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class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X79AFE7FC81C0DF3 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7A63356C85E45F6 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X83C6631B7D53AE0 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X87C50557821D478 </div></div> </div> <h3>4 <span class="Heading">Functionality</span></h3> <p><a id="X83F5AF81859D6CBF" name="X83F5AF81859D6CBF"></a></p> <h4>4.1 <span class="Heading">Individual Semigroups</span></h4> <p>The semigroups of sizes 1 to 8 are available up to isomorphism and anti-isomorphism in <strong c <p>In this section we give details about the functions relating to individual semigroups in <stro <p>If you are interested in the properties of a semigroup in the library or would like to find all th <p><a id="X8538248D78185960" name="X8538248D78185960"></a></p> <h5>4.1-1 SmallSemigroup</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Sm <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Sm <p>returns the semigroup with ID <span class="SimpleMath">\((\textit{m,n})\)</span> from the li <p>In <code class="code">SmallSemigroupNC</code> no check is performed to verify that <var class=" <p>In <code class="code">SmallSemigroup</code> an error is signalled if the semigroups of size <var <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">SmallSemigroup(8, 1353452);</span> <small semigroup of size 8> <span class="GAPprompt">gap></span> <span class="GAPinput">SmallSemigroupNC(5, 1);</span> <small semigroup of size 5> <span class="GAPprompt">gap></span> <span class="GAPinput">SmallSemigroupNC(5, 1) = SmallSemi true </pre></div> <p><a id="X857428A57D3FFACB" name="X857428A57D3FFACB"></a></p> <h5>4.1-2 IsSmallSemigroup</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <p>returns <code class="code">true</code> if <var class="Arg">sgrp</var> is a semigroup from the libr <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">sgrp := RandomSmallSemigroup(5);</ <small semigroup of size 5> <span class="GAPprompt">gap></span> <span class="GAPinput">IsSmallSemigroup(sgrp);</span> true <span class="GAPprompt">gap></span> <span class="GAPinput">sgrp := Semigroup(Transformation( <span class="GAPprompt">gap></span> <span class="GAPinput">IsSmallSemigroup(sgrp);</span> false </pre></div> <p><a id="X7EC241FB7985775E" name="X7EC241FB7985775E"></a></p> <h5>4.1-3 IsSmallSemigroupElt</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <p>returns <code class="code">true</code> if <var class="Arg">x</var> is an element of a semigroup fro <p><var class="Arg">IsSmallSemigroupElt</var> is a representation satisfying <var class="Arg">I <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">IsSmallSemigroupElt(Transformat false <span class="GAPprompt">gap></span> <span class="GAPinput">sgrp := RandomSmallSemigroup(5);;< <span class="GAPprompt">gap></span> <span class="GAPinput">IsSmallSemigroupElt(Random(sgrp true </pre></div> <p><a id="X793FF84E80B334D1" name="X793FF84E80B334D1"></a></p> <h5>4.1-4 RecoverMultiplicationTable</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Re <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Re <p>return the multiplication table of the <var class="Arg">n</var>-th semigroup with <var class="A <p>If <var class="Arg">m</var> is greater than 8 or <var class="Arg">n</var> greater than the number of s <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">RecoverMultiplicationTable(10, 2 fail <span class="GAPprompt">gap></span> <span class="GAPinput">RecoverMultiplicationTable(1, 2) fail <span class="GAPprompt">gap></span> <span class="GAPinput">RecoverMultiplicationTable(2, 1) [ [ 1, 1 ], [ 1, 1 ] ] <span class="GAPprompt">gap></span> <span class="GAPinput">RecoverMultiplicationTable(8, 11 [ [ 1, 1, 1, 1, 1, 1, 1, 1 ], [ 1, 1, 1, 1, 1, 1, 1, 3 ], [ 3, 3, 3, 3, 3, 3, 3, 3 ], [ 1, 1, 1, 4, 4, 4, 4, 1 ], [ 1, 2, 3, 4, 5, 6, 7, 1 ], [ 1, 2, 3, 4, 5, 6, 7, 1 ], [ 1, 2, 3, 4, 5, 6, 7, 1 ], [ 8, 8, 8, 8, 8, 8, 8, 8 ] ] <span class="GAPprompt">gap></span> <span class="GAPinput">RecoverMultiplicationTable(2, 11 fail </pre></div> <p>Note that no semigroup is created calling this function but just the table is created. This mak <p>To create a semigroup with the multiplication table obtained by <code class="code">RecoverMu <p><a id="X813727FD851302B7" name="X813727FD851302B7"></a></p> <h5>4.1-5 SemigroupByMultiplicationTableNC</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Se <p>returns an object with <code class="func">IsSemigroup</code> (<a href="../../../doc/ref/cha <p>If <var class="Arg">table</var> is not associative, this can lead to errors and wrong results or m <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">s := SemigroupByMultiplicationTab <semigroup of size 2, with 2 generators> <span class="GAPprompt">gap></span> <span class="GAPinput">IsSmallSemigroup(s);</span> false </pre></div> <p>Note that this function is <em>not</em> used to create semigroups when <code class="func">SmallSe <p><a id="X788211A07D67C282" name="X788211A07D67C282"></a></p> <h5>4.1-6 IdSmallSemigroup</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Id <p>returns a pair <code class="code">[m, n]</code> such that <span class="SimpleMath">\((m,n)\)</s <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">sgrp := Semigroup(Transformation( <span class="GAPprompt">></span> <span class="GAPinput"> Transformation([1, 2, 3]));;</span> <span class="GAPprompt">gap></span> <span class="GAPinput">IdSmallSemigroup(sgrp);</span> [ 2, 3 ] </pre></div> <p><a id="X79D38A3886C7431D" name="X79D38A3886C7431D"></a></p> <h5>4.1-7 EquivalenceSmallSemigroup</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Eq <p>returns a mapping <code class="code">map</code> from <var class="Arg">sgrp</var> to the semigroup <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">sgrp := Semigroup(Transformation( <span class="GAPprompt">></span> <span class="GAPinput"> Transformation([1, 2, 3]));;</span> <span class="GAPprompt">gap></span> <span class="GAPinput">EquivalenceSmallSemigroup(sgrp) SemigroupHomomorphismByImages ( Monoid( [ Transformation( [ 1, 2, 2 ] ) ] )-><small semigroup of size 2>) </pre></div> <p><a id="X8078FC78871FA496" name="X8078FC78871FA496"></a></p> <h5>4.1-8 InfoSmallsemi</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ In <p>is the info class (see <a href="../../../doc/ref/chap7_mj.html#X7A9C902479CB6F7C"><span c <p><a id="X841BB74986A73272" name="X841BB74986A73272"></a></p> <h5>4.1-9 UnloadSmallsemiData</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Un <p>deletes most or all of the data from the <strong class="pkg">GAP</strong> workspace that was load <p>If the boolean <var class="Arg">use_later</var> is <code class="code">false</code> all data loade <p>If the boolean <var class="Arg">use_later</var> is <code class="code">true</code> only the recove <p><a id="X78274024827F306D" name="X78274024827F306D"></a></p> <h4>4.2 <span class="Heading">Properties of Semigroups</span></h4> <p>In this section we detail the <strong class="pkg">GAP</strong> functions that can be used to dete <ul> <li><p><var class="Arg">S</var> is a <em>left zero semigroup</em> if <var class="Arg">xy=x</var> for all <var </li> <li><p><var class="Arg">S</var> is a <em>right zero semigroup</em> if <var class="Arg">xy=y</var> for all <va </li> <li><p><var class="Arg">S</var> is <em>commutative</em> if <var class="Arg">xy=yx</var> for all <var class= </li> <li><p><var class="Arg">S</var> is <em>simple</em> if it has no proper two-sided ideals.</p> </li> <li><p><var class="Arg">S</var> is <em>zero simple</em> if the only 2-sided ideals are <var class="Arg">{0 </li> <li><p><var class="Arg">S</var> is <em>regular</em> if for all <var class="Arg">x</var> in <var class="Arg">S< </li> <li><p><var class="Arg">S</var> is <em>completely regular</em> if every element of <var class="Arg">S</va </li> <li><p><var class="Arg">S</var> is an <em>inverse semigroup</em> if every element <var class="Arg">x</var> </li> <li><p><var class="Arg">S</var> is a <em>Clifford semigroup</em> if it is a regular semigroup whose idemp </li> <li><p><var class="Arg">S</var> is a <em>band</em> if every element is an idempotent, that is, <var class=" </li> <li><p><var class="Arg">S</var> is a <em>Brandt semigroup</em> if it is inverse and zero simple.</p> </li> <li><p><var class="Arg">S</var> is a <em>rectangular band</em> if for all <var class="Arg">x,y,z</var> in <va </li> <li><p><var class="Arg">S</var> is a <em>semiband</em> if it is generated by its idempotent elements, tha </li> <li><p><var class="Arg">S</var> is an <em>orthodox semigroup</em> if it is regular and its idempotents (e </li> <li><p><var class="Arg">S</var> is a <em>zero semigroup</em> if there exists an element <var class="Arg">0< </li> <li><p><var class="Arg">S</var> is a <em>zero group</em> if there exists an element <var class="Arg">0</var> </li> </ul> <p>The <strong class="pkg">MONOID</strong> package was used to determined which semigroups in the <p><a id="X86A4F95783A18702" name="X86A4F95783A18702"></a></p> <h5>4.2-1 Annihilators</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ An <p>returns the set of annihilators of <var class="Arg">sgrp</var> if <var class="Arg">sgrp</var> cont <p>An element <span class="SimpleMath">\(x\)</span> in a semigroup with zero <span class="SimpleM <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">s := SmallSemigroup(5, 6);</span> <small semigroup of size 5> <span class="GAPprompt">gap></span> <span class="GAPinput">Annihilators(s);</span> [ s1, s2 ] </pre></div> <p><a id="X7E1195927C454C8A" name="X7E1195927C454C8A"></a></p> <h5>4.2-2 DiagonalOfMultiplicationTable</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Di <p>returns the diagonal of the multiplication table of the semigroup <var class="Arg">sgrp</var>.< <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">s := SmallSemigroup(8, 10101);;</sp <span class="GAPprompt">gap></span> <span class="GAPinput">DiagonalOfMultiplicationTable(s [ 1, 1, 1, 1, 1, 1, 1, 1 ] <span class="GAPprompt">gap></span> <span class="GAPinput">s := SmallSemigroup(7, 10101);;</sp <span class="GAPprompt">gap></span> <span class="GAPinput">DiagonalOfMultiplicationTable(s [ 1, 1, 1, 1, 1, 1, 1 ] </pre></div> <p><a id="X7DCF7C368665E94D" name="X7DCF7C368665E94D"></a></p> <h5>4.2-3 DisplaySmallSemigroup</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Di <p>displays all the information about the small semigroup <var class="Arg">sgrp</var> that is stor <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">s := SmallSemigroup(6, 3838);;</spa <span class="GAPprompt">gap></span> <span class="GAPinput">DisplaySmallSemigroup(s);</span> IsBand: false IsBrandtSemigroup: false IsCommutative: false IsCompletelyRegularSemigroup: false IsFullTransformationSemigroupCopy: false IsGroupAsSemigroup: false IsIdempotentGenerated: false IsInverseSemigroup: false IsMonogenicSemigroup: false IsMonoidAsSemigroup: false IsMultSemigroupOfNearRing: false IsOrthodoxSemigroup: false IsRectangularBand: false IsRegularSemigroup: false IsSelfDualSemigroup: false IsSemigroupWithClosedIdempotents: true IsSimpleSemigroup: false IsSingularSemigroupCopy: false IsZeroSemigroup: false IsZeroSimpleSemigroup: false MinimalGeneratingSet: [ s3, s4, s5, s6 ] Idempotents: [ s1, s5, s6 ] GreensRClasses: [ {s1}, {s2}, {s3}, {s4}, {s5}, {s6} ] GreensLClasses: [ {s1}, {s2}, {s3}, {s4}, {s6} ] GreensHClasses: [ {s1}, {s2}, {s3}, {s4}, {s5}, {s6} ] GreensDClasses: [ {s1}, {s2}, {s3}, {s4}, {s6} ] </pre></div> <p><a id="X86B662117F4C4C7F" name="X86B662117F4C4C7F"></a></p> <h5>4.2-4 IndexPeriod</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ In <p>returns the minimum numbers <var class="Arg">m, r</var> such that <var class="Arg">x^{m+r}=x^m</ <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">s := SmallSemigroup(5, 116);</span> <small semigroup of size 5> <span class="GAPprompt">gap></span> <span class="GAPinput">x := Elements(s)[3];</span> s3 <span class="GAPprompt">gap></span> <span class="GAPinput">IndexPeriod(x);</span> [ 2, 1 ] <span class="GAPprompt">gap></span> <span class="GAPinput">x ^ 3 = x ^ 2;</span> true <span class="GAPprompt">gap></span> <span class="GAPinput">x ^ 2 = x ^ 1;</span> false <span class="GAPprompt">gap></span> <span class="GAPinput">x ^ 3 = x ^ 1;</span> false </pre></div> <p><a id="X7C8DB14587D1B55A" name="X7C8DB14587D1B55A"></a></p> <h5>4.2-5 IsBand</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <p>returns <code class="keyw">true</code> if the small semigroup <var class="Arg">sgrp</var> is a ban <p>A semigroup <var class="Arg">sgrp</var> is a <em>band</em> if every element is an idempotent, that is <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">s := SmallSemigroup(5, 519);;</span> <span class="GAPprompt">gap></span> <span class="GAPinput">IsBand(s);</span> false <span class="GAPprompt">gap></span> <span class="GAPinput">s := OneSmallSemigroup(5, IsBand, t <small semigroup of size 5> <span class="GAPprompt">gap></span> <span class="GAPinput">IsBand(s);</span> true <span class="GAPprompt">gap></span> <span class="GAPinput">IdSmallSemigroup(s);</span> [ 5, 1010 ] </pre></div> <p><a id="X7EFDBA687DCDA6FA" name="X7EFDBA687DCDA6FA"></a></p> <h5>4.2-6 IsBrandtSemigroup</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <p>returns <code class="keyw">true</code> if the small semigroup <var class="Arg">sgrp</var> is a Bra <p>A finite semigroup <var class="Arg">sgrp</var> is a <em>Brandt semigroup</em> if it is inverse and ze <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">s := SmallSemigroup(5, 519);;</span> <span class="GAPprompt">gap></span> <span class="GAPinput">IsBrandtSemigroup(s);</span> false <span class="GAPprompt">gap></span> <span class="GAPinput">s := OneSmallSemigroup(5, IsBrandt <small semigroup of size 5> <span class="GAPprompt">gap></span> <span class="GAPinput">IsBrandtSemigroup(s);</span> true <span class="GAPprompt">gap></span> <span class="GAPinput">IdSmallSemigroup(s);</span> [ 5, 149 ] </pre></div> <p><a id="X81DE11987BB81017" name="X81DE11987BB81017"></a></p> <h5>4.2-7 IsCliffordSemigroup</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <p>returns <code class="keyw">true</code> if the small semigroup <var class="Arg">sgrp</var> is a Cli <p>A semigroup <var class="Arg">sgrp</var> is a <em>Clifford semigroup</em> if it is a regular semigrou <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">s := SmallSemigroup(5, 519);;</span> <span class="GAPprompt">gap></span> <span class="GAPinput">IsBand(s);</span> false <span class="GAPprompt">gap></span> <span class="GAPinput">s := OneSmallSemigroup(5, IsBand, t <small semigroup of size 5> <span class="GAPprompt">gap></span> <span class="GAPinput">IsBand(s);</span> true <span class="GAPprompt">gap></span> <span class="GAPinput">IdSmallSemigroup(s);</span> [ 5, 1010 ] <span class="GAPprompt">gap></span> <span class="GAPinput">s := SmallSemigroup(5, 519);;</span> <span class="GAPprompt">gap></span> <span class="GAPinput">IsCliffordSemigroup(s);</span> false <span class="GAPprompt">gap></span> <span class="GAPinput">s := OneSmallSemigroup(5, IsCliffo <small semigroup of size 5> <span class="GAPprompt">gap></span> <span class="GAPinput">IsCliffordSemigroup(s);</span> true <span class="GAPprompt">gap></span> <span class="GAPinput">IdSmallSemigroup(s);</span> [ 5, 148 ] </pre></div> <p><a id="X843EFDA4807FDC31" name="X843EFDA4807FDC31"></a></p> <h5>4.2-8 IsCommutativeSemigroup</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <p>return <code class="keyw">true</code> if the small semigroup <var class="Arg">sgrp</var> is commu <p>A semigroup <var class="Arg">sgrp</var> is <em>commutative</em> if <var class="Arg">xy=yx</var> for a <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">s := SmallSemigroup(6, 871);;</span> <span class="GAPprompt">gap></span> <span class="GAPinput">IsCommutativeSemigroup(s);</span> false <span class="GAPprompt">gap></span> <span class="GAPinput">s := OneSmallSemigroup(5, IsCommut <small semigroup of size 5> <span class="GAPprompt">gap></span> <span class="GAPinput">IsCommutativeSemigroup(s);</span> true <span class="GAPprompt">gap></span> <span class="GAPinput">IsCommutative(s);</span> true <span class="GAPprompt">gap></span> <span class="GAPinput">IdSmallSemigroup(s);</span> [ 5, 1 ] <span class="GAPprompt">gap></span> <span class="GAPinput">s := OneSmallSemigroup(5, IsCommut <small semigroup of size 5> <span class="GAPprompt">gap></span> <span class="GAPinput">IsCommutativeSemigroup(s);</span> true <span class="GAPprompt">gap></span> <span class="GAPinput">IsCommutative(s);</span> true </pre></div> <p><a id="X7AFA23AF819FBF3D" name="X7AFA23AF819FBF3D"></a></p> <h5>4.2-9 IsCompletelyRegularSemigroup</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <p>returns <code class="keyw">true</code> if the semigroup <var class="Arg">sgrp</var> is completel <p>A semigroup is <em>completely regular</em> if every element is contained in a subgroup.</p> <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">s := SmallSemigroup(6, 13131);</spa <small semigroup of size 6> <span class="GAPprompt">gap></span> <span class="GAPinput">IsCompletelyRegularSemigroup(s) false <span class="GAPprompt">gap></span> <span class="GAPinput">s := OneSmallSemigroup(6, IsComple <small semigroup of size 6> <span class="GAPprompt">gap></span> <span class="GAPinput">IsCompletelyRegularSemigroup(s) true <span class="GAPprompt">gap></span> <span class="GAPinput">IdSmallSemigroup(s);</span> [ 6, 3164 ] </pre></div> <p><a id="X7D51FD2B7AA3E8DF" name="X7D51FD2B7AA3E8DF"></a></p> <h5>4.2-10 IsFullTransformationSemigroupCopy</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <p>returns <code class="keyw">true</code> if the semigroup <var class="Arg">sgrp</var> is isomorphi <p>The size of the full transformation semigroup on an <var class="Arg">n</var> element set is <span c <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">s := SmallSemigroup(1, 1);</span> <small semigroup of size 1> <span class="GAPprompt">gap></span> <span class="GAPinput">IsFullTransformationSemigroupCo true <span class="GAPprompt">gap></span> <span class="GAPinput">s := OneSmallSemigroup(4, IsFullTr <small semigroup of size 4> <span class="GAPprompt">gap></span> <span class="GAPinput">IsFullTransformationSemigroupCo true <span class="GAPprompt">gap></span> <span class="GAPinput">IdSmallSemigroup(s);</span> [ 4, 96 ] <span class="GAPprompt">gap></span> <span class="GAPinput">s := OneSmallSemigroup(6, IsFullTr fail </pre></div> <p><a id="X852F29E8795FA489" name="X852F29E8795FA489"></a></p> <h5>4.2-11 IsGroupAsSemigroup</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <p>returns <code class="keyw">true</code> if the small semigroup <var class="Arg">sgrp</var> is math <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">s := SmallSemigroup(7, 7);</span> <small semigroup of size 7> <span class="GAPprompt">gap></span> <span class="GAPinput">IsGroupAsSemigroup(s);</span> false <span class="GAPprompt">gap></span> <span class="GAPinput">s := SmallSemigroup(4, 37);;</span> <span class="GAPprompt">gap></span> <span class="GAPinput">IsGroupAsSemigroup(s);</span> true </pre></div> <p><a id="X835484C481CF3DDD" name="X835484C481CF3DDD"></a></p> <h5>4.2-12 <span class="Heading">IsIdempotentGenerated</span></h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <p>returns <code class="keyw">true</code> if the semigroup <var class="Arg">sgrp</var> is a semiband <p>A semigroup <var class="Arg">sgrp</var> is <em>idempotent generated</em> or equivalently a <em>semi <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">s := SmallSemigroup(3, 13);</span> <small semigroup of size 3> <span class="GAPprompt">gap></span> <span class="GAPinput">IsIdempotentGenerated(s);</span> true <span class="GAPprompt">gap></span> <span class="GAPinput">s := OneSmallSemigroup(3, IsIdempo <small semigroup of size 3> <span class="GAPprompt">gap></span> <span class="GAPinput">IsIdempotentGenerated(s);</span> false <span class="GAPprompt">gap></span> <span class="GAPinput">IdSmallSemigroup(s);</span> [ 3, 1 ] <span class="GAPprompt">gap></span> <span class="GAPinput">s := OneSmallSemigroup(4, IsIdempo <span class="GAPprompt">></span> <span class="GAPinput">IsSingularSemigroupCopy, true);</spa fail <span class="GAPprompt">gap></span> <span class="GAPinput">s := OneSmallSemigroup(2, IsIdempo <span class="GAPprompt">></span> <span class="GAPinput">IsSingularSemigroupCopy, true);</spa <small semigroup of size 2> </pre></div> <p><a id="X83F1529479D56665" name="X83F1529479D56665"></a></p> <h5>4.2-13 IsInverseSemigroup</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <p>returns <code class="keyw">true</code> if the semigroup <var class="Arg">sgrp</var> is an inverse <p>A semigroup <var class="Arg">sgrp</var> is an <em>inverse semigroup</em> if every element <var class <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">s := OneSmallSemigroup(7, IsInvers <small semigroup of size 7> <span class="GAPprompt">gap></span> <span class="GAPinput">IsInverseSemigroup(s);</span> true <span class="GAPprompt">gap></span> <span class="GAPinput">s := SmallSemigroup(7, 101324);</sp <small semigroup of size 7> <span class="GAPprompt">gap></span> <span class="GAPinput">IsInverseSemigroup(s);</span> false </pre></div> <p><a id="X7E9261367C8C52C0" name="X7E9261367C8C52C0"></a></p> <h5>4.2-14 IsLeftZeroSemigroup</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <p>returns <code class="keyw">true</code> if the semigroup <var class="Arg">sgrp</var> is a left zero <p>A semigroup <var class="Arg">sgrp</var> is a <em>left zero semigroup</em> if <var class="Arg">xy=x</v <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">s := SmallSemigroup(5, 438);</span> <small semigroup of size 5> <span class="GAPprompt">gap></span> <span class="GAPinput">IsLeftZeroSemigroup(s);</span> false <span class="GAPprompt">gap></span> <span class="GAPinput">s := SmallSemigroup(5, 1141);</span> <small semigroup of size 5> <span class="GAPprompt">gap></span> <span class="GAPinput">IsLeftZeroSemigroup(s);</span> true </pre></div> <p><a id="X79D46BAB7E327AD1" name="X79D46BAB7E327AD1"></a></p> <h5>4.2-15 IsMonogenicSemigroup</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <p>returns <code class="keyw">true</code> if the small semigroup <var class="Arg">sgrp</var> is gene <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">s := RandomSmallSemigroup(7);</spa <small semigroup of size 7> <span class="GAPprompt">gap></span> <span class="GAPinput">IsMonogenicSemigroup(s);</span> false <span class="GAPprompt">gap></span> <span class="GAPinput">s := OneSmallSemigroup(7, IsMonoge <small semigroup of size 7> <span class="GAPprompt">gap></span> <span class="GAPinput">IsMonogenicSemigroup(s);</span> true <span class="GAPprompt">gap></span> <span class="GAPinput">MinimalGeneratingSet(s);</span> [ s7 ] <span class="GAPprompt">gap></span> <span class="GAPinput">s := SmallSemigroup(7, 406945);</sp <small semigroup of size 7> <span class="GAPprompt">gap></span> <span class="GAPinput">IsMonogenicSemigroup(s);</span> false </pre></div> <p><a id="X7E4DEECD7CD9886D" name="X7E4DEECD7CD9886D"></a></p> <h5>4.2-16 IsMonoidAsSemigroup</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <p>returns <code class="keyw">true</code> if the semigroup <var class="Arg">sgrp</var> is a monoid (i <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">s := SmallSemigroup(4, 126);</span> <small semigroup of size 4> <span class="GAPprompt">gap></span> <span class="GAPinput">IsMonoidAsSemigroup(s);</span> false <span class="GAPprompt">gap></span> <span class="GAPinput">s := OneSmallSemigroup(4, IsMonoid <small semigroup of size 4> <span class="GAPprompt">gap></span> <span class="GAPinput">IsMonoidAsSemigroup(s);</span> true <span class="GAPprompt">gap></span> <span class="GAPinput">One(s);</span> s1 <span class="GAPprompt">gap></span> <span class="GAPinput">IdSmallSemigroup(s);</span> [ 4, 7 ] </pre></div> <p><a id="X79FF88207AFC8330" name="X79FF88207AFC8330"></a></p> <h5>4.2-17 IsMultSemigroupOfNearRing</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <p>returns <code class="keyw">true</code> if <var class="Arg">sgrp</var> is isomorphic (or anti-iso <p>Those semigroups in the library that have this property were identified using the <strong clas <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">s := OneSmallSemigroup(7, IsMultSe <small semigroup of size 7> <span class="GAPprompt">gap></span> <span class="GAPinput">IdSmallSemigroup(s);</span> [ 7, 1 ] <span class="GAPprompt">gap></span> <span class="GAPinput">IsMultSemigroupOfNearRing(s);</s true <span class="GAPprompt">gap></span> <span class="GAPinput">s := SmallSemigroup(2, 2);</span> <small semigroup of size 2> <span class="GAPprompt">gap></span> <span class="GAPinput">IsMultSemigroupOfNearRing(s);</s false </pre></div> <p><a id="X83D4ED237BB929AF" name="X83D4ED237BB929AF"></a></p> <h5>4.2-18 IsNGeneratedSemigroup</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <p>returns <code class="keyw">true</code> if the least size of a generating set for the small semigr <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">s := SmallSemigroup(7, 760041);</sp <small semigroup of size 7> <span class="GAPprompt">gap></span> <span class="GAPinput">IsNGeneratedSemigroup(s, 4);</spa false <span class="GAPprompt">gap></span> <span class="GAPinput">IsNGeneratedSemigroup(s, 3);</spa true <span class="GAPprompt">gap></span> <span class="GAPinput">MinimalGeneratingSet(s);</span> [ s3, s5, s7 ] <span class="GAPprompt">gap></span> <span class="GAPinput">s := OneSmallSemigroup(4, x -> Length <small semigroup of size 4> <span class="GAPprompt">gap></span> <span class="GAPinput">IsNGeneratedSemigroup(s, 4);</spa true </pre></div> <p><a id="X87BF19367BA2B9ED" name="X87BF19367BA2B9ED"></a></p> <h5>4.2-19 IsNIdempotentSemigroup</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <p>returns <code class="keyw">true</code> if the small semigroup <var class="Arg">sgrp</var> has <var <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">s := SmallSemigroup(4, 75);;</span> <span class="GAPprompt">gap></span> <span class="GAPinput">IsNIdempotentSemigroup(s, 1);</sp false <span class="GAPprompt">gap></span> <span class="GAPinput">IsNIdempotentSemigroup(s, 2);</sp false <span class="GAPprompt">gap></span> <span class="GAPinput">IsNIdempotentSemigroup(s, 3);</sp true </pre></div> <p><a id="X780ADE31828F0848" name="X780ADE31828F0848"></a></p> <h5>4.2-20 <span class="Heading">IsNilpotentSemigroup</span></h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <p>returns <code class="keyw">true</code> if the small semigroup <var class="Arg">sgrp</var> is nilp <p>A semigroup is <em>nilpotent</em> if it has a zero element and every element to some power equals th <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">s := SmallSemigroup(5, 116);</span> <small semigroup of size 5> <span class="GAPprompt">gap></span> <span class="GAPinput">IsNilpotentSemigroup(s);</span> false <span class="GAPprompt">gap></span> <span class="GAPinput">s := SmallSemigroup(7, 673768);;</s <span class="GAPprompt">gap></span> <span class="GAPinput">IsNilpotentSemigroup(s);</span> true <span class="GAPprompt">gap></span> <span class="GAPinput">s := SmallSemigroup(7, 657867);;</s <span class="GAPprompt">gap></span> <span class="GAPinput">IsNilpotent(s);</span> true </pre></div> <p><a id="X7935C476808C8773" name="X7935C476808C8773"></a></p> <h5>4.2-21 IsOrthodoxSemigroup</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <p>returns <code class="keyw">true</code> if the semigroup <var class="Arg">sgrp</var> is orthodox a <p>A semigroup is <em>orthodox</em> if it is regular and its idempotents form a subsemigroup.</p> <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">s := SmallSemigroup(6, 15858);;</sp <span class="GAPprompt">gap></span> <span class="GAPinput">IsSemigroupWithClosedIdempotent true <span class="GAPprompt">gap></span> <span class="GAPinput">IsRegularSemigroup(s);</span> true <span class="GAPprompt">gap></span> <span class="GAPinput">IsOrthodoxSemigroup(s);</span> true </pre></div> <p><a id="X7E9B674D781B072C" name="X7E9B674D781B072C"></a></p> <h5>4.2-22 IsRectangularBand</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <p>returns <code class="keyw">true</code> if the small semigroup <var class="Arg">sgrp</var> is a rec <p>A semigroup <var class="Arg">sgrp</var> is a <em>rectangular band</em> if for all <var class="Arg">x, <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">s := SmallSemigroup(5, 216);;</span> <span class="GAPprompt">gap></span> <span class="GAPinput">IsRectangularBand(s);</span> false <span class="GAPprompt">gap></span> <span class="GAPinput">s := SmallSemigroup(6, 15854);;</sp <span class="GAPprompt">gap></span> <span class="GAPinput">IsRectangularBand(s);</span> true </pre></div> <p><a id="X7C4663827C5ACEF1" name="X7C4663827C5ACEF1"></a></p> <h5>4.2-23 IsRegularSemigroup</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <p>returns <code class="keyw">true</code> if the small semigroup <var class="Arg">sgrp</var> is a reg <p>A semigroup <var class="Arg">sgrp</var> is <em>regular</em> if for all <var class="Arg">x</var> in <var c <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">s := SmallSemigroup(3, 10);;</span> <span class="GAPprompt">gap></span> <span class="GAPinput">IsRegularSemigroup(s);</span> true <span class="GAPprompt">gap></span> <span class="GAPinput">s := SmallSemigroup(3, 1);;</span> <span class="GAPprompt">gap></span> <span class="GAPinput">IsRegularSemigroup(s);</span> false <span class="GAPprompt">gap></span> <span class="GAPinput">s := OneSmallSemigroup(4, IsFullTr <small semigroup of size 4> <span class="GAPprompt">gap></span> <span class="GAPinput">IsRegularSemigroup(s);</span> true </pre></div> <p><a id="X7CB099958658F979" name="X7CB099958658F979"></a></p> <h5>4.2-24 IsRightZeroSemigroup</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <p>returns <code class="keyw">false</code> for any small semigroup <var class="Arg">sgrp</var> sinc <p>A semigroup <var class="Arg">sgrp</var> is a <em>right zero semigroup</em> if <var class="Arg">xy=y</ <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">s := SmallSemigroup(5, 438);</span> <small semigroup of size 5> <span class="GAPprompt">gap></span> <span class="GAPinput">IsRightZeroSemigroup(s);</span> false </pre></div> <p><a id="X846FC6247EE31607" name="X846FC6247EE31607"></a></p> <h5>4.2-25 IsSelfDualSemigroup</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <p>returns <code class="keyw">true</code> if the semigroup <var class="Arg">sgrp</var> is self dual a <p>A semigroup is <em>self dual</em> if it is isomorphic to its dual, that is, the semigroup <var class= <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">s := SmallSemigroup(5, 116);</span> <small semigroup of size 5> <span class="GAPprompt">gap></span> <span class="GAPinput">IsSelfDualSemigroup(s);</span> false <span class="GAPprompt">gap></span> <span class="GAPinput">s := RandomSmallSemigroup(5, IsSel <small semigroup of size 5> <span class="GAPprompt">gap></span> <span class="GAPinput">IsSelfDualSemigroup(s);</span> true </pre></div> <p><a id="X7D3ACD8A7C0AB34C" name="X7D3ACD8A7C0AB34C"></a></p> <h5>4.2-26 IsSemigroupWithClosedIdempotents</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <p>returns <code class="keyw">true</code> if the idempotent elements of the semigroup <var class=" <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">s := SmallSemigroup(5, 677);;</span> <span class="GAPprompt">gap></span> <span class="GAPinput">IsSemigroupWithClosedIdempotent true <span class="GAPprompt">gap></span> <span class="GAPinput">s := SmallSemigroup(5, 659);;</span> <span class="GAPprompt">gap></span> <span class="GAPinput">IsSemigroupWithClosedIdempotent true <span class="GAPprompt">gap></span> <span class="GAPinput">s := SmallSemigroup(5, 327);;</span> <span class="GAPprompt">gap></span> <span class="GAPinput">IsSemigroupWithClosedIdempotent false </pre></div> <p><a id="X82D229727C6278EC" name="X82D229727C6278EC"></a></p> <h5>4.2-27 IsSemigroupWithZero</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <p>returns <code class="keyw">true</code> if the semigroup <var class="Arg">sgrp</var> has a zero ele <p>An element <span class="SimpleMath">\(z\)</span> is a <em>zero</em> if <span class="SimpleMath">\( <div class="example"><pre> --> -------------------- --> maximum size reached --> -------------------- ¤ Die Informationen auf dieser Webseite wurden nach bestem Wissen sorgfältig zusammengestellt. 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2026-03-28