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<div class="ChapSects"><a href="chap51_mj.html#X8665D8737FDD5B10">51 <span class="Heading">Semigroups and Monoids</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap51_mj.html#X80AF5F307DBDC2B4">51.1 <span class="Heading">Semigroups</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51_mj.html#X7B412E5B8543E9B7">51.1-1 IsSemigroup</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51_mj.html#X8678D40878CC09A1">51.1-3 Subsemigroup</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51_mj.html#X7BFE938E857CA27D">51.2-10 MonoidByMultiplicationTable</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap51_mj.html#X840847B6810BD0E1">51.3 <span class="Heading">Inverse semigroups and monoids</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51_mj.html#X87C373597F787250">51.3-3 GeneratorsOfInverseSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51_mj.html#X7A3B262C85B6D475">51.3-4 GeneratorsOfInverseMonoid</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51_mj.html#X7C4C6EE681E7A57E">51.3-5 IsInverseSubsemigroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap51_mj.html#X78274024827F306D">51.4 <span class="Heading">Properties of Semigroups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51_mj.html#X7C4663827C5ACEF1">51.4-1 IsRegularSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51_mj.html#X87532A76854347E0">51.4-2 IsRegularSemigroupElement</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51_mj.html#X7AFDE0F17AE516C5">51.4-3 InversesOfSemigroupElement</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51_mj.html#X836F4692839F4874">51.4-4 IsSimpleSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51_mj.html#X8193A60F839C064E">51.4-5 IsZeroSimpleSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51_mj.html#X85F7E5CD86F0643B">51.4-6 IsZeroGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51_mj.html#X7FFEC81F7F2C4EAA">51.4-7 IsReesCongruenceSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51_mj.html#X83F1529479D56665">51.4-8 IsInverseSemigroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap51_mj.html#X7BB32D508183C0F1">51.5 <span class="Heading">Ideals of semigroups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51_mj.html#X7D5CEE4D7D4318ED">51.5-1 SemigroupIdealByGenerators</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51_mj.html#X7F01FFB18125DED5">51.5-2 ReesCongruenceOfSemigroupIdeal</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51_mj.html#X7A3FF85984345540">51.5-3 IsLeftSemigroupIdeal</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap51_mj.html#X7914691E7DFFE27A">51.6 <span class="Heading">Congruences on semigroups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51_mj.html#X78E34B737F0E009F">51.6-1 IsSemigroupCongruence</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51_mj.html#X822DB78579BCB7B5">51.6-2 IsReesCongruence</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap51_mj.html#X87CE9EAB7EE3A128">51.7 <span class="Heading">Quotients</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51_mj.html#X80EF3E6F842BE64E">51.7-1 IsQuotientSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51_mj.html#X7CAD3D1687956F7F">51.7-2 HomomorphismQuotientSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51_mj.html#X87120C46808F7289">51.7-3 QuotientSemigroupPreimage</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap51_mj.html#X80C6C718801855E9">51.8 <span class="Heading">Green's Relations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51_mj.html#X786CEDD4814A9079">51.8-1 GreensRRelation</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51_mj.html#X8364D69987D49DE1">51.8-2 IsGreensRelation</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51_mj.html#X82A11A087AFB3EB0">51.8-3 IsGreensClass</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51_mj.html#X7AA204C8850F9070">51.8-4 IsGreensLessThanOrEqual</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51_mj.html#X86FE5F5585EBCF13">51.8-5 RClassOfHClass</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51_mj.html#X78C56F4A78E0088A">51.8-6 EggBoxOfDClass</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51_mj.html#X803237F17ACD44E3">51.8-7 DisplayEggBoxOfDClass</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51_mj.html#X87C75A9D86122D93">51.8-8 GreensRClassOfElement</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51_mj.html#X844D20467A644811">51.8-9 GreensRClasses</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51_mj.html#X7CB4A18685B850E2">51.8-10 GroupHClassOfGreensDClass</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51_mj.html#X79D740EF7F0E53BD">51.8-11 IsGroupHClass</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51_mj.html#X7F5860927CAD920F">51.8-12 IsRegularDClass</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51_mj.html#X81AF2EAB7CEF8C19">51.8-13 DisplaySemigroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap51_mj.html#X8225A9EC87A255E6">51.9 <span class="Heading">Rees Matrix Semigroups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51_mj.html#X8526AA557CDF6C49">51.9-1 ReesMatrixSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51_mj.html#X78D2A48C87FC8E38">51.9-2 ReesMatrixSubsemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51_mj.html#X7964B5C97FB9C07D">51.9-3 IsomorphismReesMatrixSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51_mj.html#X7F6B852B81488C86">51.9-4 IsReesMatrixSemigroupElement</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51_mj.html#X7A0DE1F28470295E">51.9-5 ReesMatrixSemigroupElement</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51_mj.html#X7F03BE707AC7F8A0">51.9-6 IsReesMatrixSubsemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51_mj.html#X780BB78A79275244">51.9-7 IsReesMatrixSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51_mj.html#X7CACF4D686AF1D19">51.9-8 Matrix</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51_mj.html#X82FC5D6980C66AC4">51.9-9 <span class="Heading">Rows and columns</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51_mj.html#X7D9719F887AFCF8F">51.9-10 UnderlyingSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51_mj.html#X7D1D9A0382064B8F">51.9-11 AssociatedReesMatrixSemigroupOfDClass</a></span>
</div></div>
</div>
<h3>51 <span class="Heading">Semigroups and Monoids</span></h3>
<p>This chapter describes functions for creating semigroups and monoids and determining information about them.</p>
<p><a id="X80AF5F307DBDC2B4" name="X80AF5F307DBDC2B4"></a></p>
<h4>51.1 <span class="Heading">Semigroups</span></h4>
<p><a id="X7B412E5B8543E9B7" name="X7B412E5B8543E9B7"></a></p>
<h5>51.1-1 IsSemigroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsSemigroup</code>( <var class="Arg">D</var> )</td><td class="tdright">( synonym )</td></tr></table></div>
<p>returns <code class="keyw">true</code> if the object <var class="Arg">D</var> is a semigroup. A <em>semigroup</em> is a magma (see <a href="chap35_mj.html#X873E502F7D21C39C"><span class="RefLink">35</span></a>) with associative multiplication.</p>
<p><a id="X7F55D28F819B2817" name="X7F55D28F819B2817"></a></p>
<h5>51.1-2 <span class="Heading">Semigroup</span></h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Semigroup</code>( <var class="Arg">gen1</var>, <var class="Arg">gen2</var>, <var class="Arg">...</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Semigroup</code>( <var class="Arg">gens</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>In the first form, <code class="func">Semigroup</code> returns the semigroup generated by the arguments <var class="Arg">gen1</var>, <var class="Arg">gen2</var>, <span class="SimpleMath">\(\ldots\)</span>, that is, the closure of these elements under multiplication. In the second form, <code class="func">Semigroup</code> returns the semigroup generated by the elements in the homogeneous list <var class="Arg">gens</var>; a square matrix as only argument is treated as one generator, not as a list of generators.</p>
<p>It is <em>not</em> checked whether the underlying multiplication is associative, use <code class="func">Magma</code> (<a href="chap35_mj.html#X839147CF813312D6"><span class="RefLink">35.2-1</span></a>) and <code class="func">IsAssociative</code> (<a href="chap35_mj.html#X7C83B5A47FD18FB7"><span class="RefLink">35.4-7</span></a>) if you want to check whether a magma is in fact a semigroup.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">a:= Transformation( [ 2, 3, 4, 1 ] );</span>
Transformation( [ 2, 3, 4, 1 ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">b:= Transformation( [ 2, 2, 3, 4 ] );</span>
Transformation( [ 2, 2 ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">s:= Semigroup(a, b);</span>
<transformation semigroup of degree 4 with 2 generators>
</pre></div>
<p><a id="X8678D40878CC09A1" name="X8678D40878CC09A1"></a></p>
<h5>51.1-3 Subsemigroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Subsemigroup</code>( <var class="Arg">S</var>, <var class="Arg">gens</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SubsemigroupNC</code>( <var class="Arg">S</var>, <var class="Arg">gens</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>are just synonyms of <code class="func">Submagma</code> (<a href="chap35_mj.html#X8268EAA47E4A3A64"><span class="RefLink">35.2-7</span></a>) and <code class="func">SubmagmaNC</code> (<a href="chap35_mj.html#X8268EAA47E4A3A64"><span class="RefLink">35.2-7</span></a>), respectively.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">a:=GeneratorsOfSemigroup(s)[1];</span>
Transformation( [ 2, 3, 4, 1 ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">t:=Subsemigroup(s,[a]);</span>
<commutative transformation semigroup of degree 4 with 1 generator>
</pre></div>
<p><a id="X782B7BDD8252581C" name="X782B7BDD8252581C"></a></p>
<h5>51.1-4 IsSubsemigroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsSubsemigroup</code>( <var class="Arg">S</var>, <var class="Arg">T</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: <code class="keyw">true</code> or <code class="keyw">false</code>.</p>
<p>This operation returns <code class="keyw">true</code> if the semigroup <var class="Arg">T</var> is a subsemigroup of the semigroup <var class="Arg">S</var> and <code class="keyw">false</code> if it is not.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">f := Transformation([5, 6, 7, 1, 4, 3, 2, 7]);</span>
Transformation( [ 5, 6, 7, 1, 4, 3, 2, 7 ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">T := Semigroup(f);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsSubsemigroup(FullTransformationSemigroup(4), T);</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">S := Semigroup(f);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">T := Semigroup(f ^ 2);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsSubsemigroup(S, T);</span>
true</pre></div>
<p><a id="X79FBBEC9841544F3" name="X79FBBEC9841544F3"></a></p>
<h5>51.1-5 SemigroupByGenerators</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SemigroupByGenerators</code>( <var class="Arg">gens</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>is the underlying operation of <code class="func">Semigroup</code> (<a href="chap51_mj.html#X7F55D28F819B2817"><span class="RefLink">51.1-2</span></a>).</p>
<p><a id="X80ED104F85AE5134" name="X80ED104F85AE5134"></a></p>
<h5>51.1-6 AsSemigroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AsSemigroup</code>( <var class="Arg">C</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>If <var class="Arg">C</var> is a collection whose elements form a semigroup under <code class="func">\*</code> (<a href="chap31_mj.html#X8481C9B97B214C23"><span class="RefLink">31.12-1</span></a>) (see <code class="func">IsSemigroup</code> (<a href="chap51_mj.html#X7B412E5B8543E9B7"><span class="RefLink">51.1-1</span></a>)) then <code class="func">AsSemigroup</code> returns this semigroup. Otherwise <code class="keyw">fail</code> is returned.</p>
<p><a id="X7B1EEA3E82BFE09F" name="X7B1EEA3E82BFE09F"></a></p>
<h5>51.1-7 AsSubsemigroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AsSubsemigroup</code>( <var class="Arg">D</var>, <var class="Arg">C</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Let <var class="Arg">D</var> be a domain and <var class="Arg">C</var> a collection. If <var class="Arg">C</var> is a subset of <var class="Arg">D</var> that forms a semigroup then <code class="func">AsSubsemigroup</code> returns this semigroup, with parent <var class="Arg">D</var>. Otherwise <code class="keyw">fail</code> is returned.</p>
<p><a id="X78147A247963F23B" name="X78147A247963F23B"></a></p>
<h5>51.1-8 GeneratorsOfSemigroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GeneratorsOfSemigroup</code>( <var class="Arg">S</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Semigroup generators of a semigroup <var class="Arg">D</var> are the same as magma generators, see <code class="func">GeneratorsOfMagma</code> (<a href="chap35_mj.html#X872E05B478EC20CA"><span class="RefLink">35.4-1</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">GeneratorsOfSemigroup(s);</span>
[ Transformation( [ 2, 3, 4, 1 ] ), Transformation( [ 2, 2 ] ) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">GeneratorsOfSemigroup(t);</span>
[ Transformation( [ 2, 3, 4, 1 ] ) ]
</pre></div>
<p><a id="X79776D7C8399F2CF" name="X79776D7C8399F2CF"></a></p>
<h5>51.1-9 IsGeneratorsOfSemigroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsGeneratorsOfSemigroup</code>( <var class="Arg">C</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>This property reflects whether the list or collection <var class="Arg">C</var> generates a semigroup. <code class="func">IsAssociativeElementCollection</code> (<a href="chap31_mj.html#X7979AFAA80FF795A"><span class="RefLink">31.15-1</span></a>) implies  <code class="func">IsGeneratorsOfSemigroup</code>, but is not used directly in semigroup code, because of conflicts with matrices.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsGeneratorsOfSemigroup([Transformation([2,3,1])]);</span>
true
</pre></div>
<p><a id="X7C72E4747BF642BB" name="X7C72E4747BF642BB"></a></p>
<h5>51.1-10 <span class="Heading">FreeSemigroup</span></h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FreeSemigroup</code>( [<var class="Arg">wfilt</var>, ]<var class="Arg">rank</var>[, <var class="Arg">name</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FreeSemigroup</code>( [<var class="Arg">wfilt</var>, ]<var class="Arg">name1</var>[, <var class="Arg">name2</var>[, <var class="Arg">...</var>]] )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FreeSemigroup</code>( [<var class="Arg">wfilt</var>, ]<var class="Arg">names</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FreeSemigroup</code>( [<var class="Arg">wfilt</var>, ]<var class="Arg">infinity</var>[, <var class="Arg">name</var>][, <var class="Arg">init</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p><code class="code">FreeSemigroup</code> returns a free semigroup. The number of generators, and the labels given to the generators, can be specified in several different ways. Warning: the labels of generators are only an aid for printing, and do not necessarily distinguish generators; see the examples at the end for more information.</p>
<dl>
<dt><strong class="Mark">
1: For a given rank, and an optional generator name prefix
</strong></dt>
<dd><p>Called with a positive integer <var class="Arg">rank</var>, <code class="func">FreeSemigroup</code> returns a free semigroup on <var class="Arg">rank</var> generators. The optional argument <var class="Arg">name</var> must be a string; its default value is <code class="code">"s"</code>.</p>
<p>If <var class="Arg">name</var> is not given but the <code class="code">generatorNames</code> option is, then this option is respected as described in Section <a href="chap50_mj.html#X7D0FFDA4793995FC"><span class="RefLink">50.1-16</span></a>.</p>
<p>Otherwise, the generators of the returned free semigroup are labelled <var class="Arg">name</var><code class="code">1</code>, ..., <var class="Arg">name</var><code class="code">k</code>, where <code class="code">k</code> is the value of <var class="Arg">rank</var>.</p>
</dd>
<dt><strong class="Mark">2: For given generator names</strong></dt>
<dd><p>Called with various (at least one) nonempty strings, <code class="func">FreeSemigroup</code> returns a free semigroup on as many generators as arguments, which are labelled <var class="Arg">name1</var>, <var class="Arg">name2</var>, etc.</p>
</dd>
<dt><strong class="Mark">3: For a given list of generator names</strong></dt>
<dd><p>Called with a nonempty finite list <var class="Arg">names</var> of nonempty strings, <code class="func">FreeSemigroup</code> returns a free semigroup on <code class="code">Length(<var class="Arg">names</var>)</code> generators, whose <code class="code">i</code>-th generator is labelled <var class="Arg">names</var><code class="code">[i]</code>.</p>
</dd>
<dt><strong class="Mark">
4: For the rank <code class="keyw">infinity</code>,
an optional default generator name prefix,
and an optional finite list of generator names
</strong></dt>
<dd><p>Called in the fourth form, <code class="func">FreeSemigroup</code> returns a free semigroup on infinitely many generators. The optional argument <var class="Arg">name</var> must be a string; its default value is <code class="code">"s"</code>, and the optional argument <var class="Arg">init</var> must be a finite list of nonempty strings; its default value is an empty list. The generators are initially labelled according to the list <var class="Arg">init</var>, followed by <var class="Arg">name</var><code class="code">i</code> for each <code class="code">i</code> in the range from <code class="code">Length(<var class="Arg">init</var>)+1</code> to <code class="keyw">infinity</code>; such a label is not allowed to appear in <var class="Arg">init</var>.</p>
</dd>
</dl>
<p>If the optional first argument <var class="Arg">wfilt</var> is given, then it must be either <code class="code">IsSyllableWordsFamily</code>, <code class="code">IsLetterWordsFamily</code>, <code class="code">IsWLetterWordsFamily</code>, or <code class="code">IsBLetterWordsFamily</code>. This filter specifies the representation used for the elements of the free semigroup (see <a href="chap37_mj.html#X80A9F39582ED296E"><span class="RefLink">37.6</span></a>). If no such filter is given, a letter representation is used.</p>
<p>For more on associative words see Chapter <a href="chap37_mj.html#X78C56A0A87CE380E"><span class="RefLink">37</span></a>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">f1 := FreeSemigroup( 3 );</span>
<free semigroup on the generators [ s1, s2, s3 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">f2 := FreeSemigroup( 3 , "generator" );</span>
<free semigroup on the generators
[ generator1, generator2, generator3 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">f3 := FreeSemigroup( "gen1" , "gen2" );</span>
<free semigroup on the generators [ gen1, gen2 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">f4 := FreeSemigroup( ["gen1" , "gen2"] );</span>
<free semigroup on the generators [ gen1, gen2 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">FreeSemigroup( 3 : generatorNames := "boom" );</span>
<free semigroup on the generators [ boom1, boom2, boom3 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">FreeSemigroup( 2 : generatorNames := [ "u", "v", "w" ] );</span>
<free semigroup on the generators [ u, v ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">FreeSemigroup( infinity ) ;</span>
<free semigroup on the generators [ s1, s2, ... ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">F := FreeSemigroup( infinity, "g", [ "a", "b" ]);</span>
<free semigroup on the generators [ a, b, ... ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">GeneratorsOfSemigroup( F ){[1..4]};</span>
[ a, b, g3, g4 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">GeneratorsOfSemigroup( FreeSemigroup( infinity, "gen" ) ){[1..3]};</span>
[ gen1, gen2, gen3 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">GeneratorsOfSemigroup( FreeSemigroup( infinity, [ "f" ] ) ){[1..3]};</span>
[ f, s2, s3 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">FreeSemigroup(IsSyllableWordsFamily, 5);</span>
<free semigroup on the generators [ s1, s2, s3, s4, s5 ]>
</pre></div>
<p>Each free object defines a unique alphabet (and a unique family of words). Its generators are the letters of this alphabet, thus words of length one.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">FreeSemigroup( 5 );</span>
<free semigroup on the generators [ s1, s2, s3, s4, s5 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">FreeMonoid( "a", "b" );</span>
<free monoid on the generators [ a, b ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">FreeGroup( infinity );</span>
<free group with infinity generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">FreeSemigroup( "x", "y" );</span>
<free semigroup on the generators [ x, y ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">FreeMonoid( 7 );</span>
<free monoid on the generators [ m1, m2, m3, m4, m5, m6, m7 ]>
</pre></div>
<p>Remember that names are just a help for printing and do not necessarily distinguish letters. It is possible to create arbitrarily weird situations by choosing strange names for the letters.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">f := FreeGroup( "x", "x" );</span>
<free group on the generators [ x, x ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">gens := GeneratorsOfGroup( f );</span>
[ x, x ]
<span class="GAPprompt">gap></span> <span class="GAPinput">gens[1] = gens[2];</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">f:= FreeGroup( "f1*f2", "f2^-1", "Group( [ f1, f2 ] )" );</span>
<free group on the generators [ f1*f2, f2^-1, Group( [ f1, f2 ] ) ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">gens:= GeneratorsOfGroup( f );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">gens[1] * gens[2];</span>
f1*f2*f2^-1
<span class="GAPprompt">gap></span> <span class="GAPinput">gens[1] / gens[3];</span>
f1*f2*Group( [ f1, f2 ] )^-1
<span class="GAPprompt">gap></span> <span class="GAPinput">gens[3] / gens[1] / gens[2];</span>
Group( [ f1, f2 ] )*f1*f2^-1*f2^-1^-1
</pre></div>
<p><a id="X7E67E13F7A01F8D3" name="X7E67E13F7A01F8D3"></a></p>
<h5>51.1-11 SemigroupByMultiplicationTable</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SemigroupByMultiplicationTable</code>( <var class="Arg">A</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns the semigroup whose multiplication is defined by the square matrix <var class="Arg">A</var> (see <code class="func">MagmaByMultiplicationTable</code> (<a href="chap35_mj.html#X85CD1E7678295CA6"><span class="RefLink">35.3-1</span></a>)) if such a semigroup exists. Otherwise <code class="keyw">fail</code> is returned.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">SemigroupByMultiplicationTable([[1,2,3],[2,3,1],[3,1,2]]);</span>
<semigroup of size 3, with 3 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">SemigroupByMultiplicationTable([[1,2,3],[2,3,1],[3,2,1]]);</span>
fail
</pre></div>
<p><a id="X872FE34A7814C0DC" name="X872FE34A7814C0DC"></a></p>
<h4>51.2 <span class="Heading">Monoids</span></h4>
<p><a id="X861C523483C6248C" name="X861C523483C6248C"></a></p>
<h5>51.2-1 IsMonoid</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsMonoid</code>( <var class="Arg">D</var> )</td><td class="tdright">( synonym )</td></tr></table></div>
<p>A <em>monoid</em> is a magma-with-one (see <a href="chap35_mj.html#X873E502F7D21C39C"><span class="RefLink">35</span></a>) with associative multiplication.</p>
<p><a id="X7F95328B7C7E49EA" name="X7F95328B7C7E49EA"></a></p>
<h5>51.2-2 <span class="Heading">Monoid</span></h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Monoid</code>( <var class="Arg">gen1</var>, <var class="Arg">gen2</var>, <var class="Arg">...</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Monoid</code>( <var class="Arg">gens</var>[, <var class="Arg">id</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p>In the first form, <code class="func">Monoid</code> returns the monoid generated by the arguments <var class="Arg">gen1</var>, <var class="Arg">gen2</var>, <span class="SimpleMath">\(\ldots\)</span>, that is, the closure of these elements under multiplication and taking the 0-th power. In the second form, <code class="func">Monoid</code> returns the monoid generated by the elements in the homogeneous list <var class="Arg">gens</var>; a square matrix as only argument is treated as one generator, not as a list of generators. In the second form, the identity element <var class="Arg">id</var> may be given as the second argument.</p>
<p>It is <em>not</em> checked whether the underlying multiplication is associative, use <code class="func">MagmaWithOne</code> (<a href="chap35_mj.html#X7854B23286B17321"><span class="RefLink">35.2-2</span></a>) and <code class="func">IsAssociative</code> (<a href="chap35_mj.html#X7C83B5A47FD18FB7"><span class="RefLink">35.4-7</span></a>) if you want to check whether a magma-with-one is in fact a monoid.</p>
<p><a id="X8322D01E84912FD7" name="X8322D01E84912FD7"></a></p>
<h5>51.2-3 Submonoid</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Submonoid</code>( <var class="Arg">M</var>, <var class="Arg">gens</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SubmonoidNC</code>( <var class="Arg">M</var>, <var class="Arg">gens</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>are just synonyms of <code class="func">SubmagmaWithOne</code> (<a href="chap35_mj.html#X7F295EBC7A9CE87E"><span class="RefLink">35.2-8</span></a>) and <code class="func">SubmagmaWithOneNC</code> (<a href="chap35_mj.html#X7F295EBC7A9CE87E"><span class="RefLink">35.2-8</span></a>), respectively.</p>
<p><a id="X85129EE387CC4D28" name="X85129EE387CC4D28"></a></p>
<h5>51.2-4 MonoidByGenerators</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MonoidByGenerators</code>( <var class="Arg">gens</var>[, <var class="Arg">one</var>] )</td><td class="tdright">( operation )</td></tr></table></div>
<p>is the underlying operation of <code class="func">Monoid</code> (<a href="chap51_mj.html#X7F95328B7C7E49EA"><span class="RefLink">51.2-2</span></a>).</p>
<p><a id="X7B22038F832B9C0F" name="X7B22038F832B9C0F"></a></p>
<h5>51.2-5 AsMonoid</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AsMonoid</code>( <var class="Arg">C</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>If <var class="Arg">C</var> is a collection whose elements form a monoid, then <code class="func">AsMonoid</code> returns this monoid. Otherwise <code class="keyw">fail</code> is returned.</p>
<p><a id="X7C9A12DE8287B2D3" name="X7C9A12DE8287B2D3"></a></p>
<h5>51.2-6 AsSubmonoid</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AsSubmonoid</code>( <var class="Arg">D</var>, <var class="Arg">C</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Let <var class="Arg">D</var> be a domain and <var class="Arg">C</var> a collection. If <var class="Arg">C</var> is a subset of <var class="Arg">D</var> that forms a monoid then <code class="func">AsSubmonoid</code> returns this monoid, with parent <var class="Arg">D</var>. Otherwise <code class="keyw">fail</code> is returned.</p>
<p><a id="X83CA2E7279C44718" name="X83CA2E7279C44718"></a></p>
<h5>51.2-7 GeneratorsOfMonoid</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GeneratorsOfMonoid</code>( <var class="Arg">M</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Monoid generators of a monoid <var class="Arg">M</var> are the same as magma-with-one generators (see <code class="func">GeneratorsOfMagmaWithOne</code> (<a href="chap35_mj.html#X87DD93EC8061DD81"><span class="RefLink">35.4-2</span></a>)).</p>
<p><a id="X7EC77C0184587181" name="X7EC77C0184587181"></a></p>
<h5>51.2-8 TrivialSubmonoid</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TrivialSubmonoid</code>( <var class="Arg">M</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>is just a synonym for <code class="func">TrivialSubmagmaWithOne</code> (<a href="chap35_mj.html#X837DA95883CFB985"><span class="RefLink">35.4-13</span></a>).</p>
<p><a id="X79FA3FA978CA2E43" name="X79FA3FA978CA2E43"></a></p>
<h5>51.2-9 <span class="Heading">FreeMonoid</span></h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FreeMonoid</code>( [<var class="Arg">wfilt</var>, ]<var class="Arg">rank</var>[, <var class="Arg">name</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FreeMonoid</code>( [<var class="Arg">wfilt</var>][,] [<var class="Arg">name1</var>[, <var class="Arg">name2</var>[, <var class="Arg">...</var>]]] )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FreeMonoid</code>( [<var class="Arg">wfilt</var>, ]<var class="Arg">names</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FreeMonoid</code>( [<var class="Arg">wfilt</var>, ]<var class="Arg">infinity</var>[, <var class="Arg">name</var>][, <var class="Arg">init</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p><code class="code">FreeMonoid</code> returns a free monoid. The number of generators, and the labels given to the generators, can be specified in several different ways. Warning: the labels of generators are only an aid for printing, and do not necessarily distinguish generators; see the examples at the end of <code class="func">FreeSemigroup</code> (<a href="chap51_mj.html#X7C72E4747BF642BB"><span class="RefLink">51.1-10</span></a>) for more information.</p>
<dl>
<dt><strong class="Mark">
1: For a given rank, and an optional generator name prefix
</strong></dt>
<dd><p>Called with a nonnegative integer <var class="Arg">rank</var>, <code class="func">FreeMonoid</code> returns a free monoid on <var class="Arg">rank</var> generators. The optional argument <var class="Arg">name</var> must be a string; its default value is <code class="code">"m"</code>.</p>
<p>If <var class="Arg">name</var> is not given but the <code class="code">generatorNames</code> option is, then this option is respected as described in Section <a href="chap50_mj.html#X7D0FFDA4793995FC"><span class="RefLink">50.1-16</span></a>.</p>
<p>Otherwise, the generators of the returned free monoid are labelled <var class="Arg">name</var><code class="code">1</code>, ..., <var class="Arg">name</var><code class="code">k</code>, where <code class="code">k</code> is the value of <var class="Arg">rank</var>.</p>
</dd>
<dt><strong class="Mark">2: For given generator names</strong></dt>
<dd><p>Called with various nonempty strings, <code class="func">FreeMonoid</code> returns a free monoid on as many generators as arguments, which are labelled <var class="Arg">name1</var>, <var class="Arg">name2</var>, etc.</p>
</dd>
<dt><strong class="Mark">3: For a given list of generator names</strong></dt>
<dd><p>Called with a finite list <var class="Arg">names</var> of nonempty strings, <code class="func">FreeMonoid</code> returns a free monoid on <code class="code">Length(<var class="Arg">names</var>)</code> generators, whose <code class="code">i</code>-th generator is labelled <var class="Arg">names</var><code class="code">[i]</code>.</p>
</dd>
<dt><strong class="Mark">
4: For the rank <code class="keyw">infinity</code>,
an optional default generator name prefix,
and an optional finite list of generator names
</strong></dt>
<dd><p>Called in the fourth form, <code class="func">FreeMonoid</code> returns a free monoid on infinitely many generators. The optional argument <var class="Arg">name</var> must be a string; its default value is <code class="code">"m"</code>, and the optional argument <var class="Arg">init</var> must be a finite list of nonempty strings; its default value is an empty list. The generators are initially labelled according to the list <var class="Arg">init</var>, followed by <var class="Arg">name</var><code class="code">i</code> for each <code class="code">i</code> in the range from <code class="code">Length(<var class="Arg">init</var>)+1</code> to <code class="keyw">infinity</code>.</p>
</dd>
</dl>
<p>If the optional first argument <var class="Arg">wfilt</var> is given, then it must be either <code class="code">IsSyllableWordsFamily</code>, <code class="code">IsLetterWordsFamily</code>, <code class="code">IsWLetterWordsFamily</code>, or <code class="code">IsBLetterWordsFamily</code>. This filter specifies the representation used for the elements of the free monoid (see <a href="chap37_mj.html#X80A9F39582ED296E"><span class="RefLink">37.6</span></a>). If no such filter is given, a letter representation is used.</p>
<p>For more on associative words see Chapter <a href="chap37_mj.html#X78C56A0A87CE380E"><span class="RefLink">37</span></a>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">FreeMonoid(5);</span>
<free monoid on the generators [ m1, m2, m3, m4, m5 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">FreeMonoid(4, "gen");</span>
<free monoid on the generators [ gen1, gen2, gen3, gen4 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">FreeMonoid(3 : generatorNames := "turbo");</span>
<free monoid on the generators [ turbo1, turbo2, turbo3 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">FreeMonoid(2 : generatorNames := ["u", "v", "w"]);</span>
<free monoid on the generators [ u, v ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">FreeMonoid(); FreeMonoid(0); FreeMonoid([]);</span>
<free monoid of rank zero>
<free monoid of rank zero>
<free monoid of rank zero>
<span class="GAPprompt">gap></span> <span class="GAPinput">FreeMonoid("a", "b", "c");</span>
<free monoid on the generators [ a, b, c ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">FreeMonoid(["x", "y"]);</span>
<free monoid on the generators [ x, y ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">FreeMonoid(infinity);</span>
<free monoid with infinity generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">F := FreeMonoid(infinity, "g", ["a", "b"]);</span>
<free monoid with infinity generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">GeneratorsOfMonoid(F){[1..4]};</span>
[ a, b, g3, g4 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">GeneratorsOfMonoid(FreeMonoid(infinity, "gen")){[1..3]};</span>
[ gen1, gen2, gen3 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">GeneratorsOfMonoid(FreeMonoid(infinity, [ "f", "g" ])){[1..3]};</span>
[ f, g, m3 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">FreeMonoid(IsSyllableWordsFamily, 50);</span>
<free monoid with 50 generators>
</pre></div>
<p><a id="X7BFE938E857CA27D" name="X7BFE938E857CA27D"></a></p>
<h5>51.2-10 MonoidByMultiplicationTable</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MonoidByMultiplicationTable</code>( <var class="Arg">A</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns the monoid whose multiplication is defined by the square matrix <var class="Arg">A</var> (see <code class="func">MagmaByMultiplicationTable</code> (<a href="chap35_mj.html#X85CD1E7678295CA6"><span class="RefLink">35.3-1</span></a>)) if such a monoid exists. Otherwise <code class="keyw">fail</code> is returned.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">MonoidByMultiplicationTable([[1,2,3],[2,3,1],[3,1,2]]);</span>
<monoid of size 3, with 3 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">MonoidByMultiplicationTable([[1,2,3],[2,3,1],[1,3,2]]);</span>
fail
</pre></div>
<p><a id="X840847B6810BD0E1" name="X840847B6810BD0E1"></a></p>
<h4>51.3 <span class="Heading">Inverse semigroups and monoids</span></h4>
<p><a id="X78B13FED7AFB4326" name="X78B13FED7AFB4326"></a></p>
<h5>51.3-1 InverseSemigroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ InverseSemigroup</code>( <var class="Arg">obj1</var>, <var class="Arg">obj2</var>, <var class="Arg">...</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: An inverse semigroup.</p>
<p>If <var class="Arg">obj1</var>, <var class="Arg">obj2</var>, ... are (any combination) of associative elements with unique semigroup inverses, semigroups of such elements, or collections of such elements, then <code class="code">InverseSemigroup</code> returns the inverse semigroup generated by the union of <var class="Arg">obj1</var>, <var class="Arg">obj2</var>, .... This equals the semigroup generated by the union of <var class="Arg">obj1</var>, <var class="Arg">obj2</var>, ... and their inverses.</p>
<p>For example if <code class="code">S</code> and <code class="code">T</code> are inverse semigroups, then <code class="code">InverseSemigroup(S, f, Idempotents(T));</code> is the inverse semigroup generated by <code class="code">Union(GeneratorsOfInverseSemigroup(S), [f], Idempotents(T)));</code>.</p>
<p>As present, the only associative elements with unique semigroup inverses, which do not always generate a group, are partial permutations; see Chapter <a href="chap54_mj.html#X7D6495F77B8A77BD"><span class="RefLink">54</span></a>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">S := InverseSemigroup(</span>
<span class="GAPprompt">></span> <span class="GAPinput">PartialPerm( [ 1, 2, 3, 6, 8, 10 ], [ 2, 6, 7, 9, 1, 5 ] ) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">f := PartialPerm( [ 1, 2, 3, 4, 5, 8, 10 ],</span>
<span class="GAPprompt">></span> <span class="GAPinput">[ 7, 1, 4, 3, 2, 6, 5 ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">S := InverseSemigroup(S, f, Idempotents(SymmetricInverseSemigroup(5)));</span>
<inverse partial perm semigroup of rank 10 with 34 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">Size(S);</span>
1233</pre></div>
<p><a id="X80D9B9A98736051B" name="X80D9B9A98736051B"></a></p>
<h5>51.3-2 InverseMonoid</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ InverseMonoid</code>( <var class="Arg">obj1</var>, <var class="Arg">obj2</var>, <var class="Arg">...</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: An inverse monoid.</p>
<p>If <var class="Arg">obj1</var>, <var class="Arg">obj2</var>, ... are (any combination) of associative elements with unique semigroup inverses, semigroups of such elements, or collections of such elements, then <code class="code">InverseMonoid</code> returns the inverse monoid generated by the union of <var class="Arg">obj1</var>, <var class="Arg">obj2</var>, .... This equals the monoid generated by the union of <var class="Arg">obj1</var>, <var class="Arg">obj2</var>, ... and their inverses.</p>
<p>As present, the only associative elements with unique semigroup inverses are partial permutations; see Chapter <a href="chap54_mj.html#X7D6495F77B8A77BD"><span class="RefLink">54</span></a>.</p>
<p>For example if <code class="code">S</code> and <code class="code">T</code> are inverse monoids, then <code class="code">InverseMonoid(S, f, Idempotents(T));</code> is the inverse monoid generated by <code class="code">Union(GeneratorsOfInverseMonoid(S), [f], Idempotents(T)));</code>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">S := InverseMonoid(</span>
<span class="GAPprompt">></span> <span class="GAPinput">PartialPerm( [ 1, 2, 3, 6, 8, 10 ], [ 2, 6, 7, 9, 1, 5 ] ) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">f := PartialPerm( [ 1, 2, 3, 4, 5, 8, 10 ],</span>
<span class="GAPprompt">></span> <span class="GAPinput">[ 7, 1, 4, 3, 2, 6, 5 ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">S := InverseMonoid(S, f, Idempotents(SymmetricInverseSemigroup(5)));</span>
<inverse partial perm monoid of rank 10 with 35 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">Size(S);</span>
1243</pre></div>
<p><a id="X87C373597F787250" name="X87C373597F787250"></a></p>
<h5>51.3-3 GeneratorsOfInverseSemigroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GeneratorsOfInverseSemigroup</code>( <var class="Arg">S</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: The generators of an inverse semigroup.</p>
<p>If <var class="Arg">S</var> is an inverse semigroup, then <code class="code">GeneratorsOfInverseSemigroup</code> returns the generators used to define <var class="Arg">S</var>, i.e. an inverse semigroup generating set for <var class="Arg">S</var>.</p>
<p>The value of <code class="code">GeneratorsOfSemigroup(<var class="Arg">S</var>)</code>, for an inverse semigroup <var class="Arg">S</var>, is the union of inverse semigroup generator and their inverses. So, <var class="Arg">S</var> is the semigroup, as opposed to inverse semigroup, generated by the elements of <code class="code">GeneratorsOfInverseSemigroup(<var class="Arg">S</var>)</code> and their inverses.</p>
<p>If <var class="Arg">S</var> is an inverse monoid, then <code class="code">GeneratorsOfInverseSemigroup</code> returns the generators used to define <var class="Arg">S</var>, as described above, and the identity of <var class="Arg">S</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">S:=InverseMonoid(</span>
<span class="GAPprompt">></span> <span class="GAPinput"> PartialPerm( [ 1, 2 ], [ 1, 4 ] ),</span>
<span class="GAPprompt">></span> <span class="GAPinput"> PartialPerm( [ 1, 2, 4 ], [ 3, 4, 1 ] ) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">GeneratorsOfSemigroup(S);</span>
[ <identity partial perm on [ 1, 2, 3, 4 ]>, [2,4](1), [2,4,1,3],
[4,2](1), [3,1,4,2] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">GeneratorsOfInverseSemigroup(S);</span>
[ [2,4](1), [2,4,1,3], <identity partial perm on [ 1, 2, 3, 4 ]> ]
<span class="GAPprompt">gap></span> <span class="GAPinput">GeneratorsOfMonoid(S);</span>
[ [2,4](1), [2,4,1,3], [4,2](1), [3,1,4,2] ]</pre></div>
<p><a id="X7A3B262C85B6D475" name="X7A3B262C85B6D475"></a></p>
<h5>51.3-4 GeneratorsOfInverseMonoid</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GeneratorsOfInverseMonoid</code>( <var class="Arg">S</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: The generators of an inverse monoid.</p>
<p>If <var class="Arg">S</var> is an inverse monoid, then <code class="code">GeneratorsOfInverseMonoid</code> returns the generators used to define <var class="Arg">S</var>, i.e. an inverse monoid generating set for <var class="Arg">S</var>.</p>
<p>There are four different possible generating sets which define an inverse monoid. More precisely, an inverse monoid can be generated as an inverse monoid, inverse semigroup, monoid, or semigroup. The different generating sets in each case can be obtained using <code class="func">GeneratorsOfInverseMonoid</code>, <code class="func">GeneratorsOfInverseSemigroup</code> (<a href="chap51_mj.html#X87C373597F787250"><span class="RefLink">51.3-3</span></a>), <code class="func">GeneratorsOfMonoid</code> (<a href="chap51_mj.html#X83CA2E7279C44718"><span class="RefLink">51.2-7</span></a>), and <code class="func">GeneratorsOfSemigroup</code> (<a href="chap51_mj.html#X78147A247963F23B"><span class="RefLink">51.1-8</span></a>), respectively.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">S:=InverseMonoid(</span>
<span class="GAPprompt">></span> <span class="GAPinput"> PartialPerm( [ 1, 2 ], [ 1, 4 ] ),</span>
<span class="GAPprompt">></span> <span class="GAPinput"> PartialPerm( [ 1, 2, 4 ], [ 3, 4, 1 ] ) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">GeneratorsOfInverseMonoid(S);</span>
[ [2,4](1), [2,4,1,3] ]</pre></div>
<p><a id="X7C4C6EE681E7A57E" name="X7C4C6EE681E7A57E"></a></p>
<h5>51.3-5 IsInverseSubsemigroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsInverseSubsemigroup</code>( <var class="Arg">S</var>, <var class="Arg">T</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: <code class="keyw">true</code> or <code class="keyw">false</code>.</p>
<p>If the semigroup <var class="Arg">T</var> is an inverse subsemigroup of the semigroup <var class="Arg">S</var>, then this operation returns <code class="keyw">true</code>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">T:=InverseSemigroup(RandomPartialPerm(4));;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsInverseSubsemigroup(SymmetricInverseSemigroup(4), T);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">T:=Semigroup(Transformation( [ 1, 2, 4, 5, 6, 3, 7, 8 ] ),</span>
<span class="GAPprompt">></span> <span class="GAPinput">Transformation( [ 3, 3, 4, 5, 6, 2, 7, 8 ] ),</span>
<span class="GAPprompt">></span> <span class="GAPinput">Transformation([ 1, 2, 5, 3, 6, 8, 4, 4 ] ));;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsInverseSubsemigroup(FullTransformationSemigroup(8), T);</span>
true</pre></div>
<p><a id="X78274024827F306D" name="X78274024827F306D"></a></p>
<h4>51.4 <span class="Heading">Properties of Semigroups</span></h4>
<p>The following functions determine information about semigroups.</p>
<p><a id="X7C4663827C5ACEF1" name="X7C4663827C5ACEF1"></a></p>
<h5>51.4-1 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>
<p>returns <code class="keyw">true</code> if <var class="Arg">S</var> is regular, i.e., if every \(\mathcal{D}\)-class of <var class="Arg">S</var> is regular.</p>
<p><a id="X87532A76854347E0" name="X87532A76854347E0"></a></p>
<h5>51.4-2 IsRegularSemigroupElement</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsRegularSemigroupElement</code>( <var class="Arg">S</var>, <var class="Arg">x</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>returns <code class="keyw">true</code> if <var class="Arg">x</var> has a general inverse in <var class="Arg">S</var>, i.e., there is an element <span class="SimpleMath">\(y \in \textit{S}\)</span> such that <span class="SimpleMath">\(\textit{x} y \textit{x} = \textit{x}\)</span> and <span class="SimpleMath">\(y \textit{x} y = y\)</span>.</p>
<p><a id="X7AFDE0F17AE516C5" name="X7AFDE0F17AE516C5"></a></p>
<h5>51.4-3 InversesOfSemigroupElement</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ InversesOfSemigroupElement</code>( <var class="Arg">S</var>, <var class="Arg">x</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: A list of the inverses of an element of a semigroup.</p>
<p><code class="code">InversesOfSemigroupElement</code> returns a list of the inverses of the element <var class="Arg">x</var> in the semigroup <var class="Arg">S</var>.</p>
<p>An element <var class="Arg">y</var> in <var class="Arg">S</var> is an <em>inverse</em> of <var class="Arg">x</var> if <code class="code"><var class="Arg">x</var>*y*<var class="Arg">x</var>=<var class="Arg">x</var></code> and <code class="code">y*<var class="Arg">x</var>*y=y</code>. The element <var class="Arg">x</var> has an inverse if and only if <var class="Arg">x</var> is a regular element of <var class="Arg">S</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">S := Semigroup([</span>
<span class="GAPprompt">></span> <span class="GAPinput"> Transformation([3, 1, 4, 2, 5, 2, 1, 6, 1]),</span>
<span class="GAPprompt">></span> <span class="GAPinput"> Transformation([5, 7, 8, 8, 7, 5, 9, 1, 9]),</span>
<span class="GAPprompt">></span> <span class="GAPinput"> Transformation([7, 6, 2, 8, 4, 7, 5, 8, 3])]);</span>
<transformation semigroup of degree 9 with 3 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">x := Transformation([3, 1, 4, 2, 5, 2, 1, 6, 1]);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">InversesOfSemigroupElement(S, x);</span>
[ ]
<span class="GAPprompt">gap></span> <span class="GAPinput">IsRegularSemigroupElement(S, x);</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">x := Transformation([1, 9, 7, 5, 5, 1, 9, 5, 1]);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Set(InversesOfSemigroupElement(S, x));</span>
[ Transformation( [ 1, 2, 3, 5, 5, 1, 3, 5, 2 ] ),
Transformation( [ 1, 5, 1, 1, 5, 1, 3, 1, 2 ] ),
Transformation( [ 1, 5, 1, 2, 5, 1, 3, 2, 2 ] ) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">IsRegularSemigroupElement(S, x);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">S := ReesZeroMatrixSemigroup(Group((1,2,3)),</span>
<span class="GAPprompt">></span> <span class="GAPinput"> [[(), ()], [(), 0], [(), (1,2,3)]]);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">x := ReesZeroMatrixSemigroupElement(S, 2, (1,2,3), 3);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">InversesOfSemigroupElement(S, x);</span>
[ (1,(1,2,3),3), (1,(1,3,2),1), (2,(),3), (2,(1,2,3),1) ]</pre></div>
<p><a id="X836F4692839F4874" name="X836F4692839F4874"></a></p>
<h5>51.4-4 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>
<p>is <code class="keyw">true</code> if and only if the semigroup <var class="Arg">S</var> has no proper ideals.</p>
<p><a id="X8193A60F839C064E" name="X8193A60F839C064E"></a></p>
<h5>51.4-5 IsZeroSimpleSemigroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsZeroSimpleSemigroup</code>( <var class="Arg">S</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>is <code class="keyw">true</code> if and only if the semigroup has no proper ideals except for 0, where <var class="Arg">S</var> is a semigroup with zero. If the semigroup does not find its zero, then a break-loop is entered.</p>
<p><a id="X85F7E5CD86F0643B" name="X85F7E5CD86F0643B"></a></p>
<h5>51.4-6 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>
<p>is <code class="keyw">true</code> if and only if the semigroup <var class="Arg">S</var> is a group with zero adjoined.</p>
<p><a id="X7FFEC81F7F2C4EAA" name="X7FFEC81F7F2C4EAA"></a></p>
<h5>51.4-7 IsReesCongruenceSemigroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsReesCongruenceSemigroup</code>( <var class="Arg">S</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>returns <code class="keyw">true</code> if <var class="Arg">S</var> is a Rees Congruence semigroup, that is, if all congruences of <var class="Arg">S</var> are Rees Congruences.</p>
<p><a id="X83F1529479D56665" name="X83F1529479D56665"></a></p>
<h5>51.4-8 IsInverseSemigroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsInverseSemigroup</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">‣ IsInverseMonoid</code>( <var class="Arg">S</var> )</td><td class="tdright">( category )</td></tr></table></div>
<p>Returns: <code class="keyw">true</code> or <code class="keyw">false</code>.</p>
<p>A semigroup <var class="Arg">S</var> is an <em>inverse semigroup</em> if every element <code class="code">x</code> in <var class="Arg">S</var> has a unique semigroup inverse, that is, a unique element <code class="code">y</code> in <var class="Arg">S</var> such that <code class="code">x*y*x=x</code> and <code class="code">y*x*y=y</code>.</p>
<p>A monoid that happens to be an inverse semigroup is called an <em>inverse monoid</em>; see <code class="func">IsMonoid</code> (<a href="chap51_mj.html#X861C523483C6248C"><span class="RefLink">51.2-1</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">S := Semigroup([</span>
<span class="GAPprompt">></span> <span class="GAPinput"> Transformation([1, 2, 4, 5, 6, 3, 7, 8]),</span>
<span class="GAPprompt">></span> <span class="GAPinput"> Transformation([3, 3, 4, 5, 6, 2, 7, 8]),</span>
<span class="GAPprompt">></span> <span class="GAPinput"> Transformation([1, 2, 5, 3, 6, 8, 4, 4])]);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsInverseSemigroup(S);</span>
true</pre></div>
<p><a id="X7BB32D508183C0F1" name="X7BB32D508183C0F1"></a></p>
<h4>51.5 <span class="Heading">Ideals of semigroups</span></h4>
<p>Ideals of semigroups are the same as ideals of the semigroup when considered as a magma. For documentation on ideals for magmas, see <code class="func">Magma</code> (<a href="chap35_mj.html#X839147CF813312D6"><span class="RefLink">35.2-1</span></a>).</p>
<p><a id="X7D5CEE4D7D4318ED" name="X7D5CEE4D7D4318ED"></a></p>
<h5>51.5-1 SemigroupIdealByGenerators</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SemigroupIdealByGenerators</code>( <var class="Arg">S</var>, <var class="Arg">gens</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p><var class="Arg">S</var> is a semigroup, <var class="Arg">gens</var> is a list of elements of <var class="Arg">S</var>. Returns the two-sided ideal of <var class="Arg">S</var> generated by <var class="Arg">gens</var>.</p>
<p><a id="X7F01FFB18125DED5" name="X7F01FFB18125DED5"></a></p>
<h5>51.5-2 ReesCongruenceOfSemigroupIdeal</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ReesCongruenceOfSemigroupIdeal</code>( <var class="Arg">I</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>A two sided ideal <var class="Arg">I</var> of a semigroup <var class="Arg">S</var> defines a congruence on <var class="Arg">S</var> given by <span class="SimpleMath">\(\Delta \cup I \times I\)</span>.</p>
<p><a id="X7A3FF85984345540" name="X7A3FF85984345540"></a></p>
<h5>51.5-3 IsLeftSemigroupIdeal</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsLeftSemigroupIdeal</code>( <var class="Arg">I</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">‣ IsRightSemigroupIdeal</code>( <var class="Arg">I</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">‣ IsSemigroupIdeal</code>( <var class="Arg">I</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Categories of semigroup ideals.</p>
<p><a id="X7914691E7DFFE27A" name="X7914691E7DFFE27A"></a></p>
<h4>51.6 <span class="Heading">Congruences on semigroups</span></h4>
<p>An equivalence or a congruence on a semigroup is the equivalence or congruence on the semigroup considered as a magma. So, to deal with equivalences and congruences on semigroups, magma functions are used. For documentation on equivalences and congruences on magmas, see <code class="func">Magma</code> (<a href="chap35_mj.html#X839147CF813312D6"><span class="RefLink">35.2-1</span></a>).</p>
<p><a id="X78E34B737F0E009F" name="X78E34B737F0E009F"></a></p>
<h5>51.6-1 IsSemigroupCongruence</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsSemigroupCongruence</code>( <var class="Arg">c</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>a magma congruence <var class="Arg">c</var> on a semigroup.</p>
<p><a id="X822DB78579BCB7B5" name="X822DB78579BCB7B5"></a></p>
<h5>51.6-2 IsReesCongruence</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsReesCongruence</code>( <var class="Arg">c</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>returns <code class="keyw">true</code> if and only if the congruence <var class="Arg">c</var> has at most one nonsingleton congruence class.</p>
<p><a id="X87CE9EAB7EE3A128" name="X87CE9EAB7EE3A128"></a></p>
<h4>51.7 <span class="Heading">Quotients</span></h4>
<p>Given a semigroup and a congruence on the semigroup, one can construct a new semigroup: the quotient semigroup. The following functions deal with quotient semigroups in <strong class="pkg">GAP</strong>. For a semigroup <span class="SimpleMath">\(S\)</span>, elements of a quotient semigroup are equivalence classes of elements of the <code class="func">QuotientSemigroupPreimage</code> (<a href="chap51_mj.html#X87120C46808F7289"><span class="RefLink">51.7-3</span></a>) value under the congruence given by the value of <code class="func">QuotientSemigroupCongruence</code> (<a href="chap51_mj.html#X87120C46808F7289"><span class="RefLink">51.7-3</span></a>).</p>
<p>It is probably most useful for calculating the elements of the equivalence classes by using <code class="func">Elements</code> (<a href="chap30_mj.html#X79B130FC7906FB4C"><span class="RefLink">30.3-11</span></a>) or by looking at the images of elements of <code class="func">QuotientSemigroupPreimage</code> (<a href="chap51_mj.html#X87120C46808F7289"><span class="RefLink">51.7-3</span></a>) under the map returned by <code class="func">QuotientSemigroupHomomorphism</code> (<a href="chap51_mj.html#X87120C46808F7289"><span class="RefLink">51.7-3</span></a>), which maps the <code class="func">QuotientSemigroupPreimage</code> (<a href="chap51_mj.html#X87120C46808F7289"><span class="RefLink">51.7-3</span></a>) value to <var class="Arg">S</var>.</p>
<p>For intensive computations in a quotient semigroup, it is probably worthwhile finding another representation as the equality test could involve enumeration of the elements of the congruence classes being compared.</p>
<p><a id="X80EF3E6F842BE64E" name="X80EF3E6F842BE64E"></a></p>
<h5>51.7-1 IsQuotientSemigroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsQuotientSemigroup</code>( <var class="Arg">S</var> )</td><td class="tdright">( category )</td></tr></table></div>
<p>is the category of semigroups constructed from another semigroup and a congruence on it.</p>
<p><a id="X7CAD3D1687956F7F" name="X7CAD3D1687956F7F"></a></p>
<h5>51.7-2 HomomorphismQuotientSemigroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ HomomorphismQuotientSemigroup</code>( <var class="Arg">cong</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>for a congruence <var class="Arg">cong</var> and a semigroup <var class="Arg">S</var>. Returns the homomorphism from <var class="Arg">S</var> to the quotient of <var class="Arg">S</var> by <var class="Arg">cong</var>.</p>
<p><a id="X87120C46808F7289" name="X87120C46808F7289"></a></p>
<h5>51.7-3 QuotientSemigroupPreimage</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ QuotientSemigroupPreimage</code>( <var class="Arg">S</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ QuotientSemigroupCongruence</code>( <var class="Arg">S</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ QuotientSemigroupHomomorphism</code>( <var class="Arg">S</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>for a quotient semigroup <var class="Arg">S</var>.</p>
<p><a id="X80C6C718801855E9" name="X80C6C718801855E9"></a></p>
<h4>51.8 <span class="Heading">Green's Relations</span></h4>
<p>Green's equivalence relations play a very important role in semigroup theory. In this section we describe how they can be used in <strong class="pkg">GAP</strong>.</p>
<p>The five Green's relations are <span class="SimpleMath">\(R\)</span>, <span class="SimpleMath">\(L\)</span>, <span class="SimpleMath">\(J\)</span>, <span class="SimpleMath">\(H\)</span>, <span class="SimpleMath">\(D\)</span>: two elements <span class="SimpleMath">\(x\)</span>, <span class="SimpleMath">\(y\)</span> from a semigroup <span class="SimpleMath">\(S\)</span> are <span class="SimpleMath">\(R\)</span>-related if and only if <span class="SimpleMath">\(xS^1 = yS^1\)</span>, <span class="SimpleMath">\(L\)</span>-related if and only if <span class="SimpleMath">\(S^1 x = S^1 y\)</span> and <span class="SimpleMath">\(J\)</span>-related if and only if <span class="SimpleMath">\(S^1 xS^1 = S^1 yS^1\)</span>; finally, <span class="SimpleMath">\(H = R \wedge L\)</span>, and <span class="SimpleMath">\(D = R \circ L\)</span>.</p>
<p>Recall that relations <span class="SimpleMath">\(R\)</span>, <span class="SimpleMath">\(L\)</span> and <span class="SimpleMath">\(J\)</span> induce a partial order among the elements of the semigroup <span class="SimpleMath">\(S\)</span>: for two elements <span class="SimpleMath">\(x\)</span>, <span class="SimpleMath">\(y\)</span> from <span class="SimpleMath">\(S\)</span>, we say that <span class="SimpleMath">\(x\)</span> is less than or equal to <span class="SimpleMath">\(y\)</span> in the order on <span class="SimpleMath">\(R\)</span> if <span class="SimpleMath">\(xS^1 \subseteq yS^1\)</span>; similarly, <span class="SimpleMath">\(x\)</span> is less than or equal to <span class="SimpleMath">\(y\)</span> under <span class="SimpleMath">\(L\)</span> if <span class="SimpleMath">\(S^1x \subseteq S^1y\)</span>; finally <span class="SimpleMath">\(x\)</span> is less than or equal to <span class="SimpleMath">\(y\)</span> under <span class="SimpleMath">\(J\)</span> if <span class="SimpleMath">\(S^1 xS^1 \subseteq S^1 tS^1\)</span>. We extend this preorder to a partial order on equivalence classes in the natural way.</p>
<p><a id="X786CEDD4814A9079" name="X786CEDD4814A9079"></a></p>
<h5>51.8-1 GreensRRelation</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GreensRRelation</code>( <var class="Arg">semigroup</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GreensLRelation</code>( <var class="Arg">semigroup</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GreensJRelation</code>( <var class="Arg">semigroup</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GreensDRelation</code>( <var class="Arg">semigroup</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GreensHRelation</code>( <var class="Arg">semigroup</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>The Green's relations (which are equivalence relations) are attributes of the semigroup <var class="Arg">semigroup</var>.</p>
<p><a id="X8364D69987D49DE1" name="X8364D69987D49DE1"></a></p>
<h5>51.8-2 IsGreensRelation</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsGreensRelation</code>( <var class="Arg">bin-relation</var> )</td><td class="tdright">( filter )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsGreensRRelation</code>( <var class="Arg">equiv-relation</var> )</td><td class="tdright">( filter )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsGreensLRelation</code>( <var class="Arg">equiv-relation</var> )</td><td class="tdright">( filter )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsGreensJRelation</code>( <var class="Arg">equiv-relation</var> )</td><td class="tdright">( filter )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsGreensHRelation</code>( <var class="Arg">equiv-relation</var> )</td><td class="tdright">( filter )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsGreensDRelation</code>( <var class="Arg">equiv-relation</var> )</td><td class="tdright">( filter )</td></tr></table></div>
<p>Categories for the Green's relations.</p>
<p><a id="X82A11A087AFB3EB0" name="X82A11A087AFB3EB0"></a></p>
<h5>51.8-3 IsGreensClass</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsGreensClass</code>( <var class="Arg">equiv-class</var> )</td><td class="tdright">( filter )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsGreensRClass</code>( <var class="Arg">equiv-class</var> )</td><td class="tdright">( filter )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsGreensLClass</code>( <var class="Arg">equiv-class</var> )</td><td class="tdright">( filter )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsGreensJClass</code>( <var class="Arg">equiv-class</var> )</td><td class="tdright">( filter )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsGreensHClass</code>( <var class="Arg">equiv-class</var> )</td><td class="tdright">( filter )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsGreensDClass</code>( <var class="Arg">equiv-class</var> )</td><td class="tdright">( filter )</td></tr></table></div>
<p>return <code class="keyw">true</code> if the equivalence class <var class="Arg">equiv-class</var> is a Green's class of any type, or of <span class="SimpleMath">\(R\)</span>, <span class="SimpleMath">\(L\)</span>, <span class="SimpleMath">\(J\)</span>, <span class="SimpleMath">\(H\)</span>, <span class="SimpleMath">\(D\)</span> type, respectively, or <code class="keyw">false</code> otherwise.</p>
<p><a id="X7AA204C8850F9070" name="X7AA204C8850F9070"></a></p>
<h5>51.8-4 IsGreensLessThanOrEqual</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsGreensLessThanOrEqual</code>( <var class="Arg">C1</var>, <var class="Arg">C2</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>returns <code class="keyw">true</code> if the Green's class <var class="Arg">C1</var> is less than or equal to <var class="Arg">C2</var> under the respective ordering (as defined above), and <code class="keyw">false</code> otherwise.</p>
<p>Only defined for <span class="SimpleMath">\(R\)</span>, <span class="SimpleMath">\(L\)</span> and <span class="SimpleMath">\(J\)</span> classes.</p>
<p><a id="X86FE5F5585EBCF13" name="X86FE5F5585EBCF13"></a></p>
<h5>51.8-5 RClassOfHClass</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RClassOfHClass</code>( <var class="Arg">H</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LClassOfHClass</code>( <var class="Arg">H</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>are attributes reflecting the natural ordering over the various Green's classes. <code class="func">RClassOfHClass</code> and <code class="func">LClassOfHClass</code> return the <span class="SimpleMath">\(R\)</span> and <span class="SimpleMath">\(L\)</span> classes, respectively, in which an <span class="SimpleMath">\(H\)</span> class is contained.</p>
<p><a id="X78C56F4A78E0088A" name="X78C56F4A78E0088A"></a></p>
<h5>51.8-6 EggBoxOfDClass</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ EggBoxOfDClass</code>( <var class="Arg">Dclass</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns for a Green's <span class="SimpleMath">\(D\)</span> class <var class="Arg">Dclass</var> a matrix whose rows represent <span class="SimpleMath">\(R\)</span> classes and columns represent <span class="SimpleMath">\(L\)</span> classes. The entries are the <span class="SimpleMath">\(H\)</span> classes.</p>
<p><a id="X803237F17ACD44E3" name="X803237F17ACD44E3"></a></p>
<h5>51.8-7 DisplayEggBoxOfDClass</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DisplayEggBoxOfDClass</code>( <var class="Arg">Dclass</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>displays a <q>picture</q> of the <span class="SimpleMath">\(D\)</span> class <var class="Arg">Dclass</var>, as an array of 1s and 0s. A 1 represents a group <span class="SimpleMath">\(H\)</span> class.</p>
<p><a id="X87C75A9D86122D93" name="X87C75A9D86122D93"></a></p>
<h5>51.8-8 GreensRClassOfElement</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GreensRClassOfElement</code>( <var class="Arg">S</var>, <var class="Arg">a</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GreensLClassOfElement</code>( <var class="Arg">S</var>, <var class="Arg">a</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GreensDClassOfElement</code>( <var class="Arg">S</var>, <var class="Arg">a</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GreensJClassOfElement</code>( <var class="Arg">S</var>, <var class="Arg">a</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GreensHClassOfElement</code>( <var class="Arg">S</var>, <var class="Arg">a</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Creates the <span class="SimpleMath">\(X\)</span> class of the element <var class="Arg">a</var> in the semigroup <var class="Arg">S</var> where <span class="SimpleMath">\(X\)</span> is one of <span class="SimpleMath">\(L\)</span>, <span class="SimpleMath">\(R\)</span>, <span class="SimpleMath">\(D\)</span>, <span class="SimpleMath">\(J\)</span>, or <span class="SimpleMath">\(H\)</span>.</p>
<p><a id="X844D20467A644811" name="X844D20467A644811"></a></p>
<h5>51.8-9 GreensRClasses</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GreensRClasses</code>( <var class="Arg">S</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GreensLClasses</code>( <var class="Arg">S</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GreensHClasses</code>( <var class="Arg">S</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GreensJClasses</code>( <var class="Arg">S</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GreensDClasses</code>( <var class="Arg">S</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>If <var class="Arg">S</var> is a semigroup, then these attributes return the Green's <span class="SimpleMath">\(R\)</span>-, <span class="SimpleMath">\(L\)</span>-, <span class="SimpleMath">\(H\)</span>-, <span class="SimpleMath">\(J\)</span>-, or <span class="SimpleMath">\(D\)</span>-classes, respectively for the semigroup <var class="Arg">S</var>.</p>
<p>Additionally, if <var class="Arg">S</var> is a Green's <span class="SimpleMath">\(D\)</span>-class of a semigroup, then <code class="code">GreensRClasses</code> and <code class="code">GreensLClasses</code> return the Green's <span class="SimpleMath">\(R\)</span>- or <span class="SimpleMath">\(L-\)</span>classes of the semigroup, respectively, contained in the <span class="SimpleMath">\(D\)</span>-class <var class="Arg">S</var>; if <var class="Arg">S</var> is a Green's <span class="SimpleMath">\(D\)</span>-, <span class="SimpleMath">\(R\)</span>-, or <span class="SimpleMath">\(L\)</span>-class of a semigroup, then <code class="code">GreensHClasses</code> returns the Green's <span class="SimpleMath">\(H\)</span>-classes of the semigroup contained in the Green's class <var class="Arg">S</var>.</p>
<p><code class="func">EquivalenceClasses</code> (<a href="chap33_mj.html#X879439897EF4D728"><span class="RefLink">33.7-3</span></a>) for a Green's relation lead to one of these functions.</p>
<p><a id="X7CB4A18685B850E2" name="X7CB4A18685B850E2"></a></p>
<h5>51.8-10 GroupHClassOfGreensDClass</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GroupHClassOfGreensDClass</code>( <var class="Arg">Dclass</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>for a <span class="SimpleMath">\(D\)</span> class <var class="Arg">Dclass</var> of a semigroup, returns a group <span class="SimpleMath">\(H\)</span> class of the <span class="SimpleMath">\(D\)</span> class, or <code class="keyw">fail</code> if there is no group <span class="SimpleMath">\(H\)</span> class.</p>
<p><a id="X79D740EF7F0E53BD" name="X79D740EF7F0E53BD"></a></p>
<h5>51.8-11 IsGroupHClass</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsGroupHClass</code>( <var class="Arg">Hclass</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>returns <code class="keyw">true</code> if the Green's <span class="SimpleMath">\(H\)</span> class <var class="Arg">Hclass</var> is a group, which in turn is true if and only if <var class="Arg">Hclass</var><span class="SimpleMath">\(^2\)</span> intersects <var class="Arg">Hclass</var>.</p>
<p><a id="X7F5860927CAD920F" name="X7F5860927CAD920F"></a></p>
<h5>51.8-12 IsRegularDClass</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsRegularDClass</code>( <var class="Arg">Dclass</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>returns <code class="keyw">true</code> if the Greens <span class="SimpleMath">\(D\)</span> class <var class="Arg">Dclass</var> is regular. A <span class="SimpleMath">\(D\)</span> class is regular if and only if each of its elements is regular, which in turn is true if and only if any one element of <var class="Arg">Dclass</var> is regular. Idempotents are regular since <span class="SimpleMath">\(eee = e\)</span> so it follows that a Green's <span class="SimpleMath">\(D\)</span> class containing an idempotent is regular. Conversely, it is true that a regular <span class="SimpleMath">\(D\)</span> class must contain at least one idempotent. (See <a href="chapBib_mj.html#biBHowie76">[How76, Prop. 3.2]</a>.)</p>
<p><a id="X81AF2EAB7CEF8C19" name="X81AF2EAB7CEF8C19"></a></p>
<h5>51.8-13 DisplaySemigroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DisplaySemigroup</code>( <var class="Arg">S</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Produces a convenient display of a transformation semigroup's D-Class structure. Let <var class="Arg">S</var> be a transformation semigroup of degree <span class="SimpleMath">\(n\)</span>. Then for each <span class="SimpleMath">\(r\leq n\)</span>, we show all D-classes of rank <span class="SimpleMath">\(r\)</span>.</p>
<p>A regular D-class with a single H-class of size 120 appears as</p>
<div class="example"><pre>
*[H size = 120, 1 L-class, 1 R-class]
</pre></div>
<p>(the <code class="code">*</code> denoting regularity).</p>
<p><a id="X8225A9EC87A255E6" name="X8225A9EC87A255E6"></a></p>
<h4>51.9 <span class="Heading">Rees Matrix Semigroups</span></h4>
<p>In this section, we describe the functions in <strong class="pkg">GAP</strong> for Rees matrix and 0-matrix semigroups and their subsemigroups. The importance of these semigroups lies in the fact that Rees matrix semigroups over groups are exactly the completely simple semigroups, and Rees 0-matrix semigroups over groups are the completely 0-simple semigroups.</p>
<p>Let <span class="SimpleMath">\(I\)</span> and <span class="SimpleMath">\(J\)</span> be sets, let <span class="SimpleMath">\(S\)</span> be a semigroup, and let <span class="SimpleMath">\(P=(p_{ji})_{j\in J, i\in I}\)</span> be a <span class="SimpleMath">\(|J|\times |I|\)</span> matrix with entries in <span class="SimpleMath">\(S\)</span>. Then the <em>Rees matrix semigroup</em> with underlying semigroup <span class="SimpleMath">\(S\)</span> and matrix <span class="SimpleMath">\(P\)</span> is just the direct product <span class="SimpleMath">\(I\times S \times J\)</span> with multiplication defined by</p>
<p class="center">\[(i, s, j)(k, t, l)=(i,s\cdot p_{j,k}\cdot t, l).\]</p>
<p>Rees 0-matrix semigroups are defined as follows. If <span class="SimpleMath">\(I\)</span>, <span class="SimpleMath">\(J\)</span>, <span class="SimpleMath">\(S\)</span>, and <span class="SimpleMath">\(P\)</span> are as above and <span class="SimpleMath">\(0\)</span> denotes a new element, then the <em>Rees 0-matrix semigroup</em> with underlying semigroup <span class="SimpleMath">\(S\)</span> and matrix <span class="SimpleMath">\(P\)</span> is <span class="SimpleMath">\((I\times S\times J)\cup \{0\}\)</span> with multiplication defined by</p>
<p class="center">\[(i, s, j)(k, t, l)=(i, s\cdot p_{j,k}\cdot t, l)\]</p>
<p>when <span class="SimpleMath">\(p_{j,k}\)</span> is not <span class="SimpleMath">\(0\)</span> and <span class="SimpleMath">\(0\)</span> if <span class="SimpleMath">\(p_{j,k}\)</span> is 0.</p>
<p>If <span class="SimpleMath">\(R\)</span> is a Rees matrix or 0-matrix semigroup, then the <em>rows</em> of <span class="SimpleMath">\(R\)</span> is the index set <span class="SimpleMath">\(I\)</span>, the <em>columns</em> of <span class="SimpleMath">\(R\)</span> is the index set <span class="SimpleMath">\(J\)</span>, the semigroup <span class="SimpleMath">\(S\)</span> is the <em>underlying semigroup</em> of <span class="SimpleMath">\(R\)</span>, and the <em>matrix</em> <span class="SimpleMath">\(P\)</span> is the matrix of <span class="SimpleMath">\(S\)</span>.</p>
<p>Thoroughout this section, wherever the distinction is unimportant, we will refer to Rees matrix or 0-matrix semigroups collectively as Rees matrix semigroups.</p>
<p>Multiplication of elements of a Rees matrix semigroup obviously depends on the matrix used to create the semigroup. Hence elements of a Rees matrix semigroup can only be created with reference to the semigroup to which they belong. More specifically, every collection or semigroup of Rees matrix semigroup elements is created from a specific Rees matrix semigroup, which contains the whole family of its elements. So, it is not possible to multiply or compare elements belonging to distinct Rees matrix semigroups, since they belong to different families. For example, this situation is similar to free groups, but it is different to permutations, which belong to a single family, and where arbitrary permutations can be compared and multiplied without reference to any group containing them.</p>
<p>A subsemigroup of a Rees matrix semigroup is not necessarily a Rees matrix semigroup. Every semigroup consisting of elements of a Rees matrix semigroup satisfies the property <code class="func">IsReesMatrixSubsemigroup</code> (<a href="chap51_mj.html#X7F03BE707AC7F8A0"><span class="RefLink">51.9-6</span></a>) and every semigroup of Rees 0-matrix semigroup elements satisfies <code class="func">IsReesZeroMatrixSubsemigroup</code> (<a href="chap51_mj.html#X7F03BE707AC7F8A0"><span class="RefLink">51.9-6</span></a>).</p>
<p>Rees matrix and 0-matrix semigroups can be created using the operations <code class="func">ReesMatrixSemigroup</code> (<a href="chap51_mj.html#X8526AA557CDF6C49"><span class="RefLink">51.9-1</span></a>) and <code class="func">ReesZeroMatrixSemigroup</code> (<a href="chap51_mj.html#X8526AA557CDF6C49"><span class="RefLink">51.9-1</span></a>), respectively, from an underlying semigroup and a matrix. Rees matrix semigroups created in this way contain the whole family of their elements. Every element of a Rees matrix semigroup belongs to a unique semigroup created in this way; every subsemigroup of a Rees matrix semigroup is a subsemigroup of a unique semigroup created in this way.</p>
<p>Subsemigroups of Rees matrix semigroups can also be created by specifying generators. A subsemigroup of a Rees matrix semigroup <span class="SimpleMath">\(I\times U\times J\)</span> satisfies <code class="func">IsReesMatrixSemigroup</code> (<a href="chap51_mj.html#X780BB78A79275244"><span class="RefLink">51.9-7</span></a>) if and only if it is equal to <span class="SimpleMath">\(I'\times U'\times J'\)</span> where <span class="SimpleMath">\(I'\subseteq I\)</span>, <span class="SimpleMath">\(J'\subseteq J\)</span>, and <span class="SimpleMath">\(U'\)</span> is a subsemigroup of <span class="SimpleMath">\(U\)</span>. The analogous statements holds for Rees 0-matrix semigroups.</p>
<p>It is not necessarily the case that a simple subsemigroups of Rees matrix semigroups satisfies <code class="func">IsReesMatrixSemigroup</code> (<a href="chap51_mj.html#X780BB78A79275244"><span class="RefLink">51.9-7</span></a>). A Rees matrix semigroup is simple if and only if its underlying semigroup is simple. A finite semigroup is simple if and only if it is isomorphic to a Rees matrix semigroup over a group; this isomorphism can be obtained explicitly using <code class="func">IsomorphismReesMatrixSemigroup</code> (<a href="chap51_mj.html#X7964B5C97FB9C07D"><span class="RefLink">51.9-3</span></a>).</p>
<p>Similarly, 0-simple subsemigroups of Rees 0-matrix semigroups do not have to satisfy <code class="func">IsReesZeroMatrixSemigroup</code> (<a href="chap51_mj.html#X780BB78A79275244"><span class="RefLink">51.9-7</span></a>). A Rees 0-matrix semigroup with more than 2 elements is 0-simple if and only if every row and every column of its matrix contains a non-zero entry, and its underlying semigroup is simple. A finite semigroup is 0-simple if and only if it is isomorphic to a Rees 0-matrix semigroup over a group; again this isomorphism can be found by using <code class="func">IsomorphismReesZeroMatrixSemigroup</code> (<a href="chap51_mj.html#X7964B5C97FB9C07D"><span class="RefLink">51.9-3</span></a>).</p>
<p>Elements of a Rees matrix or 0-matrix semigroup belong to the categories <code class="func">IsReesMatrixSemigroupElement</code> (<a href="chap51_mj.html#X7F6B852B81488C86"><span class="RefLink">51.9-4</span></a>) and <code class="func">IsReesZeroMatrixSemigroupElement</code> (<a href="chap51_mj.html#X7F6B852B81488C86"><span class="RefLink">51.9-4</span></a>), respectively. Such elements can be created directly using the functions <code class="func">ReesMatrixSemigroupElement</code> (<a href="chap51_mj.html#X7A0DE1F28470295E"><span class="RefLink">51.9-5</span></a>) and <code class="func">ReesZeroMatrixSemigroupElement</code> (<a href="chap51_mj.html#X7A0DE1F28470295E"><span class="RefLink">51.9-5</span></a>).</p>
<p>A semigroup in <strong class="pkg">GAP</strong> can either satisfies <code class="func">IsReesMatrixSemigroup</code> (<a href="chap51_mj.html#X780BB78A79275244"><span class="RefLink">51.9-7</span></a>) or <code class="func">IsReesZeroMatrixSemigroup</code> (<a href="chap51_mj.html#X780BB78A79275244"><span class="RefLink">51.9-7</span></a>) but not both.</p>
<p><a id="X8526AA557CDF6C49" name="X8526AA557CDF6C49"></a></p>
<h5>51.9-1 ReesMatrixSemigroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ReesMatrixSemigroup</code>( <var class="Arg">S</var>, <var class="Arg">mat</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ReesZeroMatrixSemigroup</code>( <var class="Arg">S</var>, <var class="Arg">mat</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: A Rees matrix or 0-matrix semigroup.</p>
<p>When <var class="Arg">S</var> is a semigroup and <var class="Arg">mat</var> is an <code class="code">m</code> by <code class="code">n</code> matrix with entries in <var class="Arg">S</var>, the function <code class="func">ReesMatrixSemigroup</code> returns the <code class="code">n</code> by <code class="code">m</code> Rees matrix semigroup over <var class="Arg">S</var> with multiplication defined by <var class="Arg">mat</var>.</p>
<p>The arguments of <code class="func">ReesZeroMatrixSemigroup</code> should be a semigroup <var class="Arg">S</var> and an <code class="code">m</code> by <code class="code">n</code> matrix <var class="Arg">mat</var> with entries in <var class="Arg">S</var> or equal to the integer <code class="code">0</code>. <code class="func">ReesZeroMatrixSemigroup</code> returns the <code class="code">n</code> by <code class="code">m</code> Rees 0-matrix semigroup over <var class="Arg">S</var> with multiplication defined by <var class="Arg">mat</var>. In <strong class="pkg">GAP</strong> a Rees 0-matrix semigroup always contains a multiplicative zero element, regardless of whether there are any entries in <var class="Arg">mat</var> which are equal to <code class="code">0</code>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G:=Random(AllSmallGroups(Size, 32));;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">mat:=List([1..5], x-> List([1..3], y-> Random(G)));;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">S:=ReesMatrixSemigroup(G, mat);</span>
<Rees matrix semigroup 3x5 over <pc group of size 32 with
5 generators>>
<span class="GAPprompt">gap></span> <span class="GAPinput">mat:=[[(), 0, (), ()], [0, 0, 0, 0]];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">S:=ReesZeroMatrixSemigroup(DihedralGroup(IsPermGroup, 8), mat);</span>
<Rees 0-matrix semigroup 4x2 over Group([ (1,2,3,4), (2,4) ])>
</pre></div>
<p><a id="X78D2A48C87FC8E38" name="X78D2A48C87FC8E38"></a></p>
<h5>51.9-2 ReesMatrixSubsemigroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ReesMatrixSubsemigroup</code>( <var class="Arg">R</var>, <var class="Arg">I</var>, <var class="Arg">U</var>, <var class="Arg">J</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ReesZeroMatrixSubsemigroup</code>( <var class="Arg">R</var>, <var class="Arg">I</var>, <var class="Arg">U</var>, <var class="Arg">J</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: A Rees matrix or 0-matrix subsemigroup.</p>
<p>The arguments of <code class="func">ReesMatrixSubsemigroup</code> should be a Rees matrix semigroup <var class="Arg">R</var>, subsets <var class="Arg">I</var> and <var class="Arg">J</var> of the rows and columns of <var class="Arg">R</var>, respectively, and a subsemigroup <var class="Arg">U</var> of the underlying semigroup of <var class="Arg">R</var>. <code class="func">ReesMatrixSubsemigroup</code> returns the subsemigroup of <var class="Arg">R</var> generated by the direct product of <var class="Arg">I</var>, <var class="Arg">U</var>, and <var class="Arg">J</var>.</p>
<p>The usage and returned value of <code class="func">ReesZeroMatrixSubsemigroup</code> is analogous when <var class="Arg">R</var> is a Rees 0-matrix semigroup.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G:=CyclicGroup(IsPermGroup, 1007);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">mat:=[[(), 0, 0], [0, (), 0], [0, 0, ()],</span>
<span class="GAPprompt">></span> <span class="GAPinput">[(), (), ()], [0, 0, ()]];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">R:=ReesZeroMatrixSemigroup(G, mat);</span>
<Rees 0-matrix semigroup 3x5 over
<permutation group of size 1007 with 1 generator>>
<span class="GAPprompt">gap></span> <span class="GAPinput">ReesZeroMatrixSubsemigroup(R, [1,3], G, [1..5]);</span>
<Rees 0-matrix semigroup 2x5 over
<permutation group of size 1007 with 1 generator>>
</pre></div>
<p><a id="X7964B5C97FB9C07D" name="X7964B5C97FB9C07D"></a></p>
<h5>51.9-3 IsomorphismReesMatrixSemigroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsomorphismReesMatrixSemigroup</code>( <var class="Arg">S</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsomorphismReesZeroMatrixSemigroup</code>( <var class="Arg">S</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: An isomorphism.</p>
<p>Every finite simple semigroup is isomorphic to a Rees matrix semigroup over a group, and every finite 0-simple semigroup is isomorphic to a Rees 0-matrix semigroup over a group.</p>
<p>If the argument <var class="Arg">S</var> is a simple semigroup, then <code class="func">IsomorphismReesMatrixSemigroup</code> returns an isomorphism to a Rees matrix semigroup over a group. If <var class="Arg">S</var> is not simple, then <code class="func">IsomorphismReesMatrixSemigroup</code> returns an error.</p>
<p>If the argument <var class="Arg">S</var> is a 0-simple semigroup, then <code class="func">IsomorphismReesZeroMatrixSemigroup</code> returns an isomorphism to a Rees 0-matrix semigroup over a group. If <var class="Arg">S</var> is not 0-simple, then <code class="func">IsomorphismReesZeroMatrixSemigroup</code> returns an error.</p>
<p>See <code class="func">IsSimpleSemigroup</code> (<a href="chap51_mj.html#X836F4692839F4874"><span class="RefLink">51.4-4</span></a>) and <code class="func">IsZeroSimpleSemigroup</code> (<a href="chap51_mj.html#X8193A60F839C064E"><span class="RefLink">51.4-5</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">S := Semigroup(Transformation([2, 1, 1, 2, 1]),</span>
<span class="GAPprompt">></span> <span class="GAPinput"> Transformation([3, 4, 3, 4, 4]),</span>
<span class="GAPprompt">></span> <span class="GAPinput"> Transformation([3, 4, 3, 4, 3]),</span>
<span class="GAPprompt">></span> <span class="GAPinput"> Transformation([4, 3, 3, 4, 4]));;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsSimpleSemigroup(S);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">Range(IsomorphismReesMatrixSemigroup(S));</span>
<Rees matrix semigroup 4x2 over Group([ (1,2) ])>
<span class="GAPprompt">gap></span> <span class="GAPinput">mat := [[(), 0, 0],</span>
<span class="GAPprompt">></span> <span class="GAPinput"> [0, (), 0],</span>
<span class="GAPprompt">></span> <span class="GAPinput"> [0, 0, ()]];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">R := ReesZeroMatrixSemigroup(Group((1,2,4,5,6)), mat);</span>
<Rees 0-matrix semigroup 3x3 over Group([ (1,2,4,5,6) ])>
<span class="GAPprompt">gap></span> <span class="GAPinput">U := ReesZeroMatrixSubsemigroup(R, [1, 2], Group(()), [2, 3]);</span>
<subsemigroup of 3x3 Rees 0-matrix semigroup with 4 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsZeroSimpleSemigroup(U);</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">U := ReesZeroMatrixSubsemigroup(R, [2, 3], Group(()), [2, 3]);</span>
<subsemigroup of 3x3 Rees 0-matrix semigroup with 3 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsZeroSimpleSemigroup(U);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">Rows(U); Columns(U);</span>
[ 2, 3 ]
[ 2, 3 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">V := Range(IsomorphismReesZeroMatrixSemigroup(U));</span>
<Rees 0-matrix semigroup 2x2 over Group(())>
<span class="GAPprompt">gap></span> <span class="GAPinput">Rows(V); Columns(V);</span>
[ 1, 2 ]
[ 1, 2 ]</pre></div>
<p><a id="X7F6B852B81488C86" name="X7F6B852B81488C86"></a></p>
<h5>51.9-4 IsReesMatrixSemigroupElement</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsReesMatrixSemigroupElement</code>( <var class="Arg">elt</var> )</td><td class="tdright">( category )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsReesZeroMatrixSemigroupElement</code>( <var class="Arg">elt</var> )</td><td class="tdright">( category )</td></tr></table></div>
<p>Returns: <code class="keyw">true</code> or <code class="keyw">false</code>.</p>
<p>Every element of a Rees matrix semigroup belongs to the category <code class="func">IsReesMatrixSemigroupElement</code>, and every element of a Rees 0-matrix semigroup belongs to the category <code class="func">IsReesZeroMatrixSemigroupElement</code>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G:=Group((1,2,3));;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">mat:=[ [ (), (1,3,2) ], [ (1,3,2), () ] ];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">R:=ReesMatrixSemigroup(G, mat);</span>
<Rees matrix semigroup 2x2 over Group([ (1,2,3) ])>
<span class="GAPprompt">gap></span> <span class="GAPinput">GeneratorsOfSemigroup(R);</span>
[ (1,(1,2,3),1), (2,(),2) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">IsReesMatrixSemigroupElement(last[1]);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsReesZeroMatrixSemigroupElement(last2[1]);</span>
false</pre></div>
<p><a id="X7A0DE1F28470295E" name="X7A0DE1F28470295E"></a></p>
<h5>51.9-5 ReesMatrixSemigroupElement</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ReesMatrixSemigroupElement</code>( <var class="Arg">R</var>, <var class="Arg">i</var>, <var class="Arg">x</var>, <var class="Arg">j</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ReesZeroMatrixSemigroupElement</code>( <var class="Arg">R</var>, <var class="Arg">i</var>, <var class="Arg">x</var>, <var class="Arg">j</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: An element of a Rees matrix or <code class="code">0</code>-matrix semigroup.</p>
<p>The arguments of <var class="Arg">ReesMatrixSemigroupElement</var> should be a Rees matrix subsemigroup <var class="Arg">R</var>, elements <var class="Arg">i</var> and <var class="Arg">j</var> of the the rows and columns of <var class="Arg">R</var>, respectively, and an element <var class="Arg">x</var> of the underlying semigroup of <var class="Arg">R</var>. <code class="func">ReesMatrixSemigroupElement</code> returns the element of <var class="Arg">R</var> with row index <var class="Arg">i</var>, underlying element <var class="Arg">x</var> in the underlying semigroup of <var class="Arg">R</var>, and column index <var class="Arg">j</var>, if such an element exist, if such an element exists.</p>
<p>The usage of <code class="func">ReesZeroMatrixSemigroupElement</code> is analogous to that of <code class="func">ReesMatrixSemigroupElement</code>, when <var class="Arg">R</var> is a Rees 0-matrix semigroup.</p>
<p>The row <var class="Arg">i</var>, underlying element <var class="Arg">x</var>, and column <var class="Arg">j</var> of an element <code class="code">y</code> of a Rees matrix (or 0-matrix) semigroup can be recovered from <code class="code">y</code> using <code class="code">y[1]</code>, <code class="code">y[2]</code>, and <code class="code">y[3]</code>, respectively.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G:=Group((1,2,3));;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">mat:=[ [ 0, () ], [ (1,3,2), (1,3,2) ] ];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">R:=ReesZeroMatrixSemigroup(G, mat);</span>
<Rees 0-matrix semigroup 2x2 over Group([ (1,2,3) ])>
<span class="GAPprompt">gap></span> <span class="GAPinput">ReesZeroMatrixSemigroupElement(R, 1, (1,2,3), 2);</span>
(1,(1,2,3),2)
<span class="GAPprompt">gap></span> <span class="GAPinput">MultiplicativeZero(R);</span>
0</pre></div>
<p><a id="X7F03BE707AC7F8A0" name="X7F03BE707AC7F8A0"></a></p>
<h5>51.9-6 IsReesMatrixSubsemigroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsReesMatrixSubsemigroup</code>( <var class="Arg">R</var> )</td><td class="tdright">( synonym )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsReesZeroMatrixSubsemigroup</code>( <var class="Arg">R</var> )</td><td class="tdright">( synonym )</td></tr></table></div>
<p>Returns: <code class="keyw">true</code> or <code class="keyw">false</code>.</p>
<p>Every semigroup consisting of elements of a Rees matrix semigroup satisfies the property <code class="func">IsReesMatrixSubsemigroup</code> and every semigroup of Rees 0-matrix semigroup elements satisfies <code class="func">IsReesZeroMatrixSubsemigroup</code>.</p>
<p>Note that a subsemigroup of a Rees matrix semigroup is not necessarily a Rees matrix semigroup.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G:=DihedralGroup(32);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">mat:=List([1..2], x-> List([1..10], x-> Random(G)));;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">R:=ReesMatrixSemigroup(G, mat);</span>
<Rees matrix semigroup 10x2 over <pc group of size 32 with
5 generators>>
<span class="GAPprompt">gap></span> <span class="GAPinput">S:=Semigroup(GeneratorsOfSemigroup(R));</span>
<subsemigroup of 10x2 Rees matrix semigroup with 14 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsReesMatrixSubsemigroup(S);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">S:=Semigroup(GeneratorsOfSemigroup(R)[1]);</span>
<subsemigroup of 10x2 Rees matrix semigroup with 1 generator>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsReesMatrixSubsemigroup(S);</span>
true</pre></div>
<p><a id="X780BB78A79275244" name="X780BB78A79275244"></a></p>
<h5>51.9-7 IsReesMatrixSemigroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsReesMatrixSemigroup</code>( <var class="Arg">R</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">‣ IsReesZeroMatrixSemigroup</code>( <var class="Arg">R</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns: <code class="keyw">true</code> or <code class="keyw">false</code>.</p>
<p>A subsemigroup of a Rees matrix semigroup <span class="SimpleMath">\(I\times U\times J\)</span> satisfies <code class="func">IsReesMatrixSemigroup</code> if and only if it is equal to <span class="SimpleMath">\(I'\times U'\times J'\)</span> where <span class="SimpleMath">\(I'\subseteq I\)</span>, <span class="SimpleMath">\(J'\subseteq J\)</span>, and <span class="SimpleMath">\(U'\)</span> is a subsemigroup of <span class="SimpleMath">\(U\)</span>. It can be costly to check that a subsemigroup defined by generators satisfies <code class="func">IsReesMatrixSemigroup</code>. The analogous statements holds for Rees 0-matrix semigroups.</p>
<p>It is not necessarily the case that a simple subsemigroups of Rees matrix semigroups satisfies <code class="func">IsReesMatrixSemigroup</code>. A Rees matrix semigroup is simple if and only if its underlying semigroup is simple. A finite semigroup is simple if and only if it is isomorphic to a Rees matrix semigroup over a group; this isomorphism can be obtained explicitly using <code class="func">IsomorphismReesMatrixSemigroup</code> (<a href="chap51_mj.html#X7964B5C97FB9C07D"><span class="RefLink">51.9-3</span></a>).</p>
<p>Similarly, 0-simple subsemigroups of Rees 0-matrix semigroups do not have to satisfy <code class="func">IsReesZeroMatrixSemigroup</code>. A Rees 0-matrix semigroup with more than 2 elements is 0-simple if and only if every row and every column of its matrix contains a non-zero entry, and its underlying semigroup is simple. A finite semigroup is 0-simple if and only if it is isomorphic to a Rees 0-matrix semigroup over a group; again this isomorphism can be found by using <code class="func">IsomorphismReesMatrixSemigroup</code> (<a href="chap51_mj.html#X7964B5C97FB9C07D"><span class="RefLink">51.9-3</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G:=PSL(2,5);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">mat:=[ [ 0, (), 0, (2,6,3,5,4) ],</span>
<span class="GAPprompt">></span> <span class="GAPinput">[ (), 0, (), 0 ], [ 0, 0, 0, () ] ];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">R:=ReesZeroMatrixSemigroup(G, mat);</span>
<Rees 0-matrix semigroup 4x3 over Group([ (3,5)(4,6), (1,2,5)
(3,4,6) ])>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsReesZeroMatrixSemigroup(R);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">U:=ReesZeroMatrixSubsemigroup(R, [1..3], Group(()), [1..2]);</span>
<subsemigroup of 4x3 Rees 0-matrix semigroup with 4 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsReesZeroMatrixSemigroup(U);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">V:=Semigroup(GeneratorsOfSemigroup(U));</span>
<subsemigroup of 4x3 Rees 0-matrix semigroup with 4 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsReesZeroMatrixSemigroup(V);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">S:=Semigroup(Transformation([1,1]), Transformation([1,2]));</span>
<commutative transformation monoid of degree 2 with 1 generator>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsSimpleSemigroup(S);</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">mat:=[[0, One(S), 0, One(S)], [One(S), 0, One(S), 0],</span>
<span class="GAPprompt">></span> <span class="GAPinput">[0, 0, 0, One(S)]];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">R:=ReesZeroMatrixSemigroup(S, mat);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">U:=ReesZeroMatrixSubsemigroup(R, [1..3],</span>
<span class="GAPprompt">></span> <span class="GAPinput">Semigroup(Transformation([1,1])), [1..2]);</span>
<subsemigroup of 4x3 Rees 0-matrix semigroup with 6 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">V:=Semigroup(GeneratorsOfSemigroup(U));</span>
<subsemigroup of 4x3 Rees 0-matrix semigroup with 6 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsReesZeroMatrixSemigroup(V);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">T:=Semigroup(</span>
<span class="GAPprompt">></span> <span class="GAPinput">ReesZeroMatrixSemigroupElement(R, 3, Transformation( [ 1, 1 ] ), 3),</span>
<span class="GAPprompt">></span> <span class="GAPinput">ReesZeroMatrixSemigroupElement(R, 2, Transformation( [ 1, 1 ] ), 2));</span>
<subsemigroup of 4x3 Rees 0-matrix semigroup with 2 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsReesZeroMatrixSemigroup(T);</span>
false</pre></div>
<p><a id="X7CACF4D686AF1D19" name="X7CACF4D686AF1D19"></a></p>
<h5>51.9-8 Matrix</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Matrix</code>( <var class="Arg">R</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MatrixOfReesMatrixSemigroup</code>( <var class="Arg">R</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MatrixOfReesZeroMatrixSemigroup</code>( <var class="Arg">R</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: A matrix.</p>
<p>If <var class="Arg">R</var> is a Rees matrix or 0-matrix semigroup, then <code class="func">MatrixOfReesMatrixSemigroup</code> respectively <code class="func">MatrixOfReesZeroMatrixSemigroup</code> return the matrix used to define multiplication in <var class="Arg">R</var>. For convenience, one may also abbreviate either to <code class="func">Matrix</code>.</p>
<p>More specifically, if <var class="Arg">R</var> is a Rees matrix or 0-matrix semigroup, which is a proper subsemigroup of another such semigroup, then <code class="func">Matrix</code> returns the matrix used to define the Rees matrix (or 0-matrix) semigroup consisting of the whole family to which the elements of <var class="Arg">R</var> belong. Thus, for example, a <code class="code">1</code> by <code class="code">1</code> Rees matrix semigroup can have a <code class="code">65</code> by <code class="code">15</code> matrix.</p>
<p>Arbitrary subsemigroups of Rees matrix or 0-matrix semigroups do not have a matrix. Such a subsemigroup <var class="Arg">R</var> has a matrix if and only if it satisfies <code class="func">IsReesMatrixSemigroup</code> (<a href="chap51_mj.html#X780BB78A79275244"><span class="RefLink">51.9-7</span></a>) or <code class="func">IsReesZeroMatrixSemigroup</code> (<a href="chap51_mj.html#X780BB78A79275244"><span class="RefLink">51.9-7</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G:=AlternatingGroup(5);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">mat:=[[(), (), ()], [(), (), ()]];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">R:=ReesMatrixSemigroup(G, mat);</span>
<Rees matrix semigroup 3x2 over Alt( [ 1 .. 5 ] )>
<span class="GAPprompt">gap></span> <span class="GAPinput">Matrix(R);</span>
[ [ (), (), () ], [ (), (), () ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">R:=ReesMatrixSubsemigroup(R, [1,2], Group(()), [2]);</span>
<subsemigroup of 3x2 Rees matrix semigroup with 2 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">Matrix(R);</span>
[ [ (), (), () ], [ (), (), () ] ]
</pre></div>
<p><a id="X82FC5D6980C66AC4" name="X82FC5D6980C66AC4"></a></p>
<h5>51.9-9 <span class="Heading">Rows and columns</span></h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Rows</code>( <var class="Arg">R</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Columns</code>( <var class="Arg">R</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: The rows or columns of <var class="Arg">R</var>.</p>
<p><code class="func">Rows</code> returns the rows of the Rees matrix or 0-matrix semigroup <var class="Arg">R</var>. Note that the rows of the semigroup correspond to the columns of the matrix used to define multiplication in <var class="Arg">R</var>.</p>
<p><code class="func">Columns</code> returns the columns of the Rees matrix or 0-matrix semigroup <var class="Arg">R</var>. Note that the columns of the semigroup correspond to the rows of the matrix used to define multiplication in <var class="Arg">R</var>.</p>
<p>Arbitrary subsemigroups of Rees matrix or 0-matrix semigroups do not have rows or columns. Such a subsemigroup <var class="Arg">R</var> has rows and columns if and only if it satisfies <code class="func">IsReesMatrixSemigroup</code> (<a href="chap51_mj.html#X780BB78A79275244"><span class="RefLink">51.9-7</span></a>) or <code class="func">IsReesZeroMatrixSemigroup</code> (<a href="chap51_mj.html#X780BB78A79275244"><span class="RefLink">51.9-7</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G:=Group((1,2,3));;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">mat:=List([1..100], x-> List([1..200], x->Random(G)));;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">R:=ReesZeroMatrixSemigroup(G, mat);</span>
<Rees 0-matrix semigroup 200x100 over Group([ (1,2,3) ])>
<span class="GAPprompt">gap></span> <span class="GAPinput">Rows(R);</span>
[ 1 .. 200 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Columns(R);</span>
[ 1 .. 100 ]
</pre></div>
<p><a id="X7D9719F887AFCF8F" name="X7D9719F887AFCF8F"></a></p>
<h5>51.9-10 UnderlyingSemigroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ UnderlyingSemigroup</code>( <var class="Arg">R</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ UnderlyingSemigroup</code>( <var class="Arg">R</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: A semigroup.</p>
<p><code class="func">UnderlyingSemigroup</code> returns the underlying semigroup of the Rees matrix or 0-matrix semigroup <var class="Arg">R</var>.</p>
<p>Arbitrary subsemigroups of Rees matrix or 0-matrix semigroups do not have an underlying semigroup. Such a subsemigroup <var class="Arg">R</var> has an underlying semigroup if and only if it satisfies <code class="func">IsReesMatrixSemigroup</code> (<a href="chap51_mj.html#X780BB78A79275244"><span class="RefLink">51.9-7</span></a>) or <code class="func">IsReesZeroMatrixSemigroup</code> (<a href="chap51_mj.html#X780BB78A79275244"><span class="RefLink">51.9-7</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">S:=Semigroup(Transformation( [ 2, 1, 1, 2, 1 ] ),</span>
<span class="GAPprompt">></span> <span class="GAPinput">Transformation( [ 3, 4, 3, 4, 4 ] ), Transformation([ 3, 4, 3, 4, 3 ] ),</span>
<span class="GAPprompt">></span> <span class="GAPinput">Transformation([ 4, 3, 3, 4, 4 ] ) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">R:=Range(IsomorphismReesMatrixSemigroup(S));</span>
<Rees matrix semigroup 4x2 over Group([ (1,2) ])>
<span class="GAPprompt">gap></span> <span class="GAPinput">UnderlyingSemigroup(R);</span>
Group([ (1,2) ])
</pre></div>
<p><a id="X7D1D9A0382064B8F" name="X7D1D9A0382064B8F"></a></p>
<h5>51.9-11 AssociatedReesMatrixSemigroupOfDClass</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AssociatedReesMatrixSemigroupOfDClass</code>( <var class="Arg">D</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: A Rees matrix or 0-matrix semigroup.</p>
<p>If <var class="Arg">D</var> is a regular \(\mathcal{D}\)-class of a finite semigroup <code class="code">S</code>, then there is a standard way of associating a Rees matrix semigroup to <var class="Arg">D</var>. If <var class="Arg">D</var> is a subsemigroup of <code class="code">S</code>, then <var class="Arg">D</var> is simple and hence is isomorphic to a Rees matrix semigroup. In this case, the associated Rees matrix semigroup of <var class="Arg">D</var> is just the Rees matrix semigroup isomorphic to <var class="Arg">D</var>.</p>
<p>If <var class="Arg">D</var> is not a subsemigroup of <code class="code">S</code>, then we define a semigroup with elements <var class="Arg">D</var> and a new element <code class="code">0</code> with multiplication of <span class="SimpleMath">\(x,y\in D\)</span> defined by:</p>
<p class="center">\[
xy=\left\{\begin{array}{ll}
x*y\ (\textrm{in }S)&\textrm{if }x*y\in D\\
0&\textrm{if }xy\not\in D.
\end{array}\right.
\]</p>
<p>The semigroup thus defined is 0-simple and hence is isomorphic to a Rees 0-matrix semigroup. This semigroup can also be described as the Rees quotient of the ideal generated by <var class="Arg">D</var> by it maximal subideal. The associated Rees matrix semigroup of <var class="Arg">D</var> is just the Rees 0-matrix semigroup isomorphic to the semigroup defined above.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">S:=FullTransformationSemigroup(5);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">D:=GreensDClasses(S)[3];</span>
{Transformation( [ 1, 1, 1, 2, 3 ] )}
<span class="GAPprompt">gap></span> <span class="GAPinput">AssociatedReesMatrixSemigroupOfDClass(D);</span>
<Rees 0-matrix semigroup 25x10 over Group([ (1,2)(3,5)(4,6), (1,3)
(2,4)(5,6) ])>
</pre></div>
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