tdclasstdleft">codeclass="func">‣ ExtRepOfObj</code>( <var class="Arg">x/artdtd class=""(nbsp )t></>/></>
<>Returns: Apair lists  listsof .</>

<p>If <code class="code">n</code>

"pre
<span =GAPpromptgt/> ="">[-2 -,] [-, 2];/pan>
<span class="GAPprompt">></span> <span class="([ -11,[-,2],[[-,-11] [1 , 2] )
PBR([ [ -1, 1 ], [ -2, 2 ] ], [ [ -2, -1< class>&;/span classGAPinput(;/>
< classGAPprompt&;<spanspan=>(<span
[ [ [ -1, 1 ], [ -2, 2 ] ], [ [ -2java.lang.StringIndexOutOfBoundsException: Range [0, 1) out of bounds for length 0

<p><a id="X7F4C8A2B79E6D963" name="X7F4C8A2B79E6D963"></a></p>

<>.- PBRNumber>

< class"func>func "10%
<="">< classfuncwidth=0">trtd="< class"">&827 </ode < classArg><var/><td=""> )td/></><div
<p>Returns: A PBR, or a positive integer.</p>

These   set PBRs  classn<>and <code class">[11 . 2 4 ><var^2</><p>

<p>More precisely, if <var class="Arg">m</var>java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<p>If <var class="Arg">mat</var> is a PBR of degree <var class="Arg">n</var>, then <code class="code">NumberPBR</code> returns the number in <code class="code">[1 .. 2 ^ (4 * <var class="Arg">n</var> ^ 2)]</code> that corresponds to <var class="Arg">mat</var>.</p>

<div class="example"><pre>
<span class""gap;</panspan="GAPinputS:=FullPBRMonoid(1);
<pbr monoid of degree 1 with 4 generators>
<span class="GAPprompt">gap></span> <span class 3 15 5 7 8 1 4 1,1,16, 6, ,9 1,4 1 ]/pre<div
[ 3, 15, 5, 7, 8, 1, 4, 11,

<p><a id="X82FD0AB179ED4AFDdivclass="> =func%<< "">code=">#227 />(Arg"x/> )(bspnbsp<>/tr<>

<>4.5- IsEmptyPBR/>

<div class="func"><table class="func" width="100%"><tr><tdp>A  is <strongclass"button"">empty if itno edges. code>/   class="keyw">true</code> if the PBR <var class="Arg">x</var> is empty and <code class="keyw">false</code> if it is not.</p>
<>: code""true orcodejava.lang.StringIndexOutOfBoundsException: Range [57, 52) out of bounds for length 82

<p>A  is class""emptystrong>if  has . <code=code> codeclass""truecode  the PBR< class">>x/var>isempty and false if it is not.

<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">x := PBR([[]], [[]]);;</span>
<pan="GAPprompt">gap></span> <span class"GAPinput">IsEmptyPBR(x)/span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">x := PBR([[-2, 1], [2]], [[-1], [-2, 1]]);</span><span class="GAPpromptGAPprompt">gapgt<span<spanclassGAPinput">(x);
PBR([ [ -2, 1 ], [ 2 ] ], [ [ -1 ],
<spanclassGAPprompt>>/> < class""IsEmptyPBRx)</pan
false</pre></div>

<p><a id="X7E263B2F7B838D6E" name="X7E263B2F7B838D6E"></a></p>

<h54.5-6 IsIdentityPBR</h5>

<div class="func"><table class="func" width="100%"><tr><divclass="func">tableclass""="0%"tr class">

func>#27 IsIdentityPBR> < class"Arg"x<var</td><td class=tdright&;&;</>/

<p>Returns: <code class="keyw">true</code> or <code class="keyw">false</code>.</p>



<p>A PBR of degree <code class="code">n</code> is the <strong class="button">identity</strong> PBR of degree <code class="code">n</code> if it is the identity <p>Returns: <code class="keyw"true/code  it is the identity of the full PBR monoid of degree <code class="code">n</code>. The identity PBR of degree <code class="code">n</code> has <code class="code">2n</code> edges. Specifically, for each <code class="code">i</code> in the ranges <code class="code">[1 .=code>-. -1</>, identityPBRhas edge from class=""i<code  <codeclass">-icode.pjava.lang.StringIndexOutOfBoundsException: Index 556 out of bounds for length 556



<p><code class="code">IsIdentityPBR</code> returns <code class="keyw">true</code> is the PBRfalse





< class"prejava.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26

< ="GAPprompt">apgt/spanclass"">x : PBR([-2,[1], [1, 2];/>

PBR([ [ -2 ], [ -1 ] ], [ [ 1 ], [ 2 ] ])

<span class="GAPprompt">gap></java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

false

<span class>.- <h5

PBR([ =""< classwidth0<<tdjava.lang.StringIndexOutOfBoundsException: Range [73, 72) out of bounds for length 221

<span class="GAPprompt">gap></span> <span class="GAPinput">IsIdentityPBR(x);</span>

true</pre></div>



<p><a id="X7A280FC27BAD0EF0" name="X7A280FC27BAD0EF0"></a></p>



<h5>4.5-7 IsUniversalPBR</h5>



<div class="<>

<p>Returns:=GAPprompt&;</>< classGAPinputx := PBR[[], []];</span



<p>A PBR of degree <code class="code">n</code> is <strong class="button">universal</strong> if it has <code class="code">4 * n ^ 2</code> edges, i.e. every possible edge.</p>





<div class="example"><pre class"">&t;/span><span=""x:PBR[2 1][],[[1]-2, )<span

<span =GAPprompt>>span span class"">x =PBR[]] [])</>

PBR classGAPpromptgap>/spanspan=GAPinputIsUniversalPBR;<span

<span java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5

false

<span class="GAPprompt">gap></<<div

PBR([ [ -2, 1 ], [ 2 ] ], [ [ -1 ], [ -2, 1 ]])

< classGAPpromptgap

false

<span class="GAPprompt">gap><pIf PBR classArgx/> definesbipartition  < class">sBipartitionPBR/code> returns< class"">

PBR([ [ -1p>  < class"> definesa if and only if when considered as a boolean matrix it is an equivalence.

<span class="GAPprompt">gap></span> <span class="GAPinput">IsUniversalPBR(x);</span>

true</pre></div>



<p><a id="X81EC86397E098BC8" name="X81EC86397E098BC8"></a></



<>..58 sBipartitionPBRh5



<div class="func">span="GAPprompt"gapgt<spans =GAPinput">x= ([[-, 3],[, ],[2,, , ],

<div ="func"><table class"" width=10">tr>td =tdleft"<code class"">̻ IsBlockBijectionPBR</code> varclass=Arg><var </><td="tdright">&;property;<td/r></table></div>

<p>Returns: <code class="keyw">true</code> orspanclass="GAPprompt">>/> < classGAPinput>             - - 1 ];</>



<p>If the PBR <var   2-,2 ][2 1,2  ,  - 11 2



<p>  <var="">x/var definesbipartitionif onlyif considered as aboolean matrixitis equivalence.</p



<p>If <var class class"GAPprompt">><span< class=GAPinput            [2 -,1, -, -,] [])<span





<div class="example"><pre>

<span class="GAPprompt">&t;</pan< =GAPinputx :=BR[[13] -,java.lang.StringIndexOutOfBoundsException: Range [87, 86) out of bounds for length 112

<span ="GAPprompt">><span< class=GAPinput"            [-2 -1,2], [-2, -1, 1,2,3],

<span class="GAPprompt">></span> <span class="GAPinput">             [-2, -1, 1, 2]]);java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4

PBR(  -,3] [ 1, 3] [ 2, 1 2,3]]java.lang.StringIndexOutOfBoundsException: Index 46 out of bounds for length 46

  [ [ -2, -1, 2 ], [ -2, -1, 1, 2, 3 ], [ -2, -1, 1,false/></div

<span class="GAPprompt">gap></span> <span classp>< ="X7AF425D17BBE9023" name"AF425D17BBE9023>/a>

false <span class="GAPprompt">gap></span> <span class="GAPinput">x := PBR([[-2, -1, 1], [2, 3], [2, 3]],</span> <span class="GAPprompt">></span> <span class="GAPinput">            [[-2, -1, 1], [-2, -1, 1], [-3]]);</span> PBR([ [ -2, -1, 1 ], [ 2, 3 ], [ 2, 3 ] ], <span class="GAPprompt"> true <span classp>If the < class"xvar>definesatransformation,java.lang.StringIndexOutOfBoundsException: Range [75, 71) out of bounds for length 221 false/>/> <p><a id="X7AF425D17BBE9023"p>  definition,< =funccode> (ah=chap4_mjXCBBE6080439596span class=RefLink431/span/)and class""></code (ahrefchap4_mjhtmlX8407F516825A514Aspan class=RefLink.-2span</a>) define mutually inverse isomorphisms from the full transformation monoid of degree <code class="code">n</code> to the submonoid of the full PBR monoid of degree <code class="code">n</code>java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 <>4.-9</> <div class="func"><table class=" <pReturns<codeclass"">truecode  <code="keyw">false/>java.lang.StringIndexOutOfBoundsException: Index 82 out of bounds for length 82 <p>If thePBR( [ 2 -, ,[-2 ,java.lang.StringIndexOutOfBoundsException: Index 59 out of bounds for length 59 <<pre/> <p>With this definition, <"name=X7962D03186B1AFDF"</<> <div class="example"><pre> < class"gapgt;/span>">, [-1], [-3]], [2, [], 1 3]]) PBR([ [ -3 ], [ -1 ], [ -3 ] ], [ [ 2 ], [  ], [ 1, 3 ] ]) <spanclass=GAPpromptgap;</span> <span class="GAPinput>IsTransformationPBR() true <span class="GAPprompt">gap></span> <span class="GAPinput">x := AsTransformation(x);</span> Transformation( [ 3, 1, 3 ] ) <spanp> the  <varclass=Arg"x defines a dual, then IsDualTransformationPBR/code> returns < class"">true and ifnot,then code="keyw<code  returned<p> true <span class="GAPprompt">gap&java.lang.StringIndexOutOfBoundsException: Range [0, 30) out of bounds for length 0 1 <span class="GAPprompt">gap></span> <span class="GAPinput">x := PBR([[-2, -1, 2], [-2, 1, 2]], [[-1, 1], [-2]]);</span> PBR([ [ -2, -1, 2 ], [ -2, 1, 2 ] ], [ [ -1, 1 ], [ -2 ] ]) <span class="GAPprompt">gap></span> <span class="GAPinput">IsTransformationPBR(x);</span> false</pre></div> <p><a id="X7962D03186B1AFDF" name="X7962D03186B1AFDF"></a></p> <h5>4.5-10 IsDualTransformationPBR</h5> <div class="func"><table class="" width=10%>tr =tdleft>codeclass"func"ã IsDualTransformationPBR</code>( <var class="Arg">x</var> )</td><tdclass"tdright"( propertynbsp<td</></table></> <Returns<code ="keyw"></> orcodeclasskeyw">false.

<p>If the PBR <var class="Arg">x</var> PBR([ [ -3 1,3],, ,[ -,1, 3  ]

<p>A PBR <var class="Arg">x</var> defines a dual transformation if and only if <code class="code">Star(<var class="Arg">x</var>)</code> satisfies <code class="func">IsTransformationPBRspanclass="GAPprompt>>IsDualTransformationPBR(x)

divclass=examplepre
<span classpre>
<span class="GAPprompt">>
BR [-3,, 3],[ -1  ] [-3 1 3],
  [ [ -1, 2 ], [ -2 ], [ -3, 1, 3 ] ])
<span class="GAPprompt">gap></span> <span class="GAPinput">IsDualTransformationPBR(x);</span>
falseh5.-1 IsPartialPermPBR<h5
<span class="GAPprompt">gap></span> <span class="GAPinput">IsDualTransformationPBR(Star(x));</spandiv class"func"><classfuncwidth=10">x</var> )</td><td class="tdright">( property )</td></tr></table></div> true <span class="GAPprompt">gap></span> <span class="GAPinput">Number(FullPBRMonoid(1), IsDualTransformationPBR);<p>f thePBR <varclass"Arg"x<var  a partial permutation, <codejava.lang.StringIndexOutOfBoundsException: Range [117, 86) out of bounds for length 223 1</pre></div> <p><a id="X7883CD5D824CC236" name="X7883CD5D824CC236"></a></p> <h5>4.5-11 IsPartialPermPBR</h5> <div class<p>With this definition,<code class="func">AsPBR/> (ahref"chap4_mj.html#X81CBBE6080439596">< ="RefLink">4.-1/></> and class="func"></>(a hrefchap4_mjhtmlX795B1C16819905E8>span =RefLink>33<spana)definemutuallyinverseisomorphismsfrom symmetric    <code class"code>nto submonoid of hefullPBR monoid of degree n ofall the elements satisfying IsPartialPermPBR.



< class"example>pre>



<p>If the PBR <var class="Arg">x</var> defines a partial permutation, then <code class="code">IsPartialPermPBR</code> ([ [-,1 ,  ]]  [ -1  ]  -  ]java.lang.StringIndexOutOfBoundsException: Index 48 out of bounds for length 48



<p>A PBR <var class="Arg">x</var> defines a partial perm if and only if it satisfies <code class="func">IsBipartitionPBR</code> (<a href="chap4_mj.html#X81EC86397E098BC8"><span class="RefLink">4.5-8</span></a>) and and when it is considered as a bipartition it satisfies <code class="func">IsPartialPermBipartition</code> (<a href="chap3_mj.html#X87C771D37B1FE95C"><span class="RefLink">3.5-15</span></a>).</p>



<p>With this definition, <code class="func">AsPBR</code> (<a href="chap4_mj.html#X81CBBE6080439596"><span class="RefLink">4.3-1</span>2/pre><div





<div class="example"><pre>

<span class="GAPprompt">gapdiv="unc"< =""0%>td">codeclass=""&827 IsPermPBR(< class"Arg/<class"&;propertynbsp)td>/>

SSL chap4_mj.html Sprache: HTML

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    PartitionedPartitioned  binary PBRs
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<div=""><spanclass"ocline>">&;ahref=chap4_mjhtml#java.lang.StringIndexOutOfBoundsException: Range [118, 117) out of bounds for length 189
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X82CCBADC80AE2D15">4.1-1 IsPBR</a></span>
<span class=ContSSbr /< classnocss>nbsp;</pan="chap4_mj.html##X854A9CEA7AC14C0A>4.1-2 IsPBRCollectionjava.lang.StringIndexOutOfBoundsException: Index 140 out of bounds for length 140
div
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/>
<div class="ContSSBlock">
classContSS<classnocss"&; /span/pan
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X82FE736F7F11B157">4.2-2 RandomPBR</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X8646781B7EAE04C0">4.2-3 EmptyPBR</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X80D20EA3816DC862">4.2-4 IdentityPBR</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X847BA0177D90E9D7">4.2-5 UniversalPBR</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X86B714987C01895F">4.3 <span class="Heading">Changing the representation of a PBR</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X81CBBE6080439596">4.3-1 AsPBR</a></span>
<span class="ContSS"><br /><span class="nocss">  <span class=""><br /><spanclassnocss>nbsp&;<span< href".html- PBR<>span
< =ContSSbr>span"nocss>  <><br /><span class="nocss">  </span><a href="chap4_mj.html#X86786B297FBCD064">4.3-4 AsPermutation</a></span>
<>/div
<div class="ContSect"><span class="tocline"><span classa href=chap4_mjX80D20EA3816DC8622 IdentityPBR</a></span>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X78EC8E597EB99730">4.5 <span class="Heading">Attributes for PBRs</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X7DFC277E80A50C2F">4.5-1 StarOp</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X785B576B7823D626">4.5-2 DegreeOfPBR</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X78302D7E81BB1E54">4.5-3 ExtRepOfObj</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X7F4C8A2B79E6D963">4.5-4 PBRNumber</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X82FD0AB179ED4AFD">4.5-5 IsEmptyPBR</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X7E263B2F7B838D6E">4.5-6 IsIdentityPBR</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X7A280FC27BAD0EF0">4.5-7 IsUniversalPBR</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X81EC86397E098BC8">4.5-8 IsBipartitionPBR</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X7AF425D17BBE9023">4.5-9 IsTransformationPBR</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X7962D03186B1AFDF">4.5-10 IsDualTransformationPBR</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X7883CD5D824CC236">4.5-11 IsPartialPermPBR</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X85B21BB0835FE166">4.5-12 IsPermPBR</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X7ECD4BBD7A0E834E">4.6 <span class="Heading">
      Semigroups of PBRs
    </span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X8554A3F878A4DC73">4.6-1 IsPBRSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X80FC004C7B65B4C0">4.6-2 DegreeOfPBRSemigroup</a></span>
</div></div>
</div>

<h3>4 <span class="Heading">
    Partitioned binary relations (PBRs)
  </span></h3>

<p>In this chapter we describe the functions in <strong class="pkg">Semigroups</strong> for creating and manipulating partitioned binary relations, henceforth referred to by their acronym PBRs. We begin by describing what these objects are.</p>

<p>PBRs were introduced in the paper <a href="chapBib_mj.html#biBMartin2011aa">[MM13]</a> as, roughly speaking, the maximum generalization of bipartitions and related objects. Although, mathematically, bipartitions are a special type of PBR, in <strong class="pkg">Semigroups</strong> bipartitions and PBRs are currently distinct types of objects. It is possible to change the representation from bipartition to PBR, and from PBR to bipartition, when appropriate; see Section <a href="chap4_mj.html#X86B714987C01895F"><span class="RefLink">4.3</span></a> for more details. The reason for this distinct is largely historical, bipartition appeared in the literature, and in the <strong class="pkg">Semigroups</strong> package, before PBRs.</p>

<p><a id="X7C40DA67826FF873" name="X7C40DA67826FF873"></a></p>

<h4>4.1 <span class="Heading">The family and categories of PBRs</span></h4>

div="">< class"""nbsphrefchap4_mj.htmlX86B714987C01895F">4.3 <span class="Heading">Changing the representationofa PBR</span</a

<h5>4.1-1 IsPBR</h5>

<div class"">< class=func=10<><td="tdleft"><codeclass"" IsPBR<code<><tdtd classtdright>&; /='color:red'>td/></table<div
<p> class=ContSSbr >span="nocss>nbsp; nbsp;span>ahref".#java.lang.StringIndexOutOfBoundsException: Range [107, 106) out of bounds for length 138

<p>Every PBR in <strong class="pkg">GAP</strong> belongs to the category <code class="java.lang.StringIndexOutOfBoundsException: Index 90 out of bounds for length 7

<p><a id="X854A9CEA7AC14C0A" name="X854A9CEA7AC14C0A"></a></p>

<h5>4.1-2 IsPBRCollection</h5>

<div class"
span class=nocss>nbsp&;4.5-3 ExtRepOfObj
<div class="func"><table class="func" width="100%"><tr><d class""><code class="func"> IsPBRCollColl<code><var="">obj> )/td>td(nbsp )<td></tr></table></div>
<pReturns < classkeyw>true</> orcode =""false>

<p>Every collection of PBRs belongs to the category <code class="code">IsPBRCollection</code>. For example, PBR semigroupsspan="ContSS"><br /<spanclass"">nbspnbsp;/><ahref=".htmlX7F4C8A2B79E6D963>.54

<p>Every collection of collections of PBRs belongs to <code class="code">IsPBRCollColl</code>. For example, a span="">< /< class>nbspnbsp<span href".#X7E263B2F7B838D6E">456IsIdentityPBR<a></span>

<p>aid" namename=="">/p>

<h44.2 <span class=HeadingCreatingPBRs/pan<h4

<> areseveral  of creating instrongclass="pkg"></> which  in section>

<p><a id="X82A8646F7C70CF3B"span=""br> ""&;&</spana =chap4_mjhtml#"4511IsPartialPermPBR/a>/>

<h5>4.2-1 PBR</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code /div><div
<p>Returns: A PBR.</p>

<p>The arguments <var class="Arg">left</var> and       Semigroups ofPBRs

<pGiven anargumentcode class"PBR<returns the code="

"x :





<ul>

<><>or <codeclass"codei/code  the range>j</code> in <var class="Arg">left[i]</var>;</p>



</li>

<li><p>forspan>/>



</li>

</ul>

<p>pPBRs introducedin   <ahrefchapBib_mjhtml"[] as roughly speaking  maximum generalization of nd  objects Although, mathematically  are a special type of PBR, in Semigroups bipartitions and PBRs are currently distinct types of objects. It is possible to change the representation from bipartition to PBR, and from PBR to bipartition, when appropriatejava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0





<div class=example"

<span class="GAPprompt">gap></span> <span class="GAPinput">PBR([[-3, -2, -1, 2, 3], [-1], [-3, -2, 1, 2]],</span>

<span class=GAPprompt&;</span< class=GAPinput"       [-, 1,1, 2 3] [3],, [, 2 1, 1, ]]);span>

PBR([ [ -3, -2, -1, 2, 3 ], [ -1 ], [ -3, -2, 1, 2 ] ],

  [ [ -2, -1, 1, 2, 3 ], [ 3 ], [ -3, -2, -1, 1, 3 ] ])</pre



<p><a id="X82FE736F7F11B157" name="X82FE736F7F11B157"></a></p>



<h5>4.2-2 RandomPBR<h541- IsPBR>



<div class"">< classfunc ="10">tr ="tdleft">< class="func">&82;RandomPBR/code <var class=Argnvar,var=Arg<var />td">(nbspoperation )/td>

<p>Returns: A PBR.</p>



<p>If <var class="Arg">n</var> 



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





<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">RandomPBR(6);</span>

PBR(

    5 1, 2  ,  -,-3, -1, 2, 5 ], [ -5, -2, 2, 3, 5 ],

      [ -6, -4, -1, 2, 3, 6 ], [ -4, -1, 2, 4 ],

      [ -5, -3, -1, 1, 2, 3, 5 ] ],

  [ [ -6, -4, -2, 1, 3, 5, 6 ], [ -5

      [ -,5,2 , 3,  ,

      [ -6, -5, -4, -2, 3, 5 ], [ -6, -4, -2, -1, 1, 2, 6 ] ])</pre></div>



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



<h5>4.2-3 EmptyPBR</h5>



<div class="func"><table class="func" width="100%"><tr><td class="tdleftjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<>eturnsA PBR.<p>



<p>If <var class="Arg">n</var> is a positive integer, then <code class="code">EmptyPBRjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0





<=""pre

<span ([ 3 , 1,2  ]  1],[3 21, ]]

[ ,]]   ,[]]java.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 47

< =">&;java.lang.StringIndexOutOfBoundsException: Range [39, 37) out of bounds for length 83

true>>



<p><a idclassfunctable=func=10%>tr>< class"87 RandomPBRn<var< class"><]<



<h5>4.2-4 IdentityPBR</h5>



<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IdentityPBR</code>( <var class="Arg">n</var> )</td><td class="tdright">( operation )</td></tr></table></div>

pReturns .pjava.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 22



<p>If <var class="Arg">n</var> is a positive integer, then <code class="code">IdentityPBR</code> returns the identity PBR of degree <var class="Arg">n</var>. This PBR has <code class="code">2</code><var class="Arg">n</var> edges: specifically, for each <code class="code">i</code> in the ranges <code class="code">[1 .. n]</code> and <code class="code">[-n .. -1]</code>, the identity PBR has an edge from <code class="code">i</code> to <code class="code">-i</code>.</p>





<div class="example"><pre>

<span class="GAPprompt">gap -, 3,1 1 2java.lang.StringIndexOutOfBoundsException: Range [28, 27) out of bounds for length 35

PBR  -1,[- ,  -  ] [[ 1,  ] [  ])

<span class="GAPprompt">gap></span>  -6 -5,-,-2 3,5], [ -6 - -,-1 1 2,6 ] ])</pre<div

true</pre></div>



<p><a id="X847BA0177D90E9D7" name="X847BA0177D90E9D7"></a></p>



<h5



<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ UniversalPBR</code>( <var classpReturns:A PBR.p>

<pReturns A./>



<p>If <var class="Arg">n</var> is a positive integer, then <code class="code">UniversalPBR</code> returns the PBR of degree <var class="Arg">n</var> with <code class="code">4 * n ^ 2</code> edges,java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0





<divspanclass=>apgt;</span spanclassGAPinputIsEmptyPBR;/>

<span class="java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 16

PBR -2 -1,1  ],2 -1, 1, 2],

  [ [ -2, -1, 1, 2

<span class="GAPprompt">gap></span> <span class="GAPinput">IsUniversalPBR(x);</span>

true</pre></div>



<p><a id="X86B714987C01895F" name="X86B714987C01895F"></a></p>



<h4>4.3 <span class="Heading">Changing the representationp>: APBRp



<p>It is possible that a PBR can be represented as another



<p>The operations <code class="func">AsPermutation</code> (span="GAPprompt">gapgt<span<panclass=GAPinput : (3)</>



<p><a id="X81CBBE6080439596" name="X81CBBE6080439596"></a></p>



<h5</pre<div



<div class"">< =func="0%<>classtdleft"< =func&82;AsPBR/>(var class"Arg>x[, Argn<var</>td=tdright>&; >;/></table<div

<p>Returns: A PBR.</p>



<p><code class="codejava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0



<p>There are several possible arguments for <code class="code">AsPBRpR: A  ./>





<dl>

<dt><strong class="Mark">bipartitions

<dd><p>If <var class="Arg">x</var> isspan=GAPprompt>/> < classGAPinputx: ()</>



<p>If the optional second argument <var class="Arg">n[[ 2 -1,12,  -2, -,, 2 ]



</dd>

<dt><strong class="Mark">boolean matrices</strong></dt>

<dd><p>If <var class class"">gap;/span> < classGAPinput>(;/span



</dd>

<dt><strong class="Mark">transformations, partial permstrue</pre></iv>

<dda id"X86B714987C01895F" =X86B714987C01895F<a</>



</dd>

</dl>



<div class="example"><pre

<span

< :[12,1, [ 3  ], [ 4, -3, -4 ]>

<span class="GAPprompt">gap></span> <span class="

java.lang.StringIndexOutOfBoundsException: Range [36, 3) out of bounds for length 61

<span class="GAPprompt">gap

PBR[-, ,  ] [ -, 1, 2 ], [ --2,3] [-4 -,4] [  ] ,

  [ [ -1, 1, 2 ], [ -2, 3 ], [ -4, -<div =""><table="func" width1%<><td="tdleft">codeclass"" AsPBR</code(var=""x<var,< class"">n<var] )</td><td="">  )<td/r></table></div>

< classGAPpromptgap&;/panspan=GAPinputAsPBRx)span

PBR([ [ -1, 1, 2 ], [ -1, 1, 2 ], [ -2, 3 ], [ -4, -3, 4 ] ],

  [ [ -11,2 ] [-, 3] [-,, ], , 3,  ] 

<span class="GAPprompt">gap&java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<span class="GAPprompt">></span> <span class="GAPinput">                                dd>pIf <var=Arg>x/> isa and classArg>/> isapositive, then <<code ="code"AsPBR/> returns a PBR corresponding to <var class="Arg">x</var> with degree <var class="Arg">n</var>. The resulting PBR has an edge from <code class="code">i</code> to <code class="code">j</code> whenever <code class="code">i</code> and <code class="code">j</code> belong to

<span="GAPprompt"&<spanspan=GAPinput>[,,1,java.lang.StringIndexOutOfBoundsException: Range [102, 101) out of bounds for length 111

<span class="GAPprompt">></span> <span class="GAPinput">                                [0, 0, 0, 1]]);;</span>

<span class="GAPprompt">gap></span> <span class="GAPinput">AsPBR(mat);</span>

PBR([ [ -2, 1 ], [ -1, 2 ] ], [ [ -2, -1, 1 ], [ -2 ] ])

<span class="GAPprompt">gap><

PBR>< class"Mark"transformationspartialperms permutations/<dt

<span class="GAPprompt"dd>p>> <var"">x</> is a, partial, orpermutationand class"">java.lang.StringIndexOutOfBoundsException: Range [108, 107) out of bounds for length 642

PBR([ 2  ,[-  ][] , [,, ]  - ,[]]java.lang.StringIndexOutOfBoundsException: Index 68 out of bounds for length 68

<span class="GAPprompt">gap></span> <span class="GAPinput">x := Transformation([2, 2, 1]);;</span>

<span class="GAPprompt">java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 5

([-] 2], 1 ] [ 3 ]  ,java.lang.StringIndexOutOfBoundsException: Range [47, 46) out of bounds for length 58

< =g&;/>java.lang.StringIndexOutOfBoundsException: Range [40, 39) out of bounds for length 81

[ 2]   ], ]  , )

<span class=span"gap&;(x );/span>

PBR [2],[ -1 ], [ -4 ] ],

  [ [ 3 ], [ 1, 2 ], [  ], [ 4 ] ])

<span class="GAPprompt">gap></span> <span class="GAPinput">x := PartialPerm([4, 3]);</span>

[1,4][2,3]

<span class="GAPprompt">gap></span> <span class="GAPinput">AsPBR(x);</span>

PBR([ [ -4 ], [ -3 ], [  ], [  ] ], [ [  ], [  ], [ 2 ], [ 1 ] ])

< classjava.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 81

PBR([ [  ], [  ] ], [ [  ], [  ] ])

<span class="GAPprompt">gap></span> <span class="GAPinput">AsPBR(x, 5);</span>

PBR([ [ -4 ], ] [ -, 3],  -,-,  ],[ -4 3,   ])

  [ [  ], [  ], [ 2 ], [ 1 ], [  ] ])

<span class=<spanclass="GAPprompt>gap&; < class=GAPinput>mat := MatrixIsBooleanMat, [[10, ,],spanjava.lang.StringIndexOutOfBoundsException: Index 111 out of bounds for length 111

(,)2,java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 10

<span class="GAPprompt">gap></span> <span class="GAPinput">AsPBR(x);</span>

PBR([ [ -3, 1 ], [ -4, 2 ], [ -1, 3 ], [ -2, 4 ] ],

  [ [ -1, 3 ], [ -2, 4 ], [span class="GAPprompt">>>spanclassGAPinput"                                0, , , ]]);;

< class""&</span span""x,;</java.lang.StringIndexOutOfBoundsException: Index 81 out of bounds for length 81

[[-33  , [ -,  ], -1, 3] -,4 ] [55],

  [ [ -1, 3 ], [ -2, 4 ], [ -3, 1 ], [ -4, 2 ], [ -5, 5 ] ])</pre></div>



<p><a id="X8407F516825A514A" name="X8407F516825A514A"></a></p>



<h5>4.3-2 AsTransformation</ class=GAPprompt>gapgt;/spanspan="GAPinput">(mat )<span



<div class="func"><span="GAPprompt">gap&t;<//span> <spanclass"GAPinput"> :=Transformation[,2,1);/>

<p>Returns A .</p>



<p>When the argument





<div class="example"><pre>

<span class="GAPprompt">></span<classGAPinput"> :=PBR[-3] [-], [-]],[],[3,[1 2]]]);;<>

<span class="GAPprompt">gap></span> <span class="GAPinput">PBR([  -2] [-  ][[  ,1  ]java.lang.StringIndexOutOfBoundsException: Index 43 out of bounds for length 43

true

< ="">&;span< =GAPinput

Transformation( 3,3, 2 ] java.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 29

<span class="GAPprompt">gap></span> <span([ java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35

<span="gapgt ": (,)(2 4);

Error, the argument (a pbr) does not define a transformation</pre></div>



<p>[ -[-,] , ][ , ]]java.lang.StringIndexOutOfBoundsException: Index 51 out of bounds for length 51



<h5([[ [-3,   -4, 2 ], [ -1, 3 ], [ -2, 4 ], [ -5, 5 ] ],



<

<p>Returns: A partial perm.</p>



<Whentheargument< class"">xvar PBR  code="func">IsPartialPermPBR</code> (<a href="chap4_mj.html#X7883CD5D824CC236"><span class="RefLink">4.-1</span></a), thenthis returnsthatpartial./p>





<div class="example"><pre>

<span class="GAPprompt">gap>

< class"">  ""            ,1,[2] [3 ]-3[5];;/>

<span class class>>

true

< classGAPprompt>ap;>< class()<span

[2,3,4](1)</pre></div>



<p><a id="X86786B297FBCD064" name="X86786B297FBCD064"></a></p>



<h5>4Transformation( [3,, ]java.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 29



<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AsPermutation</code>( <var class="span class=GAPprompt"gt<span ">x: ([1,1, 2]] [2 -] [, -11]]);

<p>Returns: A permutation.</p>



<p>When <var=""></> isa  which satisfies <code ="func"><> ( href="chap4_mj.#">< class"">451<spana>) then this attribute returns thatpermutation./p>





< class>pre

<span class="GAPprompt">gap></span> <span class="GAPinput">x := PBR([[-Returns  perm</java.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 31

< classGAPprompt>;<spanspan=GAPinput            -,], [-2 3 [,] -4,])<span

<span class="GAPprompt">gap></span> <span class="GAPinput">IsPermPBR(x);</span>

true

<span lass"example"><re

(



<p><a id="X872B5817878660E5" namespanclass"">><span span="GAPinput"            -1,] [2,[-,2], -4 3,[5];</span



<h4>4.4 <span class="Heading">Operators for java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4





<

<dt><strong.4 />

<dd><p>returns the product of <var class="Arg">x</var> and <var class="Arg">y</var when <var class=Arg>/>and< > are.>



</dd>

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

<>>returnsclass">x/> is lessthanthe < class="">y, or the degrees are equal and the out-neighbours of x (as a list of list of positive integers) is lexicographically less than the out-neighbours of y.



</dd>

<dt><strong class="Mark"><code class="code"><var class="Arg">x</var> = <var class="Arg">y</var></code></strong></dt>

<dd> <codeclass="keyw"truecode



</dd>

</dl>

<p><a id="X78EC8E597EB99730" name="X78EC8E597EB99730span class=="GAPprompt>&;</span> span=GAPinput>x;/span



<h4>4.5 <span class



<p>In this section we>



<<>    varArgvarand class"yvar< Arg"/>and=Arg> .p



h5- StarOp>



<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">Ran style='color:red'>ddp>eturnscodeclass"">true<code   degree <varclass"">x</var is  thanthe degreeof < class"Arg">y/var, orthedegrees are equalandthe out-neighboursof<var class""x>( alistof of integers lexicographically thanthe   < classArgy>p

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Star</code>( <var class="Arg">x</var> )dd<> <code"truecode>if  the java.lang.StringIndexOutOfBoundsException: Range [57, 56) out of bounds for length 185

<p>Returns: A PBR.</p>



<pInthis we variousattributesthat PBR  possess</>





<div class="example"><pre>

<span =""<table"" =0%<td=""c classfunc#822;</>var"x/> )td< =dright>&;operation />

span="GAPprompt"gap/span""()/span

PBR([ [ : A ./>



<p>aid"" =""><a>/p>



<h5 class""gap/><java.lang.StringIndexOutOfBoundsException: Range [45, 44) out of bounds for length 77



<div class="func"><table class

<div class="func"><table class="func" width="100%"><tr

<



<p>The degree of a PBR isdiv=func""0%<class">#27 /var &bspattribute )td>//>



<p>The  PBR  ,the  pointsit defined , ifvar ="">x/var a PBR defined on <code class="code">2 * n</code> points, then   of <var class"Arg> is code><code>





<div class

<span class="GAPprompt"spanGAPprompt>spanspan="x: (-,23, - 1 2 3,/spanjava.lang.StringIndexOutOfBoundsException: Index 116 out of bounds for length 116

<span class="GAPprompt">></span> <span class="GAPinput">            [[-1, 1], [2, 3], [-3, 2, 3]]);</span>

PBR] ,1,2 3],12   ,

  [ [ -1, 1 ], [ 2, 3 ], [ -3, 2, 3 ] ])

span">gt;GAPinput()<span

3

<java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 91

&; monoid    1 generators;

<spanpre>

2</pre></div>



<p><a idjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0



<h5>4.5-3 ExtRepOfObj</h5>



< class"width0"

divjava.lang.StringIndexOutOfBoundsException: Index 216 out of bounds for length 216

PBR[-,1,[2   [ 11, -  ]

<span class="GAPprompt">gap></span> <span class="GAPinput">IsPartialPermPBR(x);</span>

true

<span class="GAPprompt">gap></span> <span class="GAPinput">x := PartialPerm([3, 1]);</spanp>With thisdefinition <code class"func">AsPBR<>(<a href".htmlX81CBBE6080439596">span=RefLink.-1/span<a and< class=funcAsPermutationcode><=chap4_mj< class"RefLink"434/></a>) define mutually isomorphisms the group degree ="">n/>to ofthefull monoidof  <codeclass"">n<code>   all  satisfying =code<>(..the codeclass"">GroupOfUnitscode < href".htmlX811AEDD88280C277>sclass="RefLink">11.9-1) of the full PBR monoid of degree n).


[2,1,3]

<span class="GAPprompt">gap></panspan=GAPinputAsPBR) (x)AsPBR(^2)<span

java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4

<span classclass="">gapgt/span> <span class"GAPinput">IsPermPBR()</>

2</></>



<p><a id="X85B21BB0835FE166" name="X85B21BB0835FE166"></a></p>



<h5>4.5-12 IsPermPBR</h5>



<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsPermPBR</code>( <var java.lang.StringIndexOutOfBoundsException: Index 129 out of bounds for length 12

<>Returns class<code < class></code>./>



<p>If the PBR <var class="Arg">x</var> defines a permutation, then <code class="code">IsPermPBR</code> returns <code class="keyw">true</code>, and if not, then <



<p> PBRvarclass">x<span=RefLink5-<><a)andandit considered    satisfiesclass"code(a ".>span=RefLink54<span/>./>



<p>java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 13





<div class="example"><preSemigroups of PBRs

<span class="GAPprompt">gap></span> <span class="

<span class

< =GAPprompt&;/span =GAPinput(;/>

true

< =""gap;>< class""> =(,5(2,, );</span>

(1,5)(2,4,3)

<span class="GAPprompt">gap></span> <span class="GAPinput">y := (1, 4, 3)(2, 5);</span>

(1,4,3)(2,5)

<span class="GAPprompt">gap></span> <span class="GAPinput">AsPBR(x) * AsPBR(y) = AsPBR(x * y);</span>

true

<span class="GAPprompt">gap></span> <span class="GAPinput">Number(FullPBRMonoid(1), IsPermPBR

1</pre></div



<p><a id="X7ECD4BBD7A0E834E" name<ivclassfunc< class="unc width="10%>tr"tdleft"><class">de(/>



<h4>: < classkeywtrue/>  < class=keywfalse><pjava.lang.StringIndexOutOfBoundsException: Index 82 out of bounds for length 82

      Semigroups PBRs

    </span></h4>



<p>Semigroups and monoids of PBRs can be created in the usual way in <strong class="pkg">GAP</strong> using the functions <code class="func">Semigroup</code> (<a href="../../../doc/



<p>It is possible to create inverse



<p>Note that every PBR semigroup in <strong class="pkg">Semigroups</strong



<p><a id="X8554A3F878A4DC73" name="X8554A3F878A4DC73"></a></p>



<h5>4.6-1 IsPBRSemigroupspan=""gap/panclass">S:= Semigroup(x, (x))



<div=""><table class"" width"00">trtd class"tdleft"><codeclass"">̶IsPBRSemigroup</code( <var classArgS<var>) class=tdright filter;<td</></table>

<div class

<p>Returns class>&;<span =""IsPBRMonoidGAPprompt;/>  =""S: [/>



<p>A <em>PBR semigroup</em> is simply a semigroup consisting of PBRs. An object <var<pan="GAPprompt"&;span class     [,3 -, ][-3, ] -,][]],



<p>A <em>PBR monoid</em> is a span=GAPprompt;span =GAPinput[1, -,1,3 ] -, 4,-]),</span>



<p>Note that itspan="GAPprompt">gt/panspan class=GAPinput ([[1 ,3] 2, 2,[-1 ,] -,4 ,-, 





div=">>

<span class="GAPprompt">gap></span> <span class="GAPinput">x := PBR([[-2, -1, 3], [

<span class="GAPprompt">></span> <span class="GAPinput">            [  11, -,2, 33 ]  - ,5] [-, ,  ,

<span class="GAPprompt">gap></span[[ 1  ,  2  ]  3,3,[ -,4 5] [-  )

<commutative monoid of 3 with1generatorgt

<span class="GAPprompt">gap></span> <span class=alse</pre></iv

true

<span class="GAPprompt">gap></span> <span class="GAPinput">IsPBRMonoid(S);</span>

true

<span class="GAPprompt">gap></span

< classGAPprompt&;</pan classGAPinput(-2 1,[-,2,-,3], [-4,4 5,[-,,]],span

<span java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 0

<span class="GAPprompt">></span> <span class="GAPinput"> PBR([[-2, 1], [-1, 2], [-3, 3], [-4, 4, 5], [-4, 4, 5]],</span>

<span class="GAPprompt">></span> <span class="GAPinput">     [[-1, 2], [-2, 1], [-3, 3], [-4, 4, 5], [-5]]),</span>

<span class="GAPprompt">></span> <span class

<span class="GAPprompt">>

<pbr semigroup of degree 5 with 3 generators>

<span class="GAPprompt">gap></span> <span class="GAPinput">One(S);</span>

fail

<span class="GAPprompt">gap></span> <span class="GAPinput">MultiplicativeNeutralElement(S);</span>

PBR([ [ -1, 1 ], [ -2, 2 ], [ -3, 3 ], < classchlinkbot => :<span"chap0_mjhtmlTopahref=chap1_mjhtml>/>  < ="chap2_mj>a>  <href.html>/  ahrefchap4_mj.html></  ahrefchap5_mj">5chap6_mj.html">6a>a href=".">7<>    < href".">< ="chap10_mj"10/>  a ="chap11_mjhtml><a><ajava.lang.StringIndexOutOfBoundsException: Range [452, 451) out of bounds for length 751

  [ [ -1, 1 ], [ -2, 2 ], [ -3, 3 ], [ -4, 4, 5 ], [ -5 ] ])

<span class="GAPprompt">gap></span> <span class="GAPinput">IsPBRMonoid(S)</html>

false</pre></div>



<p>In this example <code class="code">S</code> cannot be converted into a monoid using <code class="func">AsMonoid</code> (<a href="../../../doc/ref/chap51_mj.html#X7B22038F832B9C0F"><span class="RefLink">Reference: AsMonoid</span></a>) since the <code class="func">One</code> (<a href="../../../doc/ref/chap31_mj.html#X8046262384895B2A"><span class="RefLink">Reference: One</span></a>) of any element in <code class="code">S</code> differs from the multiplicative neutral element.</p>



<p>For more details see <code class="func">IsMagmaWithOne</code> (<a href="../../../doc/ref/chap35_mj.html#X86071DE7835F1C7C"><span class="RefLink">Reference: IsMagmaWithOne</span></a>).</p>



<p><a id="X80FC004C7B65B4C0" name="X80FC004C7B65B4C0"></a></p>



<h5>4.6-2 DegreeOfPBRSemigroup</h5>



<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DegreeOfPBRSemigroup</code>( <var class="Arg">S</var> )</td><td class="tdright">( attribute )</td></tr></table></div>

<p>Returns: A non-negative integer.</p>



<p>The <em>degree</em> of a PBR semigroup <var class="Arg">S</var> is just the degree of any (and every) element of <var class="Arg">S</var>.</p>





<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">S := Semigroup(</span>

<span class="GAPprompt">></span> <span class="GAPinput"> PBR([[-1, 1], [-2, 2], [-3, 3]],</span>

<span class="GAPprompt">></span> <span class="GAPinput">     [[-1, 1], [-2, 2], [-3, 3]]),</span>

<span class="GAPprompt">></span> <span class="GAPinput"> PBR([[1, 2], [1, 2], [-3, 3]],</span>

<span class="GAPprompt">></span> <span class="GAPinput">     [[-2, -1], [-2, -1], [-3, 3]]),</span>

<span class="GAPprompt">></span> <span class="GAPinput"> PBR([[-1, 1], [2, 3], [2, 3]],</span>

<span class="GAPprompt">></span> <span class="GAPinput">     [[-1, 1], [-3, -2], [-3, -2]]));</span>

<pbr semigroup of degree 3 with 3 generators>

<span class="GAPprompt">gap></span> <span class="GAPinput">DegreeOfPBRSemigroup(S);</span>

3</pre></div>





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