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<div class="ChapSects"><a href="chap51.html#X80AF5F307DBDC2B4">51 <span class="Heading">Semigroups</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap51.html#X82AD02117C086D6F">51.1 <span class="Heading">IsSemigroup (Filter)</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51.html#X7B412E5B8543E9B7">51.1-1 IsSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51.html#X7F55D28F819B2817">51.1-2 <span class="Heading">Semigroup</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51.html#X8678D40878CC09A1">51.1-3 Subsemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51.html#X79FBBEC9841544F3">51.1-4 SemigroupByGenerators</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51.html#X80ED104F85AE5134">51.1-5 AsSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51.html#X7B1EEA3E82BFE09F">51.1-6 AsSubsemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51.html#X78147A247963F23B">51.1-7 GeneratorsOfSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51.html#X7C72E4747BF642BB">51.1-8 <span class="Heading">FreeSemigroup</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51.html#X7E67E13F7A01F8D3">51.1-9 SemigroupByMultiplicationTable</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap51.html#X78274024827F306D">51.2 <span class="Heading">Properties of Semigroups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51.html#X7C4663827C5ACEF1">51.2-1 IsRegularSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51.html#X87532A76854347E0">51.2-2 IsRegularSemigroupElement</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51.html#X836F4692839F4874">51.2-3 IsSimpleSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51.html#X8193A60F839C064E">51.2-4 IsZeroSimpleSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51.html#X85F7E5CD86F0643B">51.2-5 IsZeroGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51.html#X7FFEC81F7F2C4EAA">51.2-6 IsReesCongruenceSemigroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap51.html#X872369257F69EA20">51.3 <span class="Heading">Making transformation semigroups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51.html#X7EAF835D7FE4026F">51.3-1 IsTransformationSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51.html#X7EA699C687952544">51.3-2 DegreeOfTransformationSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51.html#X78F29C817CF6827F">51.3-3 IsomorphismTransformationSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51.html#X85C58E1E818C838C">51.3-4 IsFullTransformationSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51.html#X7D2B0685815B4053">51.3-5 FullTransformationSemigroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap51.html#X7BB32D508183C0F1">51.4 <span class="Heading">Ideals of semigroups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51.html#X7D5CEE4D7D4318ED">51.4-1 SemigroupIdealByGenerators</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51.html#X7F01FFB18125DED5">51.4-2 ReesCongruenceOfSemigroupIdeal</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51.html#X7A3FF85984345540">51.4-3 IsLeftSemigroupIdeal</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap51.html#X7C0782D57C01E327">51.5 <span class="Heading">Congruences for semigroups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51.html#X78E34B737F0E009F">51.5-1 IsSemigroupCongruence</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51.html#X822DB78579BCB7B5">51.5-2 IsReesCongruence</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap51.html#X87CE9EAB7EE3A128">51.6 <span class="Heading">Quotients</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51.html#X80EF3E6F842BE64E">51.6-1 IsQuotientSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51.html#X7CAD3D1687956F7F">51.6-2 HomomorphismQuotientSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51.html#X87120C46808F7289">51.6-3 QuotientSemigroupPreimage</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap51.html#X80C6C718801855E9">51.7 <span class="Heading">Green's Relations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51.html#X786CEDD4814A9079">51.7-1 GreensRRelation</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51.html#X8364D69987D49DE1">51.7-2 IsGreensRelation</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51.html#X82A11A087AFB3EB0">51.7-3 IsGreensClass</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51.html#X7AA204C8850F9070">51.7-4 IsGreensLessThanOrEqual</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51.html#X86FE5F5585EBCF13">51.7-5 RClassOfHClass</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51.html#X78C56F4A78E0088A">51.7-6 EggBoxOfDClass</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51.html#X803237F17ACD44E3">51.7-7 DisplayEggBoxOfDClass</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51.html#X87C75A9D86122D93">51.7-8 GreensRClassOfElement</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51.html#X844D20467A644811">51.7-9 GreensRClasses</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51.html#X7CB4A18685B850E2">51.7-10 GroupHClassOfGreensDClass</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51.html#X79D740EF7F0E53BD">51.7-11 IsGroupHClass</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51.html#X7F5860927CAD920F">51.7-12 IsRegularDClass</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap51.html#X8225A9EC87A255E6">51.8 <span class="Heading">Rees Matrix Semigroups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51.html#X8526AA557CDF6C49">51.8-1 ReesMatrixSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51.html#X872CEF99839085B1">51.8-2 ReesZeroMatrixSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51.html#X780BB78A79275244">51.8-3 IsReesMatrixSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51.html#X7EEBAEE9857C5EBA">51.8-4 IsReesZeroMatrixSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51.html#X7A0DE1F28470295E">51.8-5 ReesMatrixSemigroupElement</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51.html#X7F6B852B81488C86">51.8-6 IsReesMatrixSemigroupElement</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51.html#X7D3366928253D5D3">51.8-7 SandwichMatrixOfReesMatrixSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51.html#X7FAF2F3E864CA202">51.8-8 RowIndexOfReesMatrixSemigroupElement</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51.html#X7FA034937CA3C41F">51.8-9 ReesZeroMatrixSemigroupElementIsZero</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51.html#X7D1D9A0382064B8F">51.8-10 AssociatedReesMatrixSemigroupOfDClass</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap51.html#X7964B5C97FB9C07D">51.8-11 IsomorphismReesMatrixSemigroup</a></span>
</div></div>
</div>
<h3>51 <span class="Heading">Semigroups</span></h3>
<p>This chapter describes functions for creating semigroups and determining information about them.</p>
<p><a id="X82AD02117C086D6F" name="X82AD02117C086D6F"></a></p>
<h4>51.1 <span class="Heading">IsSemigroup (Filter)</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">( property )</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.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">...</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.html#X839147CF813312D6"><span class="RefLink">35.2-1</span></a>) and <code class="func">IsAssociative</code> (<a href="chap35.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, 3, 4 ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">s:= Semigroup(a, b);</span>
<semigroup 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.html#X8268EAA47E4A3A64"><span class="RefLink">35.2-7</span></a>) and <code class="func">SubmagmaNC</code> (<a href="chap35.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>
<semigroup with 1 generator>
</pre></div>
<p><a id="X79FBBEC9841544F3" name="X79FBBEC9841544F3"></a></p>
<h5>51.1-4 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.html#X7F55D28F819B2817"><span class="RefLink">51.1-2</span></a>).</p>
<p><a id="X80ED104F85AE5134" name="X80ED104F85AE5134"></a></p>
<h5>51.1-5 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">( attribute )</td></tr></table></div>
<p>If <var class="Arg">C</var> is a collection whose elements form a semigroup (see <code class="func">IsSemigroup</code> (<a href="chap51.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>51.1-6 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>51.1-7 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.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, 3, 4 ] ) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">GeneratorsOfSemigroup(t);</span>
[ Transformation( [ 2, 3, 4, 1 ] ) ]
</pre></div>
<p><a id="X7C72E4747BF642BB" name="X7C72E4747BF642BB"></a></p>
<h5>51.1-8 <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>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. If the optional argument <var class="Arg">name</var> is given then the generators are printed as <var class="Arg">name</var><code class="code">1</code>, <var class="Arg">name</var><code class="code">2</code> etc., that is, each name is the concatenation of the string <var class="Arg">name</var> and an integer from <code class="code">1</code> to <var class="Arg">range</var>. The default for <var class="Arg">name</var> is the string <code class="code">"s"</code>.</p>
<p>Called in the second form, <code class="func">FreeSemigroup</code> returns a free semigroup on as many generators as arguments, printed as <var class="Arg">name1</var>, <var class="Arg">name2</var> etc.</p>
<p>Called in the third form, <code class="func">FreeSemigroup</code> returns a free semigroup on as many generators as the length of the list <var class="Arg">names</var>, the <span class="SimpleMath">i</span>-th generator being printed as <var class="Arg">names</var><span class="SimpleMath">[i]</span>.</p>
<p>Called in the fourth form, <code class="func">FreeSemigroup</code> returns a free semigroup on infinitely many generators, where the first generators are printed by the names in the list <var class="Arg">init</var>, and the other generators by <var class="Arg">name</var> and an appended number.</p>
<p>If the extra argument <var class="Arg">wfilt</var> is given, it must be either <code class="func">IsSyllableWordsFamily</code> (<a href="chap37.html#X7869716C84EA9D81"><span class="RefLink">37.6-6</span></a>) or <code class="func">IsLetterWordsFamily</code> (<a href="chap37.html#X7E36F7897D82417F"><span class="RefLink">37.6-2</span></a>) or <code class="func">IsWLetterWordsFamily</code> (<a href="chap37.html#X8719E7F27CDA1995"><span class="RefLink">37.6-4</span></a>) or <code class="func">IsBLetterWordsFamily</code> (<a href="chap37.html#X8719E7F27CDA1995"><span class="RefLink">37.6-4</span></a>). This filter then specifies the representation used for the elements of the free semigroup (see <a href="chap37.html#X80A9F39582ED296E"><span class="RefLink">37.6</span></a>). If no such filter is given, a letter representation is used.</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 ]>
</pre></div>
<p>Also see Chapter <a href="chap51.html#X80AF5F307DBDC2B4"><span class="RefLink">51</span></a>.</p>
<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">FreeGroup( 5 );</span>
<free group on the generators [ f1, f2, f3, f4, f5 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">FreeGroup( "a", "b" );</span>
<free group 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" ); gens:= GeneratorsOfGroup( f );;</span>
<free group on the generators [ 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>51.1-9 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.html#X85CD1E7678295CA6"><span class="RefLink">35.3-1</span></a>)) if such a semigroup exists. Otherwise <code class="keyw">fail</code> is returned.</p>
<p><a id="X78274024827F306D" name="X78274024827F306D"></a></p>
<h4>51.2 <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>51.2-1 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 D class of <var class="Arg">S</var> is regular.</p>
<p><a id="X87532A76854347E0" name="X87532A76854347E0"></a></p>
<h5>51.2-2 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 ∈ <var class="Arg">S</var></span> such that <span class="SimpleMath"><var class="Arg">x</var> y <var class="Arg">x</var> = <var class="Arg">x</var></span> and <span class="SimpleMath">y <var class="Arg">x</var> y = y</span>.</p>
<p><a id="X836F4692839F4874" name="X836F4692839F4874"></a></p>
<h5>51.2-3 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>51.2-4 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>51.2-5 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>51.2-6 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="X872369257F69EA20" name="X872369257F69EA20"></a></p>
<h4>51.3 <span class="Heading">Making transformation semigroups</span></h4>
<p>Cayley's Theorem gives special status to semigroups of transformations, and accordingly there are special functions to deal with them, and to create them from other finite semigroups.</p>
<p><a id="X7EAF835D7FE4026F" name="X7EAF835D7FE4026F"></a></p>
<h5>51.3-1 IsTransformationSemigroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsTransformationSemigroup</code>( <var class="Arg">obj</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">‣ IsTransformationMonoid</code>( <var class="Arg">obj</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>A transformation semigroup (resp. monoid) is a subsemigroup (resp. submonoid) of the full transformation monoid. Note that for a transformation semigroup to be a transformation monoid we necessarily require the identity transformation to be an element.</p>
<p><a id="X7EA699C687952544" name="X7EA699C687952544"></a></p>
<h5>51.3-2 DegreeOfTransformationSemigroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DegreeOfTransformationSemigroup</code>( <var class="Arg">S</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>The number of points the semigroup <var class="Arg">S</var> acts on.</p>
<p><a id="X78F29C817CF6827F" name="X78F29C817CF6827F"></a></p>
<h5>51.3-3 IsomorphismTransformationSemigroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsomorphismTransformationSemigroup</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">‣ HomomorphismTransformationSemigroup</code>( <var class="Arg">S</var>, <var class="Arg">r</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p><code class="func">IsomorphismTransformationSemigroup</code> is a generic attribute which is a transformation semigroup isomorphic to <var class="Arg">S</var> (if such can be computed). In the case of an fp-semigroup, a Todd-Coxeter approach will be attempted. For a semigroup of endomorphisms of a finite domain of <span class="SimpleMath">n</span> elements, it will be to a semigroup of transformations of <span class="SimpleMath">{ 1, 2, ..., n }</span>. Otherwise, it will be the right regular representation on <var class="Arg">S</var> or <span class="SimpleMath"><var class="Arg">S</var>^1</span> if <var class="Arg">S</var> has no multiplicative neutral element, see <code class="func">MultiplicativeNeutralElement</code> (<a href="chap35.html#X7EE2EA5F7EB7FEC2"><span class="RefLink">35.4-10</span></a>).</p>
<p><code class="func">HomomorphismTransformationSemigroup</code> finds a representation of <var class="Arg">S</var> as transformations of the set of equivalence classes of the right congruence <var class="Arg">r</var>.</p>
<p><a id="X85C58E1E818C838C" name="X85C58E1E818C838C"></a></p>
<h5>51.3-4 IsFullTransformationSemigroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsFullTransformationSemigroup</code>( <var class="Arg">obj</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>checks whether <var class="Arg">obj</var> is a full transformation semigroup.</p>
<p><a id="X7D2B0685815B4053" name="X7D2B0685815B4053"></a></p>
<h5>51.3-5 FullTransformationSemigroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FullTransformationSemigroup</code>( <var class="Arg">degree</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns the full transformation semigroup of degree <var class="Arg">degree</var>.</p>
<p><a id="X7BB32D508183C0F1" name="X7BB32D508183C0F1"></a></p>
<h4>51.4 <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.html#X839147CF813312D6"><span class="RefLink">35.2-1</span></a>).</p>
<p><a id="X7D5CEE4D7D4318ED" name="X7D5CEE4D7D4318ED"></a></p>
<h5>51.4-1 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>51.4-2 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">∆ ∪ I × I</span>.</p>
<p><a id="X7A3FF85984345540" name="X7A3FF85984345540"></a></p>
<h5>51.4-3 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="X7C0782D57C01E327" name="X7C0782D57C01E327"></a></p>
<h4>51.5 <span class="Heading">Congruences for 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 for magmas, see <code class="func">Magma</code> (<a href="chap35.html#X839147CF813312D6"><span class="RefLink">35.2-1</span></a>).</p>
<p><a id="X78E34B737F0E009F" name="X78E34B737F0E009F"></a></p>
<h5>51.5-1 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>51.5-2 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>51.6 <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.html#X87120C46808F7289"><span class="RefLink">51.6-3</span></a>) value under the congruence given by the value of <code class="func">QuotientSemigroupCongruence</code> (<a href="chap51.html#X87120C46808F7289"><span class="RefLink">51.6-3</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.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.html#X87120C46808F7289"><span class="RefLink">51.6-3</span></a>) under the map returned by <code class="func">QuotientSemigroupHomomorphism</code> (<a href="chap51.html#X87120C46808F7289"><span class="RefLink">51.6-3</span></a>), which maps the <code class="func">QuotientSemigroupPreimage</code> (<a href="chap51.html#X87120C46808F7289"><span class="RefLink">51.6-3</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>51.6-1 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>51.6-2 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>51.6-3 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>51.7 <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 ∧ L</span>, and <span class="SimpleMath">D = R ∘ 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 ⊆ 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 ⊆ 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 ⊆ 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>51.7-1 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>51.7-2 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">( property )</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">( property )</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">( property )</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">( property )</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">( property )</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">( property )</td></tr></table></div>
<p>Categories for the Green's relations.</p>
<p><a id="X82A11A087AFB3EB0" name="X82A11A087AFB3EB0"></a></p>
<h5>51.7-3 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">( property )</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">( property )</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">( property )</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">( property )</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">( property )</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">( property )</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>51.7-4 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>51.7-5 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>51.7-6 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>51.7-7 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 "picture" 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>51.7-8 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>51.7-9 GreensRClasses</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GreensRClasses</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">‣ GreensLClasses</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">‣ GreensJClasses</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">‣ GreensDClasses</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">‣ GreensHClasses</code>( <var class="Arg">semigroup</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>return the <span class="SimpleMath">R</span>, <span class="SimpleMath">L</span>, <span class="SimpleMath">J</span>, <span class="SimpleMath">H</span>, or <span class="SimpleMath">D</span> Green's classes, respectively for semigroup <var class="Arg">semigroup</var>. <code class="func">EquivalenceClasses</code> (<a href="chap33.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>51.7-10 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>51.7-11 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>51.7-12 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.html#biBHowie76">[How76, Prop. 3.2]</a>.)</p>
<p><a id="X8225A9EC87A255E6" name="X8225A9EC87A255E6"></a></p>
<h4>51.8 <span class="Heading">Rees Matrix Semigroups</span></h4>
<p>In this section we describe <strong class="pkg">GAP</strong> functions for Rees matrix semigroups and Rees 0-matrix semigroups. The importance of this construction is 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>Recall that a Rees Matrix semigroup is constructed from a semigroup (the underlying semigroup), and a matrix. A Rees Matrix semigroup element is a triple <span class="SimpleMath">(s, i, λ)</span> where <span class="SimpleMath">s</span> is an element of the underlying semigroup <span class="SimpleMath">S</span> and <span class="SimpleMath">i</span>, <span class="SimpleMath">λ</span> are indices. This can be thought of as a matrix with zero everywhere except for an occurrence of <span class="SimpleMath">s</span> at row <span class="SimpleMath">i</span> and column <span class="SimpleMath">λ</span>. The multiplication is defined by <span class="SimpleMath">(i, s, λ)*(j, t, μ) = (i, s P_{λ j} t, μ)</span> where <span class="SimpleMath">P</span> is the defining matrix of the semigroup. In the case that the underlying semigroup has a zero we can create the <code class="func">ReesZeroMatrixSemigroup</code> (<a href="chap51.html#X872CEF99839085B1"><span class="RefLink">51.8-2</span></a>) value, wherein all elements whose <span class="SimpleMath">s</span> entry is the zero of the underlying semigroup are identified to the unique zero of the Rees 0-matrix semigroup.</p>
<p><a id="X8526AA557CDF6C49" name="X8526AA557CDF6C49"></a></p>
<h5>51.8-1 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">matrix</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>for a semigroup <var class="Arg">S</var> and <var class="Arg">matrix</var> whose entries are in <var class="Arg">S</var>. Returns the Rees Matrix semigroup with multiplication defined by <var class="Arg">matrix</var>.</p>
<p><a id="X872CEF99839085B1" name="X872CEF99839085B1"></a></p>
<h5>51.8-2 ReesZeroMatrixSemigroup</h5>
<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">matrix</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>for a semigroup <var class="Arg">S</var> with zero, and <var class="Arg">matrix</var> over <var class="Arg">S</var> returns the Rees 0-Matrix semigroup such that all elements <span class="SimpleMath">(i, 0, λ)</span> are identified to zero.</p>
<p>The zero in <var class="Arg">S</var> is found automatically. If one cannot be found, an error is signalled.</p>
<p><a id="X780BB78A79275244" name="X780BB78A79275244"></a></p>
<h5>51.8-3 IsReesMatrixSemigroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsReesMatrixSemigroup</code>( <var class="Arg">T</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>returns <code class="keyw">true</code> if the object <var class="Arg">T</var> is a (whole) Rees matrix semigroup.</p>
<p><a id="X7EEBAEE9857C5EBA" name="X7EEBAEE9857C5EBA"></a></p>
<h5>51.8-4 IsReesZeroMatrixSemigroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsReesZeroMatrixSemigroup</code>( <var class="Arg">T</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>returns <code class="keyw">true</code> if the object <var class="Arg">T</var> is a (whole) Rees 0-matrix semigroup.</p>
<p><a id="X7A0DE1F28470295E" name="X7A0DE1F28470295E"></a></p>
<h5>51.8-5 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">a</var>, <var class="Arg">lambda</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">a</var>, <var class="Arg">lambda</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>for a Rees matrix semigroup <var class="Arg">R</var>, <var class="Arg">a</var> in <code class="code">UnderlyingSemigroup(<var class="Arg">R</var>)</code>, <var class="Arg">i</var> and <var class="Arg">lambda</var> in the row (resp. column) ranges of <var class="Arg">R</var>, returns the element of <var class="Arg">R</var> corresponding to the matrix with zero everywhere and <var class="Arg">a</var> in row <var class="Arg">i</var> and column <var class="Arg">x</var>.</p>
<p><a id="X7F6B852B81488C86" name="X7F6B852B81488C86"></a></p>
<h5>51.8-6 IsReesMatrixSemigroupElement</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsReesMatrixSemigroupElement</code>( <var class="Arg">e</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">e</var> )</td><td class="tdright">( category )</td></tr></table></div>
<p>is the category of elements of a Rees (0-) matrix semigroup. Returns true if <var class="Arg">e</var> is an element of a Rees Matrix semigroup.</p>
<p><a id="X7D3366928253D5D3" name="X7D3366928253D5D3"></a></p>
<h5>51.8-7 SandwichMatrixOfReesMatrixSemigroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SandwichMatrixOfReesMatrixSemigroup</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">‣ SandwichMatrixOfReesZeroMatrixSemigroup</code>( <var class="Arg">R</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>each return the defining matrix of the Rees (0-) matrix semigroup.</p>
<p><a id="X7FAF2F3E864CA202" name="X7FAF2F3E864CA202"></a></p>
<h5>51.8-8 RowIndexOfReesMatrixSemigroupElement</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RowIndexOfReesMatrixSemigroupElement</code>( <var class="Arg">x</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">‣ RowIndexOfReesZeroMatrixSemigroupElement</code>( <var class="Arg">x</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">‣ ColumnIndexOfReesMatrixSemigroupElement</code>( <var class="Arg">x</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">‣ ColumnIndexOfReesZeroMatrixSemigroupElement</code>( <var class="Arg">x</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">‣ UnderlyingElementOfReesMatrixSemigroupElement</code>( <var class="Arg">x</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">‣ UnderlyingElementOfReesZeroMatrixSemigroupElement</code>( <var class="Arg">x</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>For an element <var class="Arg">x</var> of a Rees Matrix semigroup, of the form <span class="SimpleMath">(i, s, λ)</span>, the row index is <span class="SimpleMath">i</span>, the column index is <span class="SimpleMath">λ</span> and the underlying element is <span class="SimpleMath">s</span>. If we think of an element as a matrix then this corresponds to the row where the non-zero entry is, the column where the non-zero entry is and the entry at that position, respectively.</p>
<p><a id="X7FA034937CA3C41F" name="X7FA034937CA3C41F"></a></p>
<h5>51.8-9 ReesZeroMatrixSemigroupElementIsZero</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ReesZeroMatrixSemigroupElementIsZero</code>( <var class="Arg">x</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>returns <code class="keyw">true</code> if <var class="Arg">x</var> is the zero of the Rees 0-matrix semigroup.</p>
<p><a id="X7D1D9A0382064B8F" name="X7D1D9A0382064B8F"></a></p>
<h5>51.8-10 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>Given a regular <var class="Arg">D</var> class of a finite semigroup, it can be viewed as a Rees matrix semigroup by identifying products which do not lie in the <var class="Arg">D</var> class with zero, and this is what it is returned.</p>
<p>Formally, let <span class="SimpleMath">I_1</span> be the ideal of all J classes less than or equal to <var class="Arg">D</var>, <span class="SimpleMath">I_2</span> the ideal of all J classes <em>strictly</em> less than <var class="Arg">D</var>, and <span class="SimpleMath">ρ</span> the Rees congruence associated with <span class="SimpleMath">I_2</span>. Then <span class="SimpleMath">I/ρ</span> is zero-simple. Then <code class="code">AssociatedReesMatrixSemigroupOfDClass( <var class="Arg">D</var> )</code> returns this zero-simple semigroup as a Rees matrix semigroup.</p>
<p><a id="X7964B5C97FB9C07D" name="X7964B5C97FB9C07D"></a></p>
<h5>51.8-11 IsomorphismReesMatrixSemigroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsomorphismReesMatrixSemigroup</code>( <var class="Arg">obj</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>If <var class="Arg">S</var> is a completely simple (resp. zero simple) semigroup, returns an isomorphism to a Rees matrix semigroup over a group (resp. zero group).</p>
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