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products/Sources/formale Sprachen/GAP/pkg/utils/lib/ (Algebra von RWTH Aachen Version 4.15.1^©) Datei vom 11.8.2025 mit Größe 3 kB

td =tdleft"><odeclass=func>82;java.lang.StringIndexOutOfBoundsException: Range [107, 102) out of bounds for length 213
<p>Returns: a boolean</p>

p>a="X7EEC27307C4B06E8" =""><a<p>

<p>a ="X8332653E7EA2EB72" name=8363E7EA2EB72>a>p>

<h4>2.5 <span class="Heading">Random Objects</span></h4>

<p>CAP

<ul>
<li><p><em>By

</li>
<li><p><em>By lists</em>: The list is used java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

/>
</ul>
<p><a id="X85A6C3687C46085F" name="X85A6C3687C46085F"></a></p>

<h5>2.5-1 RandomObjectByInteger</h5>

<div="unc"><="func" ="00">>< =tdleftclass="#22 >/var<classArg>/>)tdjava.lang.StringIndexOutOfBoundsException: Range [192, 191) out of bounds for length 255
<p>Returns: an object

<p>The arguments

<p><a id="X7EEC27307C4B06E8" name="X7EEC27307C4B06E8"></a></p>

<java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<
<>objectin <<pan"Math">C/pan/p>

<p>The arguments are a category <span class="Math">C</span> and a list <span class="Math">L</spanh5>261 IsEqualForCacheForObjects/5>

<p><a id="X7E0974E58188E6FB" name="X7E0974E58188E6FB"></a></p>

<h5.53 RandomObjecth5>

<div class=">

Quellcode-Bibliothek chap2.html Sprache: HTML

products/Sources/formale Sprachen/GAP/pkg/cap/doc/chap2.html

<p>Returns: a boolean</p>

<p>The argument isan object <spanclass="Math">a</pan of acategory<span ="">m{C}/>.Theoutput is<codeclass=code></code>if spanclass"">a</span isisomorphic to terminal bjectof<panclass="Math">\mathbf{C}/> theoutputis< class=">false<>.pjava.lang.StringIndexOutOfBoundsException: Index 323 out of bounds for length 323

<p><a id="X83F6095A80DCCDC3" name="X83F6095A80DCCDC3"></a></p>

<h5>2.4-5 IsInitial</h5>

<div class="func"><table class="func" width>Returns:boolean<>
<p>Returns: a boolean</p>

<pTheargumentis anobjectspan class="ath"a/>ofa < ="ath>mathbf{CThe outputoutput is

></codeif << ="ath"><span is isomorphicto initialobjectof <span="">mathbfC}/span otherwise theoutputiscode="code">/code</p>



<p><a id="X856FA5E084753312" name="X856FA5E084753312"></a></p



<h5>2.4



<div class="func"><table class="func"<li<em>By<emThe integer is a  that canbe  tocreate a randomobject</>

<p>Returnsli>p>m>y</m: Thelist is whencreatinga randomobject  need than .Lists moreflexibility at  expenseof   of methods  becauselists thatarevalid  inputin category may  notvalidforother categories ,heseoperations  not thoughttobeusedin generic algorithms<p>



<p>The argument is an object <span class="Math">p><id="" name"X85A6C3687C46085F>



<p><id=""nameX824765AD78E027CC</<



<h5>2.4-7 IsZeroReturns: anobject  <span ="">C</span>p>



<divclass=func< class="func width=00>

="0%><> class=tdleft">code="unc>&2; /code(class=>C/var, < class=A>n/var )tdright"(nbsp;operation )<

<p>These are convenient methods and

<p><a id="X7AF3D62C7F8F40AE" name="X7AF3D62C7F8F40AE"></a><java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

< class class""width"00"<<dclass">< class="func>&82;ObjectConstructorcode> < classArg>C/var,<varclass="a)< class"tdright>(&;operation </td<java.lang.StringIndexOutOfBoundsException: Index 251 out of bounds for length 251

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RandomObject</code>( <var class="Arg">C</var
pa=X7DD668627932185C="">a>/p>

<p>Returns: an <>

<p><a id="X7BAE9BBC874635CB" name="X7BAE9BBC874635CB"></<pShorthandfor code="">(C,a )<>.p>

<h5261 IsEqualForCacheForObjectsh5>

<div class=>.73 ObjectDatum</5>
<p>Returns: true or false</p>

<pBy ,  caches  the of . To   ofthecache,one to theinput of basic  withits  input To objectsin the,IsEqualForCacheForObjects used  , IsEqualForCacheForObjects back IsEqualForCache(see)  inturndefaults torecursivecomparison for andcode class=code">IsIdenticalObj/code other cases. If you add a function via < class=code"AddIsEqualForCacheForObjects</ode  function used .A function <panclass="ath"F a,b mapstobool/pan>  there output to trueorfalseFailisnot  in context/>

<p><a id="X824C0CC579E8DD7F" <p>Returnsdepends onthecategory</p

<h4>2.7 <span class="Heading

<p><a id="X825D050A8674D377" name="X825D050A8674D377"java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<h5>2.7-1 ObjectConstructor</h5>

<div class="func"><table class="<5>28- IsWellDefinedForObjects
<p>Returnsdivclass""><tableclass"" ="00"<tr< classtdleft">code class=""&822 < class=">var>)/tdclass=">&operationnbsp)/>tr/>/div>

<p>The arguments are a category <span class="Math">C</span> and an object datum <span class="Math">a</span><p>Returns  </p>

<p><a id="X7C6745F27FBA23F7" name=

<h5>2.7-2 /</h5>

classfunc"width=1"><r> ="func>87 /code < "Arg>a/var>, < ="></ar>/d>< =tdright>(  )/><tr</able><div>
<p>Returns: an object</p>

<p>Shorthand for <aid=" ="X8047E157804C680E>a>/

<p>< id"81 ="851"/a/pjava.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62

<h5>2.7-3 ObjectDatum</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ObjectDatum</code>( <var class="Arg">obj</var> )</tdReturns:listof < class"CAP<> ategory /pjava.lang.StringIndexOutOfBoundsException: Index 75 out of bounds for length 75
<p>Returns: depends java.lang.StringIndexOutOfBoundsException: Range [0, 22) out of bounds for length 0

<pTheargumentis  CAPcategoryobjectvar =""<>  outputa datum betojava.lang.StringIndexOutOfBoundsException: Range [122, 121) out of bounds for length 652

<p><a id="X7820B09C781A90A8" name="X7820B09C781A90A8"></a></p>

<h4>2.8 <span class="Heading">Well-Definedness of Objects</span></h4>

<p><a id="X7F9A38A187D9ABFA" name="X7F9A38A187D9ABFA"></a></p>

<h5>2.8-1 IsWellDefinedForObjects</h5>

<div class="func"><table class="func" width="1/strong> category objects

<p>Returns: a boolean</p>

<p>The argument is an object <span class="

<p><a id="X866BE10883381E75" name="

<h4>2.9 <java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<p><a id

<h5>2.9<

<div class="func"><table class="func" width="100
<p>java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 0

<p>Return a duplicate free list of objects of the category <var class="Arg">C</var>. The orderingp>eturns an </>

<p><a id="X7B77708881BFE682" name="X7B77708881BFE682"></a></p>

<h52.9-2 SetOfObjects</h5

<div class="func"><table class="func" width</h5
<p>Returns a of <trong class="">CAP</trong> ategory objects</p>

<p>Return a duplicate free listjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<p><a id="X7EFAF5CD83C3298F" name="X7EFAF5CD83C3298F"></a></p>

<h4>2.0<spanclass"">Projectivesspan></h4

<p>For a given object<spanclass="">A</pan inan abeliancategoryhaving enough, followingcommands usto computesome objectspan="ath>/ togetherwith epimorphism < class=ath">\pi P rightarrowA<span>./>

<p><a id="X7FED5E898784C314" name="X7FED5E898784C314"></a></p>

<h5>2.10-1 SomeProjectiveObject</h5>

<div class="func"><table class="func" width="10

Returns amorphism in athrm{Hom/>
<p>ReturnspThe are anobjectspan class=>A/spanand a projectiveobject <panclass"Math>Pthat equals output of >a/

<p><a id="X7D5951AA7EBD7750" name="X7D5951AA7EBD7750"></a></p>

<h5>2.10-2 EpimorphismFromSomeProjectiveObject<Returns a  < class=>\{Hom}P,</><pjava.lang.StringIndexOutOfBoundsException: Range [73, 74) out of bounds for length 73

<div class="func"><table
<p>Returns: a morphismjava.lang.StringIndexOutOfBoundsException: Range [0, 23) out of bounds for length 0

< argumentspan="<. <=Math"p:P\rightarrowA/panwithspan="ath>P aprojective object the output of >mathrm{SomeProjectiveObject}(A)</pan></>

<p><a id="X81F04D8186B94570" name="X81F04D8186B94570"></a></p>

<h5>2.10-3 p><a id="X7C06E1807D8775AD="X7C06E1807D8775AD" nameX7C06E1807D8775AD<a>p>

div="< class="func width10"<">&2; </code>( <arclass=""><var <var="ArgP) classtdright(&operation /table>
<p>Returns: a morphism in <span class="Math">\mathrm{Hom}(P,A)</span></p>

<p>The arguments are an object <span class="Math">A</span> and a projective object <span class>Returns  object</>

<p><a id="X7BFFD7FE82DCB5F6" name="X7BFFD7FE82DCB5F6"></a></p>
p<a =""name"">/><pjava.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62
h51 <h5

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ProjectiveLift</code>( <var class="Arg">pi</var>, <var class="Arg">epsilon</var> )</td><td<pReturns: a morphismin class=>mathrmHom(,)/>/java.lang.StringIndexOutOfBoundsException: Index 73 out of bounds for length 73
<p>Returns: a morphism in <span class="Math">\mathrm{Hom}(P,B)</span></p>

p>  are  morphism span="ath">piP rightarrowA</panwith class="><> projective,and anepimorphism \epsilon: B \rightarrow A. The output is a morphism \lambda: P \rightarrow B such that >\epsilon\ \lambda \pi<span./p>

<p>a id"810AFB7A202D2F name="">

<h4>2.11 <span class="Heading">Injectives</span></h4>

<p>For a given

<>aid"7C06E1807D8775AD" name""><a>/>

<h5.1-1h5

<div class="func"><table class="func" width="100%"><p>a id=X7CCBB0A384F37EC6name="CCBB0A384F37EC6>/
<p>Returns: an bject</>

<p>The argument is an object <span class

<p><a id="X8502893C82A38864" name="X8502893C82A38864"></a></p>

<h5>2.1- MonomorphismIntoSomeInjectiveObject</h5

<div class="func"><table class="func" width
<p>Returns:amorphism in< ="Math">mathrmHom(I,)<span<p>

<p>The argument is an object <span class="Math">A</span>.

<p><a id="X7A7C030382397A2B" name="X7A7C030382397A2B"></a></p>

<h5>2.11-3 MonomorphismIntoSomeInjectiveObjectWithGivenSomeInjectiveObject</h5>

<div class
<p>Returns a morphismin <span =Math>\{Hom}(IA)</pan/pjava.lang.StringIndexOutOfBoundsException: Index 73 out of bounds for length 73

<p>hearguments arean < class=Math>A<pan and injectiveobject spanclass=Math"I/span> equals the output .The output isa monomorphism Math">\:A \ I

<p><a id

<h5>2.11-4 ><>an  classMath">\iota_A: A rightarrow A_i.

</i>
<p>Returns: a /ul>

<Theargumentsarea < class=>\iota:  \ A<span> and m spanclass=ath\beta:  \ I/>w <spanclass"Math"><span is an .  is a morphism<pan=Math>\lambda: A\ </> uch  <spanclass=Math">lambda circ iota \beta./pjava.lang.StringIndexOutOfBoundsException: Index 351 out of bounds for length 351

<p><a id="X7DEC418578FAA1C0" name="X7DEC418578FAA1C0"></a></p>

<h4>2.12 <span class="Heading">Simplified Objects</span></h4>

<p>Let <span class="Math">i</span> be a positive integer or <span class="Math">\infty</span>. For a given object <span class="Math">A</span

<ul>
<li><p>an objectp>: object/p>

</li>
<lijava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

</li>
</ul>
<p>The idea is that the greater the <span class="Math">i</span>

<p>a ="X7BA160B382F8D742"="X7BA160B382F8D742"</><p>

<>221 </h5

<div class="func"><table class="java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 25
<Returns anobject<p>

<p>The argument is an object <span class="Math">A</span>. The output is a simplified object <span class="Math">A_{\infty}</span>.</p>

<p><a id="X85B54852792DEDD7" name="X85B54852792DEDD7"></a></p>

>.22SimplifyObject/5>

<div
<p>Returns  object/pjava.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25

<p>The arguments are an object <span class="Math">A</span> and a positive integer

<p><a id="X78C0D2CB867C5C90" name="X78C0D2CB867C5C90"></a></p>

<h5>2.12-3 SimplifyObject_IsoFromInputObject>Returns a  in<pan class=Math\{}A_i)s/p>

<div classfunc>table="unc "1%<>< classtdleft>codeclass"">82;SimplifyObject_IsoFromInputObject<code(var="""A, < class=""i/var> )

( operation )

<p>Returns: a morphism in <span class="Math">\mathrm{Hom}(Ajava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<p>The arguments are an object <span class="Math">A</span

<p><a id="X835832A685F651C8" name="X835832A685F651C8"></a></p>

<h5>2.12-4 SimplifyObject_IsoToInputObject

<div class="func"><table class="func" width="100%"><tr><td
<p>Returns: a morphism in <span class="Math">\mathrm{Hom}(A_i,A)</span></p>

<p>The arguments are an object <span class="Math">A</span> and a positive integer <span class="Math">i</span> or <code class="code">infinity</code>. The output is an isomorphism from a simplified object <span class="Math">(\iota_A^i)^{-1}: A_i \rightarrow class="func"table = ="10"<r>d class=tdleft>< class=funcProjectiveDimension</code> <ar class="">A</var )</td> class"tdright">(&bspattribute;)</td>/tr<table</>

<p><a id="X814E7CE2792896ED" name="X814E7CE2792896ED"></a></p<p>a id""name><a>/pjava.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62

<4.3span=""></span</4>

<p><a id

<h5>2.13-1 ProjectiveDimension</h5>

<div class=
<p>Returns: a nonnegative integerdiv class="chlinkbot"><span classchlinkbot<pan class=">Goto :< ="."Top a ="hap2.html"> a href">a>  <href"hap4html>a href"chap5html5/>< href="chap6.html">6</a>  <a href="chap7.html">7</a>  <a href="chap8.html">8</a>  <a href="chap9.html">9</a>  <a href="chap10.html">10</a>  <a href="chap11.html">11</a>  <a href="chap12.html">12</a>  <a href="chap13.html">13</a>  <a href="chap14.html">14</a>  <a href="chap15.html">15</a>  <a href="chap16.html">16<java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<p>The argument is an object <span class="Math">A</span>. The output is a the projective dimension of <span class="Math">A</span>.</p>

<p><a id="X823062CF792E3359" name="X823062CF792E3359"></a></p>

<h5>2.13-2 InjectiveDimension</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ InjectiveDimension</code>( <var class="Arg">A</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a nonnegative integer or infinity</p>

<p>The argument is an object <span class="Math">A</span>. The output is a the injective dimension of <span class="Math">A</span>.</p>

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