D7954784F5C73" >34 .4 -9 IsWeightLexOrdering</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap34_mj.html#X7E7FAEA484148947" >34 .4 -10 WeightOfGenerators</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap34_mj.html#X79D1019E7C3E575E" >34 .4 -11 BasicWreathProductOrdering</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap34_mj.html#X7CB765477FBC3383" >34 .4 -12 IsBasicWreathProductOrdering</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap34_mj.html#X7E6DF1B17F53642E" >34 .4 -13 WreathProductOrdering</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap34_mj.html#X7F0EE6E987148C96" >34 .4 -14 IsWreathProductOrdering</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap34_mj.html#X7901AA4479EDBE72" >34 .4 -15 LevelsOfGenerators</a></span >
</div ></div >
</div >
<div class="ContChap" ><a href="chap35_mj.html#X873E502F7D21C39C" >35 <span class="Heading" >Magmas</span ></a>
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap35_mj.html#X7E1248B186E7BB44" >35 .1 <span class="Heading" >Magma Categories</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap35_mj.html#X87D3F38B7EAB13FA" >35 .1 -1 IsMagma</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap35_mj.html#X86071DE7835F1C7C" >35 .1 -2 IsMagmaWithOne</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap35_mj.html#X83E4903D7FBB2E24" >35 .1 -3 IsMagmaWithInversesIfNonzero</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap35_mj.html#X82CBFF648574B830" >35 .1 -4 IsMagmaWithInverses</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap35_mj.html#X808F1A148398733D" >35 .2 <span class="Heading" >Magma Generation</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap35_mj.html#X839147CF813312D6" >35 .2 -1 Magma</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap35_mj.html#X7854B23286B17321" >35 .2 -2 MagmaWithOne</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap35_mj.html#X7A2B51F67EF4DA28" >35 .2 -3 MagmaWithInverses</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap35_mj.html#X7F629A498383A0AD" >35 .2 -4 MagmaByGenerators</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap35_mj.html#X84DABBEB803107EB" >35 .2 -5 MagmaWithOneByGenerators</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap35_mj.html#X82C08CFB854E3F1A" >35 .2 -6 MagmaWithInversesByGenerators</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap35_mj.html#X8268EAA47E4A3A64" >35 .2 -7 Submagma</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap35_mj.html#X7F295EBC7A9CE87E" >35 .2 -8 SubmagmaWithOne</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap35_mj.html#X79441F1F7A277E28" >35 .2 -9 SubmagmaWithInverses</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap35_mj.html#X84ED076D7E46AB79" >35 .2 -10 AsMagma</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap35_mj.html#X87EEEC018129F0F4" >35 .2 -11 AsSubmagma</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap35_mj.html#X8553F44D8123B2C6" >35 .2 -12 IsMagmaWithZeroAdjoined</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap35_mj.html#X8620878D7FD98823" >35 .2 -13 InjectionZeroMagma</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap35_mj.html#X7B353674859BF659" >35 .2 -14 UnderlyingInjectionZeroMagma</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap35_mj.html#X782215B982F2F01C" >35 .3 <span class="Heading" >Magmas Defined by Multiplication Tables</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap35_mj.html#X85CD1E7678295CA6" >35 .3 -1 MagmaByMultiplicationTable</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap35_mj.html#X865526C881645D65" >35 .3 -2 MagmaWithOneByMultiplicationTable</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap35_mj.html#X7EDAFB987EE8A770" >35 .3 -3 MagmaWithInversesByMultiplicationTable</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap35_mj.html#X828BED4580D28FB8" >35 .3 -4 MagmaElement</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap35_mj.html#X849BDCC27C4C3191" >35 .3 -5 <span class="Heading" >MultiplicationTable</span ></a>
</span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap35_mj.html#X87036FCE868FFEE9" >35 .4 <span class="Heading" >Attributes and Properties for Magmas</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap35_mj.html#X872E05B478EC20CA" >35 .4 -1 GeneratorsOfMagma</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap35_mj.html#X87DD93EC8061DD81" >35 .4 -2 GeneratorsOfMagmaWithOne</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap35_mj.html#X83A901B1857C8489" >35 .4 -3 GeneratorsOfMagmaWithInverses</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap35_mj.html#X7A2BF4527E08803C" >35 .4 -4 <span class="Heading" >Centralizer</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap35_mj.html#X847ABE6F781C7FE8" >35 .4 -5 Centre</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap35_mj.html#X7C651C9C78398FFF" >35 .4 -6 Idempotents</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap35_mj.html#X7C83B5A47FD18FB7" >35 .4 -7 IsAssociative</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap35_mj.html#X857B0E507D745ADB" >35 .4 -8 IsCentral</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap35_mj.html#X830A4A4C795FBC2D" >35 .4 -9 IsCommutative</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap35_mj.html#X7EE2EA5F7EB7FEC2" >35 .4 -10 MultiplicativeNeutralElement</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap35_mj.html#X7B39F93C8136D642" >35 .4 -11 MultiplicativeZero</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap35_mj.html#X867DB05A8218FB1E" >35 .4 -12 SquareRoots</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap35_mj.html#X837DA95883CFB985" >35 .4 -13 TrivialSubmagmaWithOne</a></span >
</div ></div >
</div >
<div class="ContChap" ><a href="chap36_mj.html#X7CB0D2F780D15136" >36 <span class="Heading" >Words</span ></a>
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap36_mj.html#X79AEC832815B9317" >36 .1 <span class="Heading" >Categories of Words and Nonassociative Words</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap36_mj.html#X843F5C3A82239398" >36 .1 -1 IsWord</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap36_mj.html#X804B616579F223D8" >36 .1 -2 IsWordCollection</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap36_mj.html#X808FA6F97E16502F" >36 .1 -3 IsNonassocWord</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap36_mj.html#X7F81276C80F690DC" >36 .1 -4 IsNonassocWordCollection</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap36_mj.html#X852C815F85DBE4BD" >36 .2 <span class="Heading" >Comparison of Words</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap36_mj.html#X7CA51DD7874115DF" ><code >36 .2 -1 \=</code ></a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap36_mj.html#X82D4C7BE803166D6" ><code >36 .2 -2 \<</code ></a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap36_mj.html#X7A60A8E57AF13901" >36 .3 <span class="Heading" >Operations for Words</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap36_mj.html#X7EC17930781D104A" >36 .3 -1 MappedWord</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap36_mj.html#X7F51B17983019D3E" >36 .4 <span class="Heading" >Free Magmas</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap36_mj.html#X7CFFD9027DDD1555" >36 .4 -1 <span class="Heading" >FreeMagma</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap36_mj.html#X86DB748080B4A9B9" >36 .4 -2 <span class="Heading" >FreeMagmaWithOne</span ></a>
</span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap36_mj.html#X84C2F9037EEE9CED" >36 .5 <span class="Heading" >External Representation for Nonassociative Words</span ></a>
</span >
</div >
</div >
<div class="ContChap" ><a href="chap37_mj.html#X78C56A0A87CE380E" >37 <span class="Heading" >Associative Words</span ></a>
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap37_mj.html#X7AB546CB7B929253" >37 .1 <span class="Heading" >Categories of Associative Words</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap37_mj.html#X7FA8DA728773BA89" >37 .1 -1 IsAssocWord</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap37_mj.html#X82E7EA7F7FD31EC3" >37 .2 <span class="Heading" >Free Groups, Monoids and Semigroups</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap37_mj.html#X8215999E835290F0" >37 .2 -1 <span class="Heading" >FreeGroup</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap37_mj.html#X8601654A7C4AF1E7" >37 .2 -2 IsFreeGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap37_mj.html#X814203E281F3272E" >37 .2 -3 AssignGeneratorVariables</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap37_mj.html#X8405BECB7AC4EB61" >37 .3 <span class="Heading" >Comparison of Associative Words</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap37_mj.html#X8206153078E97B90" ><code >37 .3 -1 \=</code ></a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap37_mj.html#X7BB12B9D7F990899" ><code >37 .3 -2 \<</code ></a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap37_mj.html#X805C519682B0A7ED" >37 .3 -3 IsShortLexLessThanOrEqual</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap37_mj.html#X84875E08847B39E1" >37 .3 -4 IsBasicWreathLessThanOrEqual</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap37_mj.html#X79AF6C757A3547BD" >37 .4 <span class="Heading" >Operations for Associative Words</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap37_mj.html#X87CD4C6978A7936A" >37 .4 -1 Length</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap37_mj.html#X7F5ED4357A9C12E6" >37 .4 -2 ExponentSumWord</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap37_mj.html#X82CC92C17AF6FFA0" >37 .4 -3 Subword</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap37_mj.html#X8509A0A4851981BB" >37 .4 -4 PositionWord</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap37_mj.html#X79186218787C224A" >37 .4 -5 <span class="Heading" >SubstitutedWord</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap37_mj.html#X8486BFE1844CFE59" >37 .4 -6 EliminatedWord</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap37_mj.html#X7D357E047ABD2C6B" >37 .5 <span class="Heading" >Operations for Associative Words by their Syllables</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap37_mj.html#X842D0B547CE93CF2" >37 .5 -1 NumberSyllables</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap37_mj.html#X7E91575F848F4526" >37 .5 -2 ExponentSyllable</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap37_mj.html#X7F2A8CFD811C73B1" >37 .5 -3 GeneratorSyllable</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap37_mj.html#X7B4F7A167E844FA5" >37 .5 -4 SubSyllables</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap37_mj.html#X80A9F39582ED296E" >37 .6 <span class="Heading" >Representations for Associative Words</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap37_mj.html#X7E3612247B3E241B" >37 .6 -1 IsLetterAssocWordRep</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap37_mj.html#X7E36F7897D82417F" >37 .6 -2 IsLetterWordsFamily</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap37_mj.html#X7C84789D7BB161E9" >37 .6 -3 IsBLetterAssocWordRep</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap37_mj.html#X8719E7F27CDA1995" >37 .6 -4 IsBLetterWordsFamily</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap37_mj.html#X7886F8BD83CD8081" >37 .6 -5 IsSyllableAssocWordRep</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap37_mj.html#X7869716C84EA9D81" >37 .6 -6 IsSyllableWordsFamily</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap37_mj.html#X83F669828481FC32" >37 .6 -7 Is16BitsFamily</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap37_mj.html#X7BD7647C7B088389" >37 .6 -8 LetterRepAssocWord</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap37_mj.html#X7AC8EC757CFB9A51" >37 .6 -9 AssocWordByLetterRep</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap37_mj.html#X7934D3D5797102EC" >37 .7 <span class="Heading" >The External Representation for Associative Words</span ></a>
</span >
</div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap37_mj.html#X7DC99E4284093FBB" >37 .8 <span class="Heading" >Straight Line Programs</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap37_mj.html#X7F69FF3F7C6694CB" >37 .8 -1 IsStraightLineProgram</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap37_mj.html#X7AECA57280DA3195" >37 .8 -2 StraightLineProgram</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap37_mj.html#X81A8AFC47F8E4B91" >37 .8 -3 LinesOfStraightLineProgram</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap37_mj.html#X820A592881D57802" >37 .8 -4 NrInputsOfStraightLineProgram</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap37_mj.html#X7847D32B863E822F" >37 .8 -5 ResultOfStraightLineProgram</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap37_mj.html#X8098EAAF7D344466" >37 .8 -6 StringOfResultOfStraightLineProgram</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap37_mj.html#X8274C7948248C053" >37 .8 -7 CompositionOfStraightLinePrograms</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap37_mj.html#X7A582FA97C786640" >37 .8 -8 IntegratedStraightLineProgram</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap37_mj.html#X7C9CABD17BE4850F" >37 .8 -9 RestrictOutputsOfSLP</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap37_mj.html#X7EF202F17DCA5D1C" >37 .8 -10 IntermediateResultOfSLP</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap37_mj.html#X8085CF79856B2889" >37 .8 -11 IntermediateResultOfSLPWithoutOverwrite</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap37_mj.html#X873244F37FAA717A" >37 .8 -12 IntermediateResultsOfSLPWithoutOverwrite</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap37_mj.html#X837101F982C35035" >37 .8 -13 ProductOfStraightLinePrograms</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap37_mj.html#X84C83CE98194FD03" >37 .8 -14 SlotUsagePattern</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap37_mj.html#X8188799182D82A92" >37 .9 <span class="Heading" >Straight Line Program Elements</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap37_mj.html#X85A5838482944FA5" >37 .9 -1 IsStraightLineProgElm</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap37_mj.html#X78889E5B7E1B3BFF" >37 .9 -2 StraightLineProgElm</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap37_mj.html#X81BC263A7E45E775" >37 .9 -3 StraightLineProgGens</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap37_mj.html#X7BEAE8AC809B27DC" >37 .9 -4 EvalStraightLineProgElm</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap37_mj.html#X7D85D1DF84DC68E3" >37 .9 -5 StretchImportantSLPElement</a></span >
</div ></div >
</div >
<div class="ContChap" ><a href="chap38_mj.html#X7CA8FCFD81AA1890" >38 <span class="Heading" >Rewriting Systems</span ></a>
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap38_mj.html#X8287CBE183EBE5D7" >38 .1 <span class="Heading" >Operations on rewriting systems</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap38_mj.html#X842C0ED87986F7AA" >38 .1 -1 IsRewritingSystem</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap38_mj.html#X833EAA8C86356F42" >38 .1 -2 Rules</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap38_mj.html#X7C38C2EF817F9E0A" >38 .1 -3 OrderOfRewritingSystem</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap38_mj.html#X8340EB2280DE6CCC" >38 .1 -4 ReducedForm</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap38_mj.html#X8006790B86328CE8" >38 .1 -5 <span class="Heading" >IsConfluent</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap38_mj.html#X870A1E1C7FB45A55" >38 .1 -6 ConfluentRws</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap38_mj.html#X8134689C7B576946" >38 .1 -7 IsReduced</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap38_mj.html#X864C82FD7FBA31A6" >38 .1 -8 ReduceRules</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap38_mj.html#X81E6B5CB789A7C3A" >38 .1 -9 AddRule</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap38_mj.html#X7FA0B54D7C533DDC" >38 .1 -10 AddRuleReduced</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap38_mj.html#X7BD6299E85561DC3" >38 .1 -11 MakeConfluent</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap38_mj.html#X795DC25886007DFE" >38 .1 -12 GeneratorsOfRws</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap38_mj.html#X81B812C778CB1E4E" >38 .2 <span class="Heading" >Operations on elements of the algebra</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap38_mj.html#X81BB38CC793F7CE2" >38 .2 -1 ReducedProduct</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap38_mj.html#X8318649681DF783B" >38 .3 <span class="Heading" >Properties of rewriting systems</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap38_mj.html#X7B647DB77D138A49" >38 .3 -1 IsBuiltFromAdditiveMagmaWithInverses</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap38_mj.html#X7F8B7848851784DF" >38 .4 <span class="Heading" >Rewriting in Groups and Monoids</span ></a>
</span >
</div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap38_mj.html#X8751F8FA7DC989A2" >38 .5 <span class="Heading" >Developing rewriting systems</span ></a>
</span >
</div >
</div >
<div class="ContChap" ><a href="chap39_mj.html#X8716635F7951801B" >39 <span class="Heading" >Groups</span ></a>
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap39_mj.html#X822370B47DEA37B1" >39 .1 <span class="Heading" >Group Elements</span ></a>
</span >
</div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap39_mj.html#X86A022F9800121F8" >39 .2 <span class="Heading" >Creating Groups</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X7D7B075385435151" >39 .2 -1 <span class="Heading" >Group</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X7F81960287F3E32A" >39 .2 -2 GroupByGenerators</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X8589EF9C7B658B94" >39 .2 -3 GroupWithGenerators</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X79C44528864044C5" >39 .2 -4 GeneratorsOfGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X7A0747F17B50D967" >39 .2 -5 AsGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X7E4143A08040BB47" >39 .2 -6 ConjugateGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X7939B3177BBD61E4" >39 .2 -7 IsGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X845874BA82E1A11F" >39 .2 -8 InfoGroup</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap39_mj.html#X7BA181CA81D785BB" >39 .3 <span class="Heading" >Subgroups</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X7C82AA387A42DCA0" >39 .3 -1 Subgroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X842AD37E79CE953E" >39 .3 -2 <span class="Heading" >Index (<strong class="pkg" >GAP</strong > operation)</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X8014135884DCC53E" >39 .3 -3 IndexInWholeGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X7904AC9D7E9A3BB7" >39 .3 -4 AsSubgroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X7839D8927E778334" >39 .3 -5 IsSubgroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X838186F9836F678C" >39 .3 -6 IsNormal</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X8390B5117A10CC52" >39 .3 -7 IsCharacteristicSubgroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X84F5464983655590" >39 .3 -8 ConjugateSubgroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X7D9990EB837075A4" >39 .3 -9 ConjugateSubgroups</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X82ABF80780CC27AF" >39 .3 -10 IsSubnormal</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X829766158665FB54" >39 .3 -11 SubgroupByProperty</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X7E95101F80583E77" >39 .3 -12 SubgroupShell</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap39_mj.html#X7B855B0485C3C6C5" >39 .4 <span class="Heading" >Closures of (Sub )groups</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X7D13FC1F8576FFD8" >39 .4 -1 ClosureGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X81A20A397C308483" >39 .4 -2 ClosureGroupAddElm</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X82F59F6680D1B0D5" >39 .4 -3 ClosureGroupDefault</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X7A7AF14A8052F055" >39 .4 -4 ClosureSubgroup</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap39_mj.html#X7E19F92284F6684E" >39 .5 <span class="Heading" >Expressing Group Elements as Words in Generators</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X7FE8A3B08458A1BF" >39 .5 -1 EpimorphismFromFreeGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X8357294D7B164106" >39 .5 -2 Factorization</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X871508DD808EB487" >39 .5 -3 GrowthFunctionOfGroup</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap39_mj.html#X87BF1B887C91CA2E" >39 .6 <span class="Heading" >Structure Descriptions</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X8199B74B84446971" >39 .6 -1 StructureDescription</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap39_mj.html#X81002AA87DDBC02F" >39 .7 <span class="Heading" >Cosets</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X8412ABD57986B9FC" >39 .7 -1 RightCoset</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X835F48248571364F" >39 .7 -2 RightCosets</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X85884F177B5D98AE" >39 .7 -3 CanonicalRightCosetElement</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X7D7625A1861D9DAB" >39 .7 -4 IsRightCoset</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X78F4F0D8838F5ABF" >39 .7 -5 IsBiCoset</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X82F6ABE378B928D1" >39 .7 -6 CosetDecomposition</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap39_mj.html#X83C723878230D616" >39 .8 <span class="Heading" >Transversals</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X85C65D06822E716F" >39 .8 -1 RightTransversal</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap39_mj.html#X78B98B257E981046" >39 .9 <span class="Heading" >Double Cosets</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X7E51ED757D17254B" >39 .9 -1 DoubleCoset</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X7F53DABD79BA4F72" >39 .9 -2 RepresentativesContainedRightCosets</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X7A5EFABB86E6D4D5" >39 .9 -3 DoubleCosets</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X85ED464F878EF24C" >39 .9 -4 IsDoubleCoset</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X7A25B1C886CF8C6A" >39 .9 -5 DoubleCosetRepsAndSizes</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X84AE7EE77E5FB30E" >39 .9 -6 InfoCoset</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap39_mj.html#X7D474F8F87E4E5D9" >39 .10 <span class="Heading" >Conjugacy Classes</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X7B2F207F7F85F5B8" >39 .10 -1 ConjugacyClass</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X871B570284BBA685" >39 .10 -2 ConjugacyClasses</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X7D6ED84C86C2979B" >39 .10 -3 ConjugacyClassesByRandomSearch</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X852B3634789D770E" >39 .10 -4 ConjugacyClassesByOrbits</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X8733F87B7E4C9903" >39 .10 -5 NrConjugacyClasses</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X7BD2A4427B7FE248" >39 .10 -6 RationalClass</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X81E9EF0A811072E8" >39 .10 -7 RationalClasses</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X877691247DE23386" >39 .10 -8 GaloisGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X83DD148D7DA2ABA9" >39 .10 -9 <span class="Heading" >IsConjugate</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X81A92F828400FC8A" >39 .10 -10 NthRootsInGroup</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap39_mj.html#X804F0F037F06E25E" >39 .11 <span class="Heading" >Normal Structure</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X87B5370C7DFD401D" >39 .11 -1 <span class="Heading" >Normalizer</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X7C4E00297E37AA44" >39 .11 -2 Core</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X7CF497C77B1E8938" >39 .11 -3 PCore</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X7BDEA0A98720D1BB" >39 .11 -4 NormalClosure</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X7D25E7DC7834A703" >39 .11 -5 NormalIntersection</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X811B8A4683DDE1F9" >39 .11 -6 ComplementClassesRepresentatives</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X8581F4E77B11C610" >39 .11 -7 InfoComplement</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap39_mj.html#X7C39EE3E836D6BC6" >39 .12 <span class="Heading" >Specific and Parametrized Subgroups</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X829759F67D4247CA" >39 .12 -1 TrivialSubgroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X7A9A3D5578CE33A0" >39 .12 -2 CommutatorSubgroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X7CC17CF179ED7EF2" >39 .12 -3 DerivedSubgroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X7B10B58F83DDE56E" >39 .12 -4 CommutatorLength</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X780552B57C30DD8F" >39 .12 -5 FittingSubgroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X788C856C82243274" >39 .12 -6 FrattiniSubgroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X81D86CCE84193E4F" >39 .12 -7 PrefrattiniSubgroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X83D5C8B8865C85F1" >39 .12 -8 PerfectResiduum</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X8250D99A830DA832" >39 .12 -9 SolvableRadical</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X81F647FA83D8854F" >39 .12 -10 Socle</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X8440C61080CDAA14" >39 .12 -11 SupersolvableResiduum</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X796DA805853FAC90" >39 .12 -12 PRump</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap39_mj.html#X7FF0BBDD80E8F6BF" >39 .13 <span class="Heading" >Sylow Subgroups and Hall Subgroups</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X7AA351308787544C" >39 .13 -1 SylowSubgroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X8605F3FE7A3B8E12" >39 .13 -2 SylowComplement</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X7EDBA19E828CD584" >39 .13 -3 HallSubgroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X832E8E6B8347B13F" >39 .13 -4 SylowSystem</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X87A245E180D27147" >39 .13 -5 ComplementSystem</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X82FE5DFD84F8A3C6" >39 .13 -6 HallSystem</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap39_mj.html#X87AF37E980382499" >39 .14 <span class="Heading" >Subgroups characterized by prime powers</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X7F069ACC83DB3374" >39 .14 -1 Omega</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X83DB33747F069ACC" >39 .14 -2 Agemo</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap39_mj.html#X7B75879B8085120A" >39 .15 <span class="Heading" >Group Properties</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X7DA27D338374FD28" >39 .15 -1 IsCyclic</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X813C952F80E775D4" >39 .15 -2 IsElementaryAbelian</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X87D062608719F2CD" >39 .15 -3 IsNilpotentGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X7E3056237C6A5D43" >39 .15 -4 NilpotencyClassOfGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X8755147280C84DBB" >39 .15 -5 IsPerfectGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X809C78D5877D31DF" >39 .15 -6 IsSolvableGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X7D7456077D3D1B86" >39 .15 -7 IsPolycyclicGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X7AADF2E88501B9FF" >39 .15 -8 IsSupersolvableGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X83977EB97A8E2290" >39 .15 -9 IsMonomialGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X7A6685D7819AEC32" >39 .15 -10 IsSimpleGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X78CC9764803601E7" >39 .15 -11 IsAlmostSimpleGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X7C1709A986B00F97" >39 .15 -12 IsQuasisimpleGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X7C6AA6897C4409AC" >39 .15 -13 <span class="Heading" >IsomorphismTypeInfoFiniteSimpleGroup</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X8492B05B822AC58C" >39 .15 -14 SimpleGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X839CDD8C7AE39FD6" >39 .15 -15 SimpleGroupsIterator</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X872E93F586F54FCE" >39 .15 -16 SmallSimpleGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X7EB47BF27D8CBF72" >39 .15 -17 AllSmallNonabelianSimpleGroups</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X81E22D07871DF37E" >39 .15 -18 IsFinitelyGeneratedGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X8648EDA287829755" >39 .15 -19 IsSubsetLocallyFiniteGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X8089F18C810B7E3E" >39 .15 -20 IsPGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X7F232B3F8261CE25" >39 .15 -21 IsPowerfulPGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X7ED4A14F7A235617" >39 .15 -22 IsRegularPGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X87356BAA7E9E2142" >39 .15 -23 PrimePGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X863434AD7DDE514B" >39 .15 -24 PClassPGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X840A4F937ABF15E1" >39 .15 -25 RankPGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X81130F9A7CFCF6BF" >39 .15 -26 IsPSolvable</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X87415A8485FCF510" >39 .15 -27 IsPNilpotent</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap39_mj.html#X7F8264FA796B2B7D" >39 .16 <span class="Heading" >Numerical Group Attributes</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X812827937F403300" >39 .16 -1 AbelianInvariants</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X7D44470C7DA59C1C" >39 .16 -2 Exponent</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X843E0CCA8351FDF4" >39 .16 -3 EulerianFunction</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap39_mj.html#X7AEDEDF67CFED672" >39 .17 <span class="Heading" >Subgroup Series</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X7BDD116F7833800F" >39 .17 -1 ChiefSeries</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X7AC93E977AC9ED58" >39 .17 -2 ChiefSeriesThrough</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X8724E15F81B51173" >39 .17 -3 ChiefSeriesUnderAction</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X7A0E7A8B8495B79D" >39 .17 -4 SubnormalSeries</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X81CDCBD67BC98A5A" >39 .17 -5 CompositionSeries</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X82C0D0217ACB2042" >39 .17 -6 DisplayCompositionSeries</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X7A879948834BD889" >39 .17 -7 DerivedSeriesOfGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X7A9AA1577CEC891F" >39 .17 -8 DerivedLength</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X83F057E5791944D6" >39 .17 -9 <span class="Heading" >ElementaryAbelianSeries</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X782BD7A47D6B6503" >39 .17 -10 InvariantElementaryAbelianSeries</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X879D55A67DB42676" >39 .17 -11 LowerCentralSeriesOfGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X8428592E8773CD7B" >39 .17 -12 UpperCentralSeriesOfGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X7809B7ED792669F3" >39 .17 -13 PCentralSeries</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X82A34BD681F24A94" >39 .17 -14 JenningsSeries</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X7C08A8B77EC09CFF" >39 .17 -15 DimensionsLoewyFactors</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X84112774812180DD" >39 .17 -16 AscendingChain</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X7C5029EE86D7FC96" >39 .17 -17 IntermediateGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X781661FB78DC83B5" >39 .17 -18 IntermediateSubgroups</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X783CDAA67BDD8195" >39 .17 -19 StructuralSeriesOfGroup</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap39_mj.html#X84091B0A7E401E2B" >39 .18 <span class="Heading" >Factor Groups</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X80FC390C7F38A13F" >39 .18 -1 NaturalHomomorphismByNormalSubgroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X7E6EED0185B27C48" >39 .18 -2 FactorGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X7816FA867BF1B8ED" >39 .18 -3 CommutatorFactorGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X7BB93B9778C5A0B2" >39 .18 -4 MaximalAbelianQuotient</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X7FC83E4C783572E7" >39 .18 -5 HasAbelianFactorGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X7FAC018680B766B7" >39 .18 -6 HasElementaryAbelianFactorGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X822A3AB27919BC1E" >39 .18 -7 CentralizerModulo</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap39_mj.html#X7D8EFB2F85AA24EE" >39 .19 <span class="Heading" >Sets of Subgroups</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X7DDE67C67E871336" >39 .19 -1 ConjugacyClassSubgroups</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X7C5BBF487977B8CD" >39 .19 -2 IsConjugacyClassSubgroupsRep</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X7E986BF48393113A" >39 .19 -3 ConjugacyClassesSubgroups</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X8486C25380853F9B" >39 .19 -4 ConjugacyClassesMaximalSubgroups</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X798BF55C837DB188" >39 .19 -5 MaximalSubgroupClassReps</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X85DAFB7582A88463" >39 .19 -6 LowIndexSubgroups</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X80399CD4870FFC4B" >39 .19 -7 AllSubgroups</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X861CD8DA790D81C2" >39 .19 -8 MaximalSubgroups</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X80237A847E24E6CF" >39 .19 -9 NormalSubgroups</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X82ECAA427C987318" >39 .19 -10 MaximalNormalSubgroups</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X86FDD9BA819F5644" >39 .19 -11 MinimalNormalSubgroups</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X7A823C5A810910C3" >39 .19 -12 CharacteristicSubgroups</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap39_mj.html#X7FA267497CFC0550" >39 .20 <span class="Heading" >Subgroup Lattice</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X7B104E2C86166188" >39 .20 -1 LatticeSubgroups</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X78928A3582882BFD" >39 .20 -2 ClassElementLattice</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X7E5DF287825EE7BA" >39 .20 -3 DotFileLatticeSubgroups</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X815CDA447C5DB285" >39 .20 -4 MaximalSubgroupsLattice</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X8138997C871EDF96" >39 .20 -5 MinimalSupergroupsLattice</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X87BE970D7B18E2C5" >39 .20 -6 LowLayerSubgroups</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X87FABD5F87AD2568" >39 .20 -7 ContainedConjugates</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X79C3619C849F97B8" >39 .20 -8 ContainingConjugates</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X8111F50C798B0D76" >39 .20 -9 MinimalFaithfulPermutationDegree</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X7BA3484E7AE0A0E1" >39 .20 -10 RepresentativesPerfectSubgroups</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X7B2233D180DF77A1" >39 .20 -11 ConjugacyClassesPerfectSubgroups</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X7BFE573187B4BEF8" >39 .20 -12 Zuppos</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X82C12E2C81963B23" >39 .20 -13 InfoLattice</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap39_mj.html#X85E613D57F28AEFF" >39 .21 <span class="Heading" >Specific Methods for Subgroup Lattice Computations</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X86462A567DDBA6BC" >39 .21 -1 LatticeByCyclicExtension</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X78918D83835A0EDF" >39 .21 -2 InvariantSubgroupsElementaryAbelianGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X7AD7804A803910AC" >39 .21 -3 SubgroupsSolvableGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X7F60BBB8874DFE40" >39 .21 -4 SizeConsiderFunction</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X833C51BD7E7812C4" >39 .21 -5 ExactSizeConsiderFunction</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X7A2C774B7CFF3E07" >39 .21 -6 InfoPcSubgroup</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap39_mj.html#X79F894537D526B61" >39 .22 <span class="Heading" >Special Generating Sets</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X82FD78AF7F80A0E2" >39 .22 -1 GeneratorsSmallest</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X7A258CCF79552198" >39 .22 -2 LargestElementGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X81D15723804771E2" >39 .22 -3 MinimalGeneratingSet</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X814DBABC878D5232" >39 .22 -4 SmallGeneratingSet</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X7D1574457B152333" >39 .22 -5 IndependentGeneratorsOfAbelianGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X86F835DA8264A0CE" >39 .22 -6 IndependentGeneratorExponents</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap39_mj.html#X7CA0B6A27E0BE6B8" >39 .23 <span class="Heading" >1 -Cohomology</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X847BEC137A49BAF4" >39 .23 -1 <span class="Heading" >OneCocycles</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X7E6438D5834ACCDA" >39 .23 -2 OneCoboundaries</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X80400ABD7F40FAA0" >39 .23 -3 OCOneCocycles</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X811E1CF07DABE924" >39 .23 -4 ComplementClassesRepresentativesEA</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X8199B1D27D487897" >39 .23 -5 InfoCoh</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap39_mj.html#X80A4B0F282977074" >39 .24 <span class="Heading" >Schur Covers and Multipliers</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X7F619DDA7DD6C43B" >39 .24 -1 EpimorphismSchurCover</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X7DD1E37987612042" >39 .24 -2 SchurCover</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X792BC39D7CEB1D27" >39 .24 -3 AbelianInvariantsMultiplier</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X819E8AEC835F8CD1" >39 .24 -4 Epicentre</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X8739CD4686301A0E" >39 .24 -5 NonabelianExteriorSquare</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X7E1C8CD77CDB9F71" >39 .24 -6 EpimorphismNonabelianExteriorSquare</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X7BF8DB3D8300BB3F" >39 .24 -7 IsCentralFactor</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X7F4240CD782B6032" >39 .24 -8 <span class="Heading" >Covering groups of symmetric groups</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X7DDA6BC1824F78FD" >39 .24 -9 BasicSpinRepresentationOfSymmetricGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X844CFFDE80F6AD15" >39 .24 -10 SchurCoverOfSymmetricGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X7E0F4896795E34FC" >39 .24 -11 DoubleCoverOfAlternatingGroup</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap39_mj.html#X7BD95B8D879B73A3" >39 .25 <span class="Heading" >2 -Cohomology</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X7A1EBC3A7AB0D614" >39 .25 -1 TwoCohomologyGeneric</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X7A65366879BB3977" >39 .25 -2 FpGroupCocycle</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap39_mj.html#X865722987E0E19B6" >39 .26 <span class="Heading" >Tests for the Availability of Methods</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X798F13EA810FB215" >39 .26 -1 CanEasilyTestMembership</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X7C2A89607BDFD920" >39 .26 -2 CanEasilyComputeWithIndependentGensAbelianGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X83245C82835D496C" >39 .26 -3 CanComputeSize</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X8268965487364912" >39 .26 -4 CanComputeSizeAnySubgroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X82DDE00D82A32083" >39 .26 -5 CanComputeIndex</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X7BE7C36B84C23511" >39 .26 -6 CanComputeIsSubset</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X87D62C2C7C375E2D" >39 .26 -7 KnowsHowToDecompose</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap39_mj.html#X83A9997586694DC0" >39 .27 <span class="Heading" >Specific functions for Normalizer calculation</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap39_mj.html#X84ABCA997D294B36" >39 .27 -1 NormalizerViaRadical</a></span >
</div ></div >
</div >
<div class="ContChap" ><a href="chap40_mj.html#X83702FC27B3C3098" >40 <span class="Heading" >Group Homomorphisms</span ></a>
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap40_mj.html#X81A7BB0F7D81B247" >40 .1 <span class="Heading" >Creating Group Homomorphisms</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap40_mj.html#X7F348F497C813BE0" >40 .1 -1 GroupHomomorphismByImages</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap40_mj.html#X7AB15AF5830F2A6B" >40 .1 -2 GroupHomomorphismByImagesNC</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap40_mj.html#X7A59F2C47BD41DC8" >40 .1 -3 GroupGeneralMappingByImages</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap40_mj.html#X7BC6C20E7CEDBFC5" >40 .1 -4 <span class="Heading" >GroupHomomorphismByFunction</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap40_mj.html#X785AB6057F736344" >40 .1 -5 AsGroupGeneralMappingByImages</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap40_mj.html#X794043AC7E4FAF9E" >40 .2 <span class="Heading" >Operations for Group Homomorphisms</span ></a>
</span >
</div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap40_mj.html#X7A121B9E7F78138A" >40 .3 <span class="Heading" >Efficiency of Homomorphisms</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap40_mj.html#X84CFBB577BAFFD4D" >40 .3 -1 <span class="Heading" >Mappings given on generators</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap40_mj.html#X86C2BE2481FDC8EE" >40 .3 -2 <span class="Heading" >Action homomorphisms</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap40_mj.html#X802C5A887D8A7CC4" >40 .3 -3 <span class="Heading" >Mappings given by functions</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap40_mj.html#X87497C207B7D7511" >40 .3 -4 <span class="Heading" >Other operations</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap40_mj.html#X80B8ABEC7CC20DFB" >40 .3 -5 ImagesSmallestGenerators</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap40_mj.html#X7BA90DA481A1C6D6" >40 .4 <span class="Heading" >Homomorphism for very large groups</span ></a>
</span >
</div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap40_mj.html#X7FFD731684606BC6" >40 .5 <span class="Heading" >Nice Monomorphisms</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap40_mj.html#X78849F81804C44B3" >40 .5 -1 IsHandledByNiceMonomorphism</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap40_mj.html#X7965086E82ABCF41" >40 .5 -2 NiceMonomorphism</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap40_mj.html#X7B47BE0983E93A83" >40 .5 -3 NiceObject</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap40_mj.html#X8652149F7F291EE3" >40 .5 -4 IsCanonicalNiceMonomorphism</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap40_mj.html#X783030917CB43959" >40 .6 <span class="Heading" >Group Automorphisms</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap40_mj.html#X7E52E99487562F3A" >40 .6 -1 ConjugatorIsomorphism</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap40_mj.html#X79ED68CF8139F08A" >40 .6 -2 ConjugatorAutomorphism</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap40_mj.html#X7E937A947856D9DA" >40 .6 -3 InnerAutomorphism</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap40_mj.html#X7F31FECC7A3D4A8A" >40 .6 -4 IsConjugatorIsomorphism</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap40_mj.html#X78FE7E307E86525A" >40 .6 -5 ConjugatorOfConjugatorIsomorphism</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap40_mj.html#X79640F3682BDBFC1" >40 .7 <span class="Heading" >Groups of Automorphisms</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap40_mj.html#X87677B0787B4461A" >40 .7 -1 AutomorphismGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap40_mj.html#X7FC631B786C1DC8B" >40 .7 -2 IsGroupOfAutomorphisms</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap40_mj.html#X7B767B9D827EB0FC" >40 .7 -3 AutomorphismDomain</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap40_mj.html#X7F87D5957D9B991E" >40 .7 -4 IsAutomorphismGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap40_mj.html#X8476738A7BF9BADA" >40 .7 -5 InnerAutomorphismsAutomorphismGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap40_mj.html#X7957AC21782B6C8C" >40 .7 -6 InnerAutomorphismGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap40_mj.html#X7FC9B6EA7CAADC0A" >40 .7 -7 InducedAutomorphism</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap40_mj.html#X7A8E961C7F1A57B3" >40 .8 <span class="Heading" >Calculating with Group Automorphisms</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap40_mj.html#X85691E8386107403" >40 .8 -1 AssignNiceMonomorphismAutomorphismGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap40_mj.html#X7C9FB0A57BFF6CC0" >40 .8 -2 NiceMonomorphismAutomGroup</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap40_mj.html#X81B79CC27F47D429" >40 .9 <span class="Heading" >Searching for Homomorphisms</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap40_mj.html#X7B536A32827788C6" >40 .9 -1 IsomorphismGroups</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap40_mj.html#X7D0C3D5E864CE954" >40 .9 -2 AllHomomorphismClasses</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap40_mj.html#X791D12B7845610CE" >40 .9 -3 AllHomomorphisms</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap40_mj.html#X790C261184EEAB94" >40 .9 -4 GQuotients</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap40_mj.html#X83B417BE7C508DC4" >40 .9 -5 IsomorphicSubgroups</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap40_mj.html#X7AABA9A27E30BF2B" >40 .9 -6 MorClassLoop</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap40_mj.html#X81FC3CEF85CED3DC" >40 .10 <span class="Heading" >Representations for Group Homomorphisms</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap40_mj.html#X82B77A5F7F9EDBC7" >40 .10 -1 IsGroupGeneralMappingByImages</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap40_mj.html#X863805187A24B5E3" >40 .10 -2 MappingGeneratorsImages</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap40_mj.html#X7DFBBAB18126B4D9" >40 .10 -3 IsGroupGeneralMappingByAsGroupGeneralMappingByImages</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap40_mj.html#X78707DF57C3927EB" >40 .10 -4 IsPreimagesByAsGroupGeneralMappingByImages</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap40_mj.html#X83E10338798F552B" >40 .10 -5 IsPermGroupGeneralMapping</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap40_mj.html#X83DADD9F7CAD829B" >40 .10 -6 IsToPermGroupGeneralMappingByImages</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap40_mj.html#X798E72E77EC85D4A" >40 .10 -7 IsGroupGeneralMappingByPcgs</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap40_mj.html#X86FF63B784FB8F85" >40 .10 -8 IsPcGroupGeneralMappingByImages</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap40_mj.html#X79A853B579B250C0" >40 .10 -9 IsToPcGroupGeneralMappingByImages</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap40_mj.html#X7BE2A2EB80DC5CFF" >40 .10 -10 IsFromFpGroupGeneralMappingByImages</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap40_mj.html#X81090C207F4F6423" >40 .10 -11 IsFromFpGroupStdGensGeneralMappingByImages</a></span >
</div ></div >
</div >
<div class="ContChap" ><a href="chap41_mj.html#X87115591851FB7F4" >41 <span class="Heading" >Group Actions</span ></a>
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap41_mj.html#X83661AFD7B7BD1D9" >41 .1 <span class="Heading" >About Group Actions</span ></a>
</span >
</div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap41_mj.html#X81B8F9CD868CD953" >41 .2 <span class="Heading" >Basic Actions</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap41_mj.html#X7FE417DD837987B4" >41 .2 -1 OnPoints</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap41_mj.html#X7960924D84B5B18F" >41 .2 -2 OnRight</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap41_mj.html#X832DF5327ECA0E44" >41 .2 -3 OnLeftInverse</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap41_mj.html#X85AA04347CD117F9" >41 .2 -4 OnSets</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap41_mj.html#X832CC5F87EEA4A7E" >41 .2 -5 OnTuples</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap41_mj.html#X80DAA1D2855B1456" >41 .2 -6 OnPairs</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap41_mj.html#X7C10492081D72376" >41 .2 -7 OnSetsSets</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap41_mj.html#X7E23686E7A9D3A20" >41 .2 -8 OnSetsDisjointSets</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap41_mj.html#X7ADE244E819035FF" >41 .2 -9 OnSetsTuples</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap41_mj.html#X7FF556CD7E6739A9" >41 .2 -10 OnTuplesSets</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap41_mj.html#X844E902382EB4151" >41 .2 -11 OnTuplesTuples</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap41_mj.html#X86DC2DD5829CAD9A" >41 .2 -12 OnLines</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap41_mj.html#X7FA394D27E721E2B" >41 .2 -13 OnIndeterminates</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap41_mj.html#X7BA8D76586F1F06E" >41 .2 -14 Permuted</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap41_mj.html#X85124D197F0F9C4D" >41 .2 -15 OnSubspacesByCanonicalBasis</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap41_mj.html#X82181CA07A5B2056" >41 .3 <span class="Heading" >Action on canonical representatives</span ></a>
</span >
</div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap41_mj.html#X81E0FF0587C54543" >41 .4 <span class="Heading" >Orbits</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap41_mj.html#X80E0234E7BD79409" >41 .4 -1 Orbit</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap41_mj.html#X86BCAE17869BBEAA" >41 .4 -2 Orbits</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap41_mj.html#X86BC8B958123F953" >41 .4 -3 <span class="Heading" >OrbitsDomain</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap41_mj.html#X799910CF832EDC45" >41 .4 -4 OrbitLength</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap41_mj.html#X8032F73078DF2DDB" >41 .4 -5 <span class="Heading" >OrbitLengths</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap41_mj.html#X8520E2487F7E98AF" >41 .4 -6 <span class="Heading" >OrbitLengthsDomain</span ></a>
</span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap41_mj.html#X797BD60E7ACEF1B1" >41 .5 <span class="Heading" >Stabilizers</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap41_mj.html#X7C34EC437EF598BF" >41 .5 -1 OrbitStabilizer</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap41_mj.html#X86FB962786397E02" >41 .5 -2 Stabilizer</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap41_mj.html#X78C3A8568414BC44" >41 .5 -3 OrbitStabilizerAlgorithm</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap41_mj.html#X7A9389097BAF670D" >41 .6 <span class="Heading" >Elements with Prescribed Images</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap41_mj.html#X857DC7B085EB0539" >41 .6 -1 RepresentativeAction</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap41_mj.html#X87F73CCA7921DE65" >41 .7 <span class="Heading" >The Permutation Image of an Action</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap41_mj.html#X78E6A002835288A4" >41 .7 -1 <span class="Heading" >ActionHomomorphism</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap41_mj.html#X85A8E93D786C3C9C" >41 .7 -2 Action</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap41_mj.html#X86FF54A383B73967" >41 .7 -3 SparseActionHomomorphism</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap41_mj.html#X7FED50ED7ACA5FB2" >41 .8 <span class="Heading" >Action of a group on itself</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap41_mj.html#X784D417D87F4E58D" >41 .8 -1 FactorCosetAction</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap41_mj.html#X8561DEBA79E01ABD" >41 .8 -2 RegularActionHomomorphism</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap41_mj.html#X835317A7847477D4" >41 .8 -3 AbelianSubfactorAction</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap41_mj.html#X807AA91E841D132B" >41 .9 <span class="Heading" >Permutations Induced by Elements and Cycles</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap41_mj.html#X7807A33381DCAB26" >41 .9 -1 <span class="Heading" >Permutation</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap41_mj.html#X81D4EA42810974A0" >41 .9 -2 PermutationCycle</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap41_mj.html#X80AF6E0683CA7F14" >41 .9 -3 Cycle</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap41_mj.html#X7F559E897B333758" >41 .9 -4 CycleLength</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap41_mj.html#X7F3B387A7FD8AE5E" >41 .9 -5 Cycles</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap41_mj.html#X83040A6080C2C6C6" >41 .9 -6 CycleLengths</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap41_mj.html#X87FDA6838065CDCB" >41 .9 -7 <span class="Heading" >CycleIndex</span ></a>
</span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap41_mj.html#X850A84618421392A" >41 .10 <span class="Heading" >Tests for Actions</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap41_mj.html#X79B15750851828CB" >41 .10 -1 <span class="Heading" >IsTransitive</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap41_mj.html#X8295D733796B7A37" >41 .10 -2 <span class="Heading" >Transitivity</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap41_mj.html#X8166A6A17C8D6E73" >41 .10 -3 <span class="Heading" >RankAction</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap41_mj.html#X7B77040F8543CD6E" >41 .10 -4 <span class="Heading" >IsSemiRegular</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap41_mj.html#X7CF02C4785F0EAB5" >41 .10 -5 <span class="Heading" >IsRegular</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap41_mj.html#X7CB1D74280F92AFC" >41 .10 -6 <span class="Heading" >Earns</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap41_mj.html#X84C19AD68247B760" >41 .10 -7 <span class="Heading" >IsPrimitive</span ></a>
</span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap41_mj.html#X7E9D3D0B7A9A8572" >41 .11 <span class="Heading" >Block Systems</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap41_mj.html#X84FE699F85371643" >41 .11 -1 <span class="Heading" >Blocks</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap41_mj.html#X79936EB97AAD1144" >41 .11 -2 <span class="Heading" >MaximalBlocks</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap41_mj.html#X7941DB6380B74510" >41 .11 -3 <span class="Heading" >RepresentativesMinimalBlocks</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap41_mj.html#X835658B07B28EF3B" >41 .11 -4 AllBlocks</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap41_mj.html#X7FD3D2D2788709B7" >41 .12 <span class="Heading" >External Sets</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap41_mj.html#X8264C3C479FF0A8B" >41 .12 -1 IsExternalSet</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap41_mj.html#X7C90F648793E47DD" >41 .12 -2 ExternalSet</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap41_mj.html#X7B9DB15D80CE28B4" >41 .12 -3 ActingDomain</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap41_mj.html#X86153CB087394DC1" >41 .12 -4 FunctionAction</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap41_mj.html#X86A0CC1479A5932A" >41 .12 -5 HomeEnumerator</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap41_mj.html#X879DE63C7858453C" >41 .12 -6 IsExternalSubset</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap41_mj.html#X87D1EA1486D86233" >41 .12 -7 ExternalSubset</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap41_mj.html#X7E081F568407317F" >41 .12 -8 IsExternalOrbit</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap41_mj.html#X7FB656AE7A066C35" >41 .12 -9 ExternalOrbit</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap41_mj.html#X7BAFF02B7D6DF9F2" >41 .12 -10 StabilizerOfExternalSet</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap41_mj.html#X867262FA82FDD592" >41 .12 -11 <span class="Heading" >ExternalOrbits</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap41_mj.html#X7A64EF807CE8893E" >41 .12 -12 <span class="Heading" >ExternalOrbitsStabilizers</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap41_mj.html#X8048AE727A7F1A2F" >41 .12 -13 CanonicalRepresentativeOfExternalSet</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap41_mj.html#X8071A8D784DC8325" >41 .12 -14 CanonicalRepresentativeDeterminatorOfExternalSet</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap41_mj.html#X85E9A6A77B8D00B8" >41 .12 -15 ActorOfExternalSet</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap41_mj.html#X8190A8247F29A5C7" >41 .12 -16 UnderlyingExternalSet</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap41_mj.html#X7A3D87DE809FBFD4" >41 .12 -17 SurjectiveActionHomomorphismAttr</a></span >
</div ></div >
</div >
<div class="ContChap" ><a href="chap42_mj.html#X80F808307A2D5AB8" >42 <span class="Heading" >Permutations</span ></a>
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap42_mj.html#X80F07BE2811D4BAC" >42 .1 <span class="Heading" >IsPerm (Filter)</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap42_mj.html#X7AA69C6686FC49EA" >42 .1 -1 IsPerm</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap42_mj.html#X82069E437D2DF9AA" >42 .1 -2 IsPermCollection</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap42_mj.html#X819628B083B3939B" >42 .1 -3 PermutationsFamily</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap42_mj.html#X83C711557DEB4B36" >42 .1 -4 PERM_INVERSE_THRESHOLD</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap42_mj.html#X7A21DE5779D21A6D" >42 .2 <span class="Heading" >Comparison of Permutations</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap42_mj.html#X7CEC03A9808E2E7C" ><code >42 .2 -1 \=</code ></a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap42_mj.html#X7BC944F57A04AFF2" >42 .2 -2 DistancePerms</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap42_mj.html#X83A917F67D45C7EA" >42 .2 -3 SmallestGeneratorPerm</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap42_mj.html#X82C255E2821C0721" >42 .3 <span class="Heading" >Moved Points of Permutations</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap42_mj.html#X84EF0A697F7A87DC" >42 .3 -1 SmallestMovedPoint</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap42_mj.html#X84AA603987C94AC0" >42 .3 -2 LargestMovedPoint</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap42_mj.html#X85E61B9C7A6B0CCA" >42 .3 -3 MovedPoints</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap42_mj.html#X85E7B1E28430F49E" >42 .3 -4 NrMovedPoints</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap42_mj.html#X79BE80267F4AA2B0" >42 .4 <span class="Heading" >Sign and Cycle Structure</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap42_mj.html#X7BE5011B7C0DB704" >42 .4 -1 SignPerm</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap42_mj.html#X7944D1447804A69A" >42 .4 -2 CycleStructurePerm</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap42_mj.html#X7B3194EC869D936D" >42 .5 <span class="Heading" >Creating Permutations</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap42_mj.html#X7A9DCFD986958C1E" >42 .5 -1 ListPerm</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap42_mj.html#X78D611D17EA6E3BC" >42 .5 -2 PermList</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap42_mj.html#X8087DCC780B9656A" >42 .5 -3 MappingPermListList</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap42_mj.html#X7EF8388E7DA8E600" >42 .5 -4 RestrictedPerm</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap42_mj.html#X80665A5D800CAFE1" >42 .5 -5 CycleFromList</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap42_mj.html#X8353AB8987E35DF3" >42 .5 -6 AsPermutation</a></span >
</div ></div >
</div >
<div class="ContChap" ><a href="chap43_mj.html#X85ED46007CED6191" >43 <span class="Heading" >Permutation Groups</span ></a>
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap43_mj.html#X7F38777E7BBE12AE" >43 .1 <span class="Heading" >IsPermGroup (Filter)</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap43_mj.html#X7879877482F59676" >43 .1 -1 IsPermGroup</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap43_mj.html#X85D769FF85545AAB" >43 .2 <span class="Heading" >The Natural Action</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap43_mj.html#X84CFA16D858B00B8" >43 .2 -1 OrbitPerms</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap43_mj.html#X81F98222818DA35B" >43 .2 -2 OrbitsPerms</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap43_mj.html#X7E468B64860D5604" >43 .3 <span class="Heading" >Computing a Permutation Representation</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap43_mj.html#X80B7B1C783AA1567" >43 .3 -1 IsomorphismPermGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap43_mj.html#X8086628878AFD3EA" >43 .3 -2 SmallerDegreePermutationRepresentation</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap43_mj.html#X834208CD7C2956A3" >43 .4 <span class="Heading" >Symmetric and Alternating Groups</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap43_mj.html#X8129BE59781478E1" >43 .4 -1 IsNaturalSymmetricGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap43_mj.html#X85CA6AD17BE90C95" >43 .4 -2 IsSymmetricGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap43_mj.html#X8514BE9E79C608E0" >43 .4 -3 IsAlternatingGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap43_mj.html#X7ED60F7E81F1B614" >43 .4 -4 SymmetricParentGroup</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap43_mj.html#X83F8D3B578A7BEEB" >43 .5 <span class="Heading" >Primitive Groups</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap43_mj.html#X7E50211A7B92455F" >43 .5 -1 ONanScottType</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap43_mj.html#X7E89A46A86A3F4A2" >43 .5 -2 SocleTypePrimitiveGroup</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap43_mj.html#X7FA58C3A8283F3BD" >43 .6 <span class="Heading" >Stabilizer Chains</span ></a>
</span >
</div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap43_mj.html#X7C2406B97E057196" >43 .7 <span class="Heading" >Randomized Methods for Permutation Groups</span ></a>
</span >
</div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap43_mj.html#X7C7EA55C80E457FA" >43 .8 <span class="Heading" >Construction of Stabilizer Chains</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap43_mj.html#X80B5CF78829495C2" >43 .8 -1 StabChain</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap43_mj.html#X790C27B8783EDE68" >43 .8 -2 StabChainOptions</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap43_mj.html#X87E1292E85A5D31C" >43 .8 -3 DefaultStabChainOptions</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap43_mj.html#X86D64D2B81D58431" >43 .8 -4 StabChainBaseStrongGenerators</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap43_mj.html#X7BEC5F5A7851CAAB" >43 .8 -5 MinimalStabChain</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap43_mj.html#X81D7FCE47AC7F942" >43 .9 <span class="Heading" >Stabilizer Chain Records</span ></a>
</span >
</div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap43_mj.html#X7ECF8A4586346FD4" >43 .10 <span class="Heading" >Operations for Stabilizer Chains</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap43_mj.html#X7FBE6EB57EBE8B7D" >43 .10 -1 BaseStabChain</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap43_mj.html#X7D2A190D8308ED39" >43 .10 -2 BaseOfGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap43_mj.html#X7EF36DC78465026A" >43 .10 -3 SizeStabChain</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap43_mj.html#X8384170881B9B531" >43 .10 -4 StrongGeneratorsStabChain</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap43_mj.html#X87F473777EFDE867" >43 .10 -5 GroupStabChain</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap43_mj.html#X87FB6DED80692D3F" >43 .10 -6 OrbitStabChain</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap43_mj.html#X7AC8F165875906DE" >43 .10 -7 IndicesStabChain</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap43_mj.html#X7CF607BC82C2C202" >43 .10 -8 ListStabChain</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap43_mj.html#X7F40E52D7B0438BF" >43 .10 -9 ElementsStabChain</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap43_mj.html#X780875477CD2A57D" >43 .10 -10 IteratorStabChain</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap43_mj.html#X861062AE87ACF340" >43 .10 -11 InverseRepresentative</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap43_mj.html#X79D2248C8787EAF2" >43 .10 -12 SiftedPermutation</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap43_mj.html#X7B870C217D0B9997" >43 .10 -13 MinimalElementCosetStabChain</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap43_mj.html#X87435B7884D9B353" >43 .10 -14 LargestElementStabChain</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap43_mj.html#X809B2C3B7C5F77AB" >43 .10 -15 ApproximateSuborbitsStabilizerPermGroup</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap43_mj.html#X8188051F79E72A95" >43 .11 <span class="Heading" >Low Level Routines to Modify and Create Stabilizer Chains</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap43_mj.html#X86B31E6A81AE5FCB" >43 .11 -1 CopyStabChain</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap43_mj.html#X7E167E557B567C6A" >43 .11 -2 CopyOptionsDefaults</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap43_mj.html#X87FF64AB87BFC779" >43 .11 -3 ChangeStabChain</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap43_mj.html#X8778B4657D3FD97B" >43 .11 -4 ExtendStabChain</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap43_mj.html#X7E5E9F727D0B19D9" >43 .11 -5 ReduceStabChain</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap43_mj.html#X85BF290D848C4091" >43 .11 -6 RemoveStabChain</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap43_mj.html#X84E4906B86E5C089" >43 .11 -7 EmptyStabChain</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap43_mj.html#X80C7D2E87E6EE357" >43 .11 -8 InsertTrivialStabilizer</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap43_mj.html#X7B47B379824F6150" >43 .11 -9 IsFixedStabilizer</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap43_mj.html#X8373007880EBF736" >43 .11 -10 AddGeneratorsExtendSchreierTree</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap43_mj.html#X86C78160854C7F30" >43 .12 <span class="Heading" >Backtrack</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap43_mj.html#X7BE3F03C80BF8B08" >43 .12 -1 SubgroupProperty</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap43_mj.html#X7EE7DDCC87C4BC31" >43 .12 -2 ElementProperty</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap43_mj.html#X7A2D046B83DD5F5F" >43 .12 -3 TwoClosure</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap43_mj.html#X861461AB7964DC64" >43 .12 -4 InfoBckt</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap43_mj.html#X78A68F5A80ADD1B6" >43 .13 <span class="Heading" >Working with large degree permutation groups</span ></a>
</span >
</div >
</div >
<div class="ContChap" ><a href="chap44_mj.html#X7CF51CB48610A07D" >44 <span class="Heading" >Matrix Groups</span ></a>
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap44_mj.html#X86CEA60E7C04744C" >44 .1 <span class="Heading" >IsMatrixGroup (Filter)</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap44_mj.html#X7E6093FF85F1C3A1" >44 .1 -1 IsMatrixGroup</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap44_mj.html#X7FD808E386FAF9B0" >44 .2 <span class="Heading" >Attributes and Properties for Matrix Groups</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap44_mj.html#X7E55258C783C50CA" >44 .2 -1 DimensionOfMatrixGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap44_mj.html#X7D540083793CD496" >44 .2 -2 DefaultFieldOfMatrixGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap44_mj.html#X78A9F0E580DA613A" >44 .2 -3 FieldOfMatrixGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap44_mj.html#X832D18C77ED608DE" >44 .2 -4 TransposedMatrixGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap44_mj.html#X84B36A827E5EFC35" >44 .2 -5 IsFFEMatrixGroup</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap44_mj.html#X7F4B0B397AAC7659" >44 .3 <span class="Heading" >Actions of Matrix Groups</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap44_mj.html#X7BD4F38E8624735D" >44 .3 -1 ProjectiveActionOnFullSpace</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap44_mj.html#X7F8EA8D583C1E9B2" >44 .3 -2 ProjectiveActionHomomorphismMatrixGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap44_mj.html#X849C451A80B4A210" >44 .3 -3 BlowUpIsomorphism</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap44_mj.html#X7934EED77891BE6B" >44 .4 <span class="Heading" >GL and SL</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap44_mj.html#X781387AF7999EA99" >44 .4 -1 IsGeneralLinearGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap44_mj.html#X86F9A27D7AFAEB5A" >44 .4 -2 IsNaturalGL</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap44_mj.html#X816677CD821261FA" >44 .4 -3 IsSpecialLinearGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap44_mj.html#X84134F08781EB943" >44 .4 -4 IsNaturalSL</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap44_mj.html#X7ED43D4F7E993A31" >44 .4 -5 IsSubgroupSL</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap44_mj.html#X7CA4097C79F5BD90" >44 .5 <span class="Heading" >Invariant Forms</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap44_mj.html#X7C08385A81AB05E1" >44 .5 -1 InvariantBilinearForm</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap44_mj.html#X8652FBF781940AC3" >44 .5 -2 IsFullSubgroupGLorSLRespectingBilinearForm</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap44_mj.html#X82F22079852130C9" >44 .5 -3 InvariantSesquilinearForm</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap44_mj.html#X7B35A8AF7D8F0313" >44 .5 -4 IsFullSubgroupGLorSLRespectingSesquilinearForm</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap44_mj.html#X7BCACC007EB9B613" >44 .5 -5 InvariantQuadraticForm</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap44_mj.html#X84AB04A67DFC0274" >44 .5 -6 IsFullSubgroupGLorSLRespectingQuadraticForm</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap44_mj.html#X7FB0138F79E8C5E7" >44 .6 <span class="Heading" >Matrix Groups in Characteristic 0 </span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap44_mj.html#X850821F78558C829" >44 .6 -1 IsCyclotomicMatrixGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap44_mj.html#X7FEDB2E17EE02674" >44 .6 -2 IsRationalMatrixGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap44_mj.html#X7F737FC4795F3E48" >44 .6 -3 IsIntegerMatrixGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap44_mj.html#X86F9CC1E7DB97CB6" >44 .6 -4 IsNaturalGLnZ</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap44_mj.html#X7B0E70127F5D2EAF" >44 .6 -5 IsNaturalSLnZ</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap44_mj.html#X7DE412A37A6975B3" >44 .6 -6 InvariantLattice</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap44_mj.html#X7CC4D6DC81739698" >44 .6 -7 NormalizerInGLnZ</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap44_mj.html#X7DAFB71F86525DE7" >44 .6 -8 CentralizerInGLnZ</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap44_mj.html#X8217762A863F1382" >44 .6 -9 ZClassRepsQClass</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap44_mj.html#X84FD9FC97FB90795" >44 .6 -10 IsBravaisGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap44_mj.html#X7AAE301C83116451" >44 .6 -11 BravaisGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap44_mj.html#X788C7D9C7C2301C5" >44 .6 -12 BravaisSubgroups</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap44_mj.html#X7F5FF1A481E08AD5" >44 .6 -13 BravaisSupergroups</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap44_mj.html#X79B7CD797A420720" >44 .6 -14 NormalizerInGLnZBravaisGroup</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap44_mj.html#X868288377CFA8D1B" >44 .7 <span class="Heading" >Acting OnRight and OnLeft</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap44_mj.html#X7D1318A6780CD88B" >44 .7 -1 CrystGroupDefaultAction</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap44_mj.html#X792D237385977BE6" >44 .7 -2 SetCrystGroupDefaultAction</a></span >
</div ></div >
</div >
<div class="ContChap" ><a href="chap45_mj.html#X86007B0083F60470" >45 <span class="Heading" >Polycyclic Groups</span ></a>
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap45_mj.html#X7F18A01785DBAC4E" >45 .1 <span class="Heading" >Polycyclic Generating Systems</span ></a>
</span >
</div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap45_mj.html#X87F7E31879AFA06C" >45 .2 <span class="Heading" >Computing a Pcgs</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap45_mj.html#X84C3750C7A4EEC34" >45 .2 -1 Pcgs</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap45_mj.html#X8635E61A7DB73BA6" >45 .2 -2 IsPcgs</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap45_mj.html#X7B561B1685CEC2AB" >45 .2 -3 CanEasilyComputePcgs</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap45_mj.html#X7CAAD6D2838354D9" >45 .3 <span class="Heading" >Defining a Pcgs Yourself</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap45_mj.html#X7E139C3D80847D76" >45 .3 -1 PcgsByPcSequence</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap45_mj.html#X816C5E8E7F71C9D8" >45 .4 <span class="Heading" >Elementary Operations for a Pcgs</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap45_mj.html#X7DD0DF677AC1CF10" >45 .4 -1 RelativeOrders</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap45_mj.html#X80D526848427A5C6" >45 .4 -2 IsFiniteOrdersPcgs</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap45_mj.html#X866C3A5382FF231A" >45 .4 -3 IsPrimeOrdersPcgs</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap45_mj.html#X827A7B097A959579" >45 .4 -4 PcSeries</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap45_mj.html#X7903702E8194EF29" >45 .4 -5 GroupOfPcgs</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap45_mj.html#X878FB11887524E2C" >45 .4 -6 OneOfPcgs</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap45_mj.html#X84243AA07DA5A827" >45 .5 <span class="Heading" >Elementary Operations for a Pcgs and an Element</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap45_mj.html#X7B941D4A7CAFCD73" >45 .5 -1 RelativeOrderOfPcElement</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap45_mj.html#X78134914842E2F5F" >45 .5 -2 ExponentOfPcElement</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap45_mj.html#X848DAEBF7DC448A5" >45 .5 -3 ExponentsOfPcElement</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap45_mj.html#X829BCB267CDBC5C0" >45 .5 -4 DepthOfPcElement</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap45_mj.html#X7D47966479EA2890" >45 .5 -5 LeadingExponentOfPcElement</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap45_mj.html#X87AF746B8328F5D0" >45 .5 -6 PcElementByExponents</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap45_mj.html#X7F8BD7A87DB3933A" >45 .5 -7 LinearCombinationPcgs</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap45_mj.html#X8066B66D8069BAB4" >45 .5 -8 SiftedPcElement</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap45_mj.html#X7B52ADE7878A749A" >45 .5 -9 CanonicalPcElement</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap45_mj.html#X7A94AA357DB2F86C" >45 .5 -10 ReducedPcElement</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap45_mj.html#X8702D76D8284CF3E" >45 .5 -11 CleanedTailPcElement</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap45_mj.html#X830A0D037DBEAA97" >45 .5 -12 HeadPcElementByNumber</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap45_mj.html#X7EF61EA4822870E7" >45 .6 <span class="Heading" >Exponents of Special Products</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap45_mj.html#X868D6DB07D349460" >45 .6 -1 ExponentsConjugateLayer</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap45_mj.html#X874F70697FE7B6DF" >45 .6 -2 ExponentsOfRelativePower</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap45_mj.html#X78CAF32F864EF656" >45 .6 -3 ExponentsOfConjugate</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap45_mj.html#X875689897DD0CAFC" >45 .6 -4 ExponentsOfCommutator</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap45_mj.html#X8676397383093D1E" >45 .7 <span class="Heading" >Subgroups of Polycyclic Groups – Induced Pcgs</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap45_mj.html#X81FA878C854D63F8" >45 .7 -1 IsInducedPcgs</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap45_mj.html#X83F6759184937F1B" >45 .7 -2 InducedPcgsByPcSequence</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap45_mj.html#X86308E80843BF9E5" >45 .7 -3 ParentPcgs</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap45_mj.html#X7F0EB20080590B23" >45 .7 -4 InducedPcgs</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap45_mj.html#X8332F1197DF6FEDE" >45 .7 -5 InducedPcgsByGenerators</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap45_mj.html#X7AF82BD079D811E5" >45 .7 -6 InducedPcgsByPcSequenceAndGenerators</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap45_mj.html#X845FF8CA783D6CB3" >45 .7 -7 LeadCoeffsIGS</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap45_mj.html#X800287C680C5DEC3" >45 .7 -8 ExtendedPcgs</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap45_mj.html#X817E16D67B31389B" >45 .7 -9 SubgroupByPcgs</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap45_mj.html#X84068D2478C134C1" >45 .8 <span class="Heading" >Subgroups of Polycyclic Groups – Canonical Pcgs</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap45_mj.html#X80D122B986B42F80" >45 .8 -1 IsCanonicalPcgs</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap45_mj.html#X816F6B4187032A10" >45 .8 -2 CanonicalPcgs</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap45_mj.html#X8294F5EF81B7ABA0" >45 .9 <span class="Heading" >Factor Groups of Polycyclic Groups – Modulo Pcgs</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap45_mj.html#X7FE689A37E559F66" >45 .9 -1 ModuloPcgs</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap45_mj.html#X868207D77D09D915" >45 .9 -2 IsModuloPcgs</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap45_mj.html#X8027CC9878031D74" >45 .9 -3 NumeratorOfModuloPcgs</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap45_mj.html#X87DBE2797D51B2F1" >45 .9 -4 DenominatorOfModuloPcgs</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap45_mj.html#X7E923519832F2D08" ><code >45 .9 -5 \mod</code ></a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap45_mj.html#X876A41F97FBA7754" >45 .9 -6 CorrespondingGeneratorsByModuloPcgs</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap45_mj.html#X8480852A7D49BC3F" >45 .9 -7 CanonicalPcgsByGeneratorsWithImages</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap45_mj.html#X8254C0F485F945BD" >45 .10 <span class="Heading" >Factor Groups of Polycyclic Groups in their Own Representation</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap45_mj.html#X806C2D827E04ACF3" >45 .10 -1 ProjectedPcElement</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap45_mj.html#X82F39CCE7C928D3A" >45 .10 -2 ProjectedInducedPcgs</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap45_mj.html#X816813A078B93A6B" >45 .10 -3 LiftedPcElement</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap45_mj.html#X83C60F1587577D65" >45 .10 -4 LiftedInducedPcgs</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap45_mj.html#X83FE235E7B208EC0" >45 .11 <span class="Heading" >Pcgs and Normal Series</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap45_mj.html#X7E7E89C278DDE20D" >45 .11 -1 IsPcgsElementaryAbelianSeries</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap45_mj.html#X863A20B57EA37BAC" >45 .11 -2 PcgsElementaryAbelianSeries</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap45_mj.html#X7BCC1E2A80544CC7" >45 .11 -3 IndicesEANormalSteps</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap45_mj.html#X7FCE308887F621FC" >45 .11 -4 EANormalSeriesByPcgs</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap45_mj.html#X79675266796D7254" >45 .11 -5 IsPcgsCentralSeries</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap45_mj.html#X8187FCF483659E69" >45 .11 -6 PcgsCentralSeries</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap45_mj.html#X7FB73FEB7BED5BFA" >45 .11 -7 IndicesCentralNormalSteps</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap45_mj.html#X82266ADA86B2A689" >45 .11 -8 CentralNormalSeriesByPcgs</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap45_mj.html#X786E60AF7B61BF9E" >45 .11 -9 IsPcgsPCentralSeriesPGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap45_mj.html#X86F19DBD7D346E7F" >45 .11 -10 PcgsPCentralSeriesPGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap45_mj.html#X863968F08509E7D4" >45 .11 -11 IndicesPCentralNormalStepsPGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap45_mj.html#X7A92C9EA7BAF60CA" >45 .11 -12 PCentralNormalSeriesByPcgsPGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap45_mj.html#X7EA5BC3B7FE9D98D" >45 .11 -13 IsPcgsChiefSeries</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap45_mj.html#X7E7326947EAE4BC9" >45 .11 -14 PcgsChiefSeries</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap45_mj.html#X7C05E84A78CA405E" >45 .11 -15 IndicesChiefNormalSteps</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap45_mj.html#X83C5ABC587074B14" >45 .11 -16 ChiefNormalSeriesByPcgs</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap45_mj.html#X7A954E3887189842" >45 .11 -17 IndicesNormalSteps</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap45_mj.html#X7947B0FB87F44042" >45 .11 -18 NormalSeriesByPcgs</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap45_mj.html#X7E624B4E8224DE2D" >45 .12 <span class="Heading" >Sum and Intersection of Pcgs</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap45_mj.html#X7833DAAA7C07CFD7" >45 .12 -1 SumFactorizationFunctionPcgs</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap45_mj.html#X83039CF97D27D819" >45 .13 <span class="Heading" >Special Pcgs</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap45_mj.html#X7C8A82FA786AC021" >45 .13 -1 IsSpecialPcgs</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap45_mj.html#X827EB7767BACD023" >45 .13 -2 <span class="Heading" >SpecialPcgs</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap45_mj.html#X82DC7CE682140588" >45 .13 -3 LGWeights</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap45_mj.html#X824645C97E347EEE" >45 .13 -4 LGLayers</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap45_mj.html#X7A655F4C7D9AE130" >45 .13 -5 LGFirst</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap45_mj.html#X7C3912F77B12C8B6" >45 .13 -6 LGLength</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap45_mj.html#X814C35BF7C9A8DEF" >45 .13 -7 IsInducedPcgsWrtSpecialPcgs</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap45_mj.html#X7C14AE5C82FB0771" >45 .13 -8 InducedPcgsWrtSpecialPcgs</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap45_mj.html#X7E86EB517DC08809" >45 .14 <span class="Heading" >Action on Subfactors Defined by a Pcgs</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap45_mj.html#X7A9BB9D0817CA949" >45 .14 -1 VectorSpaceByPcgsOfElementaryAbelianGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap45_mj.html#X81FC09DD7FC06C6E" >45 .14 -2 LinearAction</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap45_mj.html#X7C2135B98732BBC3" >45 .14 -3 LinearActionLayer</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap45_mj.html#X79C2D6BF7DD69ED6" >45 .14 -4 AffineAction</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap45_mj.html#X7E4CB1358524497B" >45 .14 -5 AffineActionLayer</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap45_mj.html#X7EEA8D638492F432" >45 .15 <span class="Heading" >Orbit Stabilizer Methods for Polycyclic Groups</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap45_mj.html#X7CFCCF607A30B5EE" >45 .15 -1 StabilizerPcgs</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap45_mj.html#X7A87E72F86813132" >45 .15 -2 Pcgs_OrbitStabilizer</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap45_mj.html#X7A19DF1E7E841074" >45 .16 <span class="Heading" >Operations which have Special Methods for Groups with Pcgs</span ></a>
</span >
</div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap45_mj.html#X79DCCF6D80351859" >45 .17 <span class="Heading" >Conjugacy Classes in Solvable Groups</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap45_mj.html#X79593F667A68A21D" >45 .17 -1 ClassesSolvableGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap45_mj.html#X7B358D3B7E236973" >45 .17 -2 CentralizerSizeLimitConsiderFunction</a></span >
</div ></div >
</div >
<div class="ContChap" ><a href="chap46_mj.html#X7EAD57C97EBF7E67" >46 <span class="Heading" >Pc Groups</span ></a>
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap46_mj.html#X78E9E4D778A57A96" >46 .1 <span class="Heading" >The Family Pcgs</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap46_mj.html#X79EDB35E82C99304" >46 .1 -1 FamilyPcgs</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap46_mj.html#X80893D2A7FFC791B" >46 .1 -2 IsFamilyPcgs</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap46_mj.html#X85C1596A867BE93D" >46 .1 -3 InducedPcgsWrtFamilyPcgs</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap46_mj.html#X8333ACCB7F530406" >46 .1 -4 IsParentPcgsFamilyPcgs</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap46_mj.html#X842526BE7FEFE8BD" >46 .2 <span class="Heading" >Elements of Pc Groups</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap46_mj.html#X869DCE7D86E32337" >46 .2 -1 <span class="Heading" >Comparison of elements of pc groups</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap46_mj.html#X7D1B700882FC6C78" >46 .2 -2 <span class="Heading" >Arithmetic operations for elements of pc groups</span ></a>
</span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap46_mj.html#X87B866C386B386E4" >46 .3 <span class="Heading" >Pc Groups versus Fp Groups</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap46_mj.html#X7D1F506D7830B1D9" >46 .3 -1 IsPcGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap46_mj.html#X7D2735A18111FE39" >46 .3 -2 IsomorphismFpGroupByPcgs</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap46_mj.html#X8581887880556E0C" >46 .4 <span class="Heading" >Constructing Pc Groups</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap46_mj.html#X84C10D1F7CB5274F" >46 .4 -1 PcGroupFpGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap46_mj.html#X7E958DB281E070FD" >46 .4 -2 SingleCollector</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap46_mj.html#X86A08D887E049347" >46 .4 -3 SetConjugate</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap46_mj.html#X7B25997C7DF92B6D" >46 .4 -4 SetCommutator</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap46_mj.html#X7BC319BA8698420C" >46 .4 -5 SetPower</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap46_mj.html#X84F0521486672C3C" >46 .4 -6 GroupByRws</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap46_mj.html#X7DF4835F79667099" >46 .4 -7 IsConfluent</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap46_mj.html#X7E6226597DFE5F8F" >46 .4 -8 IsomorphismRefinedPcGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap46_mj.html#X821560A387762DD1" >46 .4 -9 RefinedPcGroup</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap46_mj.html#X83F69FE27B024E24" >46 .5 <span class="Heading" >Computing Pc Groups</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap46_mj.html#X81C55D4F825C36D4" >46 .5 -1 PcGroupWithPcgs</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap46_mj.html#X873CEB137BA1CD6E" >46 .5 -2 IsomorphismPcGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap46_mj.html#X82BE14A986FA6882" >46 .5 -3 IsomorphismSpecialPcGroup</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap46_mj.html#X85696AB9791DF047" >46 .6 <span class="Heading" >Saving a Pc Group</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap46_mj.html#X8593253380D84508" >46 .6 -1 GapInputPcGroup</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap46_mj.html#X8391EE8D782D0C9E" >46 .7 <span class="Heading" >Operations for Pc Groups</span ></a>
</span >
</div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap46_mj.html#X877AAB887D4507E3" >46 .8 <span class="Heading" ><span class="SimpleMath" >\(2 \)</span >-Cohomology and Extensions</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap46_mj.html#X78E6E11E8285E288" >46 .8 -1 TwoCoboundaries</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap46_mj.html#X784FCA207B8694A6" >46 .8 -2 TwoCocycles</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap46_mj.html#X838065F97F60468F" >46 .8 -3 TwoCohomology</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap46_mj.html#X8236AD927A5A0E5A" >46 .8 -4 Extensions</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap46_mj.html#X7B3BE908867CE4F9" >46 .8 -5 Extension</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap46_mj.html#X83DCB5AB7B6EE785" >46 .8 -6 SplitExtension</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap46_mj.html#X7EAC6B8B7ABEEB86" >46 .8 -7 ModuleOfExtension</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap46_mj.html#X824F2B2E7C11ABAF" >46 .8 -8 CompatiblePairs</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap46_mj.html#X854FFEF187C4AAB9" >46 .8 -9 ExtensionRepresentatives</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap46_mj.html#X84E2DA897FAAF6D8" >46 .8 -10 SplitExtension</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap46_mj.html#X874E4B107BD78F5A" >46 .9 <span class="Heading" >Coding a Pc Presentation</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap46_mj.html#X79948F1D7D4FF8D9" >46 .9 -1 CodePcgs</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap46_mj.html#X8041C2D88721EEA9" >46 .9 -2 CodePcGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap46_mj.html#X826BFDA07A707C54" >46 .9 -3 PcGroupCode</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap46_mj.html#X81D211D8838B875C" >46 .10 <span class="Heading" >Random Isomorphism Testing</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap46_mj.html#X84F6F9787CB2CF16" >46 .10 -1 RandomIsomorphismTest</a></span >
</div ></div >
</div >
<div class="ContChap" ><a href="chap47_mj.html#X7AA982637E90B35A" >47 <span class="Heading" >Finitely Presented Groups</span ></a>
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap47_mj.html#X7824C8167B3CFAB1" >47 .1 <span class="Heading" >IsSubgroupFpGroup and IsFpGroup</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap47_mj.html#X7AF7E2B48199452C" >47 .1 -1 IsSubgroupFpGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap47_mj.html#X850B9DF17D90C3A2" >47 .1 -2 IsFpGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap47_mj.html#X8370BF3B78D0B14D" >47 .1 -3 InfoFpGroup</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap47_mj.html#X7D55E56E790F85FD" >47 .2 <span class="Heading" >Creating Finitely Presented Groups</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap47_mj.html#X7EF4179E78BC7313" ><code >47 .2 -1 \/</code ></a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap47_mj.html#X7CE0FA5F8695241E" >47 .2 -2 FactorGroupFpGroupByRels</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap47_mj.html#X7B3D290B87B6EFE4" >47 .2 -3 ParseRelators</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap47_mj.html#X85EAA789848B528E" >47 .2 -4 StringFactorizationWord</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap47_mj.html#X84D693EC872DAA55" >47 .3 <span class="Heading" >Comparison of Elements of Finitely Presented Groups</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap47_mj.html#X797D29628203CBD6" ><code >47 .3 -1 \=</code ></a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap47_mj.html#X7B350C718573B8DF" ><code >47 .3 -2 \<</code ></a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap47_mj.html#X87512CF485CC4128" >47 .3 -3 FpElmComparisonMethod</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap47_mj.html#X82CB9EC982CDAEAC" >47 .3 -4 SetReducedMultiplication</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap47_mj.html#X7B0B2781796800AD" >47 .4 <span class="Heading" >Preimages in the Free Group</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap47_mj.html#X85CF3931849FB441" >47 .4 -1 FreeGroupOfFpGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap47_mj.html#X79C77C5184CA02B6" >47 .4 -2 FreeGeneratorsOfFpGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap47_mj.html#X87BA180287CD1F71" >47 .4 -3 RelatorsOfFpGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap47_mj.html#X8447A2397A1E524B" >47 .4 -4 UnderlyingElement</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap47_mj.html#X7F34C8017DC03FDB" >47 .4 -5 ElementOfFpGroup</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap47_mj.html#X869143D284F3379D" >47 .5 <span class="Heading" >Operations for Finitely Presented Groups</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap47_mj.html#X7AB7187779EDC9BA" >47 .5 -1 PseudoRandom</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap47_mj.html#X7BD0CEBA7B225416" >47 .6 <span class="Heading" >Coset Tables and Coset Enumeration</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap47_mj.html#X7F7F31E47D7F6EF8" >47 .6 -1 CosetTable</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap47_mj.html#X87D175757C581E62" >47 .6 -2 TracedCosetFpGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap47_mj.html#X7EC1B0EE876E478A" >47 .6 -3 FactorCosetAction</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap47_mj.html#X82926A7F8365A341" >47 .6 -4 CosetTableBySubgroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap47_mj.html#X7DE601F179E6FD09" >47 .6 -5 CosetTableFromGensAndRels</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap47_mj.html#X822B188F87E9E642" >47 .6 -6 CosetTableDefaultMaxLimit</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap47_mj.html#X7A80A00E7E088E44" >47 .6 -7 CosetTableDefaultLimit</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap47_mj.html#X829D31A981CB2AF4" >47 .6 -8 MostFrequentGeneratorFpGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap47_mj.html#X7912E6577B577A5C" >47 .6 -9 IndicesInvolutaryGenerators</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap47_mj.html#X85B882F782D7AFD0" >47 .7 <span class="Heading" >Standardization of coset tables</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap47_mj.html#X85FD1D637EF1EBE7" >47 .7 -1 CosetTableStandard</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap47_mj.html#X85FCD8DF81BA94D5" >47 .7 -2 StandardizeTable</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap47_mj.html#X87C3FA0784A85309" >47 .8 <span class="Heading" >Coset tables for subgroups in the whole group</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap47_mj.html#X846EC8AB7803114D" >47 .8 -1 CosetTableInWholeGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap47_mj.html#X857F239583AFE0B7" >47 .8 -2 SubgroupOfWholeGroupByCosetTable</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap47_mj.html#X7E17A14E823F953D" >47 .9 <span class="Heading" >Augmented Coset Tables and Rewriting</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap47_mj.html#X80F8BF1D867DA7C1" >47 .9 -1 AugmentedCosetTableInWholeGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap47_mj.html#X7AF67CFD846C1159" >47 .9 -2 AugmentedCosetTableMtc</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap47_mj.html#X7F3F09C778552811" >47 .9 -3 AugmentedCosetTableRrs</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap47_mj.html#X86B65EA186140244" >47 .9 -4 RewriteWord</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap47_mj.html#X87FBDA2B815A8776" >47 .10 <span class="Heading" >Low Index Subgroups</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap47_mj.html#X85C5151380E19122" >47 .10 -1 LowIndexSubgroupsFpGroupIterator</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap47_mj.html#X81003D217D92E342" >47 .11 <span class="Heading" >Converting Groups to Finitely Presented Groups</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap47_mj.html#X7F28268F850F454E" >47 .11 -1 IsomorphismFpGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap47_mj.html#X81B2B3B6812FD62D" >47 .11 -2 IsomorphismFpGroupByGenerators</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap47_mj.html#X826604AA7F18BFA3" >47 .12 <span class="Heading" >New Presentations and Presentations for Subgroups</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap47_mj.html#X78D87FA68233C401" >47 .12 -1 IsomorphismSimplifiedFpGroup</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap47_mj.html#X86E7CE077D82133D" >47 .13 <span class="Heading" >Preimages under Homomorphisms from an FpGroup</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap47_mj.html#X7ABC3C917D41A74B" >47 .13 -1 SubgroupOfWholeGroupByQuotientSubgroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap47_mj.html#X8047D7A37B27FEEA" >47 .13 -2 IsSubgroupOfWholeGroupByQuotientRep</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap47_mj.html#X84E6CEA28611C112" >47 .13 -3 AsSubgroupOfWholeGroupByQuotient</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap47_mj.html#X7DA1151D84289FC9" >47 .13 -4 DefiningQuotientHomomorphism</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap47_mj.html#X846072F779B51087" >47 .14 <span class="Heading" >Quotient Methods</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap47_mj.html#X7B5DDADC80F5796B" >47 .14 -1 PQuotient</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap47_mj.html#X86EB30A7867EEF16" >47 .14 -2 EpimorphismQuotientSystem</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap47_mj.html#X7CA738DB80B20D67" >47 .14 -3 EpimorphismPGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap47_mj.html#X7CA20E2582DC45FD" >47 .14 -4 EpimorphismNilpotentQuotient</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap47_mj.html#X869F70CC818C946D" >47 .14 -5 SolvableQuotient</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap47_mj.html#X79A4D3B68110F48A" >47 .14 -6 EpimorphismSolvableQuotient</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap47_mj.html#X81167847832DD3B1" >47 .14 -7 LargerQuotientBySubgroupAbelianization</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap47_mj.html#X81451C4B8463B848" >47 .15 <span class="Heading" >Abelian Invariants for Subgroups</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap47_mj.html#X83B63ED8826F4268" >47 .15 -1 AbelianInvariantsSubgroupFpGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap47_mj.html#X804F664180BA2134" >47 .15 -2 AbelianInvariantsSubgroupFpGroupMtc</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap47_mj.html#X8586137B7AAA6C10" >47 .15 -3 <span class="Heading" >AbelianInvariantsSubgroupFpGroupRrs</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap47_mj.html#X850E4CD784F6EAA8" >47 .15 -4 AbelianInvariantsNormalClosureFpGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap47_mj.html#X801635B28079E56A" >47 .15 -5 AbelianInvariantsNormalClosureFpGroupRrs</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap47_mj.html#X86C43E3B81ED25DC" >47 .16 <span class="Heading" >Testing Finiteness of Finitely Presented Groups</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap47_mj.html#X82F444F67BE0E4FE" >47 .16 -1 IsInfiniteAbelianizationGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap47_mj.html#X85C9FD548394C1E2" >47 .16 -2 NewmanInfinityCriterion</a></span >
</div ></div >
</div >
<div class="ContChap" ><a href="chap48_mj.html#X782985197BE809BF" >48 <span class="Heading" >Presentations and Tietze Transformations</span ></a>
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap48_mj.html#X867D00387957450F" >48 .1 <span class="Heading" >Creating Presentations</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap48_mj.html#X797867B287AD92F8" >48 .1 -1 PresentationFpGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap48_mj.html#X8637837A79422445" >48 .1 -2 TzSort</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap48_mj.html#X849429BC7D435F77" >48 .1 -3 GeneratorsOfPresentation</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap48_mj.html#X7D6F40A87F24D3D6" >48 .1 -4 FpGroupPresentation</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap48_mj.html#X84E056C57AFEDEA8" >48 .1 -5 PresentationViaCosetTable</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap48_mj.html#X7E1F2658827FC228" >48 .1 -6 SimplifiedFpGroup</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap48_mj.html#X8118FECE7AD1879B" >48 .2 <span class="Heading" >Subgroup Presentations</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap48_mj.html#X7DB32FA97DAC5AC8" >48 .2 -1 PresentationSubgroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap48_mj.html#X857365CD87ADC29E" >48 .2 -2 <span class="Heading" >PresentationSubgroupRrs</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap48_mj.html#X7FCE7ED581CF7897" >48 .2 -3 PrimaryGeneratorWords</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap48_mj.html#X80BA10F780EAE68E" >48 .2 -4 PresentationSubgroupMtc</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap48_mj.html#X7D6A52837BEE5C3D" >48 .2 -5 PresentationNormalClosureRrs</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap48_mj.html#X7A7E5D0084DB0B4F" >48 .2 -6 PresentationNormalClosure</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap48_mj.html#X7BC960AB7E8DE419" >48 .3 <span class="Heading" >Relators in a Presentation</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap48_mj.html#X8365BAFA785FCD8D" >48 .3 -1 TietzeWordAbstractWord</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap48_mj.html#X8573E91C838B1D13" >48 .3 -2 AbstractWordTietzeWord</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap48_mj.html#X867F64FA840B3F81" >48 .4 <span class="Heading" >Printing Presentations</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap48_mj.html#X847EA6737C21171C" >48 .4 -1 TzPrintGenerators</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap48_mj.html#X821B63DD82894443" >48 .4 -2 TzPrintRelators</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap48_mj.html#X852C52C37FAAB7DD" >48 .4 -3 TzPrintLengths</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap48_mj.html#X7D7B3F46865443E4" >48 .4 -4 TzPrintStatus</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap48_mj.html#X85F8DAE27F06C32B" >48 .4 -5 TzPrintPresentation</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap48_mj.html#X7CA8BA51802655FC" >48 .4 -6 TzPrint</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap48_mj.html#X82F6B0EE7C7C7901" >48 .4 -7 TzPrintPairs</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap48_mj.html#X82455E5885D73FFF" >48 .5 <span class="Heading" >Changing Presentations</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap48_mj.html#X7F632A6D8685855D" >48 .5 -1 AddGenerator</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap48_mj.html#X83A5667086FD538A" >48 .5 -2 TzNewGenerator</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap48_mj.html#X78D1BCE67FA852D8" >48 .5 -3 AddRelator</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap48_mj.html#X7B11E89E78A22EBF" >48 .5 -4 RemoveRelator</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap48_mj.html#X829B634286471AB7" >48 .6 <span class="Heading" >Tietze Transformations</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap48_mj.html#X7C4A30328224C466" >48 .6 -1 TzGo</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap48_mj.html#X78C3D23387DAC35A" >48 .6 -2 SimplifyPresentation</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap48_mj.html#X801D3D8984E1CA55" >48 .6 -3 TzGoGo</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap48_mj.html#X7D19E30080290FC7" >48 .7 <span class="Heading" >Elementary Tietze Transformations</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap48_mj.html#X85989AF886EC2BF6" >48 .7 -1 <span class="Heading" >TzEliminate</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap48_mj.html#X7DF4BBDF839643DD" >48 .7 -2 TzSearch</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap48_mj.html#X87F7A87A7ACF2445" >48 .7 -3 TzSearchEqual</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap48_mj.html#X80D31A0F7C2A51BD" >48 .7 -4 TzFindCyclicJoins</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap48_mj.html#X7D2FACCF79F57040" >48 .8 <span class="Heading" >Tietze Transformations that introduce new Generators</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap48_mj.html#X846DB23E8236FF8A" >48 .8 -1 <span class="Heading" >TzSubstitute</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap48_mj.html#X7ADE3B437C19B94D" >48 .8 -2 TzSubstituteCyclicJoins</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap48_mj.html#X85E703997A0212EE" >48 .9 <span class="Heading" >Tracing generator images through Tietze transformations</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap48_mj.html#X7D855FA08242898A" >48 .9 -1 TzInitGeneratorImages</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap48_mj.html#X7AB9A06F80FB3659" >48 .9 -2 OldGeneratorsOfPresentation</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap48_mj.html#X798B38F87C082C45" >48 .9 -3 TzImagesOldGens</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap48_mj.html#X7AC41B117DBB87D6" >48 .9 -4 TzPreImagesNewGens</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap48_mj.html#X7F086D0E7AD6173B" >48 .9 -5 TzPrintGeneratorImages</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap48_mj.html#X7D9E283D81CCCF1A" >48 .10 <span class="Heading" >The Decoding Tree Procedure</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap48_mj.html#X7ACBFE2F78D72A31" >48 .10 -1 DecodeTree</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap48_mj.html#X856F37537E9927EE" >48 .11 <span class="Heading" >Tietze Options</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap48_mj.html#X8178683283214D88" >48 .11 -1 TzOptions</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap48_mj.html#X7BC90B6882DE6D10" >48 .11 -2 TzPrintOptions</a></span >
</div ></div >
</div >
<div class="ContChap" ><a href="chap49_mj.html#X7D5C75647DB168F1" >49 <span class="Heading" >Group Products</span ></a>
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap49_mj.html#X7D39232A84CD8DBD" >49 .1 <span class="Heading" >Direct Products</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap49_mj.html#X861BA02C7902A4F4" >49 .1 -1 DirectProduct</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap49_mj.html#X87FE512E7DB7346C" >49 .2 <span class="Heading" >Semidirect Products</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap49_mj.html#X7D905A5778D7ACDE" >49 .2 -1 <span class="Heading" >SemidirectProduct</span ></a>
</span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap49_mj.html#X815AFC537B215D7B" >49 .3 <span class="Heading" >Subdirect Products</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap49_mj.html#X82112D768085AD98" >49 .3 -1 SubdirectProduct</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap49_mj.html#X814204E97812894C" >49 .3 -2 SubdirectProducts</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap49_mj.html#X7DF2AEBC8518FFA4" >49 .4 <span class="Heading" >Wreath Products</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap49_mj.html#X8786EFBC78D7D6ED" >49 .4 -1 WreathProduct</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap49_mj.html#X8589DCFA7C2E5FAA" >49 .4 -2 WreathProductImprimitiveAction</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap49_mj.html#X82B8DD1C868A3726" >49 .4 -3 WreathProductProductAction</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap49_mj.html#X80634C3180E0C593" >49 .4 -4 KuKGenerators</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap49_mj.html#X801A358E879A0FF0" >49 .4 -5 ListWreathProductElement</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap49_mj.html#X7ECB076E81D8D402" >49 .4 -6 WreathProductElementList</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap49_mj.html#X7AC1AD17833117DF" >49 .5 <span class="Heading" >Free Products</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap49_mj.html#X837AC5A081EECF50" >49 .5 -1 <span class="Heading" >FreeProduct</span ></a>
</span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap49_mj.html#X798FDA1386A0EAC6" >49 .6 <span class="Heading" >Embeddings and Projections for Group Products</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap49_mj.html#X784149B8847B20FF" >49 .6 -1 Embedding</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap49_mj.html#X86F275AC7C625626" >49 .6 -2 Projection</a></span >
</div ></div >
</div >
<div class="ContChap" ><a href="chap50_mj.html#X81B00B667D2BD022" >50 <span class="Heading" >Group Libraries</span ></a>
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap50_mj.html#X839981CC7D9B671B" >50 .1 <span class="Heading" >Basic Groups</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap50_mj.html#X8489BECB78664847" >50 .1 -1 TrivialGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap50_mj.html#X7A7C473D87B31F3B" >50 .1 -2 CyclicGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap50_mj.html#X81CCC3BF8005A2D7" >50 .1 -3 AbelianGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap50_mj.html#X8778256286E50743" >50 .1 -4 ElementaryAbelianGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap50_mj.html#X7F43050D8587E767" >50 .1 -5 FreeAbelianGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap50_mj.html#X838DE1AB7B3D70FF" >50 .1 -6 DihedralGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap50_mj.html#X8233A853818CAF33" >50 .1 -7 IsDihedralGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap50_mj.html#X7E9844EF7C47EEB0" >50 .1 -8 DicyclicGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap50_mj.html#X7F260D177FD4BE4C" >50 .1 -9 IsDicyclicGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap50_mj.html#X86E76B3A796BEFA8" >50 .1 -10 ExtraspecialGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap50_mj.html#X7E54D3E778E6A53E" >50 .1 -11 <span class="Heading" >AlternatingGroup</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap50_mj.html#X858666F97BD85ABB" >50 .1 -12 <span class="Heading" >SymmetricGroup</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap50_mj.html#X788FA7DE84E0FE6A" >50 .1 -13 MathieuGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap50_mj.html#X8469DBBF82F8E5C3" >50 .1 -14 SuzukiGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap50_mj.html#X87E5B0F679CA7FE4" >50 .1 -15 ReeGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap50_mj.html#X7D0FFDA4793995FC" >50 .1 -16 <span class="Heading" >Generator Names</span ></a>
</span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap50_mj.html#X8674AAA578FE4AEE" >50 .2 <span class="Heading" >Classical Groups</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap50_mj.html#X85D607DD82AF3E27" >50 .2 -1 <span class="Heading" >GeneralLinearGroup</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap50_mj.html#X7CA3F7BF83992C6B" >50 .2 -2 <span class="Heading" >SpecialLinearGroup</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap50_mj.html#X866D4E2B816BDFA5" >50 .2 -3 GeneralUnitaryGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap50_mj.html#X82A2AADE805DCDE9" >50 .2 -4 SpecialUnitaryGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap50_mj.html#X8142A8B07811CA90" >50 .2 -5 <span class="Heading" >SymplecticGroup</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap50_mj.html#X7C2051CB7B94CEB1" >50 .2 -6 GeneralOrthogonalGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap50_mj.html#X78D4EEF27AA2DCFD" >50 .2 -7 SpecialOrthogonalGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap50_mj.html#X8365E0AB8338DA3F" >50 .2 -8 Omega</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap50_mj.html#X79C3C61A7D83A6D0" >50 .2 -9 GeneralSemilinearGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap50_mj.html#X7D3779237CB5B49C" >50 .2 -10 SpecialSemilinearGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap50_mj.html#X7F0DBEB880D2D574" >50 .2 -11 ProjectiveGeneralLinearGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap50_mj.html#X86784EDA80224B74" >50 .2 -12 ProjectiveSpecialLinearGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap50_mj.html#X7E471ADE7E095604" >50 .2 -13 ProjectiveGeneralUnitaryGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap50_mj.html#X7A88FE2B7EF9C804" >50 .2 -14 ProjectiveSpecialUnitaryGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap50_mj.html#X7DEDE2537B8FFFF5" >50 .2 -15 ProjectiveSymplecticGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap50_mj.html#X87AAB20A8434356B" >50 .2 -16 ProjectiveGeneralOrthogonalGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap50_mj.html#X835E0D3384C4AB6B" >50 .2 -17 ProjectiveSpecialOrthogonalGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap50_mj.html#X7F546F907A37DF15" >50 .2 -18 ProjectiveOmega</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap50_mj.html#X824925DB7C3A2FA6" >50 .2 -19 ProjectiveGeneralSemilinearGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap50_mj.html#X86BD9AE27CCAB1A6" >50 .2 -20 ProjectiveSpecialSemilinearGroup</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap50_mj.html#X85B9F2D379616C35" >50 .3 <span class="Heading" >Conjugacy Classes in Classical Groups</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap50_mj.html#X831789117E93171E" >50 .3 -1 NrConjugacyClassesGL</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap50_mj.html#X817EBD6E841285CD" >50 .4 <span class="Heading" >Constructors for Basic Groups</span ></a>
</span >
</div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap50_mj.html#X82676ED5826E9E2E" >50 .5 <span class="Heading" >Selection Functions</span ></a>
</span >
</div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap50_mj.html#X7A884ECF813C2026" >50 .6 <span class="Heading" >Finite Perfect Groups</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap50_mj.html#X866A25F882A4E97B" >50 .6 -1 SizesPerfectGroups</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap50_mj.html#X7906BBA7818E9415" >50 .6 -2 <span class="Heading" >PerfectGroup</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap50_mj.html#X7E1CB2D18085FF9D" >50 .6 -3 PerfectIdentification</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap50_mj.html#X7D68BE547FE5C0F5" >50 .6 -4 NumberPerfectGroups</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap50_mj.html#X866356A684F6B15E" >50 .6 -5 SizeNumbersPerfectGroups</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap50_mj.html#X845419F07BB92867" >50 .6 -6 <span class="Heading" >DisplayInformationPerfectGroups</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap50_mj.html#X875C5BE67BAB7F71" >50 .6 -7 <span class="Heading" >More about the Perfect Groups Library</span ></a>
</span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap50_mj.html#X7873506D873EDB95" >50 .7 <span class="Heading" >Irreducible Maximal Finite Integral Matrix Groups</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap50_mj.html#X8693FD647EF3C53B" >50 .7 -1 ImfNumberQQClasses</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap50_mj.html#X8705F64B7E19DDC7" >50 .7 -2 DisplayImfInvariants</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap50_mj.html#X8604A2167B2E8434" >50 .7 -3 ImfInvariants</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap50_mj.html#X78935B307B909101" >50 .7 -4 ImfMatrixGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap50_mj.html#X84BF34B27CD5E85C" >50 .7 -5 IsomorphismPermGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap50_mj.html#X7CEDB6CE7BAC4518" >50 .7 -6 IsomorphismPermGroupImfGroup</a></span >
</div ></div >
</div >
<div class="ContChap" ><a href="chap51_mj.html#X8665D8737FDD5B10" >51 <span class="Heading" >Semigroups and Monoids</span ></a>
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap51_mj.html#X80AF5F307DBDC2B4" >51 .1 <span class="Heading" >Semigroups</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap51_mj.html#X7B412E5B8543E9B7" >51 .1 -1 IsSemigroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap51_mj.html#X7F55D28F819B2817" >51 .1 -2 <span class="Heading" >Semigroup</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap51_mj.html#X8678D40878CC09A1" >51 .1 -3 Subsemigroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap51_mj.html#X782B7BDD8252581C" >51 .1 -4 IsSubsemigroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap51_mj.html#X79FBBEC9841544F3" >51 .1 -5 SemigroupByGenerators</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap51_mj.html#X80ED104F85AE5134" >51 .1 -6 AsSemigroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap51_mj.html#X7B1EEA3E82BFE09F" >51 .1 -7 AsSubsemigroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap51_mj.html#X78147A247963F23B" >51 .1 -8 GeneratorsOfSemigroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap51_mj.html#X79776D7C8399F2CF" >51 .1 -9 IsGeneratorsOfSemigroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap51_mj.html#X7C72E4747BF642BB" >51 .1 -10 <span class="Heading" >FreeSemigroup</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap51_mj.html#X7E67E13F7A01F8D3" >51 .1 -11 SemigroupByMultiplicationTable</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap51_mj.html#X872FE34A7814C0DC" >51 .2 <span class="Heading" >Monoids</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap51_mj.html#X861C523483C6248C" >51 .2 -1 IsMonoid</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap51_mj.html#X7F95328B7C7E49EA" >51 .2 -2 <span class="Heading" >Monoid</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap51_mj.html#X8322D01E84912FD7" >51 .2 -3 Submonoid</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap51_mj.html#X85129EE387CC4D28" >51 .2 -4 MonoidByGenerators</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap51_mj.html#X7B22038F832B9C0F" >51 .2 -5 AsMonoid</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap51_mj.html#X7C9A12DE8287B2D3" >51 .2 -6 AsSubmonoid</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap51_mj.html#X83CA2E7279C44718" >51 .2 -7 GeneratorsOfMonoid</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap51_mj.html#X7EC77C0184587181" >51 .2 -8 TrivialSubmonoid</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap51_mj.html#X79FA3FA978CA2E43" >51 .2 -9 <span class="Heading" >FreeMonoid</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap51_mj.html#X7BFE938E857CA27D" >51 .2 -10 MonoidByMultiplicationTable</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap51_mj.html#X840847B6810BD0E1" >51 .3 <span class="Heading" >Inverse semigroups and monoids</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap51_mj.html#X78B13FED7AFB4326" >51 .3 -1 InverseSemigroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap51_mj.html#X80D9B9A98736051B" >51 .3 -2 InverseMonoid</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap51_mj.html#X87C373597F787250" >51 .3 -3 GeneratorsOfInverseSemigroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap51_mj.html#X7A3B262C85B6D475" >51 .3 -4 GeneratorsOfInverseMonoid</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap51_mj.html#X7C4C6EE681E7A57E" >51 .3 -5 IsInverseSubsemigroup</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap51_mj.html#X78274024827F306D" >51 .4 <span class="Heading" >Properties of Semigroups</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap51_mj.html#X7C4663827C5ACEF1" >51 .4 -1 IsRegularSemigroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap51_mj.html#X87532A76854347E0" >51 .4 -2 IsRegularSemigroupElement</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap51_mj.html#X7AFDE0F17AE516C5" >51 .4 -3 InversesOfSemigroupElement</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap51_mj.html#X836F4692839F4874" >51 .4 -4 IsSimpleSemigroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap51_mj.html#X8193A60F839C064E" >51 .4 -5 IsZeroSimpleSemigroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap51_mj.html#X85F7E5CD86F0643B" >51 .4 -6 IsZeroGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap51_mj.html#X7FFEC81F7F2C4EAA" >51 .4 -7 IsReesCongruenceSemigroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap51_mj.html#X83F1529479D56665" >51 .4 -8 IsInverseSemigroup</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap51_mj.html#X7BB32D508183C0F1" >51 .5 <span class="Heading" >Ideals of semigroups</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap51_mj.html#X7D5CEE4D7D4318ED" >51 .5 -1 SemigroupIdealByGenerators</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap51_mj.html#X7F01FFB18125DED5" >51 .5 -2 ReesCongruenceOfSemigroupIdeal</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap51_mj.html#X7A3FF85984345540" >51 .5 -3 IsLeftSemigroupIdeal</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap51_mj.html#X7914691E7DFFE27A" >51 .6 <span class="Heading" >Congruences on semigroups</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap51_mj.html#X78E34B737F0E009F" >51 .6 -1 IsSemigroupCongruence</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap51_mj.html#X822DB78579BCB7B5" >51 .6 -2 IsReesCongruence</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap51_mj.html#X87CE9EAB7EE3A128" >51 .7 <span class="Heading" >Quotients</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap51_mj.html#X80EF3E6F842BE64E" >51 .7 -1 IsQuotientSemigroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap51_mj.html#X7CAD3D1687956F7F" >51 .7 -2 HomomorphismQuotientSemigroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap51_mj.html#X87120C46808F7289" >51 .7 -3 QuotientSemigroupPreimage</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap51_mj.html#X80C6C718801855E9" >51 .8 <span class="Heading" >Green's Relations</span></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap51_mj.html#X786CEDD4814A9079" >51 .8 -1 GreensRRelation</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap51_mj.html#X8364D69987D49DE1" >51 .8 -2 IsGreensRelation</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap51_mj.html#X82A11A087AFB3EB0" >51 .8 -3 IsGreensClass</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap51_mj.html#X7AA204C8850F9070" >51 .8 -4 IsGreensLessThanOrEqual</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap51_mj.html#X86FE5F5585EBCF13" >51 .8 -5 RClassOfHClass</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap51_mj.html#X78C56F4A78E0088A" >51 .8 -6 EggBoxOfDClass</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap51_mj.html#X803237F17ACD44E3" >51 .8 -7 DisplayEggBoxOfDClass</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap51_mj.html#X87C75A9D86122D93" >51 .8 -8 GreensRClassOfElement</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap51_mj.html#X844D20467A644811" >51 .8 -9 GreensRClasses</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap51_mj.html#X7CB4A18685B850E2" >51 .8 -10 GroupHClassOfGreensDClass</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap51_mj.html#X79D740EF7F0E53BD" >51 .8 -11 IsGroupHClass</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap51_mj.html#X7F5860927CAD920F" >51 .8 -12 IsRegularDClass</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap51_mj.html#X81AF2EAB7CEF8C19" >51 .8 -13 DisplaySemigroup</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap51_mj.html#X8225A9EC87A255E6" >51 .9 <span class="Heading" >Rees Matrix Semigroups</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap51_mj.html#X8526AA557CDF6C49" >51 .9 -1 ReesMatrixSemigroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap51_mj.html#X78D2A48C87FC8E38" >51 .9 -2 ReesMatrixSubsemigroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap51_mj.html#X7964B5C97FB9C07D" >51 .9 -3 IsomorphismReesMatrixSemigroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap51_mj.html#X7F6B852B81488C86" >51 .9 -4 IsReesMatrixSemigroupElement</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap51_mj.html#X7A0DE1F28470295E" >51 .9 -5 ReesMatrixSemigroupElement</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap51_mj.html#X7F03BE707AC7F8A0" >51 .9 -6 IsReesMatrixSubsemigroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap51_mj.html#X780BB78A79275244" >51 .9 -7 IsReesMatrixSemigroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap51_mj.html#X7CACF4D686AF1D19" >51 .9 -8 Matrix</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap51_mj.html#X82FC5D6980C66AC4" >51 .9 -9 <span class="Heading" >Rows and columns</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap51_mj.html#X7D9719F887AFCF8F" >51 .9 -10 UnderlyingSemigroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap51_mj.html#X7D1D9A0382064B8F" >51 .9 -11 AssociatedReesMatrixSemigroupOfDClass</a></span >
</div ></div >
</div >
<div class="ContChap" ><a href="chap52_mj.html#X7DE7C52A7C4BDADE" >52 <span class="Heading" >Finitely Presented Semigroups and Monoids</span ></a>
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap52_mj.html#X78C80F1A84C58E1E" >52 .1 <span class="Heading" >IsSubsemigroupFpSemigroup (Filter)</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap52_mj.html#X8496E23C80453C33" >52 .1 -1 IsSubsemigroupFpSemigroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap52_mj.html#X8239EF2B853411E9" >52 .1 -2 IsFpSemigroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap52_mj.html#X81ABBE997A4C19B7" >52 .1 -3 IsElementOfFpSemigroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap52_mj.html#X7DC8A5D380AFE5DB" >52 .1 -4 FpGrpMonSmgOfFpGrpMonSmgElement</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap52_mj.html#X865E230B83982E66" >52 .2 <span class="Heading" >Creating Finitely Presented Semigroups and Monoids</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap52_mj.html#X84745EC6789FEB4C" ><code >52 .2 -1 \/</code ></a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap52_mj.html#X822F04B2833BE254" >52 .2 -2 FactorFreeSemigroupByRelations</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap52_mj.html#X869F966B8196F28C" >52 .2 -3 IsomorphismFpSemigroup</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap52_mj.html#X85E7C8407C9D5FBE" >52 .3 <span class="Heading" >Comparison of Elements of Finitely Presented Semigroups</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap52_mj.html#X7DD9D81F863EBE31" ><code >52 .3 -1 \=</code ></a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap52_mj.html#X7CD806CA7E0A1438" >52 .4 <span class="Heading" >Preimages in the Free Semigroup or Monoid</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap52_mj.html#X784B3DB686E7080C" >52 .4 -1 UnderlyingElement</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap52_mj.html#X847012347856C55E" >52 .4 -2 ElementOfFpSemigroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap52_mj.html#X8726523779601873" >52 .4 -3 FreeSemigroupOfFpSemigroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap52_mj.html#X79A39402806B5EB7" >52 .4 -4 FreeGeneratorsOfFpSemigroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap52_mj.html#X862BE9FA7C987CAB" >52 .4 -5 RelationsOfFpSemigroup</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap52_mj.html#X87693BDC79DC6EBF" >52 .5 <span class="Heading" >Rewriting Systems and the Knuth-Bendix Procedure</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap52_mj.html#X7D8F804E814D894D" >52 .5 -1 ReducedConfluentRewritingSystem</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap52_mj.html#X7A3F8AE285C41D80" >52 .5 -2 KB_REW</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap52_mj.html#X87A3823483E4FF86" >52 .5 -3 <span class="Heading" >KnuthBendixRewritingSystem</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap52_mj.html#X7966343587A04AFF" >52 .5 -4 SemigroupOfRewritingSystem</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap52_mj.html#X80B8115C8147F605" >52 .5 -5 FreeSemigroupOfRewritingSystem</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap52_mj.html#X812C28217F3E6720" >52 .6 <span class="Heading" >Todd-Coxeter Procedure</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap52_mj.html#X7C24508A7F677520" >52 .6 -1 CosetTableOfFpSemigroup</a></span >
</div ></div >
</div >
<div class="ContChap" ><a href="chap53_mj.html#X860026B880BCB2A5" >53 <span class="Heading" >Transformations</span ></a>
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap53_mj.html#X7CF9291C7CC42340" >53 .1 <span class="Heading" >The family and categories of transformations</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap53_mj.html#X7B6259467974FB70" >53 .1 -1 IsTransformation</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap53_mj.html#X7A6747CE85F2E6EA" >53 .1 -2 IsTransformationCollection</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap53_mj.html#X7E58AFA1832FF064" >53 .1 -3 TransformationFamily</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap53_mj.html#X80F3086F87E93DF8" >53 .2 <span class="Heading" >Creating transformations</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap53_mj.html#X86ADBDE57A20E323" >53 .2 -1 Transformation</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap53_mj.html#X8040642687531E7F" >53 .2 -2 Transformation</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap53_mj.html#X7E82EBD68455EE4A" >53 .2 -3 TransformationByImageAndKernel</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap53_mj.html#X85D1071484CE004C" >53 .2 -4 Idempotent</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap53_mj.html#X7C2A3FC9782F2099" >53 .2 -5 TransformationOp</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap53_mj.html#X7D6FCC417DE86CD1" >53 .2 -6 TransformationNumber</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap53_mj.html#X8475448F87E8CB8A" >53 .2 -7 <span class="Heading" >RandomTransformation</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap53_mj.html#X8268A58685BEFD6F" >53 .2 -8 IdentityTransformation</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap53_mj.html#X7F1E4B5184210D2B" >53 .2 -9 ConstantTransformation</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap53_mj.html#X7F81A18B813C9DF0" >53 .3 <span class="Heading" >Changing the representation of a transformation</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap53_mj.html#X7C5360B2799943F3" >53 .3 -1 AsTransformation</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap53_mj.html#X846A6F6B7B715188" >53 .3 -2 RestrictedTransformation</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap53_mj.html#X8708AE247F5B129B" >53 .3 -3 PermutationOfImage</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap53_mj.html#X812CEC008609A8A2" >53 .4 <span class="Heading" >Operators for transformations</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap53_mj.html#X853031E37F214D10" ><code >53 .4 -1 \^</code ></a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap53_mj.html#X824A8E247DE7E53E" ><code >53 .4 -2 \^</code ></a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap53_mj.html#X7F6573E67D27D822" ><code >53 .4 -3 \*</code ></a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap53_mj.html#X828CA7137F97C124" ><code >53 .4 -4 \/</code ></a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap53_mj.html#X7856D91E8709EF5B" >53 .4 -5 LeftQuotient</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap53_mj.html#X84B8E294826A9377" ><code >53 .4 -6 \<</code ></a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap53_mj.html#X7D454AAD851AE07E" ><code >53 .4 -7 \=</code ></a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap53_mj.html#X83DBA2A18719EFA8" >53 .4 -8 PermLeftQuoTransformation</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap53_mj.html#X8275DFAA8270BB59" >53 .4 -9 IsInjectiveListTrans</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap53_mj.html#X834A313B7DAF06D5" >53 .4 -10 ComponentTransformationInt</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap53_mj.html#X82F5DEEC837B60A3" >53 .4 -11 PreImagesOfTransformation</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap53_mj.html#X86DE4F7A7C535820" >53 .5 <span class="Heading" >Attributes for transformations</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap53_mj.html#X78A209C87CF0E32B" >53 .5 -1 DegreeOfTransformation</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap53_mj.html#X7AEC9E6687B3505A" >53 .5 -2 ImageListOfTransformation</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap53_mj.html#X839A6D6082A21D1F" >53 .5 -3 ImageSetOfTransformation</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap53_mj.html#X818EBB167C7EA37B" >53 .5 -4 RankOfTransformation</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap53_mj.html#X844F00F982D5BD3C" >53 .5 -5 MovedPoints</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap53_mj.html#X7FA6A4B57FDA003D" >53 .5 -6 NrMovedPoints</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap53_mj.html#X86C0DDDC7881273A" >53 .5 -7 SmallestMovedPoint</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap53_mj.html#X8383A7727AC97724" >53 .5 -8 LargestMovedPoint</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap53_mj.html#X7CCFE27E83676572" >53 .5 -9 SmallestImageOfMovedPoint</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap53_mj.html#X7E7172567C3A3E63" >53 .5 -10 LargestImageOfMovedPoint</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap53_mj.html#X8083794579274E87" >53 .5 -11 FlatKernelOfTransformation</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap53_mj.html#X80FCB5048789CF75" >53 .5 -12 KernelOfTransformation</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap53_mj.html#X860306EB7FAAD2D4" >53 .5 -13 InverseOfTransformation</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap53_mj.html#X7BB9DB6E8558356D" >53 .5 -14 Inverse</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap53_mj.html#X863216CB7AF88BED" >53 .5 -15 IndexPeriodOfTransformation</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap53_mj.html#X85FE9F20810BCC70" >53 .5 -16 SmallestIdempotentPower</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap53_mj.html#X858E944481F6B591" >53 .5 -17 ComponentsOfTransformation</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap53_mj.html#X8640AE1C79201470" >53 .5 -18 NrComponentsOfTransformation</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap53_mj.html#X784650B583CEAF7D" >53 .5 -19 ComponentRepsOfTransformation</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap53_mj.html#X7EAA15557D55D93B" >53 .5 -20 CyclesOfTransformation</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap53_mj.html#X786EB02A829260DB" >53 .5 -21 CycleTransformationInt</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap53_mj.html#X845869E0815A6AA6" >53 .5 -22 LeftOne</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap53_mj.html#X7F19C9C77F9F8981" >53 .5 -23 TrimTransformation</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap53_mj.html#X810D23017A5527B7" >53 .6 <span class="Heading" >Displaying transformations</span ></a>
</span >
</div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap53_mj.html#X7B51CE257B814B09" >53 .7 <span class="Heading" >Semigroups of transformations</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap53_mj.html#X7EAF835D7FE4026F" >53 .7 -1 IsTransformationSemigroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap53_mj.html#X7EA699C687952544" >53 .7 -2 DegreeOfTransformationSemigroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap53_mj.html#X7D2B0685815B4053" >53 .7 -3 FullTransformationSemigroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap53_mj.html#X85C58E1E818C838C" >53 .7 -4 IsFullTransformationSemigroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap53_mj.html#X78F29C817CF6827F" >53 .7 -5 IsomorphismTransformationSemigroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap53_mj.html#X820ECE00846E480F" >53 .7 -6 AntiIsomorphismTransformationSemigroup</a></span >
</div ></div >
</div >
<div class="ContChap" ><a href="chap54_mj.html#X7D6495F77B8A77BD" >54 <span class="Heading" >Partial permutations</span ></a>
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap54_mj.html#X87B0D6657A3F2B0E" >54 .1 <span class="Heading" >The family and categories of partial permutations</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap54_mj.html#X7EECE133792B30FC" >54 .1 -1 IsPartialPerm</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap54_mj.html#X8262A827790DD1CC" >54 .1 -2 IsPartialPermCollection</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap54_mj.html#X7E63D17780F64FBA" >54 .1 -3 PartialPermFamily</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap54_mj.html#X7B9D451D7FDA1DD8" >54 .2 <span class="Heading" >Creating partial permutations</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap54_mj.html#X8538BAE77F2FB2F8" >54 .2 -1 PartialPerm</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap54_mj.html#X81188D9F83F64222" >54 .2 -2 PartialPermOp</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap54_mj.html#X80ABBF4883C79060" >54 .2 -3 RestrictedPartialPerm</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap54_mj.html#X849668DD7B0B9E3B" >54 .2 -4 JoinOfPartialPerms</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap54_mj.html#X81E2B6977E28CD00" >54 .2 -5 MeetOfPartialPerms</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap54_mj.html#X80EFB142817A0A9F" >54 .2 -6 EmptyPartialPerm</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap54_mj.html#X7E6ADC8583C31530" >54 .2 -7 <span class="Heading" >RandomPartialPerm</span ></a>
</span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap54_mj.html#X8779F0997D0FDA78" >54 .3 <span class="Heading" >Attributes for partial permutations</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap54_mj.html#X8612A4DC864E7959" >54 .3 -1 DegreeOfPartialPerm</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap54_mj.html#X8413D0EF7DEE1FFF" >54 .3 -2 CodegreeOfPartialPerm</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap54_mj.html#X7C1ABD8A80E95B39" >54 .3 -3 RankOfPartialPerm</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap54_mj.html#X784A14F787E041D7" >54 .3 -4 DomainOfPartialPerm</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap54_mj.html#X7CD84B107831E0FC" >54 .3 -5 ImageOfPartialPermCollection</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap54_mj.html#X8333293F87F654FA" >54 .3 -6 ImageListOfPartialPerm</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap54_mj.html#X7F0724A07A14DCF7" >54 .3 -7 ImageSetOfPartialPerm</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap54_mj.html#X82AAFF938623422E" >54 .3 -8 FixedPointsOfPartialPerm</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap54_mj.html#X82FE981A87FAA2DC" >54 .3 -9 MovedPoints</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap54_mj.html#X7FAF969C84CDC742" >54 .3 -10 NrFixedPoints</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap54_mj.html#X81F5C64E7DAD27A7" >54 .3 -11 NrMovedPoints</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap54_mj.html#X84A49C977E1E29AA" >54 .3 -12 SmallestMovedPoint</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap54_mj.html#X7D4290A785ABC86D" >54 .3 -13 LargestMovedPoint</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap54_mj.html#X85280F1A7B1014BA" >54 .3 -14 SmallestImageOfMovedPoint</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap54_mj.html#X7A95CD437BC1CB1A" >54 .3 -15 LargestImageOfMovedPoint</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap54_mj.html#X873A9F717DA75CBC" >54 .3 -16 IndexPeriodOfPartialPerm</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap54_mj.html#X7C04AA377F080722" >54 .3 -17 SmallestIdempotentPower</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap54_mj.html#X8185065E788BDD0D" >54 .3 -18 ComponentsOfPartialPerm</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap54_mj.html#X7CB51EB67FFA95E9" >54 .3 -19 NrComponentsOfPartialPerm</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap54_mj.html#X7AAAAE4082B30E18" >54 .3 -20 ComponentRepsOfPartialPerm</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap54_mj.html#X7A8FB86C78C49F85" >54 .3 -21 LeftOne</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap54_mj.html#X857FC10C81507E8B" >54 .3 -22 One</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap54_mj.html#X7D90CF497D58D759" >54 .3 -23 MultiplicativeZero</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap54_mj.html#X8585AA8B78E9CDFB" >54 .4 <span class="Heading" >Changing the representation of a partial permutation</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap54_mj.html#X81B32CB182489ACA" >54 .4 -1 AsPartialPerm</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap54_mj.html#X87EC67747B260E98" >54 .4 -2 AsPartialPerm</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap54_mj.html#X848CD855802C6CE1" >54 .5 <span class="Heading" >Operators and operations for partial permutations</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap54_mj.html#X7B8630027B7F0BCC" >54 .5 -1 Inverse</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap54_mj.html#X792D3BA278DAB869" ><code >54 .5 -2 \^</code ></a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap54_mj.html#X7DC25BC47AAA9C73" ><code >54 .5 -3 \/</code ></a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap54_mj.html#X8213CD6E7C461169" ><code >54 .5 -4 \^</code ></a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap54_mj.html#X86469D597F8BC7CE" ><code >54 .5 -5 \*</code ></a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap54_mj.html#X869DBDF67FA3817B" ><code >54 .5 -6 \/</code ></a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap54_mj.html#X82E3A3E186A4F2D2" >54 .5 -7 LeftQuotient</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap54_mj.html#X8659E9E57AC8D9CE" ><code >54 .5 -8 \<</code ></a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap54_mj.html#X7828338C7DB8AAC7" ><code >54 .5 -9 \=</code ></a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap54_mj.html#X8382A0F8875CEB08" >54 .5 -10 PermLeftQuoPartialPerm</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap54_mj.html#X7C7F5EAB7E9A381D" >54 .5 -11 PreImagePartialPerm</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap54_mj.html#X797A6CC084068219" >54 .5 -12 ComponentPartialPermInt</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap54_mj.html#X87B1ED93785257C1" >54 .5 -13 NaturalLeqPartialPerm</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap54_mj.html#X81BD69307D294A1C" >54 .5 -14 ShortLexLeqPartialPerm</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap54_mj.html#X83560BE678ACF855" >54 .5 -15 TrimPartialPerm</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap54_mj.html#X7849595B81D063EE" >54 .6 <span class="Heading" >Displaying partial permutations</span ></a>
</span >
</div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap54_mj.html#X7CCC82E07A73EB55" >54 .7 <span class="Heading" >Semigroups and inverse semigroups of partial permutations</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap54_mj.html#X7D161674800B50E0" >54 .7 -1 IsPartialPermSemigroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap54_mj.html#X7D7F0BAB82F0D820" >54 .7 -2 DegreeOfPartialPermSemigroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap54_mj.html#X81D271B380995F8A" >54 .7 -3 SymmetricInverseSemigroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap54_mj.html#X7C8AEA50834060DD" >54 .7 -4 IsSymmetricInverseSemigroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap54_mj.html#X7EA51F087CF7621F" >54 .7 -5 NaturalPartialOrder</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap54_mj.html#X7FE18EBE79B9C17C" >54 .7 -6 IsomorphismPartialPermSemigroup</a></span >
</div ></div >
</div >
<div class="ContChap" ><a href="chap55_mj.html#X7D0D096B81365B02" >55 <span class="Heading" >Additive Magmas</span ></a>
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap55_mj.html#X82A4AB7B812B063B" >55 .1 <span class="Heading" >(Near-)Additive Magma Categories</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap55_mj.html#X8129E95D83227658" >55 .1 -1 IsNearAdditiveMagma</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap55_mj.html#X7DADE4577D0A7208" >55 .1 -2 IsNearAdditiveMagmaWithZero</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap55_mj.html#X7FC3A9C178185942" >55 .1 -3 IsNearAdditiveGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap55_mj.html#X8565FD0C847BAA3A" >55 .1 -4 IsAdditiveMagma</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap55_mj.html#X785B41A67D791783" >55 .1 -5 IsAdditiveMagmaWithZero</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap55_mj.html#X7B8FBD9082CE271B" >55 .1 -6 IsAdditiveGroup</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap55_mj.html#X7C39F9DE7CA22688" >55 .2 <span class="Heading" >(Near-)Additive Magma Generation</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap55_mj.html#X79C947CF8060335A" >55 .2 -1 NearAdditiveMagma</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap55_mj.html#X80F57FB47E1DB380" >55 .2 -2 NearAdditiveMagmaWithZero</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap55_mj.html#X872307537ECC5755" >55 .2 -3 NearAdditiveGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap55_mj.html#X85122CFD7BDAD668" >55 .2 -4 NearAdditiveMagmaByGenerators</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap55_mj.html#X81880460851DEFBC" >55 .2 -5 NearAdditiveMagmaWithZeroByGenerators</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap55_mj.html#X85F120B68576B267" >55 .2 -6 NearAdditiveGroupByGenerators</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap55_mj.html#X7AA6092683FC0F9C" >55 .2 -7 SubnearAdditiveMagma</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap55_mj.html#X784859197D89A548" >55 .2 -8 SubnearAdditiveMagmaWithZero</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap55_mj.html#X844C49BA807AB99F" >55 .2 -9 SubnearAdditiveGroup</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap55_mj.html#X799E6CC28737BF1B" >55 .3 <span class="Heading" >Attributes and Properties for (Near-)Additive Magmas</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap55_mj.html#X82D471327A9CA960" >55 .3 -1 IsAdditivelyCommutative</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap55_mj.html#X804B178884002A40" >55 .3 -2 GeneratorsOfNearAdditiveMagma</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap55_mj.html#X7EB9ABF880DCAE01" >55 .3 -3 GeneratorsOfNearAdditiveMagmaWithZero</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap55_mj.html#X7EA15714795D71CF" >55 .3 -4 GeneratorsOfNearAdditiveGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap55_mj.html#X851EA2E67F0C9A75" >55 .3 -5 AdditiveNeutralElement</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap55_mj.html#X78FB0A5C86DC86F9" >55 .3 -6 TrivialSubnearAdditiveMagmaWithZero</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap55_mj.html#X7BB03781863BE4EB" >55 .4 <span class="Heading" >Operations for (Near-)Additive Magmas</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap55_mj.html#X845E915B87D2AC16" >55 .4 -1 <span class="Heading" >ClosureNearAdditiveGroup</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap55_mj.html#X8142D994794B700A" >55 .4 -2 ShowAdditionTable</a></span >
</div ></div >
</div >
<div class="ContChap" ><a href="chap56_mj.html#X81897F6082CACB59" >56 <span class="Heading" >Rings</span ></a>
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap56_mj.html#X839FC48687C25FCD" >56 .1 <span class="Heading" >Generating Rings</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap56_mj.html#X80FD843C8221DAC9" >56 .1 -1 IsRing</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap56_mj.html#X820B172A860A5B1A" >56 .1 -2 <span class="Heading" >Ring</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap56_mj.html#X83AFFCC77DE6ABDA" >56 .1 -3 <span class="Heading" >DefaultRing</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap56_mj.html#X7D736E027DFD8961" >56 .1 -4 RingByGenerators</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap56_mj.html#X839E609480495E27" >56 .1 -5 DefaultRingByGenerators</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap56_mj.html#X7D0428D87E63288C" >56 .1 -6 GeneratorsOfRing</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap56_mj.html#X860E4AC78520D27E" >56 .1 -7 Subring</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap56_mj.html#X819B0AFE79C78C34" >56 .1 -8 <span class="Heading" >ClosureRing</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap56_mj.html#X8350500B8576F833" >56 .1 -9 Quotient</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap56_mj.html#X8776C3F97A731E70" >56 .2 <span class="Heading" >Ideals of Rings</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap56_mj.html#X7C486A7C821D79F0" >56 .2 -1 TwoSidedIdeal</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap56_mj.html#X7C8E196478C7431A" >56 .2 -2 TwoSidedIdealNC</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap56_mj.html#X7DF623847B338850" >56 .2 -3 IsTwoSidedIdeal</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap56_mj.html#X86C998178690DAE0" >56 .2 -4 TwoSidedIdealByGenerators</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap56_mj.html#X82D8B07281EB0AC7" >56 .2 -5 LeftIdealByGenerators</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap56_mj.html#X858EAEAF87751428" >56 .2 -6 RightIdealByGenerators</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap56_mj.html#X86AAF5F9800E97EE" >56 .2 -7 GeneratorsOfTwoSidedIdeal</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap56_mj.html#X7B20BD2B7FAFBD64" >56 .2 -8 GeneratorsOfLeftIdeal</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap56_mj.html#X80F2239F8653FF74" >56 .2 -9 GeneratorsOfRightIdeal</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap56_mj.html#X81D81D027C2F8D06" >56 .2 -10 LeftActingRingOfIdeal</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap56_mj.html#X83D9D7408706B69A" >56 .2 -11 AsLeftIdeal</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap56_mj.html#X790DD00586F9B8B8" >56 .3 <span class="Heading" >Rings With One</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap56_mj.html#X7E601FBD8020A0F3" >56 .3 -1 IsRingWithOne</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap56_mj.html#X80942A318417366E" >56 .3 -2 <span class="Heading" >RingWithOne</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap56_mj.html#X851115EC79B8C393" >56 .3 -3 RingWithOneByGenerators</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap56_mj.html#X7F9F122C831BCDD1" >56 .3 -4 GeneratorsOfRingWithOne</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap56_mj.html#X7D0BADF178D4DDF8" >56 .3 -5 SubringWithOne</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap56_mj.html#X797F5869874BDBFB" >56 .4 <span class="Heading" >Properties of Rings</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap56_mj.html#X87A7D5B584713B52" >56 .4 -1 IsIntegralRing</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap56_mj.html#X789A917085DB7527" >56 .4 -2 IsUniqueFactorizationRing</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap56_mj.html#X7D4BB44187C55BF2" >56 .4 -3 IsLDistributive</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap56_mj.html#X79A5AEE786AED315" >56 .4 -4 IsRDistributive</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap56_mj.html#X86716D4F7B968604" >56 .4 -5 IsDistributive</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap56_mj.html#X82DECD237D49D937" >56 .4 -6 IsAnticommutative</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap56_mj.html#X7EC0FEC88535E8CC" >56 .4 -7 IsZeroSquaredRing</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap56_mj.html#X799BEF8581971A13" >56 .4 -8 IsJacobianRing</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap56_mj.html#X8130085978A9B3C4" >56 .5 <span class="Heading" >Units and Factorizations</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap56_mj.html#X85CBFBAE78DE72E8" >56 .5 -1 IsUnit</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap56_mj.html#X853C045B7BA6A580" >56 .5 -2 Units</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap56_mj.html#X7B307F217DDC7E20" >56 .5 -3 IsAssociated</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap56_mj.html#X7A69C9097E17D161" >56 .5 -4 Associates</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap56_mj.html#X7B1A9A4C7C59FB36" >56 .5 -5 StandardAssociate</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap56_mj.html#X7EB6803C789E027D" >56 .5 -6 StandardAssociateUnit</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap56_mj.html#X7CD7C64A7D961A18" >56 .5 -7 IsIrreducibleRingElement</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap56_mj.html#X7AA107AE7F79C6D8" >56 .5 -8 IsPrime</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap56_mj.html#X82D6EDC685D12AE2" >56 .5 -9 Factors</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap56_mj.html#X8559CC7B80C479F1" >56 .5 -10 PadicValuation</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap56_mj.html#X7F12BB99865EB7BF" >56 .6 <span class="Heading" >Euclidean Rings</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap56_mj.html#X808B8E8E80D48E4A" >56 .6 -1 IsEuclideanRing</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap56_mj.html#X784234088350D4E4" >56 .6 -2 EuclideanDegree</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap56_mj.html#X7A93FA788318B147" >56 .6 -3 EuclideanQuotient</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap56_mj.html#X7B5E9639865E91BA" >56 .6 -4 EuclideanRemainder</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap56_mj.html#X876B7532801A1B35" >56 .6 -5 QuotientRemainder</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap56_mj.html#X7E9CF2C07C4A6CEE" >56 .7 <span class="Heading" >Gcd and Lcm</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap56_mj.html#X7DE207718456F98F" >56 .7 -1 <span class="Heading" >Gcd</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap56_mj.html#X7836D50F8341D6E1" >56 .7 -2 GcdOp</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap56_mj.html#X7ABB91EF838075EF" >56 .7 -3 <span class="Heading" >GcdRepresentation</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap56_mj.html#X81392E7F84956341" >56 .7 -4 GcdRepresentationOp</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap56_mj.html#X836DB8B47A0219FB" >56 .7 -5 ShowGcd</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap56_mj.html#X7ABA92057DD6C7AF" >56 .7 -6 <span class="Heading" >Lcm</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap56_mj.html#X7FB6C5A67AC1E8C1" >56 .7 -7 LcmOp</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap56_mj.html#X8555913A83D716A4" >56 .7 -8 QuotientMod</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap56_mj.html#X805A35D684B7A952" >56 .7 -9 PowerMod</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap56_mj.html#X87711E6F8024A358" >56 .7 -10 InterpolatedPolynomial</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap56_mj.html#X7B13484581169439" >56 .8 <span class="Heading" >Homomorphisms of Rings</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap56_mj.html#X7DE9CC5B877C91DA" >56 .8 -1 RingGeneralMappingByImages</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap56_mj.html#X78C1016284F08026" >56 .8 -2 RingHomomorphismByImages</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap56_mj.html#X7D01646A7CCBEDBB" >56 .8 -3 RingHomomorphismByImagesNC</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap56_mj.html#X83D53D98809EC461" >56 .8 -4 NaturalHomomorphismByIdeal</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap56_mj.html#X81D526A57B375AAD" >56 .9 <span class="Heading" >Small Rings</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap56_mj.html#X7E86DCB7812DF04C" >56 .9 -1 SmallRing</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap56_mj.html#X7F2EE9AF83DCE641" >56 .9 -2 NumberSmallRings</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap56_mj.html#X8070D20B86148929" >56 .9 -3 Subrings</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap56_mj.html#X83629803819C4A6F" >56 .9 -4 Ideals</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap56_mj.html#X82AD6F187B550060" >56 .9 -5 DirectSum</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap56_mj.html#X7E7B1B727EA434CF" >56 .9 -6 RingByStructureConstants</a></span >
</div ></div >
</div >
<div class="ContChap" ><a href="chap57_mj.html#X8183A6857B0C3633" >57 <span class="Heading" >Modules</span ></a>
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap57_mj.html#X87A33EFD7CC179C1" >57 .1 <span class="Heading" >Generating modules</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap57_mj.html#X7C62FE5282E9C505" >57 .1 -1 IsLeftOperatorAdditiveGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap57_mj.html#X7ED323027B291BDF" >57 .1 -2 IsLeftModule</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap57_mj.html#X7F76B1FD84775025" >57 .1 -3 GeneratorsOfLeftOperatorAdditiveGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap57_mj.html#X7C7684EF867323C2" >57 .1 -4 GeneratorsOfLeftModule</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap57_mj.html#X7EB3E46D7BC4A35C" >57 .1 -5 AsLeftModule</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap57_mj.html#X7F19AD3D799D0469" >57 .1 -6 IsRightOperatorAdditiveGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap57_mj.html#X8479A5AA7DF25F50" >57 .1 -7 IsRightModule</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap57_mj.html#X7DBC4BCB876EEE1C" >57 .1 -8 GeneratorsOfRightOperatorAdditiveGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap57_mj.html#X8586A83B85F176F6" >57 .1 -9 GeneratorsOfRightModule</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap57_mj.html#X79ED1D7D7F0AE59A" >57 .1 -10 LeftModuleByGenerators</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap57_mj.html#X86F070E0807DC34E" >57 .1 -11 LeftActingDomain</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap57_mj.html#X7934FAE97B6D2AD8" >57 .2 <span class="Heading" >Submodules</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap57_mj.html#X8465103F874BC07B" >57 .2 -1 Submodule</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap57_mj.html#X83CF3AD18050C982" >57 .2 -2 SubmoduleNC</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap57_mj.html#X7C68C4E287481EC0" >57 .2 -3 ClosureLeftModule</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap57_mj.html#X7980BC20856B2B7D" >57 .2 -4 TrivialSubmodule</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap57_mj.html#X85BD57F27F513D3E" >57 .3 <span class="Heading" >Free Modules</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap57_mj.html#X7C4832187F3D9228" >57 .3 -1 IsFreeLeftModule</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap57_mj.html#X7C043E307E344AEE" >57 .3 -2 FreeLeftModule</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap57_mj.html#X7E6926C6850E7C4E" >57 .3 -3 Dimension</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap57_mj.html#X802DB9FB824B0167" >57 .3 -4 IsFiniteDimensional</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap57_mj.html#X7909E8E785420F0E" >57 .3 -5 UseBasis</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap57_mj.html#X7C8F844783F4FA09" >57 .3 -6 IsRowModule</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap57_mj.html#X81FCC1D780435CF1" >57 .3 -7 IsMatrixModule</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap57_mj.html#X853E085C868196EF" >57 .3 -8 IsFullRowModule</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap57_mj.html#X848041A47BC4B038" >57 .3 -9 FullRowModule</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap57_mj.html#X814CEA62842CF5BB" >57 .3 -10 IsFullMatrixModule</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap57_mj.html#X7A0C871B7C446F1F" >57 .3 -11 FullMatrixModule</a></span >
</div ></div >
</div >
<div class="ContChap" ><a href="chap58_mj.html#X80A8E676814A19FD" >58 <span class="Heading" >Fields and Division Rings</span ></a>
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap58_mj.html#X82B74B458705B3CE" >58 .1 <span class="Heading" >Generating Fields</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap58_mj.html#X7F2CAA9E7A16913D" >58 .1 -1 IsDivisionRing</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap58_mj.html#X7A5AE30E7C0F457C" >58 .1 -2 IsField</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap58_mj.html#X871AA7D58263E9AC" >58 .1 -3 Field</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap58_mj.html#X7D9F7FD4786691EE" >58 .1 -4 DefaultField</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap58_mj.html#X7C298A40852C2AFF" >58 .1 -5 DefaultFieldByGenerators</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap58_mj.html#X7EF624958648D0FA" >58 .1 -6 GeneratorsOfDivisionRing</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap58_mj.html#X7AA715317A81261B" >58 .1 -7 GeneratorsOfField</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap58_mj.html#X8641861A8550F8BE" >58 .1 -8 DivisionRingByGenerators</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap58_mj.html#X7C193B7D7AFB29BE" >58 .1 -9 AsDivisionRing</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap58_mj.html#X7C53566A839B57F6" >58 .2 <span class="Heading" >Subfields of Fields</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap58_mj.html#X7FE1FA217A08DCE5" >58 .2 -1 Subfield</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap58_mj.html#X82A0E79A7B9799E0" >58 .2 -2 FieldOverItselfByGenerators</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap58_mj.html#X86DB31B57FB4F570" >58 .2 -3 PrimitiveElement</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap58_mj.html#X7DD27F927BD57FDE" >58 .2 -4 PrimeField</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap58_mj.html#X84B6F1E67AD0E33D" >58 .2 -5 IsPrimeField</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap58_mj.html#X7845CECE86A83219" >58 .2 -6 DegreeOverPrimeField</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap58_mj.html#X7ADDCBF47E2ED3D4" >58 .2 -7 DefiningPolynomial</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap58_mj.html#X8173DA4982DB1E8A" >58 .2 -8 RootOfDefiningPolynomial</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap58_mj.html#X82718B3B818DC699" >58 .2 -9 FieldExtension</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap58_mj.html#X83490C65819D85FE" >58 .2 -10 Subfields</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap58_mj.html#X7D9A02B07D08FA40" >58 .3 <span class="Heading" >Galois Action</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap58_mj.html#X80CAA5BA82F09ED2" >58 .3 -1 GaloisGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap58_mj.html#X8738C6687D784BB5" >58 .3 -2 MinimalPolynomial</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap58_mj.html#X80FE7E017C2D255C" >58 .3 -3 TracePolynomial</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap58_mj.html#X838515278587FF01" >58 .3 -4 Norm</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap58_mj.html#X7DD17EB581200AD6" >58 .3 -5 <span class="Heading" >Traces of field elements and matrices</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap58_mj.html#X837A4A5781F8EE92" >58 .3 -6 Conjugates</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap58_mj.html#X8236A8B47E6AAD93" >58 .3 -7 NormalBase</a></span >
</div ></div >
</div >
<div class="ContChap" ><a href="chap59_mj.html#X7893ABF67A028802" >59 <span class="Heading" >Finite Fields</span ></a>
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap59_mj.html#X7B9DCCCC83400B47" >59 .1 <span class="Heading" >Finite Field Elements</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap59_mj.html#X7D3DF32C84FEBD25" >59 .1 -1 IsFFE</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap59_mj.html#X7AA52FAF7EDEDD56" >59 .1 -2 Z</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap59_mj.html#X8612BCEA816CF1B9" >59 .1 -3 IsLexOrderedFFE</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap59_mj.html#X7A79399283EF78D0" >59 .2 <span class="Heading" >Operations for Finite Field Elements</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap59_mj.html#X828E846E7C1EA3DD" >59 .2 -1 DegreeFFE</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap59_mj.html#X7B049A3478B369E4" >59 .2 -2 LogFFE</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap59_mj.html#X79F48E337FC2746A" >59 .2 -3 IntFFE</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap59_mj.html#X7DABD827848BCC2A" >59 .2 -4 IntFFESymm</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap59_mj.html#X8009968782F18888" >59 .2 -5 IntVecFFE</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap59_mj.html#X807959EE82CED148" >59 .2 -6 AsInternalFFE</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap59_mj.html#X794AEB148410825B" >59 .2 -7 RootFFE</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap59_mj.html#X81B54A8378734C33" >59 .3 <span class="Heading" >Creating Finite Fields</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap59_mj.html#X7979F51D7C43AB05" >59 .3 -1 DefaultField</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap59_mj.html#X8592DBB086A8A9BE" >59 .3 -2 GaloisField</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap59_mj.html#X788B1ECD83C70516" >59 .3 -3 PrimitiveRoot</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap59_mj.html#X7A5F075185CE5B06" >59 .4 <span class="Heading" >Frobenius Automorphisms</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap59_mj.html#X8758E4AB7D0A1955" >59 .4 -1 FrobeniusAutomorphism</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap59_mj.html#X869919BB7EBE5741" >59 .5 <span class="Heading" >Conway Polynomials</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap59_mj.html#X7C2425A786F09054" >59 .5 -1 ConwayPolynomial</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap59_mj.html#X78A7C1247E129AD9" >59 .5 -2 IsCheapConwayPolynomial</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap59_mj.html#X7ECC593583E68A6C" >59 .5 -3 RandomPrimitivePolynomial</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap59_mj.html#X78EE3656879C3B88" >59 .6 <span class="Heading" >Printing, Viewing and Displaying Finite Field Elements</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap59_mj.html#X80DAAA5E7C79C94C" >59 .6 -1 ViewObj</a></span >
</div ></div >
</div >
<div class="ContChap" ><a href="chap60_mj.html#X80510B5880521FDC" >60 <span class="Heading" >Abelian Number Fields</span ></a>
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap60_mj.html#X7D4E43E5799753B5" >60 .1 <span class="Heading" >Construction of Abelian Number Fields</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap60_mj.html#X80D21D80850EFA4B" >60 .1 -1 CyclotomicField</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap60_mj.html#X80E5AD028143E11E" >60 .1 -2 AbelianNumberField</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap60_mj.html#X82F53C65802FF551" >60 .1 -3 GaussianRationals</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap60_mj.html#X81B5FE06781DB824" >60 .2 <span class="Heading" >Operations for Abelian Number Fields</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap60_mj.html#X7B0AB0FB7A4136C4" >60 .2 -1 Factors</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap60_mj.html#X87D78F5E875F2E8A" >60 .2 -2 IsNumberField</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap60_mj.html#X7D202D707D5708FA" >60 .2 -3 IsAbelianNumberField</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap60_mj.html#X84CAE4627F0CD639" >60 .2 -4 IsCyclotomicField</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap60_mj.html#X87E7313D8070B9CC" >60 .2 -5 GaloisStabilizer</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap60_mj.html#X7D2421AC8491D2BE" >60 .3 <span class="Heading" >Integral Bases of Abelian Number Fields</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap60_mj.html#X7F52BEA0862E06F2" >60 .3 -1 ZumbroichBase</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap60_mj.html#X87DB9C2C858B722A" >60 .3 -2 LenstraBase</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap60_mj.html#X7E4AB4B17C7BA10C" >60 .4 <span class="Heading" >Galois Groups of Abelian Number Fields</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap60_mj.html#X7B55A90582E818F3" >60 .4 -1 GaloisGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap60_mj.html#X8643D4B47A827D9D" >60 .4 -2 ANFAutomorphism</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap60_mj.html#X85E9E90D7FE877CC" >60 .5 <span class="Heading" >Gaussians</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap60_mj.html#X80BD5EAB879F096E" >60 .5 -1 GaussianIntegers</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap60_mj.html#X7BFD33D27BFB7C5A" >60 .5 -2 IsGaussianIntegers</a></span >
</div ></div >
</div >
<div class="ContChap" ><a href="chap61_mj.html#X7DAD6700787EC845" >61 <span class="Heading" >Vector Spaces</span ></a>
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap61_mj.html#X8754F7207CFDA38B" >61 .1 <span class="Heading" >IsLeftVectorSpace (Filter)</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap61_mj.html#X80290A908241706B" >61 .1 -1 IsLeftVectorSpace</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap61_mj.html#X87AD06FE873619EA" >61 .2 <span class="Heading" >Constructing Vector Spaces</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap61_mj.html#X805413157CE9BECF" >61 .2 -1 VectorSpace</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap61_mj.html#X78C9826780BC9AE6" >61 .2 -2 Subspace</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap61_mj.html#X7B001BAF7D5FD5D0" >61 .2 -3 AsVectorSpace</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap61_mj.html#X7D4F84C27EDAC89B" >61 .2 -4 AsSubspace</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap61_mj.html#X789FB2D883E53662" >61 .3 <span class="Heading" >Operations and Attributes for Vector Spaces</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap61_mj.html#X849651C6830C94A1" >61 .3 -1 GeneratorsOfLeftVectorSpace</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap61_mj.html#X86DC71A9835430FD" >61 .3 -2 TrivialSubspace</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap61_mj.html#X8125675583357131" >61 .4 <span class="Heading" >Domains of Subspaces of Vector Spaces</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap61_mj.html#X7975E41A7B29C3FD" >61 .4 -1 Subspaces</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap61_mj.html#X7A8F5C367FAE3D1B" >61 .4 -2 IsSubspacesVectorSpace</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap61_mj.html#X828AA09B87F14FAD" >61 .5 <span class="Heading" >Bases of Vector Spaces</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap61_mj.html#X8739510881F5D862" >61 .5 -1 IsBasis</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap61_mj.html#X837BE54C80DE368E" >61 .5 -2 Basis</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap61_mj.html#X7C8EBFF5805F8C51" >61 .5 -3 CanonicalBasis</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap61_mj.html#X8786D40B84120F38" >61 .5 -4 RelativeBasis</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap61_mj.html#X839B9C4880EBFB5F" >61 .6 <span class="Heading" >Operations for Vector Space Bases</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap61_mj.html#X7B1F17AE8027A590" >61 .6 -1 BasisVectors</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap61_mj.html#X81E8AE88843B70FF" >61 .6 -2 UnderlyingLeftModule</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap61_mj.html#X80B32F667BF6AFD8" >61 .6 -3 Coefficients</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap61_mj.html#X7D305AB3834889BF" >61 .6 -4 LinearCombination</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap61_mj.html#X7EB0D16A7EC2DEE3" >61 .6 -5 EnumeratorByBasis</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap61_mj.html#X855625D47979005D" >61 .6 -6 IteratorByBasis</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap61_mj.html#X82809D6C82DE4EC2" >61 .7 <span class="Heading" >Operations for Special Kinds of Bases</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap61_mj.html#X7CC2B3DD81628CE9" >61 .7 -1 IsCanonicalBasis</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap61_mj.html#X86DE147F8606B739" >61 .7 -2 IsIntegralBasis</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap61_mj.html#X7FC051C579D61223" >61 .7 -3 IsNormalBasis</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap61_mj.html#X7C11B9C3819F3EA2" >61 .8 <span class="Heading" >Mutable Bases</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap61_mj.html#X7F466FB47F7E9F00" >61 .8 -1 IsMutableBasis</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap61_mj.html#X8115C061819E5172" >61 .8 -2 MutableBasis</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap61_mj.html#X7EC90F4F7BCAF8D4" >61 .8 -3 NrBasisVectors</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap61_mj.html#X7BA87512823A8CFD" >61 .8 -4 ImmutableBasis</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap61_mj.html#X85B50AC77A22108B" >61 .8 -5 IsContainedInSpan</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap61_mj.html#X7B52C99B84316F61" >61 .8 -6 CloseMutableBasis</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap61_mj.html#X7D937EBC7DE2819B" >61 .9 <span class="Heading" >Row and Matrix Spaces</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap61_mj.html#X79B305CE87511C4B" >61 .9 -1 IsRowSpace</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap61_mj.html#X7A2BBBA07B2BE8F8" >61 .9 -2 IsMatrixSpace</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap61_mj.html#X83724C157F4FDFB4" >61 .9 -3 IsGaussianSpace</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap61_mj.html#X80209A8785126AAB" >61 .9 -4 FullRowSpace</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap61_mj.html#X876B66C37A7B749F" >61 .9 -5 FullMatrixSpace</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap61_mj.html#X8534A750878478D0" >61 .9 -6 DimensionOfVectors</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap61_mj.html#X865A540F85FAE2DF" >61 .9 -7 IsSemiEchelonized</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap61_mj.html#X87DCA09579589106" >61 .9 -8 SemiEchelonBasis</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap61_mj.html#X7C3CC5F97FA048A4" >61 .9 -9 IsCanonicalBasisFullRowModule</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap61_mj.html#X83D282697C1A3148" >61 .9 -10 IsCanonicalBasisFullMatrixModule</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap61_mj.html#X7D6537F87E940344" >61 .9 -11 NormedRowVectors</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap61_mj.html#X815C69A57C042C34" >61 .9 -12 SiftedVector</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap61_mj.html#X7F61CECA84CEF39D" >61 .10 <span class="Heading" >Vector Space Homomorphisms</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap61_mj.html#X82013D328645E370" >61 .10 -1 LeftModuleGeneralMappingByImages</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap61_mj.html#X85F5293983E47B5A" >61 .10 -2 LeftModuleHomomorphismByImages</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap61_mj.html#X8477E6C3872A6DBB" >61 .10 -3 LeftModuleHomomorphismByMatrix</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap61_mj.html#X8494AA5D7C3B88AD" >61 .10 -4 NaturalHomomorphismBySubspace</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap61_mj.html#X80015C78876B4F1E" >61 .10 -5 Hom</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap61_mj.html#X8680ADD381ECF879" >61 .10 -6 End</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap61_mj.html#X7A9A08EA79259659" >61 .10 -7 IsFullHomModule</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap61_mj.html#X7C4737687E76A24A" >61 .10 -8 IsPseudoCanonicalBasisFullHomModule</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap61_mj.html#X84F87C327A1856F2" >61 .10 -9 IsLinearMappingsModule</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap61_mj.html#X81503EB77FCE648D" >61 .11 <span class="Heading" >Vector Spaces Handled By Nice Bases</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap61_mj.html#X826FD4BC7BA0559D" >61 .11 -1 NiceFreeLeftModule</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap61_mj.html#X807B8032780C59A4" >61 .11 -2 NiceVector</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap61_mj.html#X79350786800C2DD8" >61 .11 -3 NiceFreeLeftModuleInfo</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap61_mj.html#X8388E0248690C214" >61 .11 -4 NiceBasis</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap61_mj.html#X82BC30A487967F5B" >61 .11 -5 IsBasisByNiceBasis</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap61_mj.html#X79D1DEA679AEDA40" >61 .11 -6 IsHandledByNiceBasis</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap61_mj.html#X8238195B851D3C44" >61 .12 <span class="Heading" >How to Implement New Kinds of Vector Spaces</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap61_mj.html#X7DE34C3E837FCBC3" >61 .12 -1 DeclareHandlingByNiceBasis</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap61_mj.html#X7E6077F0830A28DA" >61 .12 -2 NiceBasisFiltersInfo</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap61_mj.html#X7A374553786DF5E7" >61 .12 -3 CheckForHandlingByNiceBasis</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap61_mj.html#X78515F448644204E" >61 .13 <span class="Heading" >Tensor Products and Exterior and Symmetric Powers</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap61_mj.html#X81B2276A7EBA8ED1" >61 .13 -1 TensorProduct</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap61_mj.html#X787BB7FF85F0AD68" >61 .13 -2 ExteriorPower</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap61_mj.html#X79E2C2AF842E8419" >61 .13 -3 SymmetricPower</a></span >
</div ></div >
</div >
<div class="ContChap" ><a href="chap62_mj.html#X7DDBF6F47A2E021C" >62 <span class="Heading" >Algebras</span ></a>
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap62_mj.html#X830EDB5F85645FFB" >62 .1 <span class="Heading" >InfoAlgebra (Info Class)</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap62_mj.html#X8665F459841AAD53" >62 .1 -1 InfoAlgebra</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap62_mj.html#X8686DEBA85D3F3B6" >62 .2 <span class="Heading" >Constructing Algebras by Generators</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap62_mj.html#X7B213851791A594B" >62 .2 -1 Algebra</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap62_mj.html#X80FE16EA84EE56CD" >62 .2 -2 AlgebraWithOne</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap62_mj.html#X7A7B00127DC9DD40" >62 .3 <span class="Heading" >Constructing Algebras as Free Algebras</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap62_mj.html#X83484C917D8F7A1A" >62 .3 -1 FreeAlgebra</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap62_mj.html#X7FBD04B07B85623D" >62 .3 -2 FreeAlgebraWithOne</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap62_mj.html#X87835FFE79D2E068" >62 .3 -3 FreeAssociativeAlgebra</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap62_mj.html#X845A777584A7D711" >62 .3 -4 FreeAssociativeAlgebraWithOne</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap62_mj.html#X7E8F45547CC07CE5" >62 .4 <span class="Heading" >Constructing Algebras by Structure Constants</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap62_mj.html#X7CC58DFD816E6B65" >62 .4 -1 AlgebraByStructureConstants</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap62_mj.html#X80D528A57FD64873" >62 .4 -2 AlgebraWithOneByStructureConstants</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap62_mj.html#X804ADF0280F67CDC" >62 .4 -3 StructureConstantsTable</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap62_mj.html#X7F1203A1793411DF" >62 .4 -4 EmptySCTable</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap62_mj.html#X817BD086876EC1C4" >62 .4 -5 SetEntrySCTable</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap62_mj.html#X7F333822780B6731" >62 .4 -6 GapInputSCTable</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap62_mj.html#X7C23ED85814C0371" >62 .4 -7 TestJacobi</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap62_mj.html#X78B633CE7A5B9F9A" >62 .4 -8 IdentityFromSCTable</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap62_mj.html#X7F2A71608602635D" >62 .4 -9 QuotientFromSCTable</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap62_mj.html#X79B7C3078112E7E1" >62 .5 <span class="Heading" >Some Special Algebras</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap62_mj.html#X83DF4BCC7CE494FC" >62 .5 -1 QuaternionAlgebra</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap62_mj.html#X7B807702782F56FF" >62 .5 -2 ComplexificationQuat</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap62_mj.html#X78C88A38853A8443" >62 .5 -3 OctaveAlgebra</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap62_mj.html#X7D88E42B7DE087B0" >62 .5 -4 FullMatrixAlgebra</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap62_mj.html#X78B8BA77869DAA13" >62 .5 -5 NullAlgebra</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap62_mj.html#X7DF5989886BE611E" >62 .6 <span class="Heading" >Subalgebras</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap62_mj.html#X8396643D7A49EEAD" >62 .6 -1 Subalgebra</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap62_mj.html#X7C6B08657BD836C3" >62 .6 -2 SubalgebraNC</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap62_mj.html#X83ECF489846F00B0" >62 .6 -3 SubalgebraWithOne</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap62_mj.html#X7A11B177868E76AA" >62 .6 -4 SubalgebraWithOneNC</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap62_mj.html#X7FDD953A84CFC3D2" >62 .6 -5 TrivialSubalgebra</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap62_mj.html#X81EE8C1F7D7A7CF8" >62 .7 <span class="Heading" >Ideals of Algebras</span ></a>
</span >
</div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap62_mj.html#X7DC95D6982C9D7B0" >62 .8 <span class="Heading" >Categories and Properties of Algebras</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap62_mj.html#X7FEDFAA383AB20D2" >62 .8 -1 IsFLMLOR</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap62_mj.html#X85C1E13A877DF2C8" >62 .8 -2 IsFLMLORWithOne</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap62_mj.html#X801ED693808F6C84" >62 .8 -3 IsAlgebra</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap62_mj.html#X80B21AC27DE6D068" >62 .8 -4 IsAlgebraWithOne</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap62_mj.html#X839BAC687B4E1A1D" >62 .8 -5 IsLieAlgebra</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap62_mj.html#X877DF13387831A6A" >62 .8 -6 IsSimpleAlgebra</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap62_mj.html#X7C5AECE87D79D075" >62 .8 -7 IsFiniteDimensional</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap62_mj.html#X82B3A9077D0CB453" >62 .8 -8 IsQuaternion</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap62_mj.html#X7E9273E47CF38CF1" >62 .9 <span class="Heading" >Attributes and Operations for Algebras</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap62_mj.html#X83B055F37EBF2438" >62 .9 -1 GeneratorsOfAlgebra</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap62_mj.html#X7FA408307A5A420E" >62 .9 -2 GeneratorsOfAlgebraWithOne</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap62_mj.html#X7D309FD37D94B196" >62 .9 -3 ProductSpace</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap62_mj.html#X875CD2B37EE9A8A2" >62 .9 -4 PowerSubalgebraSeries</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap62_mj.html#X788F4E6184E5C863" >62 .9 -5 AdjointBasis</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap62_mj.html#X800A410B8536E6DD" >62 .9 -6 IndicesOfAdjointBasis</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap62_mj.html#X7BA35CB28062D407" >62 .9 -7 AsAlgebra</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap62_mj.html#X878323367D0B68EB" >62 .9 -8 AsAlgebraWithOne</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap62_mj.html#X7A922D26805AFF99" >62 .9 -9 AsSubalgebra</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap62_mj.html#X7B964BC37A975E48" >62 .9 -10 AsSubalgebraWithOne</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap62_mj.html#X7C280DAC7F840B60" >62 .9 -11 MutableBasisOfClosureUnderAction</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap62_mj.html#X7BA1739D7F8B3A2B" >62 .9 -12 MutableBasisOfNonassociativeAlgebra</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap62_mj.html#X8467B687823371F9" >62 .9 -13 MutableBasisOfIdealInNonassociativeAlgebra</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap62_mj.html#X7C591B7C7DEA7EEB" >62 .9 -14 DirectSumOfAlgebras</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap62_mj.html#X7D0EB1437D3D9495" >62 .9 -15 FullMatrixAlgebraCentralizer</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap62_mj.html#X850C29907A509533" >62 .9 -16 RadicalOfAlgebra</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap62_mj.html#X82571785846CF05C" >62 .9 -17 CentralIdempotentsOfAlgebra</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap62_mj.html#X7CFB230582C26DAA" >62 .9 -18 DirectSumDecomposition</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap62_mj.html#X85C58364833E014C" >62 .9 -19 LeviMalcevDecomposition</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap62_mj.html#X7DCA2568870A2D34" >62 .9 -20 Grading</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap62_mj.html#X7E94B857847F95C1" >62 .10 <span class="Heading" >Homomorphisms of Algebras</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap62_mj.html#X83CE798C7D39E368" >62 .10 -1 AlgebraGeneralMappingByImages</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap62_mj.html#X7A7F97ED8608C882" >62 .10 -2 AlgebraHomomorphismByImages</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap62_mj.html#X8326D1BD79725462" >62 .10 -3 AlgebraHomomorphismByImagesNC</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap62_mj.html#X8057E55B864567AD" >62 .10 -4 AlgebraWithOneGeneralMappingByImages</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap62_mj.html#X866F32B5846E5857" >62 .10 -5 AlgebraWithOneHomomorphismByImages</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap62_mj.html#X80BF4D6A7FDC959A" >62 .10 -6 AlgebraWithOneHomomorphismByImagesNC</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap62_mj.html#X825149467C57DEFC" >62 .10 -7 AlgebraHomomorphismByFunction</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap62_mj.html#X8712E5C1861CC32C" >62 .10 -8 NaturalHomomorphismByIdeal</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap62_mj.html#X8705A9C68102FEA3" >62 .10 -9 OperationAlgebraHomomorphism</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap62_mj.html#X7B249E8E86D895F0" >62 .10 -10 NiceAlgebraMonomorphism</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap62_mj.html#X79D770777D873F80" >62 .10 -11 IsomorphismFpAlgebra</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap62_mj.html#X7FB760F9813B0789" >62 .10 -12 IsomorphismMatrixAlgebra</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap62_mj.html#X7F8D3DF2863EC50D" >62 .10 -13 IsomorphismSCAlgebra</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap62_mj.html#X7F34244B81979696" >62 .10 -14 RepresentativeLinearOperation</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap62_mj.html#X818DE6C57D1A4B33" >62 .11 <span class="Heading" >Representations of Algebras</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap62_mj.html#X8055B87F7ADBD66B" >62 .11 -1 LeftAlgebraModuleByGenerators</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap62_mj.html#X8026B99B7955A355" >62 .11 -2 RightAlgebraModuleByGenerators</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap62_mj.html#X7F28A47E876427E0" >62 .11 -3 BiAlgebraModuleByGenerators</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap62_mj.html#X852524F581613359" >62 .11 -4 LeftAlgebraModule</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap62_mj.html#X8222F2B67D753036" >62 .11 -5 RightAlgebraModule</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap62_mj.html#X84517770868DDA02" >62 .11 -6 BiAlgebraModule</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap62_mj.html#X79AAB50D83A14A43" >62 .11 -7 GeneratorsOfAlgebraModule</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap62_mj.html#X82B708BD84F3DAB1" >62 .11 -8 IsAlgebraModuleElement</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap62_mj.html#X80E786467F9163F9" >62 .11 -9 IsLeftAlgebraModuleElement</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap62_mj.html#X863756787E2B6E75" >62 .11 -10 IsRightAlgebraModuleElement</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap62_mj.html#X85654EF07F708AC3" >62 .11 -11 LeftActingAlgebra</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap62_mj.html#X826298B37E1B1520" >62 .11 -12 RightActingAlgebra</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap62_mj.html#X8308408D86CFC3C9" >62 .11 -13 ActingAlgebra</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap62_mj.html#X7C325A507EC9BA18" >62 .11 -14 IsBasisOfAlgebraModuleElementSpace</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap62_mj.html#X789863037B0E35D2" >62 .11 -15 MatrixOfAction</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap62_mj.html#X8742A7D27F26AFAB" >62 .11 -16 SubAlgebraModule</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap62_mj.html#X86E0515987192F0E" >62 .11 -17 LeftModuleByHomomorphismToMatAlg</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap62_mj.html#X7EE41297867E41A8" >62 .11 -18 RightModuleByHomomorphismToMatAlg</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap62_mj.html#X8729F0A678A4A09C" >62 .11 -19 AdjointModule</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap62_mj.html#X84813BCD80BDF3C4" >62 .11 -20 FaithfulModule</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap62_mj.html#X7E16630185CE2C10" >62 .11 -21 ModuleByRestriction</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap62_mj.html#X7885AAC87FDCF649" >62 .11 -22 NaturalHomomorphismBySubAlgebraModule</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap62_mj.html#X85D0F3758551DADC" >62 .11 -23 DirectSumOfAlgebraModules</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap62_mj.html#X7D7A6486803B15CE" >62 .11 -24 TranslatorSubalgebra</a></span >
</div ></div >
</div >
<div class="ContChap" ><a href="chap63_mj.html#X85A22A8286596D02" >63 <span class="Heading" >Finitely Presented Algebras</span ></a>
</div >
<div class="ContChap" ><a href="chap64_mj.html#X78559D4C800AF58A" >64 <span class="Heading" >Lie Algebras</span ></a>
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap64_mj.html#X80A607C47B7A2E69" >64 .1 <span class="Heading" >Lie Objects</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap64_mj.html#X87F121978775AF48" >64 .1 -1 LieObject</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap64_mj.html#X83E5DD4381D9A65D" >64 .1 -2 IsLieObject</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap64_mj.html#X8725993C7BF386EE" >64 .1 -3 LieFamily</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap64_mj.html#X81D9F5C6876FE93B" >64 .1 -4 UnderlyingFamily</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap64_mj.html#X874B2B2A7F5A9A78" >64 .1 -5 UnderlyingRingElement</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap64_mj.html#X789A44F283C16B2B" >64 .2 <span class="Heading" >Constructing Lie algebras</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap64_mj.html#X7D362350824FA115" >64 .2 -1 LieAlgebraByStructureConstants</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap64_mj.html#X7EEB79EE855E124C" >64 .2 -2 RestrictedLieAlgebraByStructureConstants</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap64_mj.html#X7C840A9F85D28C81" >64 .2 -3 LieAlgebra</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap64_mj.html#X7F7B34BD80F0F1C8" >64 .2 -4 FreeLieAlgebra</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap64_mj.html#X8735EE937A0081F0" >64 .2 -5 FullMatrixLieAlgebra</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap64_mj.html#X821B6C197C08878B" >64 .2 -6 RightDerivations</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap64_mj.html#X7933F05F7DE342AB" >64 .2 -7 SimpleLieAlgebra</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap64_mj.html#X798391F47E835F85" >64 .3 <span class="Heading" >Distinguished Subalgebras</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap64_mj.html#X8111F58E7DE3E25C" >64 .3 -1 LieCentre</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap64_mj.html#X811444717EEDCC34" >64 .3 -2 LieCentralizer</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap64_mj.html#X7E62B6B37A75E09D" >64 .3 -3 LieNormalizer</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap64_mj.html#X7C95C0057C977747" >64 .3 -4 LieDerivedSubalgebra</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap64_mj.html#X7D072F6D7A3D0BAF" >64 .3 -5 LieNilRadical</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap64_mj.html#X8445C9F17F7CBEA1" >64 .3 -6 LieSolvableRadical</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap64_mj.html#X86114F157DFF6523" >64 .3 -7 CartanSubalgebra</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap64_mj.html#X7A72840882F7A9B6" >64 .4 <span class="Heading" >Series of Ideals</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap64_mj.html#X7DEF89A8869809F5" >64 .4 -1 LieDerivedSeries</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap64_mj.html#X7900D17E7BA26A48" >64 .4 -2 LieLowerCentralSeries</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap64_mj.html#X86A8701C868828C7" >64 .4 -3 LieUpperCentralSeries</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap64_mj.html#X8208CE5F8286155F" >64 .5 <span class="Heading" >Properties of a Lie Algebra</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap64_mj.html#X7F97D08F7B738ADE" >64 .5 -1 IsLieAbelian</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap64_mj.html#X78452F4E875A62A8" >64 .5 -2 IsLieNilpotent</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap64_mj.html#X859FF1B3812B8FCC" >64 .5 -3 IsLieSolvable</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap64_mj.html#X83F829017D46C544" >64 .6 <span class="Heading" >Semisimple Lie Algebras and Root Systems</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap64_mj.html#X8401CDC2859F8A85" >64 .6 -1 SemiSimpleType</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap64_mj.html#X82EBF10A7B3B6F6E" >64 .6 -2 ChevalleyBasis</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap64_mj.html#X79B5D27681193625" >64 .6 -3 IsRootSystem</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap64_mj.html#X7D64D49479CBB203" >64 .6 -4 IsRootSystemFromLieAlgebra</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap64_mj.html#X80D15C027BB8029B" >64 .6 -5 RootSystem</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap64_mj.html#X7CA021E28527763E" >64 .6 -6 UnderlyingLieAlgebra</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap64_mj.html#X7B6B0BBD8035D7E5" >64 .6 -7 PositiveRoots</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap64_mj.html#X81F9E0E67DD2688F" >64 .6 -8 NegativeRoots</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap64_mj.html#X829C78427A442C23" >64 .6 -9 PositiveRootVectors</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap64_mj.html#X7AB374DC87A39349" >64 .6 -10 NegativeRootVectors</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap64_mj.html#X7DBD179E7CCF6699" >64 .6 -11 SimpleSystem</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap64_mj.html#X84E3FEF587CB66C3" >64 .6 -12 CartanMatrix</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap64_mj.html#X878644D68571BF44" >64 .6 -13 BilinearFormMat</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap64_mj.html#X7FAE45B37C5779A0" >64 .6 -14 CanonicalGenerators</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap64_mj.html#X7945D07786D1C4BB" >64 .7 <span class="Heading" >Semisimple Lie Algebras and Weyl Groups of Root Systems</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap64_mj.html#X82AA29DD7969A935" >64 .7 -1 IsWeylGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap64_mj.html#X81EF01E57E5DC18A" >64 .7 -2 SparseCartanMatrix</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap64_mj.html#X86BED5098322EBEF" >64 .7 -3 WeylGroup</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap64_mj.html#X7829BC4D7F253649" >64 .7 -4 ApplySimpleReflection</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap64_mj.html#X80A7204F7D40D80F" >64 .7 -5 LongestWeylWordPerm</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap64_mj.html#X7D4E213F82F73857" >64 .7 -6 ConjugateDominantWeight</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap64_mj.html#X7E000FA97949BFD5" >64 .7 -7 WeylOrbitIterator</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap64_mj.html#X878080BB79BE3F2E" >64 .8 <span class="Heading" >Restricted Lie algebras</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap64_mj.html#X81F28B1D830F28EB" >64 .8 -1 IsRestrictedLieAlgebra</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap64_mj.html#X7D7BD5908016461B" >64 .8 -2 PthPowerImages</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap64_mj.html#X879BB01782E7D7A9" >64 .8 -3 PthPowerImage</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap64_mj.html#X8692ADD581359CA1" >64 .8 -4 JenningsLieAlgebra</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap64_mj.html#X785251E879E1BFC6" >64 .8 -5 PCentralLieAlgebra</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap64_mj.html#X781ADBEC850C7DE7" >64 .8 -6 NaturalHomomorphismOfLieAlgebraFromNilpotentGroup</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap64_mj.html#X7C419FFA835EBE12" >64 .9 <span class="Heading" >The Adjoint Representation</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap64_mj.html#X786886D882795F78" >64 .9 -1 AdjointMatrix</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap64_mj.html#X873A64307AC6C63E" >64 .9 -2 AdjointAssociativeAlgebra</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap64_mj.html#X877CCFD5832E035D" >64 .9 -3 KillingMatrix</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap64_mj.html#X8234046083B60F6E" >64 .9 -4 KappaPerp</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap64_mj.html#X7A00601387A060CF" >64 .9 -5 IsNilpotentElement</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap64_mj.html#X86EF3E6F7BC0A8AD" >64 .9 -6 NonNilpotentElement</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap64_mj.html#X7A912D9E7B3BA874" >64 .9 -7 FindSl2</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap64_mj.html#X7875070C85DD4E8E" >64 .10 <span class="Heading" >Universal Enveloping Algebras</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap64_mj.html#X8226CD1680207A5F" >64 .10 -1 UniversalEnvelopingAlgebra</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap64_mj.html#X7B8C71E07F50B286" >64 .11 <span class="Heading" >Finitely Presented Lie Algebras</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap64_mj.html#X780A5B457A051110" >64 .11 -1 FpLieAlgebraByCartanMatrix</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap64_mj.html#X79FD70C487EA9438" >64 .11 -2 NilpotentQuotientOfFpLieAlgebra</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap64_mj.html#X7FBCB43C86BDD9C2" >64 .12 <span class="Heading" >Modules over Lie Algebras and Their Cohomology</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap64_mj.html#X82CC31CF79F59FEE" >64 .12 -1 IsCochain</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap64_mj.html#X79F3DF0D8791C2E3" >64 .12 -2 Cochain</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap64_mj.html#X7CF2919081600A3D" >64 .12 -3 CochainSpace</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap64_mj.html#X7D6760DA84683011" >64 .12 -4 ValueCochain</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap64_mj.html#X851F5EF47FA90CBC" >64 .12 -5 LieCoboundaryOperator</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap64_mj.html#X7FB815F38143939E" >64 .12 -6 Cocycles</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap64_mj.html#X7C4F372C7AE2F739" >64 .12 -7 Coboundaries</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap64_mj.html#X78A201238137E822" >64 .13 <span class="Heading" >Modules over Semisimple Lie Algebras</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap64_mj.html#X7D8522E37ED1024A" >64 .13 -1 DominantWeights</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap64_mj.html#X79AAC71E8267E9F8" >64 .13 -2 DominantCharacter</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap64_mj.html#X7BE7129384B012DF" >64 .13 -3 DecomposeTensorProduct</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap64_mj.html#X7D67A9BC7E4714D9" >64 .13 -4 DimensionOfHighestWeightModule</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap64_mj.html#X840E5FAE7D2C2702" >64 .14 <span class="Heading" >Admissible Lattices in UEA</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap64_mj.html#X86E6722379576746" >64 .14 -1 IsUEALatticeElement</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap64_mj.html#X79F4F58B7888B0A5" >64 .14 -2 LatticeGeneratorsInUEA</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap64_mj.html#X875FD1627F3B72DB" >64 .14 -3 ObjByExtRep</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap64_mj.html#X8248DB547B02B0FA" >64 .14 -4 IsWeightRepElement</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap64_mj.html#X7FB14F7F80EFF33F" >64 .14 -5 HighestWeightModule</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap64_mj.html#X8750BDBF7EA5E868" >64 .15 <span class="Heading" >Tensor Products and Exterior and Symmetric Powers of Algebra Modules</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap64_mj.html#X7A1E0AC4800E7FDA" >64 .15 -1 TensorProductOfAlgebraModules</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap64_mj.html#X7F4AB6A1863E8FB2" >64 .15 -2 ExteriorPowerOfAlgebraModule</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap64_mj.html#X842DF85687D61A56" >64 .15 -3 SymmetricPowerOfAlgebraModule</a></span >
</div ></div >
</div >
<div class="ContChap" ><a href="chap65_mj.html#X825897DC7A16E07D" >65 <span class="Heading" >Magma Rings</span ></a>
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap65_mj.html#X8398F87F8231A163" >65 .1 <span class="Heading" >Free Magma Rings</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap65_mj.html#X7B9AF0A47F44E4B4" >65 .1 -1 FreeMagmaRing</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap65_mj.html#X86D2CA90847C091B" >65 .1 -2 GroupRing</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap65_mj.html#X7A24B95C8210BD09" >65 .1 -3 IsFreeMagmaRing</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap65_mj.html#X8382ED697A28CE67" >65 .1 -4 IsFreeMagmaRingWithOne</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap65_mj.html#X82C63644805EB1EE" >65 .1 -5 IsGroupRing</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap65_mj.html#X848D60417DFF7947" >65 .1 -6 UnderlyingMagma</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap65_mj.html#X7B21DB3E7CD80983" >65 .1 -7 AugmentationIdeal</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap65_mj.html#X8402D3897F2C5955" >65 .2 <span class="Heading" >Elements of Free Magma Rings</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap65_mj.html#X827B2D7D7E41780C" >65 .2 -1 IsMagmaRingObjDefaultRep</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap65_mj.html#X7D9C684A81E66310" >65 .2 -2 IsElementOfFreeMagmaRing</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap65_mj.html#X869768AF7B444BF8" >65 .2 -3 IsElementOfFreeMagmaRingFamily</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap65_mj.html#X843D1D8578C33513" >65 .2 -4 CoefficientsAndMagmaElements</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap65_mj.html#X78C3DB417E353390" >65 .2 -5 ZeroCoefficient</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap65_mj.html#X8671DE0A81BEEFB0" >65 .2 -6 ElementOfMagmaRing</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap65_mj.html#X80366F1480ACD8DF" >65 .3 <span class="Heading" >Natural Embeddings related to Magma Rings</span ></a>
</span >
</div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap65_mj.html#X81B002EE799B5E77" >65 .4 <span class="Heading" >Magma Rings modulo Relations</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap65_mj.html#X869D54847E881848" >65 .4 -1 IsElementOfMagmaRingModuloRelations</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap65_mj.html#X875BEB1A840FFAA4" >65 .4 -2 IsElementOfMagmaRingModuloRelationsFamily</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap65_mj.html#X85956ED27FA6AC68" >65 .4 -3 NormalizedElementOfMagmaRingModuloRelations</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap65_mj.html#X804B5AAB813E184D" >65 .4 -4 IsMagmaRingModuloRelations</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap65_mj.html#X7D859DBF81DFA751" >65 .5 <span class="Heading" >Magma Rings modulo the Span of a Zero Element</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap65_mj.html#X7B3D45A6802B695C" >65 .5 -1 IsElementOfMagmaRingModuloSpanOfZeroFamily</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap65_mj.html#X872713EE84DA9B72" >65 .5 -2 IsMagmaRingModuloSpanOfZero</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap65_mj.html#X7A7F880D7D7D3722" >65 .5 -3 MagmaRingModuloSpanOfZero</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap65_mj.html#X79889F017F2EB7ED" >65 .6 <span class="Heading" >Technical Details about the Implementation of Magma Rings</span ></a>
</span >
</div >
</div >
<div class="ContChap" ><a href="chap66_mj.html#X7A14A6588268810A" >66 <span class="Heading" >Polynomials and Rational Functions</span ></a>
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap66_mj.html#X7A8FADCD875826DA" >66 .1 <span class="Heading" >Indeterminates</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap66_mj.html#X79D0380D7FA39F7D" >66 .1 -1 <span class="Heading" >Indeterminate</span ></a>
</span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap66_mj.html#X816C8D797C804380" >66 .1 -2 IndeterminateNumberOfUnivariateRationalFunction</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap66_mj.html#X7A2FA46885EF403D" >66 .1 -3 IndeterminateOfUnivariateRationalFunction</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap66_mj.html#X7FD4AC807A1C8E27" >66 .1 -4 IndeterminateName</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap66_mj.html#X791A06E67F784328" >66 .1 -5 CIUnivPols</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap66_mj.html#X86A68FD582F4F757" >66 .2 <span class="Heading" >Operations for Rational Functions</span ></a>
</span >
</div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap66_mj.html#X824B6D328643CE04" >66 .3 <span class="Heading" >Comparison of Rational Functions</span ></a>
</span >
</div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap66_mj.html#X7D871EA180E9486C" >66 .4 <span class="Heading" >Properties and Attributes of Rational Functions</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap66_mj.html#X86C92F677DA9347F" >66 .4 -1 IsPolynomialFunction</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap66_mj.html#X7D7D2667803D8D8A" >66 .4 -2 NumeratorOfRationalFunction</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap66_mj.html#X78DC1B5B866ADB6C" >66 .4 -3 DenominatorOfRationalFunction</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap66_mj.html#X7974B0707C8DAB6C" >66 .4 -4 IsPolynomial</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap66_mj.html#X7914771F7C6013EF" >66 .4 -5 AsPolynomial</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap66_mj.html#X8738F73583273FCA" >66 .4 -6 IsUnivariateRationalFunction</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap66_mj.html#X7F1F67527A35A9CE" >66 .4 -7 CoefficientsOfUnivariateRationalFunction</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap66_mj.html#X86A2546685D0016B" >66 .4 -8 IsUnivariatePolynomial</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap66_mj.html#X78C9524D7F2708C2" >66 .4 -9 CoefficientsOfUnivariatePolynomial</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap66_mj.html#X79138FF28213B6EC" >66 .4 -10 IsLaurentPolynomial</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap66_mj.html#X7F2A49208341C2A8" >66 .4 -11 IsConstantRationalFunction</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap66_mj.html#X834B54947FAADEA4" >66 .4 -12 IsPrimitivePolynomial</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap66_mj.html#X87531E03849391C1" >66 .4 -13 SplittingField</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap66_mj.html#X82E2F1707FC2E553" >66 .5 <span class="Heading" >Univariate Polynomials</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap66_mj.html#X8379F8CB7D0076BA" >66 .5 -1 UnivariatePolynomial</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap66_mj.html#X85178A3E7B4F11E0" >66 .5 -2 UnivariatePolynomialByCoefficients</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap66_mj.html#X78AF77C383245254" >66 .5 -3 DegreeOfLaurentPolynomial</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap66_mj.html#X7CBB760C87B04F75" >66 .5 -4 RootsOfPolynomial</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap66_mj.html#X80CEB10D7879767F" >66 .5 -5 RootsOfUPol</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap66_mj.html#X7887FBC78149BB0C" >66 .5 -6 QuotRemLaurpols</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap66_mj.html#X7DDADF157879EFBF" >66 .5 -7 UnivariatenessTestRationalFunction</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap66_mj.html#X7A3BC96B7A50DE98" >66 .5 -8 InfoPoly</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap66_mj.html#X81499B5A823E6EA3" >66 .6 <span class="Heading" >Polynomials as Univariate Polynomials in one Indeterminate</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap66_mj.html#X826B99B17ABD11BE" >66 .6 -1 DegreeIndeterminate</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap66_mj.html#X85646FD07F9C60F5" >66 .6 -2 PolynomialCoefficientsOfPolynomial</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap66_mj.html#X80710E9B7D8340BD" >66 .6 -3 LeadingCoefficient</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap66_mj.html#X7B3EAE41795598A5" >66 .6 -4 LeadingMonomial</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap66_mj.html#X7B57CEE2780D0E0B" >66 .6 -5 Derivative</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap66_mj.html#X7C7D790A7D6E11AD" >66 .6 -6 Discriminant</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap66_mj.html#X857AD5587EF49029" >66 .6 -7 Resultant</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap66_mj.html#X85ABC4687DF05777" >66 .7 <span class="Heading" >Multivariate Polynomials</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap66_mj.html#X7A70769C7F52CD2D" >66 .7 -1 <span class="Heading" >Value</span ></a>
</span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap66_mj.html#X7ED3E7D17C7AC732" >66 .8 <span class="Heading" >Minimal Polynomials</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap66_mj.html#X8643915A8424DAF8" >66 .8 -1 MinimalPolynomial</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap66_mj.html#X837B8E55832CDFEB" >66 .9 <span class="Heading" >Cyclotomic Polynomials</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap66_mj.html#X827FC7FE81EE4C02" >66 .9 -1 CyclotomicPolynomial</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap66_mj.html#X8551EF5187509D69" >66 .10 <span class="Heading" >Polynomial Factorization</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap66_mj.html#X83511D517B544F36" >66 .10 -1 Factors</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap66_mj.html#X7F5A4ACB7AF9E329" >66 .10 -2 FactorsSquarefree</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap66_mj.html#X7F45E9E47EA2C18B" >66 .11 <span class="Heading" >Polynomials over the Rationals</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap66_mj.html#X7E66494B7B05A055" >66 .11 -1 PrimitivePolynomial</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap66_mj.html#X7A73A3877EB73566" >66 .11 -2 PolynomialModP</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap66_mj.html#X7AB9A6257ED694EC" >66 .11 -3 GaloisType</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap66_mj.html#X7EB610D37D156DC6" >66 .11 -4 ProbabilityShapes</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap66_mj.html#X7C178AB9866FDDE5" >66 .12 <span class="Heading" >Factorization of Polynomials over the Rationals</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap66_mj.html#X8723075C81D2CCA6" >66 .12 -1 BombieriNorm</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap66_mj.html#X856D769D878AF7AE" >66 .12 -2 MinimizedBombieriNorm</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap66_mj.html#X8139BB0F87399F2C" >66 .12 -3 HenselBound</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap66_mj.html#X79CC9C8D7C9F6B6A" >66 .12 -4 OneFactorBound</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap66_mj.html#X844B3C6C87A0E7E0" >66 .13 <span class="Heading" >Laurent Polynomials</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap66_mj.html#X8467263B7EFA013E" >66 .13 -1 LaurentPolynomialByCoefficients</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap66_mj.html#X86D58AB67F86469F" >66 .13 -2 CoefficientsOfLaurentPolynomial</a></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap66_mj.html#X8381E1B582F38C85" >66 .13 -3 IndeterminateNumberOfLaurentPolynomial</a></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap66_mj.html#X7C1708D27F97B05F" >66 .14 <span class="Heading" >Univariate Rational Functions</span ></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap66_mj.html#X83DD411179888783" >66 .14 -1 UnivariateRationalFunctionByCoefficients</a></span >
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