td ="tdright>(nbsp&;</><tr/</>
<p>Returns: <code class="keyw">true</code> or <code class="keyw">false</code>.</p>

<p>If the PBR <var class="Arg">x</var> defines a permutation, then <code class="code">IsPermPBR</code> returns <code class="keyw">true</codePBR([ [ -1 1]  2 ]],[ [-1  ] [ 2 ] ]

<p>A

<With ,<code=func<code  =chap4_mj#> class"">43<span>/>)and <codeclass""></code>(a href="chap4_mj.html#X86786B297FBCD064"><spanclass=RefLink>.-4</span  inverse from symmetric of <codeclasscode></code  the subgroup the  PBRmonoid ofdegree =coden/ consistingof theelements <codeclass"">IsPermPBR/code ie the< class=func></>(ahref=chap11_mj#"

<div class/> <span class"">(x *AsPBR)= AsPBRx  2;/span>
<span classtrue
<span class="GAPprompt">></span> <span class="GAPinput">[[-1, 3], [-2, 1], [-3, 4], [-4, 2]]);;</span>
<span class=GAPprompt&;<span<span=GAPinputx;span
true/prediv
<java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
(1,5)(2,4,3)
<span class="GAPprompt">gap></span> <span class="GAPinput">y := (1, 4, 3)(2, 5);</span>
(1,4,p>Returns: <code="keyw">true/> orcode="keyw">false><pjava.lang.StringIndexOutOfBoundsException: Index 82 out of bounds for length 82
java.lang.StringIndexOutOfBoundsException: Index 161 out of bounds for length 104
true
<span class="GAPprompt">gap>  < ="Arg">/>defines if and only ifit codeclassfunc"IsBipartitionPBRchap4_mjX81EC86397E098BC8> class"RefLink">4.5-8</span/>   when it isconsidered asabipartitionit <code ="func>IsPermBipartition (<span class"">3.5-14/span><a).<p>
1</pre></div>

<p><a id="X7ECD4BBD7A0E834E" name="X7ECD4BBD7A0E834E"></a></p>

<h4>4.6 <span class="Heading">

java.lang.StringIndexOutOfBoundsException: Index 52 out of bounds for length 16

<p>Semigroupsspanclass"GAPprompt">gapgt</span> <spanclass"GAPinput">IsPermPBR(x)<span

<p>It is possible to create inverse semigroups and monoids of PBRs using <code class="func">InverseSemigroup</code> (<a href="../../../doc/ref/chap51_mj.html#X78B13FED7AFB4326"><span class="RefLink">Reference: InverseSemigroup</span></a>) and <code class="func">InverseMonoid</code> (<a href="../../../doc/ref/chap51_mj.html#X80D9B9A98736051B"><span class="RefLink">Reference: InverseMonoid</span></a>) when the argument is a collection ofspanclassGAPprompt>></span spanclass=GAPinputx: 1 )2, 43</span

<p>Note that every PBR semigroup in <strong class="pkg">Semigroups</strong> is java.lang.StringIndexOutOfBoundsException: Index 85 out of bounds for length 4

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

<h5>4.6-1 IsPBRSemigroup</h5>

< ="func">table""width0"<>

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<div class="ChapSects"><a href="chap4_mj.html#X85A717D1790B7BB5">4 <span class="Heading">
    Partitioned binary relations (PBRs)
  </span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X7C40DA67826FF873">4.1 <span class="Heading">The family and categories of PBRs</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </spanspanclass""><br/>span="nocss"&; /><a hrefhtml"a>/span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X854A9CEA7AC14C0A">4.1-2 IsPBRCollection</div></div>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span</span
</span>
<div class="
>4.2-1 PBR</>
spanContSS> =""&;nbsp/><href=chap4_mj#X82A8646F7C70CF3B">4.21 PBR/a
<span class="ContSS"><br /><span class="nocss">  </span><a href="spanclass"">< /< class="">&;&;/>a href=chap4_mj.html#X795B1C16819905E8">4.3-3 AsPartialPerm</a></span>
<span class="ContSS"><br /><span class="nocss">&/div<>
<span class="ContSS"><br /><span class="nocss">  </span><ahref"chap4_mj.html#X80D20EA3816DC862">4.-4java.lang.StringIndexOutOfBoundsException: Range [125, 115) out of bounds for length 136
<
</div></div>
< classContSectspan=tocline><span class="nocss> #X86B714987C01895F  PBR>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>
<spandiv class=functableclass="" width"1%">tr classtdleft =func>#2227 IsPBR/>( var class="Arg">obj</var )/><="">(nbspcategory;)</><tr>/>
<span""><br/< class"&&>4.3-3 AsPartialPerm</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X86786B297FBCD064">4.3-4 AsPermutation</a></span>
</div></div>
<div class="ContSect">java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
</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 java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<span class="ContSS"><br /><span class="nocss"java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<spanclass="ContSS"> /<span""&;nbspspan hrefchap4_mj.html>=tdleftclass>#2;/(  classArg></var<td< class="tdright">(&;category;)<td>:code="">true/code  <codeclasskeyw></code>.</p>
< classContSSbr>span =nocss&; <spana chap4_mj#"4.-PBRNumber>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X82FD0AB179ED4AFD">4.5-5 IsEmptyPBR</a></span>
< classContSSbr>span="nocss">&;&;/><a=chap4_mjhtmlX7E263B2F7B838D6E.- IsIdentityPBR/java.lang.StringIndexOutOfBoundsException: Range [131, 130) out of bounds for length 138
<span< ="X8758C4FB81D2C2A1" nameX8758C4FB81D2C2A1/<p>
<span class>42<span class""> PBRs</pan>/>
<pThere severalwaysofcreating PBRs < class=pkgGAPstrong, aredescribed this.</p>
<span class="ContSS"><br
< class"ContSS>< /nbsp;nbsp;/span>< href"chap4_mj.htmlX7883CD5D824CC236>.-11 <><span
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X85B21BB0835FE166">4.5-12 IsPermPBR<java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<div<>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X7ECD4BBD7A0E834E">4.6 <span class="Heading">
      Semigroups
    </span></a>
</span>
<div class="ContSSBlock">
<span class=> such , <code="code>PBR< returns PBR< classcode"x<> where<p>
<span
</ul>
</divlip> each class=""></in code"code>[1. n]> there is an from code ="">

to 



<h3>4 <span class="Heading">

java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 0

  </span<h3



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



<> were in thepaper =".#biBMartin2011aa>MM13]aas,roughly speaking,themaximumof  bipartitionsa related . Althoughmathematically,,bipartitions java.lang.StringIndexOutOfBoundsException: Range [221, 218) out of bounds for length 741



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



div="



<p><a id="X82CCBADC80AE2D15" namespan"">gt> span="[2 -, 12,],] -3-2 13]);/java.lang.StringIndexOutOfBoundsException: Range [119, 118) out of bounds for length 119



<>4.11 </h5



<div =functable=""width0%<><tdclasstdleftcodeclass=func#27 </>( <varclass=">[, ">p/var>])

< class="tdright&;;<tdtrtable>
<p>Returns: <code class="keyw">true</code> or <code class="keyw">false</code>.</p>

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

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

<h5>4.1-2 IsPBRCollection</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsPBRCollection</code>( <var class="Arg">obj</var> )</td><td class="tdright">( category )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsPBRCollColl</code>( <var class="Arg">obj</var> )</td><td class="tdright">( category )</td></tr></table></div>
<p>Returns: <code class="keyw">true</code> or <code class="keyw">false</code>.</p>

<p>Every collection of PBRs belongs to the category <code class="code">IsPBRCollection</code>. For example, PBR semigroups belong to <code class="code">IsPBRCollection</code>.</p>

<p>Every collection of collections of PBRs belongs to <code class="code">IsPBRCollColl</code>. For example, a list of PBR semigroups belongs to <code class="code">IsPBRCollColl</code>.</p>

<p><a id="X8758C4FB81D2C2A1" name="X8758C4FB81D2C2A1"></a></p>

<h4>4.2 <span class="Heading">Creating PBRs</span></h4>

<p>There are several ways of creating PBRs in <strong class="pkg">GAP</strong>, which are described in this section.</p>

<p><a id="X82A8646F7C70CF3B" name="X82A8646F7C70CF3B"></a></p>

<h5>4.2-1 PBR</h5

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PBR</code>( <var class="Arg">left</var>, <var class="Arg">right</var> )</td><td class="tdright">( operation )</
<[[-,,,3][6 3java.lang.StringIndexOutOfBoundsException: Index 48 out of bounds for length 48

<p>The arguments <var class- -,-2, 1, 5 ], [ -6, -5, -3, -,1 ,4]java.lang.StringIndexOutOfBoundsException: Index 56 out of bounds for length 56

<p>Given such an argument, <code class="code">PBR</code> returns the

<ul>
<li><p>for each <code class="code">i</code> in the range <code class="code">[1 .. n]</code> there is an edge from <code class="code">i</code> to

</li>
<li><p>for each <code class="code">i</code> in the range <code class="code">[-n .. -1]</p>: APBR<java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 22

</li>
</ul>
<p><code class="code">PBR</code> returns an error if the argument does not define a PBR.</p>

<div class="example"><pre>
<span class
<span class=<div class"example>



PBR[ -, -2- ,3,[ -1 ], -,-, 1,2] ,

  [ [ -2, -1, 1, 2, 3 ], [ 3 ], [ -3, -2, -1, 1, 3 ] ])</pre><PBR([ [  ],   ],    ], [ [  ],[  ],   ])



<p><a id="X82FE736F7F11B157" namespanclassGAPprompt"gap&t; IsEmptyPBR(x);



<h5>4.2-2 RandomPBR</h5true</pre></div



<div class=""><table class"" width"00">tr>td="tdleft>< class"" RandomPBR( >n</>[, <var="Arg>p/var> )/td>td class="tdright">( operation )

<p>Returns: A PBR.</p>

<p>If <var class="Arg">n</var> is a positive integer and <var class="Arg">p</var> is an float between <code class="code">0</code> and <code class="code">1</code>, then <code class="code">RandomPBR</code> returns a random PBR of degree <var class="Arg">n</var> where the probability of there being an edge from <code class="code">i</code> to <code class="code">j</code> is approximately <code<p>Returns: APBR</>

<p>If the optional second argument is not present, then a random value <var class="Arg">p</var> is used (chosen with uniform probability).</p>

<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">RandomPBR(6);</span>
PBR(
  [ [ -5, 1, 2, 3 ], [ -6, -3, -1, 2, 5 ], [ -5, -2, 2, 3, 5 ],
      [ -6,java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
      [5-, -,1,, 3, 5 ] ],
  [ [ -6, -4, -2, 1, 3, 5, 6 ], [ -5, -2, 1, 2, 3, 5 ],
      [ -6, -5, -2, 1, 5 ], [ -6, -5, ([ [  ]  -][ 3]], [ 1 ][ 2 ,  3])
       ,-5 4 ,  ,-,2 1,,2,  />/>

<p><a id="X8646781B7EAE04C0" name

<h5>4.2-3 EmptyPBR<

<div java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 0
<>Returns: PBR</>

<p>If <var class="Arg">n</var> is a positive integer, then <code class="code">EmptyPBR</code> returns the PBR of degree <var class="Arg"<>:  PBR<pjava.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 22

<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">x := EmptyPBR(3);</span>
PBR([ [  ], [  ], [  ] ], [ [  ], [  ], [  ] ])
<span class="APprompt">>span>< ="">(x)<span
true</pre></div>

<p><a id="X80D20EA3816DC862" name([ [-,,1, 2 ] [ -,,,   ]

<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">( java.lang.StringIndexOutOfBoundsException: Index 180 out of bounds for length 87
<Returns .</>

<p>If <var class="java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<div class="example">java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
< classGAPprompt&;/> < class"GAPinput">x:=IdentityPBR;span
PBR([ [ -1 ], [ -2 ], [ -3 ] ], [ [ 1 ], [ 2 ], [ 3 ]
<span class="GAPprompt"
true>/>

<p><a iddivclass=functableclass"" width10">tr

>codeclass"">#27 </code(< =""/>[  ="">/>] )td< class"">nbspoperationnbsp;)</td<tr>/>

<h5>4.2-5 UniversalPBR</h5>

<div class="func"><table class=java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<>Returns A PBR<pjava.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 22

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

<div class="example"><pre>
< class"">gap&;<spanspan="GAPinput"> =UniversalPBR2);span
PBR([ [ -2, -1java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
   [-2,1 1,  ] [ -1 1 2 ])
<span=GAPprompt><spanspan="GAPinput"IsUniversalPBRx)<span>
true<>

<a id=X86B714987C01895Fname"8B714987C01895F">>/>pjava.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62

<h4>4.3 <span class="Heading">Changing the representation of a PBR</span></h4>

<p>It is possible;blockbijection  ,  - ] [ 3,-2java.lang.StringIndexOutOfBoundsException: Range [44, 43) out of bounds for length 63

<p>The operations <code class="func"[ [ -1, 1, 2 ], [ -2 ] ])

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

<h5( [ 11,2,12]  , , ,-,  ,[]]

<div classfunc< classfunc ="00">tr classtdleft< class=func>82;AsPBR>(< classArg>/>[,  <var=Argn</> classtdright(;operation;)/></java.lang.StringIndexOutOfBoundsException: Range [227, 226) out of bounds for length 241
<p>Returns: A PBR.</p>

<p><code class="code">AsPBR</code> returns the boolean matrix, bipartition, transformation, partial permutation, or permutation <var class="Arg">span="">gapgt<s> < class"GAPinput">();/>

<p>There[-1, 1 2, [ 2, 3, 4 -3,4 ],[-4 -3, 4 ] ])

<dl>
<dt><strong class="Mark">bipartitions</strong></dt>
<<p>If < class""><varis a bipartition <var=""n<var is  ositive integer then codeclass><codejava.lang.StringIndexOutOfBoundsException: Range [148, 142) out of bounds for length 455

<p>If the optional second argument <var class="Arg">n</var> is not specified, then degree of the bipartition <var class="Arg">x</var> is used by default.</p>

</dd>
<dt><strong class="Mark">boolean matrices</strong></dt>
<dd><p>If <var class="Arg">x</var> is a boolean matrix of even dimension <code class="code">2 * m</code> and <var class="Arg">n</var> is a positive integer, then <code class="code">AsPBR</code> returns a PBR corresponding to <var class="Arg">x</var> with degree <var class="Arg">n</var>. If the optional second argument <var class="Arg">n</var> is not class"gt;/> < class"" 1, 0 1 1],

</dd>
<dtstrong=Mark>,  , </strong/>
<>pIfvar class=Arg><varis  transformationpartial permor   <var=Argn</var> is a positive integer, then <code class="code">AsPBR</code> is a synonym for <code class="code">AsPBR(AsBipartition(<var class="Arg">x</var>, <var class="Arg">n</var>)[ [-,1]  -,2]    ] [ -2 -1,1],[ 2]    ])

</dd>
</dl>

<div class="example"><pre>
<span class="GAPprompt"PBR[  2 ,[ - ] [ -1 ] ,  [ ],[1 2 ], [  ] ])
<block bijection: [ 1, 2, -1 ], [ 3, -2 ], [ 4, -3, -4 ]>
<spanclass="GAPprompt">apgt<span <span class="GAPinput">AsPBR(x, 2);</span>
PBR([ [ -1, 1, 2 ], [ -1, 1, 2 ] ], [ [ -1,PBR( [- ,[-2 ] [[  , [1 2] ])
<span class="GAPprompt">gt><spanclass"APinput">sPBRx,5;/span
PBR(([ [ -2 ],[ - ] java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 39
  [ [ -1,java.lang.StringIndexOutOfBoundsException: Index 78 out of bounds for length 78
<span class="GAPprompt">span="GAPprompt">gap></span> <span class="GAPinput">AsPBR(x, 2);</span>
PBR([ [ -1, 1, 2 ], [ -1, 1, 2 ], [ -2, 3 ], java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35
,-,  ,[- 34 ,-, 4]java.lang.StringIndexOutOfBoundsException: Index 60 out of bounds for length 60
< class""gtspanspan""mat(,,  0 1</>
<span class="GAPprompt">></span> <span class="GAPinput">                                [0, 1, 1, 0],</span13(,4)
<span class="GAPprompt">></span> <span class="GAPinput">                                [1, 0, 1, 1],</span>
<</span < ="">[, 001];/span>
<span class="GAPprompt">gap></span> <span="GAPprompt>gapgt;/span> AsPBR(,5);
PBR([ [ -BR([  -3,1 ]4 2, [- 3 , [2 ],[ -,   ]java.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62
<span class="GAPprompt">gap></span
PBR([ [ 1 ] ], [ [ -1
<span""&</> < classAsPBR,6;/>
PBR([ [ -2, 1
< class&<span =x=(2 2 ];<span
<span class="GAPprompt">gap: transformation
PBR([ [ -2 ], [ -2 ], [ -1 ] ], [ [ 3 ], [ 1, 2 ], [  ] ])
<spanclass"gapgt;span> span ="GAPinput>=([3,[32], [ ] 1 2];;<span
PBR[2 , 2] ,  [] [ ,2 ])
<span class="GAPprompt">gap></span> <span class="GAPinput">AsPBR(x, 4);</span>
PBR([ [ -2 ], [ -2 ], [ -1 ], [ -4 ] ],
  [true
<spanclass"GAPpromptgapgt GAPinput">x := PartialPerm([4, 3]);
[1,4][2,3]
<span class="GAPprompt">gap></span> <spanTransformation [ ,  )
PBR([ [ -4 ], [ -3 ], [  ], [  ] ], [ [  ], [  ], [ 2 ], [ 1 ] ])
<span class="GAPprompt">gap></span> <span class="GAPinput">AsPBR(x, 2);</span>
PBR[  ], [  ] ], [ [  ], [  ] ])
<span class="GAPprompt">gap></span> <span class="GAPinput">AsPBR(x, 5);</span>
PBR([ [ -4 ], [ -3 ], [  ], [  ], [  ] ],
  [ [  ], [  ], [ 2 ], [ 1 ], [  ] ])
<span class=GAPprompt>&;/panspanclass="GAPinput">x =1 3)2,java.lang.StringIndexOutOfBoundsException: Range [78, 77) out of bounds for length 87
(1,3)(2,4)
<span class="GAPprompt">gap></span> <span class="GAPinput">AsPBR(x);</span>
PBR [ -3, 1 ], [ 4, 2 , [-1 3], [-2 4  ,
  [ [ -1, 3 ], [ -2, 4 ], [ -3, 1 ], [ -4, 2 ] ])
<span class="GAPprompt">gap></span> <span class="GAPinput">AsPBR(x, 5);</span>
PBR([  31 ],[java.lang.StringIndexOutOfBoundsException: Range [21, 19) out of bounds for length 62
  [ [ -1, 3 ], [ -2, 4 ], [ -3, 1 ], [ -4, 2 ]

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

<h5>4.3-2 AsTransformation</h5>

<div class="func"><table class="func" width="100%"><trp> the argument var=Arg</> is a whichsatisfies< class451/span>  function   perm<pjava.lang.StringIndexOutOfBoundsException: Index 248 out of bounds for length 248
<p>Returns: A transformation.</p>

<p>When the argument <var span="GAPprompt>> <pre
<span class="GAPprompt">gap></span>span="">ap></span span="GAPinput">AsPartialPermx;/>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsTransformationPBR
true
<
Transformation  ,3,2 ])
<span"GAPprompt>gap&; x :=PBR[] [1 2,[-,1,-2, -1];;/span
<span class="GAPprompt">gap></span> <span class="GAPinput">AsTransformation(x);</span>
Error, the argument (a pbr) does not define a transformation</pre></div>

<p><a id the argument< classArgxvar  PBRsatisfiesclass>sPermPBR/code><ahref"chap4_mjhtmlX85B21BB0835FE166"span=RefLink.-2/></> then  <p

<h5>4.3-3 AsPartialPerm

<div class="func"><tablediv="example"<>
<p>: Apartial.p>

<p>When the argument <var class="Arg">x</var> is a PBR which satisfies <code class="func">IsPartialPermPBR</code> (<a href="chap4_mj.html#X7883CD5D824CC236"><span class="RefLink">4.5-11</span></a>), thenspan="">/> < class"">[[1 1,[2,3,[3 4,[4 2];;/>

=example<>
<span class="GAPprompt">gap></span> <span class="
<span =GAPprompt;/>< class>[[-1 1,[-] 3 [,] -]);s>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsPartialPermPBR
true
<span class="GAPprompt">gap></span> <span class="GAPinput">AsPartialPerm(x);</span>
[2,3,4](1)</pre></div>

<p><a id="X86786B297FBCD064dl>

<h5>43- AsPermutation</h5

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AsPermutation</code>( <var class="Arg">x</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: A permutation.</p>

<p>When the argument <var>whenvar="xy

<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">x := PBR([[-1, 1], [-4, 2], <d>p> <code ="keyw"true>ifofvar""x<var   the degree of<arArg<>java.lang.StringIndexOutOfBoundsException: Index 340 out of bounds for length 340
<span class="GAPprompt">>><preturns classkeyw></> if the PBR <var class="Arg">x</var> equals the PBR <var class="Arg">y</var> and returns <code class="keyw">false</code> if it does not.</p>
<class"gapgt/ class="">IsPermPBR()
true
<span class="GAPprompt">gap
(2,4,3)</pre></div>

<p><a id="X872B5817878660E5" name="X872B5817878660E5"></a></p>

<h4

<dl
<dt><strong class
<dd><p>returnstheproductof<var class="Arg">x</var> and <var="Arg> when var classclass="Arg>x<var and <var class"Arg">y</var arePBRs</>

</dd<h5>4.5-1 StarOp</h5
<dt><strong class="Mark"><code class=java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<><p> < class=keywtrue/>ifthe of =Arg/>isless the of<var=Arg<>  degrees  the  var=Arg>x</var> (asa list  list positive) is less theout-neighboursofvar="">y</var.</p>

</dd>
<dt><strong class="Mark"><code class="code"><var class="Arg">x</var> = <var class="Arg">y</var></code></strong></dt>
<>preturns class="keyw>Arg">x equals the PBR y and returns false if it does not.

</dd>
</dl>
<p><a id="X78EC8E597EB99730" name="X78EC8E597EB99730"></a></p>

<h4>4.5 <span class="Heading">Attributes for PBRs</span></h4>

<p>>  section describe   a PBRcan.</

<p><a id="X7DFC277E80A50C2F" name="java.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 26

<h5>4.5-1 StarOp</h5>

<lass"unc>10">tr>< class=""tdleft>code( <var class="Arg"><var)/td>tdclass="dright">(nbsp)/>/tr</table/iv
<div class="func"><table class<spanpan class="GAPprompt>gap> Star(x)<>
<p>Returns:  APBR<p>

<p><codep< =X785B576B7823D626nameX785B576B7823D626/</p>

<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">x := PBR([[], [-1], []],
<an class="GAPprompt>gap&>Star(x);
PBR([ [ -3, -2, 2, 3 ], [ -1, 2 ], [  ] ], [ [  ], [ 1 ], [  ] ])</pre></div>

<p><a id="X785B576B7823D626" name="X785B576B7823D626"></a></p>

<h5>4.5-2 DegreeOfPBR</h5>

<div java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 0
<div class"func"><table class="func width="10"> width"10%<>tdclass="tdleft< class=func&82IsEmptyPBR<code  Arg><var<td class"tdright>(&bsp;property )/td

tdleft">

x<>)/tdd><td class="tdright">(&;attribute&;)</td<tr><table/>

<p>Returns: A positive integer.</p>



<p> degree ofa is,roughlyspeaking  numberof where it is. Moreprecisely <var classArg<> is thedegree ="x/is < ="n/.</p>



<p>The degree of a collection <var class="Arg">coll</var> of PBRs of equal





<div class="example"><pre>

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

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

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

  ([ [ -2 ], [-2 -, 2, ] [ -, 1, , 3 ]]java.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 50

<span class="GAPprompt">gap></span> <span class="GAPinput"<span class=GAPprompt"gap>n  class=GAPinput">DegreeOfPBRx;/>

3

<span class="span class="GAPprompt">gap> S := FullPBRMonoid(2);

<pbr monoid of degree 2 with 10 generators>

<span class="GAPprompt">gap<pbr ofdegree2with1 generators>

2</></div



<p><a id="X78302D7E81BB1E54" name="X78302D7E81BB1E54"></a></p>



<h5>4.5-3 ExtRepOfObj</h5>



<div class="funcjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<p>Returns: div="func>table class=" ="10%">< ="tdleft"< </> )</><=tdright>&;operation</td/r>table/iv



<p>If <code class="code">n</code> is thepReturns A  of of ofintegersp





<div class="

<span class="GAPprompt">gap<divclass="example>

<spanclass"GAPprompt">></span <spanclassGAPinput            [2, 1 1, [1 1,2)</pan

PBR( [-1 1] [ -   ],  2 -1,  ][-1,1,2  ]]java.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62

<pan="GAPprompt">gapgt</> <span="GAPinput">ExtRepOfObjx)<span

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



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



<h5>4.5-4 PBRNumber</h5>



<div class="func"><table<h5>454 </h5>

<div class="func">java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<p>Returns: A PBR, or a positive integer.</p>



<p>These functions implement a bijection from the set of all PBRs of degreediv class="func"table="width="0">tr>>

‣ PBRNumber( m, n )

( operation )

codeclass=func#2;NumberPBR/>(var=""mat/> )<td classtdright(nbsp;operation </><trtable/>

<p>More precisely, if <var class=p> functions implement a bijection fromtheset of all of degree <<var="Arg">n</var> the numbers="ode"[1..2^(* =Argn<> ^ )]code./

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

<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">S := FullPBRMonoid(1);</span>
<pbr
<span class="GAPprompt">gap></span> span =GAPprompt>>/> < class"> FullPBRMonoidjava.lang.StringIndexOutOfBoundsException: Index 91 out of bounds for length 91
[,1,,,,,,1 3 1, ,2 92 1,0<>/>

<p><a id="X82FD0AB179ED4AFD" name="X82FD0AB179ED4AFD"></a></p>

<h5>4.5-5 IsEmptyPBR</h5>

< class=func<tableclass"func"" width="100%">

/div
<p>Returnsh54.55IsEmptyPBR<h5

<PBR =button it has noedges =">IsEmptyPBR

returns





<div class="example"><pre>

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

or < class="keyw">false.

<span class="GAPprompt">gap></span> <span class="GAPinput">IsEmptyPBR(x);</span>
true
<p>A PBRPBR <strong=button></strong ifit noedges class"">IsEmptyPBR</code>returns< class=keyw></>ifthePBR var="Arg"x< java.lang.StringIndexOutOfBoundsException: Range [244, 190) out of bounds for length 244
PBR([ [ -2, 1 ], [ 2 ] ], [ [ -1 ], [ classgap;<span =GAPinputx)<spanjava.lang.StringIndexOutOfBoundsException: Range [83, 84) out of bounds for length 83
<=">&;/> ="GAPinputIsEmptyPBRspan
false</pre></div>

<p><a id="X7E263B2F7B838D6E" name="X7E263B2F7B838D6E"></a ="">ap;<spanspan=GAPinput>(;/>

>IsIdentityPBR

< class> ="func width10><>

tdleft<=func"&82;IsIdentityPBR(nbsppropertynbsp)td

pReturns=></>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. n]</code> and <code class"code">- .-]/code the an <codeclasscode>/>to ="code>-i

<p><code class="code">IsIdentityPBR</code> returns <code class="keyw">true</code> is the PBR <var class="Arg">x</var> is an identity PBR and <code class="keyw">false</code> if it is not.</p>

<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">x := PBR([[-2], [-1]], [[1], [2]]);</span>
PBR([ [ -2 ], [ -1 ] ], [ [ 1 ], [ 2 ] ])
<span class="GAPprompt">gap></span> <span class="GAPinput">IsIdentityPBR(x);</span>
java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
<span class="GAPprompt"div="example><>
PBR([ [ -1 ] ], [ [ 1 ] ])
<span class="GAPprompt">gap></span> <span class="GAPinput">spanclassGAPprompt">> < class="GAPinput = PBR[-] -1],[] [])</span
true</pre></div>

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

<h5>457IsUniversalPBR/>

<divclassfunc>table="func" ="10%"><trtd class="tdleft"><code class="func">‣ IsUniversalPBR</code>( <var class="Arg">x</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns: <code class

<p>A PBR of degree <code class

<div class=example">



<span class"">gapgt</span spanclass="">x:PBR(],[[]]);/>

PBR([ [  ]java.lang.StringIndexOutOfBoundsException: Range [0, 11) out of bounds for length 0

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

false

<span=GAPpromptgap&<span < classGAPinput> = PBR([-, , 2] -],, [2,1]];/span>

PBR([ [ -2, 1 ], [ 2 ] ], span class""gapgt</><=GAPinput>: ([],[];span

<span ="">gap&;</> < class"">(x);</>

false

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

PBR([ [ -1, 1 ] ], [ [ -1, 1 ] ])

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

true/pre></div>



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



<h5>4.5-8 IsBipartitionPBR</h5>



<div class="func">< )

<div class="func"><tablespanclass="GAPprompt">></span> <span class="GAPinput">IsUniversalPBR(x);</span>

<p>Returnsfalse



<p> the <var=""><var a bipartition,thencode="code"I</code  code=keywtrue>, and if not, then it returns <code class="keyw">false</code>.</p>



<>PBR<var="Arg">var  bipartitionandifconsideredjava.lang.StringIndexOutOfBoundsException: Range [110, 109) out of bounds for length 132



<p>If





<div class="h5>4.-I

< class">gap>/>  : PBR-13] -13 - 123]/

<span classclass<table=func="0%<>code=func82;IsBlockBijectionPBR>(< class""x/>)<td class"(nbspproperty 

< classGAPprompt&;<spanspan="">             [2,-11,2]);span

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

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

<span class="GAPprompt">gap&

falsepAPBR classArg<>  a   and  whenas boolean   an/

<span

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

PBR(

  [ [span"gap&/pan> spanclass""> = BR([-1 3,[1 3], [-2, 1, 2, 3]],

<span class="GAPprompt">gap></span> <span classspanclass&;/> span=GAPinput>[,-,22,-,   ,/

true

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

<pre>



<<aid" ="7"



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



<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsTransformationPBR</code>( <var class="

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



<pIf PBRvar="Arg> defines  transformation, then IsTransformationPBR returns true, and if not, then false is returned.



<p>A PBR</pre<div



<pWiththis codeclass"">AsPBR</ < href".html#81"><spanclass"RefLink">.-</></a> and <codeclass=funcAsTransformation>< =".html#"><spanclass="">432</></java.lang.StringIndexOutOfBoundsException: Range [268, 267) out of bounds for length 551





<div class="example"><pre>

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

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

<span class="GAPprompt">gap></span> <span classh5.5-IsTransformationPBR/h5

true

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

Transformation( [ 3, 1, 3 ] )

<span class="GAPprompt">gap></span> <span class="GAPinput">AsPBR(x) * AsPBR(x) = AsPBR(x ^ 2);</span>

true

<span class="GAPprompt">gap></span> <span class>: < class=keyw</>orcode classkeyw></ode.</p>

1

<span class="GAPprompt">gap></span

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

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

false/pre></div



<F "X7962D03186B1AFDF>/p>



<



<div class="func"><tablespan="GAPprompt>&<  =GAPinput>x := BR([[-3],[], [][,[,];span>

<p>Returns: <code class="keyw">true "">>spanspanclass"IsTransformationPBR(;spanjava.lang.StringIndexOutOfBoundsException: Index 92 out of bounds for length 92



<pIf PBR =Arg>var transformation class><<code=keywcode>  ,< class">false/code>is.



<p>A PBR <var class="Arg">x</var> defines a dual transformation if and only if <code class="code">Star(





<div class=func=00"<>"< =">8 =>nbsp&;)/>/tr><div

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

<span class="GAPprompt">>

PBR3,1   [ -1 2] -, 1,  ],

  [ [ -1, 2 ], java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

< class"gap&;<  =""IsDualTransformationPBRx;>

false

<span class

true

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

1</></div



<p><a id=P([[ -,1  ] -,2, ,1,  ]java.lang.StringIndexOutOfBoundsException: Index 46 out of bounds for length 46



<>451/>



< ="table ="" width"0%>>tdtdleftclass"&82; varjava.lang.StringIndexOutOfBoundsException: Range [136, 131) out of bounds for length 223

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



p>  =>/>definespermutation then class="code">IsPartialPermPBR</code> returns <code class="keyw">true</code>, and if not, then <code class="keyw">false</code> is returned.</p>



<p>A PBR <var class



<With,codeclass><code< =chap4_mj<spanclassRefLink3<spana) <codeclass">AsPartialPerm/ode ( =".#""4.- define mutually inverse isomorphisms from thesymmetric inversemonoidofdegreecodeclass=""n





div="

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

PBR[- 1] [2] ,[-1,1 ,[ 2])

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

true

<span class="GAPprompt">gap></span> <span class="GAPinput">x := PartialPerm([3, 1]);</span>

[2,1,3]

<span class="GAPprompt">gap></span> <span class="GAPinput">AsPBR(x) * AsPBR(x) = AsPBR(x ^ 2);</span>

true

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

2/pre/>



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



<java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 0



<div class"unc>tableclass"func width=10"

x )

S )>true<code>orcodeclass=""></code./>

<p>A <em>PBR semigroup</em> is simply a semigroup consisting       of

<p>A <em>PBR monoid</em> is a monoid consisting of PBRs. An object <varjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<p>Note that it is possible for a PBR semigroup to have a multiplicative neutral element

<div class> is finite.</p>
<span class="GAPprompt">gap></span> <span class="GAPinput">x := PBR([[-2, -1, 3], [-2, 2], [-3, -2, 1, 2, 3]],</span
<span class="GAPprompt">>
< class="GAPprompt>gap> >S/var </td><tdclass"">(  )/d><tr></div
<span class="GAPprompt">gap></span> <span class="GAPinput">IsMonoid(S);</span>
true
<span="GAPprompt">gapgt/> <span classGAPinput>(S);</span>
true
<span class="GAPprompt">gap></span <spanclassGAPinput> = Semigroup([</pan
<span class="GAPprompt">></span> <span class="GAPinput"> PBR([[-2, 1], [-3, 2], [-1, 3], [-4, 4,
< classGAPprompt>gt</> <span="GAPinput">[[- ],[2, 1, [2,[-, 4 5] -5])</span>
<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> <spanclass"">     [-1 2],[2 ], -3,,3, [4, 4 5] [5]
< classGAPprompt&;</> <spanclass"">PBR[-,1 3,[-,] [, 1 3,[-, ,5], [-4, 4, 5]],</span>
<span
<pbr semigroup of degree
<span class="GAPprompt">gap><div class=example">prejava.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26
fail
<span class="GAPprompt">gap></span> <span class="GAPinput">MultiplicativeNeutralElement(S);</span>
PBR([[ -1,  ], [2,  ], [ -, 3 ,[-,4  , 4,4 5]]java.lang.StringIndexOutOfBoundsException: Index 68 out of bounds for length 68
  [-,1] [-,2,[-, 3] [ 4 , ,[ 5]]
<span class="GAPprompt">gap></span; pbr  degree  &;
alsepre/>

<p>In java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4

<p>For more details see <code class="

<p><a id="X80FC004C7B65B4C0" name=span="">>/> <span="GAPinput"> PBR[[-,1] [- ] [1 , 4, ],[-4 4 5]]</span>

<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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