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

<div class="ContSSBlock">
 <a href="chap12_mj.html#X7C8DB14587D1B55A">12.1-1 IsBand</a>
 <a href="chap12_mj.html#X79659C467C8A7EBD">12.1-2 IsBlockGroup</a>
 <a href="chap12_mj.html#X843EFDA4807FDC31">12.1-3 IsCommutativeSemigroup</a>
 <a href="chap12_mj.html#X7AFA23AF819FBF3D">12.1-4 IsCompletelyRegularSemigroup</a>
 <a href="chap12_mj.html#X855088F378D8F5E1">12.1-5 IsCongruenceFreeSemigroup</a>
 <a href="chap12_mj.html#X7C9560A18348AEE3">12.1-6 IsSurjectiveSemigroup</a>
 <a href="chap12_mj.html#X852F29E8795FA489">12.1-7 IsGroupAsSemigroup</a>
 <a href="chap12_mj.html#X835484C481CF3DDD">12.1-8 IsIdempotentGenerated</a>

 <a href="chap12_mj.html#X8206D2B0809952EF">12.1-9 IsLeftSimple</a>
 <a href="chap12_mj.html#X7E9261367C8C52C0">12.1-10 IsLeftZeroSemigroup</a>
 <a href="chap12_mj.html#X79D46BAB7E327AD1">12.1-11 IsMonogenicSemigroup</a>
 <a href="chap12_mj.html#X790DC9F4798DBB09">12.1-12 IsMonogenicMonoid</a>
 <a href="chap12_mj.html#X7E4DEECD7CD9886D">12.1-13 IsMonoidAsSemigroup</a>
 <a href="chap12_mj.html#X7935C476808C8773">12.1-14 IsOrthodoxSemigroup</a>
 <a href="chap12_mj.html#X7E9B674D781B072C">12.1-15 IsRectangularBand</a>
 <a href="chap12_mj.html#X80E682BB78547F41">12.1-16 IsRectangularGroup</a>
 <a href="chap12_mj.html#X7C4663827C5ACEF1">12.1-17 IsRegularSemigroup</a>
 <a href="chap12_mj.html#X7CB099958658F979">12.1-18 IsRightZeroSemigroup</a>
 <a href="chap12_mj.html#X8752642C7F7E512B">12.1-19 IsXTrivial</a>

 <a href="chap12_mj.html#X7826DDF8808EC4D9">12.1-20 IsSemigroupWithAdjoinedZero</a>
 <a href="chap12_mj.html#X833D24AE7C900B85">12.1-21 IsSemilattice</a>
 <a href="chap12_mj.html#X836F4692839F4874">12.1-22 IsSimpleSemigroup</a>

 <a href="chap12_mj.html#X7EEC85187D315398">12.1-23 IsSynchronizingSemigroup</a>
 <a href="chap12_mj.html#X80F9A4B87997839F">12.1-24 IsUnitRegularMonoid</a>
 <a href="chap12_mj.html#X85F7E5CD86F0643B">12.1-25 IsZeroGroup</a>
 <a href="chap12_mj.html#X7C6787D07B95B450">12.1-26 IsZeroRectangularBand</a>
 <a href="chap12_mj.html#X81A1882181B75CC9">12.1-27 IsZeroSemigroup</a>
 <a href="chap12_mj.html#X8193A60F839C064E">12.1-28 IsZeroSimpleSemigroup</a>
 <a href="chap12_mj.html#X846FC6247EE31607">12.1-29 IsSelfDualSemigroup</a>
</div></div>
<div class="ContSect"> <a href="chap12_mj.html#X80F2725581B166EE">12.2 
 Inverse semigroups
 </a>

<div class="ContSSBlock">
 <a href="chap12_mj.html#X81DE11987BB81017">12.2-1 IsCliffordSemigroup</a>
 <a href="chap12_mj.html#X7EFDBA687DCDA6FA">12.2-2 IsBrandtSemigroup</a>
 <a href="chap12_mj.html#X843EA0E37C968BBF">12.2-3 IsEUnitaryInverseSemigroup</a>
 <a href="chap12_mj.html#X86F942F48158DAC3">12.2-4 IsFInverseSemigroup</a>
 <a href="chap12_mj.html#X864F1906858BB8CF">12.2-5 IsFInverseMonoid</a>
 <a href="chap12_mj.html#X8440E22681BD1EE6">12.2-6 IsFactorisableInverseMonoid</a>
 <a href="chap12_mj.html#X817F9F3984FC842C">12.2-7 IsJoinIrreducible</a>
 <a href="chap12_mj.html#X81E6D24F852A7937">12.2-8 IsMajorantlyClosed</a>
 <a href="chap12_mj.html#X7D2641AD830DEC1C">12.2-9 IsMonogenicInverseSemigroup</a>
 <a href="chap12_mj.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_mj.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 \(\mathscr{L}\)-class and every \(\mathscr{R}\)-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_mj.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_mj.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_mj.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_mj.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_mj.html#X7C651C9C78398FFF">11.10-1</a>) and <code class="func">IdempotentGeneratedSubsemigroup</code> (<a href="chap11_mj.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_mj.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 \(\mathscr{L}\)-class or one \(\mathscr{R}\)-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_mj.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_mj.html#X790DC9F4798DBB09">12.1-12</a>), <code class="func">IsMonogenicInverseSemigroup</code> (<a href="chap12_mj.html#X7D2641AD830DEC1C">12.2-9</a>), and <code class="func">IsMonogenicInverseMonoid</code> (<a href="chap12_mj.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_mj.html#X79D46BAB7E327AD1">12.1-11</a>) and <code class="func">IsMonogenicInverseMonoid</code> (<a href="chap12_mj.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_mj.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 \(I \times \Lambda\) with multiplication \((i, \lambda)(j, \mu) = (i, \mu)\). In this case, <var class="Arg">S</var> is called an \(|I| \times |\Lambda|\) 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_mj.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 \(\mathscr{R}\)-relation, \(\mathscr{L}\)-relation, \(\mathscr{H}\)-relation, \(\mathscr{D}\)-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 \(\mathscr{R}\)-trivial, \(\mathscr{L}\)-trivial, \(\mathscr{D}\)-trivial, and a semilattice are equivalent; see <code class="func">IsSemilattice</code> (<a href="chap12_mj.html#X833D24AE7C900B85">12.1-21</a>).

A semigroup is aperiodic if its contains no non-trivial subgroups (equivalently, all of its group \(\mathscr{H}\)-classes are trivial). A finite semigroup is aperiodic if and only if it is \(\mathscr{H}\)-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 \(\textit{S} \setminus \{ 0 \}\) and \(\{ 0 \}\) (where \(0\) is the <code class="func">MultiplicativeZero</code> (<a href="chap11_mj.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 \(\textit{S} \setminus \{ 0 \}\) 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));;
--> --------------------

--> maximum size reached

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

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