Ziele Untersuchung
mit Columbo Integrität von
Datenbanken Interaktion und
Portierbarkeit Ergonomie der
Schnittstellen

Angebot Produkte Projekt Beratung

Mittel Analytik Modellierung Sprachen Algebra Logik Hardware Denken Kreativität

Zusammenhänge Gesellschaft Wirtschaft Branche Firma


products/Sources/formale Sprachen/GAP/pkg/semigroups/doc/ (Algebra von RWTH Aachen Version 4.15.1^©) Datei vom 29.7.2025 mit Größe 98 kB

SSL chap12.html Sprache: HTML

products/Sources/formale Sprachen/GAP/pkg/semigroups/doc/chap12.html

<?xml version="1.0" encoding="UTF-8"?>

<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
 "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">

<html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en">
<head>
<title>GAP (Semigroups) - Chapter 12:
 Properties of semigroups
 </title>
<meta http-equiv="content-type" content="text/html; charset=UTF-8" />
<meta name="generator" content="GAPDoc2HTML" />
<link rel="stylesheet" type="text/css" href="manual.css" />
<script src="manual.js" type="text/javascript"></script>
<script type="text/javascript">overwriteStyle();</script>
</head>
<body class="chap12" onload="jscontent()">

<div class="chlinktop">Goto Chapter: <a href="chap0.html">Top</a> <a href="chap1.html">1</a> <a href="chap2.html">2</a> <a href="chap3.html">3</a> <a href="chap4.html">4</a> <a href="chap5.html">5</a> <a 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</a> <a href="chap17.html">17</a> <a href="chap18.html">18</a> <a href="chapBib.html">Bib</a> <a href="chapInd.html">Ind</a> </div>

<div class="chlinkprevnexttop"> <a href="chap0.html">[Top of Book]</a> <a href="chap0.html#contents">[Contents]</a> <a href="chap11.html">[Previous Chapter]</a> <a href="chap13.html">[Next Chapter]</a> </div>

<a href="chap12_mj.html">[MathJax on]</a>
<a id="X78274024827F306D" name="X78274024827F306D"></a>
<div class="ChapSects"><a href="chap12.html#X78274024827F306D">12 
 Properties of semigroups
 </a>
<div class="ContSect"> <a href="chap12.html#X7D297AEC827F3D4E">12.1 
 Arbitrary semigroups
 </a>

<div class="ContSSBlock">
 <a href="chap12.html#X7C8DB14587D1B55A">12.1-1 IsBand</a>
 <a href="chap12.html#X79659C467C8A7EBD">12.1-2 IsBlockGroup</a>
 <a href="chap12.html#X843EFDA4807FDC31">12.1-3 IsCommutativeSemigroup</a>
 <a href="chap12.html#X7AFA23AF819FBF3D">12.1-4 IsCompletelyRegularSemigroup</a>
 <a href="chap12.html#X855088F378D8F5E1">12.1-5 IsCongruenceFreeSemigroup</a>
 <a href="chap12.html#X7C9560A18348AEE3">12.1-6 IsSurjectiveSemigroup</a>
 <a href="chap12.html#X852F29E8795FA489">12.1-7 IsGroupAsSemigroup</a>
 <a href="chap12.html#X835484C481CF3DDD">12.1-8 IsIdempotentGenerated</a>

 <a href="chap12.html#X8206D2B0809952EF">12.1-9 IsLeftSimple</a>
 <a href="chap12.html#X7E9261367C8C52C0">12.1-10 IsLeftZeroSemigroup</a>
 <a href="chap12.html#X79D46BAB7E327AD1">12.1-11 IsMonogenicSemigroup</a>
 <a href="chap12.html#X790DC9F4798DBB09">12.1-12 IsMonogenicMonoid</a>
 <a href="chap12.html#X7E4DEECD7CD9886D">12.1-13 IsMonoidAsSemigroup</a>
 <a href="chap12.html#X7935C476808C8773">12.1-14 IsOrthodoxSemigroup</a>
 <a href="chap12.html#X7E9B674D781B072C">12.1-15 IsRectangularBand</a>
 <a href="chap12.html#X80E682BB78547F41">12.1-16 IsRectangularGroup</a>
 <a href="chap12.html#X7C4663827C5ACEF1">12.1-17 IsRegularSemigroup</a>
 <a href="chap12.html#X7CB099958658F979">12.1-18 IsRightZeroSemigroup</a>
 <a href="chap12.html#X8752642C7F7E512B">12.1-19 IsXTrivial</a>

 <a href="chap12.html#X7826DDF8808EC4D9">12.1-20 IsSemigroupWithAdjoinedZero</a>
 <a href="chap12.html#X833D24AE7C900B85">12.1-21 IsSemilattice</a>
 <a href="chap12.html#X836F4692839F4874">12.1-22 IsSimpleSemigroup</a>

 <a href="chap12.html#X7EEC85187D315398">12.1-23 IsSynchronizingSemigroup</a>
 <a href="chap12.html#X80F9A4B87997839F">12.1-24 IsUnitRegularMonoid</a>
 <a href="chap12.html#X85F7E5CD86F0643B">12.1-25 IsZeroGroup</a>
 <a href="chap12.html#X7C6787D07B95B450">12.1-26 IsZeroRectangularBand</a>
 <a href="chap12.html#X81A1882181B75CC9">12.1-27 IsZeroSemigroup</a>
 <a href="chap12.html#X8193A60F839C064E">12.1-28 IsZeroSimpleSemigroup</a>
 <a href="chap12.html#X846FC6247EE31607">12.1-29 IsSelfDualSemigroup</a>
</div></div>
<div class="ContSect"> <a href="chap12.html#X80F2725581B166EE">12.2 
 Inverse semigroups
 </a>

<div class="ContSSBlock">
 <a href="chap12.html#X81DE11987BB81017">12.2-1 IsCliffordSemigroup</a>
 <a href="chap12.html#X7EFDBA687DCDA6FA">12.2-2 IsBrandtSemigroup</a>
 <a href="chap12.html#X843EA0E37C968BBF">12.2-3 IsEUnitaryInverseSemigroup</a>
 <a href="chap12.html#X86F942F48158DAC3">12.2-4 IsFInverseSemigroup</a>
 <a href="chap12.html#X864F1906858BB8CF">12.2-5 IsFInverseMonoid</a>
 <a href="chap12.html#X8440E22681BD1EE6">12.2-6 IsFactorisableInverseMonoid</a>
 <a href="chap12.html#X817F9F3984FC842C">12.2-7 IsJoinIrreducible</a>
 <a href="chap12.html#X81E6D24F852A7937">12.2-8 IsMajorantlyClosed</a>
 <a href="chap12.html#X7D2641AD830DEC1C">12.2-9 IsMonogenicInverseSemigroup</a>
 <a href="chap12.html#X7EDFA6CA86645DBE">12.2-10 IsMonogenicInverseMonoid</a>
</div></div>
</div>

<h3>12 
 Properties of semigroups
 </h3>

In this chapter we describe the methods that are available in Semigroups for determining various properties of a semigroup or monoid.

<a id="X7D297AEC827F3D4E" name="X7D297AEC827F3D4E"></a>

<h4>12.1 
 Arbitrary semigroups
 </h4>

In this section we describe the properties of an arbitrary semigroup or monoid that can be determined using the Semigroups package.

<a id="X7C8DB14587D1B55A" name="X7C8DB14587D1B55A"></a>

<h5>12.1-1 IsBand</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsBand</code>( <var class="Arg">S</var> )</td><td class="tdright">( property )</td></tr></table></div>
Returns: <code class="keyw">true</code> or <code class="keyw">false</code>.

<code class="code">IsBand</code> returns <code class="keyw">true</code> if every element of the semigroup <var class="Arg">S</var> is an idempotent and <code class="keyw">false</code> if it is not. An inverse semigroup is band if and only if it is a semilattice; see <code class="func">IsSemilattice</code> (<a href="chap12.html#X833D24AE7C900B85">12.1-21</a>).

<div class="example"><pre>
gap> S := Semigroup(
> Transformation([1, 1, 1, 4, 4, 4, 7, 7, 7, 1]),
> Transformation([2, 2, 2, 5, 5, 5, 8, 8, 8, 2]),
> Transformation([3, 3, 3, 6, 6, 6, 9, 9, 9, 3]),
> Transformation([1, 1, 1, 4, 4, 4, 7, 7, 7, 4]),
> Transformation([1, 1, 1, 4, 4, 4, 7, 7, 7, 7]));;
gap> IsBand(S);
true
gap> S := InverseSemigroup(
> PartialPerm([1, 2, 3, 4, 8, 9], [5, 8, 7, 6, 9, 1]),
> PartialPerm([1, 3, 4, 7, 8, 9, 10], [2, 3, 8, 7, 10, 6, 1]));;
gap> IsBand(S);
false
gap> IsBand(IdempotentGeneratedSubsemigroup(S));
true
gap> S := PartitionMonoid(4);
<regular bipartition *-monoid of size 4140, degree 4 with 4
generators>
gap> M := MinimalIdeal(S);
<simple bipartition *-semigroup ideal of degree 4 with 1 generator>
gap> IsBand(M);
true</pre></div>

<a id="X79659C467C8A7EBD" name="X79659C467C8A7EBD"></a>

<h5>12.1-2 IsBlockGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsBlockGroup</code>( <var class="Arg">S</var> )</td><td class="tdright">( property )</td></tr></table></div>
Returns: <code class="keyw">true</code> or <code class="keyw">false</code>.

<code class="code">IsBlockGroup</code> returns <code class="keyw">true</code> if the semigroup <var class="Arg">S</var> is a block group and <code class="keyw">false</code> if it is not.

A semigroup <var class="Arg">S</var> is a block group if every L-class and every R-class of <var class="Arg">S</var> contains at most one idempotent. Every semigroup of partial permutations is a block group.

<div class="example"><pre>
gap> S := Semigroup(Transformation([5, 6, 7, 3, 1, 4, 2, 8]),
> Transformation([3, 6, 8, 5, 7, 4, 2, 8]));;
gap> IsBlockGroup(S);
true
gap> S := Semigroup(
> Transformation([2, 1, 10, 4, 5, 9, 7, 4, 8, 4]),
> Transformation([10, 7, 5, 6, 1, 3, 9, 7, 10, 2]));;
gap> IsBlockGroup(S);
false
gap> S := Semigroup(PartialPerm([1, 2], [5, 4]),
> PartialPerm([1, 2, 3], [1, 2, 5]),
> PartialPerm([1, 2, 3], [2, 1, 5]),
> PartialPerm([1, 3, 4], [3, 1, 2]),
> PartialPerm([1, 3, 4, 5], [5, 4, 3, 2]));;
gap> T := AsSemigroup(IsBlockBijectionSemigroup, S);
<block bijection semigroup of degree 6 with 5 generators>
gap> IsBlockGroup(T);
true
gap> IsBlockGroup(AsSemigroup(IsBipartitionSemigroup, S));
true
gap> S := Semigroup(
> Bipartition([[1, -2], [2, -3], [3, -4], [4, -1]]),
> Bipartition([[1, -2], [2, -1], [3, -3], [4, -4]]),
> Bipartition([[1, 2, -3], [3, -1, -2], [4, -4]]),
> Bipartition([[1, -1], [2, -2], [3, -3], [4, -4]]));;
gap> IsBlockGroup(S);
true</pre></div>

<a id="X843EFDA4807FDC31" name="X843EFDA4807FDC31"></a>

<h5>12.1-3 IsCommutativeSemigroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsCommutativeSemigroup</code>( <var class="Arg">S</var> )</td><td class="tdright">( property )</td></tr></table></div>
Returns: <code class="keyw">true</code> or <code class="keyw">false</code>.

<code class="code">IsCommutativeSemigroup</code> returns <code class="keyw">true</code> if the semigroup <var class="Arg">S</var> is commutative and <code class="keyw">false</code> if it is not. The function <code class="func">IsCommutative</code> (<a href="../../../doc/ref/chap35_mj.html#X830A4A4C795FBC2D">Reference: IsCommutative</a>) can also be used to test if a semigroup is commutative.

A semigroup <var class="Arg">S</var> is commutative if <code class="code">x * y = y * x</code> for all <code class="code">x, y</code> in <var class="Arg">S</var>.

<div class="example"><pre>
gap> S := Semigroup(Transformation([2, 4, 5, 3, 7, 8, 6, 9, 1]),
> Transformation([3, 5, 6, 7, 8, 1, 9, 2, 4]));;
gap> IsCommutativeSemigroup(S);
true
gap> IsCommutative(S);
true
gap> S := InverseSemigroup(
> PartialPerm([1, 2, 3, 4, 5, 6], [2, 5, 1, 3, 9, 6]),
> PartialPerm([1, 2, 3, 4, 6, 8], [8, 5, 7, 6, 2, 1]));;
gap> IsCommutativeSemigroup(S);
false
gap> S := Semigroup(
> Bipartition([[1, 2, 3, 6, 7, -1, -4, -6],
> [4, 5, 8, -2, -3, -5, -7, -8]]),
> Bipartition([[1, 2, -3, -4], [3, -5], [4, -6], [5, -7],
> [6, -8], [7, -1], [8, -2]]));;
gap> IsCommutativeSemigroup(S);
true</pre></div>

<a id="X7AFA23AF819FBF3D" name="X7AFA23AF819FBF3D"></a>

<h5>12.1-4 IsCompletelyRegularSemigroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsCompletelyRegularSemigroup</code>( <var class="Arg">S</var> )</td><td class="tdright">( property )</td></tr></table></div>
Returns: <code class="keyw">true</code> or <code class="keyw">false</code>.

<code class="code">IsCompletelyRegularSemigroup</code> returns <code class="keyw">true</code> if every element of the semigroup <var class="Arg">S</var> is contained in a subgroup of <var class="Arg">S</var>.

An inverse semigroup is completely regular if and only if it is a Clifford semigroup; see <code class="func">IsCliffordSemigroup</code> (<a href="chap12.html#X81DE11987BB81017">12.2-1</a>).

<div class="example"><pre>
gap> S := Semigroup(Transformation([1, 2, 4, 3, 6, 5, 4]),
> Transformation([1, 2, 5, 6, 3, 4, 5]),
> Transformation([2, 1, 2, 2, 2, 2, 2]));;
gap> IsCompletelyRegularSemigroup(S);
true
gap> IsInverseSemigroup(S);
true
gap> T := Range(IsomorphismPartialPermSemigroup(S));;
gap> IsCompletelyRegularSemigroup(T);
true
gap> IsCliffordSemigroup(T);
true
gap> S := Semigroup(
> Bipartition([[1, 3, -4], [2, 4, -1, -2], [-3]]),
> Bipartition([[1, -1], [2, 3, 4, -3], [-2, -4]]));;
gap> IsCompletelyRegularSemigroup(S);
false</pre></div>

<a id="X855088F378D8F5E1" name="X855088F378D8F5E1"></a>

<h5>12.1-5 IsCongruenceFreeSemigroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsCongruenceFreeSemigroup</code>( <var class="Arg">S</var> )</td><td class="tdright">( property )</td></tr></table></div>
Returns: <code class="keyw">true</code> or <code class="keyw">false</code>.

<code class="code">IsCongruenceFreeSemigroup</code> returns <code class="keyw">true</code> if the semigroup <var class="Arg">S</var> is a congruence-free semigroup and <code class="keyw">false</code> if it is not.

A semigroup <var class="Arg">S</var> is congruence-free if it has no non-trivial proper congruences.

A semigroup with zero is congruence-free if and only if it is isomorphic to a regular Rees 0-matrix semigroup <code class="code">R</code> whose underlying semigroup is the trivial group, no two rows of the matrix of <code class="code">R</code> are identical, and no two columns are identical; see Theorem 3.7.1 in <a href="chapBib.html#biBHowie1995aa">[How95]</a>.

A semigroup without zero is congruence-free if and only if it is a simple group or has order 2; see Theorem 3.7.2 in <a href="chapBib.html#biBHowie1995aa">[How95]</a>.

<div class="example"><pre>
gap> S := Semigroup(Transformation([4, 2, 3, 3, 4]));;
gap> IsCongruenceFreeSemigroup(S);
true
gap> S := Semigroup(Transformation([2, 2, 4, 4]),
> Transformation([5, 3, 4, 4, 6, 6]));;
gap> IsCongruenceFreeSemigroup(S);
false</pre></div>

<a id="X7C9560A18348AEE3" name="X7C9560A18348AEE3"></a>

<h5>12.1-6 IsSurjectiveSemigroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsSurjectiveSemigroup</code>( <var class="Arg">S</var> )</td><td class="tdright">( property )</td></tr></table></div>
Returns: <code class="keyw">true</code> or <code class="keyw">false</code>.

A semigroup is surjective if each of its elements can be written as a product of two elements in the semigroup. <code class="code">IsSurjectiveSemigroup(<var class="Arg">S</var>)</code> returns <code class="keyw">true</code> if the semigroup <var class="Arg">S</var> is surjective, and <code class="keyw">false</code> if it is not.

See also <code class="func">IndecomposableElements</code> (<a href="chap11.html#X7B4CD8937858A895">11.7-6</a>).

Note that every monoid, and every regular semigroup, is surjective.

<div class="example"><pre>
gap> S := FullTransformationMonoid(100);
<full transformation monoid of degree 100>
gap> IsSurjectiveSemigroup(S);
true
gap> F := FreeSemigroup(3);;
gap> P := F / [[F.1, F.2 * F.1], [F.3 ^ 3, F.3]];
<fp semigroup with 3 generators and 2 relations of length 10>
gap> IsSurjectiveSemigroup(P);
false
gap> I := SingularTransformationMonoid(5);
<regular transformation semigroup ideal of degree 5 with 1 generator>
gap> IsSurjectiveSemigroup(I);
true
gap> M := MonogenicSemigroup(IsBipartitionSemigroup, 3, 2);
<commutative non-regular block bijection semigroup of size 4,
degree 6 with 1 generator>
gap> IsSurjectiveSemigroup(M);
false</pre></div>

<a id="X852F29E8795FA489" name="X852F29E8795FA489"></a>

<h5>12.1-7 IsGroupAsSemigroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsGroupAsSemigroup</code>( <var class="Arg">S</var> )</td><td class="tdright">( property )</td></tr></table></div>
Returns: <code class="keyw">true</code> or <code class="keyw">false</code>.

<code class="code">IsGroupAsSemigroup</code> returns <code class="keyw">true</code> if and only if the semigroup <var class="Arg">S</var> is mathematically a group.

<div class="example"><pre>
gap> S := Semigroup(Transformation([2, 4, 5, 3, 7, 8, 6, 9, 1]),
> Transformation([3, 5, 6, 7, 8, 1, 9, 2, 4]));;
gap> IsGroupAsSemigroup(S);
true
gap> G := SymmetricGroup(5);;
gap> IsGroupAsSemigroup(G);
true
gap> S := AsSemigroup(IsPartialPermSemigroup, G);
<partial perm group of size 120, rank 5 with 2 generators>
gap> IsGroupAsSemigroup(S);
true
gap> G := SymmetricGroup([1, 2, 10]);;
gap> T := AsSemigroup(IsBlockBijectionSemigroup, G);
<inverse block bijection semigroup of size 6, degree 11 with 2
generators>
gap> IsGroupAsSemigroup(T);
true</pre></div>

<a id="X835484C481CF3DDD" name="X835484C481CF3DDD"></a>

<h5>12.1-8 IsIdempotentGenerated</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsIdempotentGenerated</code>( <var class="Arg">S</var> )</td><td class="tdright">( property )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsSemiband</code>( <var class="Arg">S</var> )</td><td class="tdright">( property )</td></tr></table></div>
Returns: <code class="keyw">true</code> or <code class="keyw">false</code>.

<code class="code">IsIdempotentGenerated</code> and <code class="code">IsSemiband</code> return <code class="keyw">true</code> if the semigroup <var class="Arg">S</var> is generated by its idempotents and <code class="keyw">false</code> if it is not. See also <code class="func">Idempotents</code> (<a href="chap11.html#X7C651C9C78398FFF">11.10-1</a>) and <code class="func">IdempotentGeneratedSubsemigroup</code> (<a href="chap11.html#X83970D028143B79B">11.10-3</a>).

An inverse semigroup is idempotent-generated if and only if it is a semilattice; see <code class="func">IsSemilattice</code> (<a href="chap12.html#X833D24AE7C900B85">12.1-21</a>).

The terms semiband and idempotent-generated are synonymous in this context.

<div class="example"><pre>
gap> S := SingularTransformationSemigroup(4);
<regular transformation semigroup ideal of degree 4 with 1 generator>
gap> IsIdempotentGenerated(S);
true
gap> S := SingularBrauerMonoid(5);
<regular bipartition *-semigroup ideal of degree 5 with 1 generator>
gap> IsIdempotentGenerated(S);
true</pre></div>

<a id="X8206D2B0809952EF" name="X8206D2B0809952EF"></a>

<h5>12.1-9 IsLeftSimple</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsLeftSimple</code>( <var class="Arg">S</var> )</td><td class="tdright">( property )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsRightSimple</code>( <var class="Arg">S</var> )</td><td class="tdright">( property )</td></tr></table></div>
Returns: <code class="keyw">true</code> or <code class="keyw">false</code>.

<code class="code">IsLeftSimple</code> and <code class="code">IsRightSimple</code> returns <code class="keyw">true</code> if the semigroup <var class="Arg">S</var> has only one L-class or one R-class, respectively, and returns <code class="keyw">false</code> if it has more than one.

An inverse semigroup is left simple if and only if it is right simple if and only if it is a group; see <code class="func">IsGroupAsSemigroup</code> (<a href="chap12.html#X852F29E8795FA489">12.1-7</a>).

<div class="example"><pre>
gap> S := Semigroup(Transformation([6, 7, 9, 6, 8, 9, 8, 7, 6]),
> Transformation([6, 8, 9, 6, 8, 8, 7, 9, 6]),
> Transformation([6, 8, 9, 7, 8, 8, 7, 9, 6]),
> Transformation([6, 9, 8, 6, 7, 9, 7, 8, 6]),
> Transformation([6, 9, 9, 6, 8, 8, 7, 9, 6]),
> Transformation([6, 9, 9, 7, 8, 8, 6, 9, 7]),
> Transformation([7, 8, 8, 7, 9, 9, 7, 8, 6]),
> Transformation([7, 9, 9, 7, 6, 9, 6, 8, 7]),
> Transformation([8, 7, 6, 9, 8, 6, 8, 7, 9]),
> Transformation([9, 6, 6, 7, 8, 8, 7, 6, 9]),
> Transformation([9, 6, 6, 7, 9, 6, 9, 8, 7]),
> Transformation([9, 6, 7, 9, 6, 6, 9, 7, 8]),
> Transformation([9, 6, 8, 7, 9, 6, 9, 8, 7]),
> Transformation([9, 7, 6, 8, 7, 7, 9, 6, 8]),
> Transformation([9, 7, 7, 8, 9, 6, 9, 7, 8]),
> Transformation([9, 8, 8, 9, 6, 7, 6, 8, 9]));;
gap> IsRightSimple(S);
false
gap> IsLeftSimple(S);
true
gap> IsGroupAsSemigroup(S);
false
gap> NrRClasses(S);
16
gap> S := BrauerMonoid(6);;
gap> S := Semigroup(RClass(S, Random(MinimalDClass(S))));;
gap> IsLeftSimple(S);
false
gap> IsRightSimple(S);
true</pre></div>

<a id="X7E9261367C8C52C0" name="X7E9261367C8C52C0"></a>

<h5>12.1-10 IsLeftZeroSemigroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsLeftZeroSemigroup</code>( <var class="Arg">S</var> )</td><td class="tdright">( property )</td></tr></table></div>
Returns: <code class="keyw">true</code> or <code class="keyw">false</code>.

<code class="code">IsLeftZeroSemigroup</code> returns <code class="keyw">true</code> if the semigroup <var class="Arg">S</var> is a left zero semigroup and <code class="keyw">false</code> if it is not.

A semigroup is a left zero semigroup if <code class="code">x*y=x</code> for all <code class="code">x,y</code>. An inverse semigroup is a left zero semigroup if and only if it is trivial.

<div class="example"><pre>
gap> S := Semigroup(Transformation([2, 1, 4, 3, 5]),
> Transformation([3, 2, 3, 1, 1]));;
gap> IsRightZeroSemigroup(S);
false
gap> S := Semigroup(Transformation([1, 2, 3, 3, 1]),
> Transformation([1, 2, 3, 3, 3]));;
gap> IsLeftZeroSemigroup(S);
true</pre></div>

<a id="X79D46BAB7E327AD1" name="X79D46BAB7E327AD1"></a>

<h5>12.1-11 IsMonogenicSemigroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsMonogenicSemigroup</code>( <var class="Arg">S</var> )</td><td class="tdright">( property )</td></tr></table></div>
Returns: <code class="keyw">true</code> or <code class="keyw">false</code>.

<code class="code">IsMonogenicSemigroup</code> returns <code class="keyw">true</code> if the semigroup <var class="Arg">S</var> is monogenic and it returns <code class="keyw">false</code> if it is not.

A semigroup is monogenic if it is generated by a single element. See also <code class="func">IsMonogenicMonoid</code> (<a href="chap12.html#X790DC9F4798DBB09">12.1-12</a>), <code class="func">IsMonogenicInverseSemigroup</code> (<a href="chap12.html#X7D2641AD830DEC1C">12.2-9</a>), and <code class="func">IsMonogenicInverseMonoid</code> (<a href="chap12.html#X7EDFA6CA86645DBE">12.2-10</a>).

<div class="example"><pre>
gap> S := Semigroup(
> Transformation(
> [2, 2, 2, 11, 10, 8, 10, 11, 2, 11, 10, 2, 11, 11, 10]),
> Transformation(
> [2, 2, 2, 8, 11, 15, 11, 10, 2, 10, 11, 2, 10, 4, 7]),
> Transformation(
> [2, 2, 2, 11, 10, 8, 10, 11, 2, 11, 10, 2, 11, 11, 10]),
> Transformation(
> [2, 2, 12, 7, 8, 14, 8, 11, 2, 11, 10, 2, 11, 15, 4]));;
gap> IsMonogenicSemigroup(S);
true
gap> S := Semigroup(
> Bipartition([[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, -2, -5, -7, -9],
> [-1, -10], [-3, -4, -6, -8]]),
> Bipartition([[1, 4, 7, 8, -2], [2, 3, 5, 10, -5],
> [6, 9, -7, -9], [-1, -10], [-3, -4, -6, -8]]));;
gap> IsMonogenicSemigroup(S);
true
gap> S := FullTransformationSemigroup(5);;
gap> IsMonogenicSemigroup(S);
false</pre></div>

<a id="X790DC9F4798DBB09" name="X790DC9F4798DBB09"></a>

<h5>12.1-12 IsMonogenicMonoid</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsMonogenicMonoid</code>( <var class="Arg">S</var> )</td><td class="tdright">( property )</td></tr></table></div>
Returns: <code class="keyw">true</code> or <code class="keyw">false</code>.

<code class="code">IsMonogenicMonoid</code> returns <code class="keyw">true</code> if the monoid <var class="Arg">S</var> is a monogenic monoid and it returns <code class="keyw">false</code> if it is not.

A monoid is monogenic if it is generated as a monoid by a single element. See also <code class="func">IsMonogenicSemigroup</code> (<a href="chap12.html#X79D46BAB7E327AD1">12.1-11</a>) and <code class="func">IsMonogenicInverseMonoid</code> (<a href="chap12.html#X7EDFA6CA86645DBE">12.2-10</a>).

<div class="example"><pre>
gap> x := PartialPerm([1, 2, 3, 6, 8, 10], [2, 6, 7, 9, 1, 5]);;
gap> S := Monoid(x, x ^ 2, x ^ 3);;
gap> IsMonogenicSemigroup(S);
false
gap> IsMonogenicMonoid(S);
true
gap> S := FullTransformationMonoid(5);;
gap> IsMonogenicMonoid(S);
false</pre></div>

<a id="X7E4DEECD7CD9886D" name="X7E4DEECD7CD9886D"></a>

<h5>12.1-13 IsMonoidAsSemigroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsMonoidAsSemigroup</code>( <var class="Arg">S</var> )</td><td class="tdright">( property )</td></tr></table></div>
Returns: <code class="keyw">true</code> or <code class="keyw">false</code>.

<code class="code">IsMonoidAsSemigroup</code> returns <code class="keyw">true</code> if and only if the semigroup <var class="Arg">S</var> is mathematically a monoid, i.e. if and only if it contains a <code class="func">MultiplicativeNeutralElement</code> (<a href="../../../doc/ref/chap35_mj.html#X7EE2EA5F7EB7FEC2">Reference: MultiplicativeNeutralElement</a>).

It is possible that a semigroup which satisfies <code class="code">IsMonoidAsSemigroup</code> is not in the GAP category <code class="func">IsMonoid</code> (<a href="../../../doc/ref/chap51_mj.html#X861C523483C6248C">Reference: IsMonoid</a>). This is possible if the <code class="func">MultiplicativeNeutralElement</code> (<a href="../../../doc/ref/chap35_mj.html#X7EE2EA5F7EB7FEC2">Reference: MultiplicativeNeutralElement</a>) of <var class="Arg">S</var> is not equal to the <code class="func">One</code> (<a href="../../../doc/ref/chap31_mj.html#X8046262384895B2A">Reference: One</a>) of any element in <var class="Arg">S</var>. Therefore a semigroup satisfying <code class="code">IsMonoidAsSemigroup</code> may not possess the attributes of a monoid (such as, <code class="func">GeneratorsOfMonoid</code> (<a href="../../../doc/ref/chap51_mj.html#X83CA2E7279C44718">Reference: GeneratorsOfMonoid</a>)).

See also <code class="func">One</code> (<a href="../../../doc/ref/chap31_mj.html#X8046262384895B2A">Reference: One</a>), <code class="func">IsInverseMonoid</code> (<a href="../../../doc/ref/chap51_mj.html#X83F1529479D56665">Reference: IsInverseMonoid</a>) and <code class="func">IsomorphismTransformationMonoid</code> (<a href="../../../doc/ref/chap53_mj.html#X78F29C817CF6827F">Reference: IsomorphismTransformationMonoid</a>).

<div class="example"><pre>
gap> S := Semigroup(Transformation([1, 4, 6, 2, 5, 3, 7, 8, 9, 9]),
> Transformation([6, 3, 2, 7, 5, 1, 8, 8, 9, 9]));;
gap> IsMonoidAsSemigroup(S);
true
gap> IsMonoid(S);
false
gap> MultiplicativeNeutralElement(S);
Transformation( [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 9 ] )
gap> T := AsSemigroup(IsBipartitionSemigroup, S);;
gap> IsMonoidAsSemigroup(T);
true
gap> IsMonoid(T);
false
gap> One(T);
fail
gap> S := Monoid(Transformation([8, 2, 8, 9, 10, 6, 2, 8, 7, 8]),
> Transformation([9, 2, 6, 3, 6, 4, 5, 5, 3, 2]));;
gap> IsMonoidAsSemigroup(S);
true</pre></div>

<a id="X7935C476808C8773" name="X7935C476808C8773"></a>

<h5>12.1-14 IsOrthodoxSemigroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsOrthodoxSemigroup</code>( <var class="Arg">S</var> )</td><td class="tdright">( property )</td></tr></table></div>
Returns: <code class="keyw">true</code> or <code class="keyw">false</code>.

<code class="code">IsOrthodoxSemigroup</code> returns <code class="keyw">true</code> if the semigroup <var class="Arg">S</var> is orthodox and <code class="keyw">false</code> if it is not.

A semigroup is orthodox if it is regular and its idempotent elements form a subsemigroup. Every inverse semigroup is also an orthodox semigroup.

See also <code class="func">IsRegularSemigroup</code> (<a href="chap12.html#X7C4663827C5ACEF1">12.1-17</a>) and <code class="func">IsRegularSemigroup</code> (<a href="../../../doc/ref/chap51_mj.html#X7C4663827C5ACEF1">Reference: IsRegularSemigroup</a>).

<div class="example"><pre>
gap> S := Semigroup(Transformation([1, 1, 1, 4, 5, 4]),
> Transformation([1, 2, 3, 1, 1, 2]),
> Transformation([1, 2, 3, 1, 1, 3]),
> Transformation([5, 5, 5, 5, 5, 5]));;
gap> IsOrthodoxSemigroup(S);
true
gap> S := DualSymmetricInverseMonoid(5);;
gap> S := Semigroup(GeneratorsOfSemigroup(S));;
gap> IsOrthodoxSemigroup(S);
true</pre></div>

<a id="X7E9B674D781B072C" name="X7E9B674D781B072C"></a>

<h5>12.1-15 IsRectangularBand</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsRectangularBand</code>( <var class="Arg">S</var> )</td><td class="tdright">( property )</td></tr></table></div>
Returns: <code class="keyw">true</code> or <code class="keyw">false</code>.

<code class="code">IsRectangularBand</code> returns <code class="keyw">true</code> if the semigroup <var class="Arg">S</var> is a rectangular band and <code class="keyw">false</code> if it is not.

A semigroup <var class="Arg">S</var> is a rectangular band if for all x, y, z in <var class="Arg">S</var> we have that x ^ 2 = x and xyz = xz.

Equivalently, <var class="Arg">S</var> is a rectangular band if <var class="Arg">S</var> is isomorphic to a semigroup of the form I × Λ with multiplication (i, λ)(j, μ) = (i, μ). In this case, <var class="Arg">S</var> is called an |I| × |Λ| rectangular band.

An inverse semigroup is a rectangular band if and only if it is a group.

<div class="example"><pre>
gap> S := Semigroup(
> Transformation([1, 1, 1, 4, 4, 4, 7, 7, 7, 1]),
> Transformation([2, 2, 2, 5, 5, 5, 8, 8, 8, 2]),
> Transformation([3, 3, 3, 6, 6, 6, 9, 9, 9, 3]),
> Transformation([1, 1, 1, 4, 4, 4, 7, 7, 7, 4]),
> Transformation([1, 1, 1, 4, 4, 4, 7, 7, 7, 7]));;
gap> IsRectangularBand(S);
true
gap> IsRectangularBand(MinimalIdeal(PartitionMonoid(4)));
true</pre></div>

<a id="X80E682BB78547F41" name="X80E682BB78547F41"></a>

<h5>12.1-16 IsRectangularGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsRectangularGroup</code>( <var class="Arg">S</var> )</td><td class="tdright">( property )</td></tr></table></div>
Returns: <code class="keyw">true</code> or <code class="keyw">false</code>.

A semigroup is rectangular group if it is the direct product of a group and a rectangular band. Or equivalently, if it is orthodox and simple.

<div class="example"><pre>
gap> G := AsSemigroup(IsTransformationSemigroup, MathieuGroup(11));
<transformation group of size 7920, degree 11 with 2 generators>
gap> R := RectangularBand(3, 2);
<regular transformation semigroup of size 6, degree 6 with 3
generators>
gap> S := DirectProduct(G, R);;
gap> IsRectangularGroup(R);
true
gap> IsRectangularGroup(G);
true
gap> IsRectangularGroup(S);
true
gap> IsRectangularGroup(JonesMonoid(3));
false</pre></div>

<a id="X7C4663827C5ACEF1" name="X7C4663827C5ACEF1"></a>

<h5>12.1-17 IsRegularSemigroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsRegularSemigroup</code>( <var class="Arg">S</var> )</td><td class="tdright">( property )</td></tr></table></div>
Returns: <code class="keyw">true</code> or <code class="keyw">false</code>.

<code class="code">IsRegularSemigroup</code> returns <code class="keyw">true</code> if the semigroup <var class="Arg">S</var> is regular and <code class="keyw">false</code> if it is not.

A semigroup <code class="code">S</code> is regular if for all <code class="code">x</code> in <code class="code">S</code> there exists <code class="code">y</code> in <code class="code">S</code> such that <code class="code">x * y * x = x</code>. Every inverse semigroup is regular, and a semigroup of partial permutations is regular if and only if it is an inverse semigroup.

See also <code class="func">IsRegularDClass</code> (<a href="../../../doc/ref/chap51_mj.html#X7F5860927CAD920F">Reference: IsRegularDClass</a>), <code class="func">IsRegularGreensClass</code> (<a href="chap10.html#X859DD1C079C80DCC">10.3-2</a>), and <code class="func">IsRegularSemigroupElement</code> (<a href="../../../doc/ref/chap51_mj.html#X87532A76854347E0">Reference: IsRegularSemigroupElement</a>).

<div class="example"><pre>
gap> IsRegularSemigroup(FullTransformationSemigroup(5));
true
gap> IsRegularSemigroup(JonesMonoid(5));
true</pre></div>

<a id="X7CB099958658F979" name="X7CB099958658F979"></a>

<h5>12.1-18 IsRightZeroSemigroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsRightZeroSemigroup</code>( <var class="Arg">S</var> )</td><td class="tdright">( property )</td></tr></table></div>
Returns: <code class="keyw">true</code> or <code class="keyw">false</code>.

<code class="code">IsRightZeroSemigroup</code> returns <code class="keyw">true</code> if the <var class="Arg">S</var> is a right zero semigroup and <code class="keyw">false</code> if it is not.

A semigroup <code class="code">S</code> is a right zero semigroup if <code class="code">x * y = y</code> for all <code class="code">x, y</code> in <code class="code">S</code>. An inverse semigroup is a right zero semigroup if and only if it is trivial.

<div class="example"><pre>
gap> S := Semigroup(Transformation([2, 1, 4, 3, 5]),
> Transformation([3, 2, 3, 1, 1]));;
gap> IsRightZeroSemigroup(S);
false
gap> S := Semigroup(Transformation([1, 2, 3, 3, 1]),
> Transformation([1, 2, 4, 4, 1]));;
gap> IsRightZeroSemigroup(S);
true</pre></div>

<a id="X8752642C7F7E512B" name="X8752642C7F7E512B"></a>

<h5>12.1-19 IsXTrivial</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsRTrivial</code>( <var class="Arg">S</var> )</td><td class="tdright">( property )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsLTrivial</code>( <var class="Arg">S</var> )</td><td class="tdright">( property )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsHTrivial</code>( <var class="Arg">S</var> )</td><td class="tdright">( property )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsDTrivial</code>( <var class="Arg">S</var> )</td><td class="tdright">( property )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsAperiodicSemigroup</code>( <var class="Arg">S</var> )</td><td class="tdright">( property )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsCombinatorialSemigroup</code>( <var class="Arg">S</var> )</td><td class="tdright">( property )</td></tr></table></div>
Returns: <code class="keyw">true</code> or <code class="keyw">false</code>.

<code class="code">IsXTrivial</code> returns <code class="keyw">true</code> if Green's R-relation, L-relation, H-relation, D-relation, respectively, on the semigroup S is trivial and false if it is not. These properties can also be applied to a Green's class instead of a semigroup where applicable.

For inverse semigroups, the properties of being R-trivial, L-trivial, D-trivial, and a semilattice are equivalent; see <code class="func">IsSemilattice</code> (<a href="chap12.html#X833D24AE7C900B85">12.1-21</a>).

A semigroup is aperiodic if its contains no non-trivial subgroups (equivalently, all of its group H-classes are trivial). A finite semigroup is aperiodic if and only if it is H-trivial.

Combinatorial is a synonym for aperiodic in this context.

<div class="example"><pre>
gap> S := Semigroup(
> Transformation([1, 5, 1, 3, 7, 10, 6, 2, 7, 10]),
> Transformation([4, 4, 5, 6, 7, 7, 7, 4, 3, 10]));;
gap> IsHTrivial(S);
true
gap> Size(S);
108
gap> IsRTrivial(S);
false
gap> IsLTrivial(S);
false</pre></div>

<a id="X7826DDF8808EC4D9" name="X7826DDF8808EC4D9"></a>

<h5>12.1-20 IsSemigroupWithAdjoinedZero</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsSemigroupWithAdjoinedZero</code>( <var class="Arg">S</var> )</td><td class="tdright">( property )</td></tr></table></div>
Returns: <code class="keyw">true</code> or <code class="keyw">false</code>.

<code class="code">IsSemigroupWithAdjoinedZero</code> returns <code class="keyw">true</code> if the semigroup <var class="Arg">S</var> can be expressed as the disjoint union of subsemigroups <var class="Arg">S</var> ∖ { 0 } and { 0 } (where 0 is the <code class="func">MultiplicativeZero</code> (<a href="chap11.html#X7B39F93C8136D642">11.8-3</a>) of <var class="Arg">S</var>).

If this is not the case, then either <var class="Arg">S</var> lacks a multiplicative zero, or the set <var class="Arg">S</var> ∖ { 0 } is not a subsemigroup of <var class="Arg">S</var>, and so <code class="code">IsSemigroupWithAdjoinedZero</code> returns <code class="keyw">false</code>.

<div class="example"><pre>
gap> S := Semigroup(Transformation([2, 3, 4, 5, 1, 6]),
> Transformation([2, 1, 3, 4, 5, 6]),
> Transformation([6, 6, 6, 6, 6, 6]));
<transformation semigroup of degree 6 with 3 generators>
gap> IsZeroGroup(S);
true
gap> IsSemigroupWithAdjoinedZero(S);
true
gap> S := FullTransformationMonoid(4);;
gap> IsSemigroupWithAdjoinedZero(S);
false</pre></div>

<a id="X833D24AE7C900B85" name="X833D24AE7C900B85"></a>

<h5>12.1-21 IsSemilattice</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsSemilattice</code>( <var class="Arg">S</var> )</td><td class="tdright">( property )</td></tr></table></div>
Returns: <code class="keyw">true</code> or <code class="keyw">false</code>.

<code class="code">IsSemilattice</code> returns <code class="keyw">true</code> if the semigroup <var class="Arg">S</var> is a semilattice and <code class="keyw">false</code> if it is not.

A semigroup is a semilattice if it is commutative and every element is an idempotent. The idempotents of an inverse semigroup form a semilattice.

<div class="example"><pre>
gap> S := Semigroup(Transformation([2, 5, 1, 7, 3, 7, 7]),
> Transformation([3, 6, 5, 7, 2, 1, 7]));;
gap> Size(S);
631
gap> IsInverseSemigroup(S);
true
gap> A := Semigroup(Idempotents(S));
<transformation semigroup of degree 7 with 32 generators>
gap> IsSemilattice(A);
true
gap> S := FactorisableDualSymmetricInverseMonoid(5);;
gap> S := IdempotentGeneratedSubsemigroup(S);;
gap> IsSemilattice(S);
true</pre></div>

<a id="X836F4692839F4874" name="X836F4692839F4874"></a>

<h5>12.1-22 IsSimpleSemigroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsSimpleSemigroup</code>( <var class="Arg">S</var> )</td><td class="tdright">( property )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsCompletelySimpleSemigroup</code>( <var class="Arg">S</var> )</td><td class="tdright">( property )</td></tr></table></div>
Returns: <code class="keyw">true</code> or <code class="keyw">false</code>.

<code class="code">IsSimpleSemigroup</code> returns <code class="keyw">true</code> if the semigroup <var class="Arg">S</var> is simple and <code class="keyw">false</code> if it is not.

A semigroup is simple if it has no proper 2-sided ideals. A semigroup is completely simple if it is simple and possesses minimal left and right ideals. A finite semigroup is simple if and only if it is completely simple. An inverse semigroup is simple if and only if it is a group.

<div class="example"><pre>
gap> S := Semigroup(
> Transformation([2, 2, 4, 4, 6, 6, 8, 8, 10, 10, 12, 12, 2]),
> Transformation([1, 1, 3, 3, 5, 5, 7, 7, 9, 9, 11, 11, 3]),
> Transformation([1, 7, 3, 9, 5, 11, 7, 1, 9, 3, 11, 5, 5]),
> Transformation([7, 7, 9, 9, 11, 11, 1, 1, 3, 3, 5, 5, 7]));;
gap> IsSimpleSemigroup(S);
true
gap> IsCompletelySimpleSemigroup(S);
true
gap> IsSimpleSemigroup(MinimalIdeal(BrauerMonoid(6)));
true
gap> R := Range(IsomorphismReesMatrixSemigroup(
> MinimalIdeal(BrauerMonoid(6))));
<Rees matrix semigroup 15x15 over Group(())></pre></div>

<a id="X7EEC85187D315398" name="X7EEC85187D315398"></a>

<h5>12.1-23 IsSynchronizingSemigroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsSynchronizingSemigroup</code>( <var class="Arg">S</var> )</td><td class="tdright">( property )</td></tr></table></div>
Returns: <code class="keyw">true</code> or <code class="keyw">false</code>.

For a positive integer <var class="Arg">n</var>, <code class="code">IsSynchronizingSemigroup</code> returns <code class="keyw">true</code> if the semigroup of transformations <var class="Arg">S</var> contains a transformation with constant value on <code class="code">[1 .. <var class="Arg">n</var>]</code> where <code class="code">n</code> is the degree of the semigroup. See also <code class="func">ConstantTransformation</code> (<a href="../../../doc/ref/chap53_mj.html#X7F1E4B5184210D2B">Reference: ConstantTransformation</a>).

Note that the semigroup consisting of the identity transformation is the unique transformation semigroup with degree <code class="code">0</code>. In this special case, the function <code class="code">IsSynchronizingSemigroup</code> will return <code class="keyw">false</code>.

<div class="example"><pre>
gap> S := Semigroup(
> Transformation([1, 1, 8, 7, 6, 6, 4, 1, 8, 9]),
> Transformation([5, 8, 7, 6, 10, 8, 7, 6, 9, 7]));;
gap> IsSynchronizingSemigroup(S);
true
gap> S := Semigroup(
> Transformation([3, 8, 1, 1, 9, 9, 8, 7, 9, 6]),
> Transformation([7, 6, 8, 7, 5, 6, 8, 7, 8, 9]));;
gap> IsSynchronizingSemigroup(S);
false
gap> Representative(MinimalIdeal(S));
Transformation( [ 8, 7, 7, 7, 8, 8, 7, 8, 8, 8 ] )</pre></div>

<a id="X80F9A4B87997839F" name="X80F9A4B87997839F"></a>

<h5>12.1-24 IsUnitRegularMonoid</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsUnitRegularMonoid</code>( <var class="Arg">S</var> )</td><td class="tdright">( property )</td></tr></table></div>
Returns: <code class="keyw">true</code> if the semigroup <var class="Arg">S</var> is unit regular and <code class="keyw">false</code> if it is not.

A monoid is unit regular if and only if for every <code class="code">>x</code> in <var class="Arg">S</var> there exists an element <code class="code">y</code> in the group of units of <var class="Arg">S</var> such that <code class="code">x*y*x=x</code>.

<div class="example"><pre>
gap> IsUnitRegularMonoid(FullTransformationMonoid(3));
true</pre></div>

<a id="X85F7E5CD86F0643B" name="X85F7E5CD86F0643B"></a>

<h5>12.1-25 IsZeroGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsZeroGroup</code>( <var class="Arg">S</var> )</td><td class="tdright">( property )</td></tr></table></div>
Returns: <code class="keyw">true</code> or <code class="keyw">false</code>.

<code class="code">IsZeroGroup</code> returns <code class="keyw">true</code> if the semigroup <var class="Arg">S</var> is a zero group and <code class="keyw">false</code> if it is not.

A semigroup <code class="code">S</code> is a zero group if there exists an element <code class="code">z</code> in <code class="code">S</code> such that <code class="code">S</code> without <code class="code">z</code> is a group and <code class="code">x*z=z*x=z</code> for all <code class="code">x</code> in <code class="code">S</code>. Every zero group is an inverse semigroup.

<div class="example"><pre>
gap> S := Semigroup(Transformation([2, 2, 3, 4, 6, 8, 5, 5, 9]),
> Transformation([3, 3, 8, 2, 5, 6, 4, 4, 9]),
> ConstantTransformation(9, 9));;
gap> IsZeroGroup(S);
true
gap> T := Range(IsomorphismPartialPermSemigroup(S));;
gap> IsZeroGroup(T);
true
gap> IsZeroGroup(JonesMonoid(2));
true</pre></div>

<a id="X7C6787D07B95B450" name="X7C6787D07B95B450"></a>

<h5>12.1-26 IsZeroRectangularBand</h5>

--> --------------------

--> maximum size reached

--> --------------------

quality95%

¤ Dauer der Verarbeitung: 0.27 Sekunden (vorverarbeitet) ¤

Wurzel

Suchen

Beweissystem der NASA

Beweissystem Isabelle

NIST Cobol Testsuite

Cephes Mathematical Library

Wiener Entwicklungsmethode

Haftungshinweis

Die Informationen auf dieser Webseite wurden nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit, noch Qualität der bereit gestellten Informationen zugesichert.