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/><span class="nocss"> </span><a href="chap51.html#X87C75A9D86122D93"> <span class="ContSS"><br /><span class="nocss"> </span><a href="chap51.html#X844D20467A644811"> <span class="ContSS"><br /><span class="nocss"> </span><a href="chap51.html#X7CB4A18685B850E2"> <span class="ContSS"><br /><span class="nocss"> </span><a href="chap51.html#X79D740EF7F0E53BD"> <span class="ContSS"><br /><span class="nocss"> </span><a href="chap51.html#X7F5860927CAD920F"> <span class="ContSS"><br /><span class="nocss"> </span><a href="chap51.html#X81AF2EAB7CEF8C19"> </div></div> <div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap51.html#X </span> <div class="ContSSBlock"> <span class="ContSS"><br /><span class="nocss"> </span><a href="chap51.html#X8526AA557CDF6C49"> <span class="ContSS"><br /><span class="nocss"> </span><a href="chap51.html#X78D2A48C87FC8E38"> <span class="ContSS"><br /><span class="nocss"> </span><a href="chap51.html#X7964B5C97FB9C07D"> <span class="ContSS"><br /><span class="nocss"> </span><a href="chap51.html#X7F6B852B81488C86"> <span class="ContSS"><br /><span class="nocss"> </span><a href="chap51.html#X7A0DE1F28470295E"> <span class="ContSS"><br /><span class="nocss"> </span><a href="chap51.html#X7F03BE707AC7F8A0"> <span class="ContSS"><br /><span class="nocss"> </span><a href="chap51.html#X780BB78A79275244"> <span class="ContSS"><br /><span class="nocss"> </span><a href="chap51.html#X7CACF4D686AF1D19"> <span class="ContSS"><br /><span class="nocss"> </span><a href="chap51.html#X82FC5D6980C66AC4"> </span> <span class="ContSS"><br /><span class="nocss"> </span><a href="chap51.html#X7D9719F887AFCF8F"> <span class="ContSS"><br /><span class="nocss"> </span><a href="chap51.html#X7D1D9A0382064B8F"> </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 infor <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">‣ Is <p>returns <code class="keyw">true</code> if the object <var class="Arg">D</var> is a semigroup. A <em>s <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">‣ Se <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Se <p>In the first form, <code class="func">Semigroup</code> returns the semigroup generated by the a <p>It is <em>not</em> checked whether the underlying multiplication is associative, use <code class <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">a:= Transformation( [ 2, 3, 4, 1 ] );</s Transformation( [ 2, 3, 4, 1 ] ) <span class="GAPprompt">gap></span> <span class="GAPinput">b:= Transformation( [ 2, 2, 3, 4 ] );</s 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">‣ Su <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Su <p>are just synonyms of <code class="func">Submagma</code> (<a href="chap35.html#X8268EAA47E4A <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">a:=GeneratorsOfSemigroup(s)[1];< 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">‣ Is <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> i <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">f := Transformation([5, 6, 7, 1, 4, 3, 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(FullTransformati 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">‣ Se <p>is the underlying operation of <code class="func">Semigroup</code> (<a href="chap51.html#X7F <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">‣ As <p>If <var class="Arg">C</var> is a collection whose elements form a semigroup under <code class="fu <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">‣ As <p>Let <var class="Arg">D</var> be a domain and <var class="Arg">C</var> a collection. If <var class="Ar <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">‣ Ge <p>Semigroup generators of a semigroup <var class="Arg">D</var> are the same as magma generators, s <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">‣ Is <p>This property reflects whether the list or collection <var class="Arg">C</var> generates a semi <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">IsGeneratorsOfSemigroup([Transf 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">‣ Fr <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Fr <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Fr <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Fr <p><code class="code">FreeSemigroup</code> returns a free semigroup. The number of generators, a <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">FreeSemigrou <p>If <var class="Arg">name</var> is not given but the <code class="code">generatorNames</code> opti <p>Otherwise, the generators of the returned free semigroup are labelled <var class="Arg">name</ </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</co </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 cla </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 </dd> </dl> <p>If the optional first argument <var class="Arg">wfilt</var> is given, then it must be either <code <p>For more on associative words see Chapter <a href="chap37.html#X78C56A0A87CE380E"><span cla <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" <free semigroup on the generators [ generator1, generator2, generator3 ]> <span class="GAPprompt">gap></span> <span class="GAPinput">f3 := FreeSemigroup( "gen1" , "gen2" <free semigroup on the generators [ gen1, gen2 ]> <span class="GAPprompt">gap></span> <span class="GAPinput">f4 := FreeSemigroup( ["gen1" , "gen2 <free semigroup on the generators [ gen1, gen2 ]> <span class="GAPprompt">gap></span> <span class="GAPinput">FreeSemigroup( 3 : generatorNames : <free semigroup on the generators [ boom1, boom2, boom3 ]> <span class="GAPprompt">gap></span> <span class="GAPinput">FreeSemigroup( 2 : generatorNames : <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", [ <free semigroup on the generators [ a, b, ... ]> <span class="GAPprompt">gap></span> <span class="GAPinput">GeneratorsOfSemigroup( F ){[1..4] [ a, b, g3, g4 ] <span class="GAPprompt">gap></span> <span class="GAPinput">GeneratorsOfSemigroup( FreeSemig [ gen1, gen2, gen3 ] <span class="GAPprompt">gap></span> <span class="GAPinput">GeneratorsOfSemigroup( FreeSemig [ f, s2, s3 ] <span class="GAPprompt">gap></span> <span class="GAPinput">FreeSemigroup(IsSyllableWordsFa <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 t <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. I <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 );</spa [ 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", "G <free group on the generators [ f1*f2, f2^-1, Group( [ f1, f2 ] ) ]> <span class="GAPprompt">gap></span> <span class="GAPinput">gens:= GeneratorsOfGroup( f );;</sp <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">‣ Se <p>returns the semigroup whose multiplication is defined by the square matrix <var class="Arg">A< <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">SemigroupByMultiplicationTable( <semigroup of size 3, with 3 generators> <span class="GAPprompt">gap></span> <span class="GAPinput">SemigroupByMultiplicationTable( 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">‣ Is <p>A <em>monoid</em> is a magma-with-one (see <a href="chap35.html#X873E502F7D21C39C"><span class <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">‣ Mo <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Mo <p>In the first form, <code class="func">Monoid</code> returns the monoid generated by the argumen <p>It is <em>not</em> checked whether the underlying multiplication is associative, use <code class <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">‣ Su <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Su <p>are just synonyms of <code class="func">SubmagmaWithOne</code> (<a href="chap35.html#X7F295 <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">‣ Mo <p>is the underlying operation of <code class="func">Monoid</code> (<a href="chap51.html#X7F953 <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">‣ As <p>If <var class="Arg">C</var> is a collection whose elements form a monoid, then <code class="func"> <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">‣ As <p>Let <var class="Arg">D</var> be a domain and <var class="Arg">C</var> a collection. If <var class="Ar <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">‣ Ge <p>Monoid generators of a monoid <var class="Arg">M</var> are the same as magma-with-one generator <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">‣ Tr <p>is just a synonym for <code class="func">TrivialSubmagmaWithOne</code> (<a href="chap35.html <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">‣ Fr <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Fr <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Fr <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Fr <p><code class="code">FreeMonoid</code> returns a free monoid. The number of generators, and the l <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">FreeMonoi <p>If <var class="Arg">name</var> is not given but the <code class="code">generatorNames</code> opti <p>Otherwise, the generators of the returned free monoid are labelled <var class="Arg">name</var>< </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 m </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 </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 infi </dd> </dl> <p>If the optional first argument <var class="Arg">wfilt</var> is given, then it must be either <code <p>For more on associative words see Chapter <a href="chap37.html#X78C56A0A87CE380E"><span cla <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 := "t <free monoid on the generators [ turbo1, turbo2, turbo3 ]> <span class="GAPprompt">gap></span> <span class="GAPinput">FreeMonoid(2 : generatorNames := [" <free monoid on the generators [ u, v ]> <span class="GAPprompt">gap></span> <span class="GAPinput">FreeMonoid(); FreeMonoid(0); Free <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" <free monoid with infinity generators> <span class="GAPprompt">gap></span> <span class="GAPinput">GeneratorsOfMonoid(F){[1..4]};</ [ a, b, g3, g4 ] <span class="GAPprompt">gap></span> <span class="GAPinput">GeneratorsOfMonoid(FreeMonoid(i [ gen1, gen2, gen3 ] <span class="GAPprompt">gap></span> <span class="GAPinput">GeneratorsOfMonoid(FreeMonoid(i [ f, g, m3 ] <span class="GAPprompt">gap></span> <span class="GAPinput">FreeMonoid(IsSyllableWordsFamil <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">‣ Mo <p>returns the monoid whose multiplication is defined by the square matrix <var class="Arg">A</va <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">MonoidByMultiplicationTable([[1 <monoid of size 3, with 3 generators> <span class="GAPprompt">gap></span> <span class="GAPinput">MonoidByMultiplicationTable([[1 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">‣ In <p>Returns: An inverse semigroup.</p> <p>If <var class="Arg">obj1</var>, <var class="Arg">obj2</var>, ... are (any combination) of associa <p>For example if <code class="code">S</code> and <code class="code">T</code> are inverse semigroups <p>As present, the only associative elements with unique semigroup inverses, which do not alway <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, <span class="GAPprompt">gap></span> <span class="GAPinput">f := PartialPerm( [ 1, 2, 3, 4, 5, 8, 10 ] <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, Idempot <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">‣ In <p>Returns: An inverse monoid.</p> <p>If <var class="Arg">obj1</var>, <var class="Arg">obj2</var>, ... are (any combination) of associa <p>As present, the only associative elements with unique semigroup inverses are partial permut <p>For example if <code class="code">S</code> and <code class="code">T</code> are inverse monoids, th |
