Quelle semifp.xml Sprache: XML

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#W semifp.xml
#Y Copyright (C) 2020-2022 Luke Elliott
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## Licensing information can be found in the README file of this package.
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<#GAPDoc Label="ParseRelations">
<ManSection>
 <Oper Name="ParseRelations" Arg="gens, rels"/>
 <Returns>A list of pairs of semigroup or monoid elements.</Returns>
 <Description>
 <C>ParseRelations</C> converts a string describing relations for a
 semigroup or monoid to the list of pairs of semigroup or monoid elements it
 represents. Any white space given is ignored. The output list is then
 compatible with other &GAP; functions. In the below examples we see free
 semigroups and monoids being directly quotiented by the output of the
 <C>ParseRelations</C> function.
 

 The argument <A>gens</A> must be a list of generators for a free semigroup,
 each being a single alphabet letter (in upper or lower case). The argument
 <A>rels</A> must be string that lists the equalities desired.

 To take a quotient of a free monoid, it is necessary to use
 <Ref Attr="GeneratorsOfMonoid" BookName="ref"/> as the 1st argument to
 <C>ParseRelations</C> and the identity may appear as <C>1</C> in the
 specified relations.

 <Example><![CDATA[
gap> f := FreeSemigroup("x", "y", "z");;
gap> AssignGeneratorVariables(f);
gap> ParseRelations([x, y, z], " x=(y^2z) ^2x, y=xxx , z=y^3");
[ [ x, (y^2*z)^2*x ], [ y, x^3 ], [ z, y^3 ] ]
gap> r := ParseRelations([x, y, z], " x=(y^2z)^2x, y=xxx=z , z=y^3");
[ [ x, (y^2*z)^2*x ], [ y, x^3 ], [ x^3, z ], [ z, y^3 ] ]
gap> f / r;
<fp semigroup with 3 generators and 4 relations of length 23>
gap> f2 := FreeSemigroup("a");
<free semigroup on the generators [ a ]>
gap> f2 / ParseRelations(GeneratorsOfSemigroup(f2), "a = a^2");
<fp semigroup with 1 generator and 1 relation of length 4>
gap> FreeMonoidAndAssignGeneratorVars("a", "b")
> / ParseRelations([a, b], "a^2=1,b^3=1,(ab)^3=1");
<fp monoid with 2 generators and 3 relations of length 13>
]]></Example>
 </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="ElementOfFpSemigroup">
<ManSection>
 <Oper Name="ElementOfFpSemigroup" Arg="S, word"/>
 <Returns>An element of the fp semigroup <A>S</A>.</Returns>
 <Description>
 When <A>S</A> is a finitely presented semigroup and <A>word</A> is an
 associative word in the associated free semigroup (see
 <Ref Filt="IsAssocWord" BookName="ref"/>), this returns the
 fp semigroup element with representative <A>word</A>.
 

 This function is just a short form of the &GAP; library implementation of
 <Ref Oper="ElementOfFpSemigroup" BookName="ref"/> which does not require
 retrieving an element family.

 <Example><![CDATA[
gap> f := FreeSemigroup("x", "y");;
gap> AssignGeneratorVariables(f);
gap> s := f / [[x * x, x], [y * y, y]];
<fp semigroup with 2 generators and 2 relations of length 8>
gap> a := ElementOfFpSemigroup(s, x * y);
x*y
gap> b := ElementOfFpSemigroup(s, x * y * y);
x*y^2
gap> a in s;
true
gap> a = b;
true]]></Example>
 </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="ElementOfFpMonoid">
<ManSection>
 <Oper Name="ElementOfFpMonoid" Arg="M, word"/>
 <Returns>An element of the fp monoid <A>M</A>.</Returns>
 <Description>
 When <A>M</A> is a finitely presented monoid and <A>word</A> is an
 associative word in the associated free monoid (see
 <Ref Filt="IsAssocWord" BookName="ref"/>), this returns the
 fp monoid element with representative <A>word</A>.
 

 This is analogous to <Ref Oper="ElementOfFpSemigroup"/>.
 </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="FreeMonoidAndAssignGeneratorVars">
<ManSection>
 <Func Name="FreeMonoidAndAssignGeneratorVars" Arg="arg..."/>
 <Func Name="FreeSemigroupAndAssignGeneratorVars" Arg="arg..."/>
 <Returns>A free semigroup or monoid.</Returns>
 <Description>
 <C>FreeMonoidAndAssignGeneratorVars</C> is synonym with:
 <Log>
 FreeMonoid(arg...);
 AssignGeneratorVariables(last);</Log>
 These functions exist so that the <C>String</C> method for a finitely
 presented semigroup or monoid to be valid &GAP; input which can be used to
 reconstruct the semigroup or monoid.
 <Example><![CDATA[
gap> F := FreeSemigroupAndAssignGeneratorVars("x", "y");;
gap> IsBound(x);
true
gap> IsBound(y);
true]]></Example>
 </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="AntiIsomorphismDualFpSemigroup">
<ManSection>
 <Attr Name="AntiIsomorphismDualFpSemigroup" Arg="S"/>
 <Attr Name="AntiIsomorphismDualFpMonoid" Arg="S"/>
 <Returns>A finitely presented semigroup or monoid.</Returns>
 <Description>
 <C>AntiIsomorphismDualFpSemigroup</C> returns an anti-isomorphism (<Ref
 Func="MappingByFunction" BookName="ref"/>) from the
 finitely presented semigroup <A>S</A> to another finitely presented
 semigroup. The range finitely presented semigroup is obtained from <A>S</A>
 by reversing the relations of <A>S</A>.

 <C>AntiIsomorphismDualFpMonoid</C> works analogously when <A>S</A> is a
 finitely presented monoid, and the range of the returned anti-isomorphism
 is a finitely presented monoid.

 <Example><![CDATA[
gap> F := FreeSemigroup("a", "b");
<free semigroup on the generators [ a, b ]>
gap> AssignGeneratorVariables(F);
gap> R := [[a ^ 3, a], [b ^ 2, b], [(a * b) ^ 2, a]];
[ [ a^3, a ], [ b^2, b ], [ (a*b)^2, a ] ]
gap> S := F / R;
<fp semigroup with 2 generators and 3 relations of length 14>
gap> map := AntiIsomorphismDualFpSemigroup(S);
MappingByFunction( <fp semigroup with 2 generators and
 3 relations of length 14>, <fp semigroup with 2 generators and
 3 relations of length 14>
, function( x ) ... end, function( x ) ... end )
gap> RelationsOfFpSemigroup(Range(map));
[ [ a^3, a ], [ b^2, b ], [ (b*a)^2, a ] ]
]]></Example>
 </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="EmbeddingFpMonoid">
<ManSection>
 <Attr Name="EmbeddingFpMonoid" Arg="S"/>
 <Returns>A finitely presented monoid.</Returns>
 <Description>
 <C>EmbeddingFpMonoid</C> returns an embedding (<Ref
 Oper="SemigroupHomomorphismByImages" Label="for two semigroups and two
 lists"/>) from the finitely presented semigroup S into a finitely
 presented monoid. If <A>S</A> satisfies <Ref Prop="IsMonoidAsSemigroup"/>,
 then the mapping returned by this function is an isomorphism (the same
 isomorphism as <Ref Attr="IsomorphismFpMonoid" BookName="ref"/>). If
 <A>S</A> is not a monoid, then the range is isomorphic to <A>S</A> with an
 identity adjoined (a new element not previously in <A>S</A>). The embedded
 copy of <A>S</A> in the range can be recovered using <Ref Oper="Image"
 BookName="ref"/>.

 <Example><![CDATA[
gap> F := FreeSemigroup("a", "b");
<free semigroup on the generators [ a, b ]>
gap> AssignGeneratorVariables(F);
gap> R := [[a ^ 3, a], [b ^ 2, b], [(a * b) ^ 2, a]];
[ [ a^3, a ], [ b^2, b ], [ (a*b)^2, a ] ]
gap> S := F / R;
<fp semigroup with 2 generators and 3 relations of length 14>
gap> Size(S);
3
gap> IsMonoidAsSemigroup(S);
false
gap> map := EmbeddingFpMonoid(S);
<fp semigroup with 2 generators and 3 relations of length 14> ->
<fp monoid with 2 generators and 3 relations of length 14>
gap> Size(Range(map));
4
]]></Example>
 </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="IsSubsemigroupOfFpMonoid">
<ManSection>
 <Prop Name="IsSubsemigroupOfFpMonoid" Arg="S"/>
 <Returns><K>true</K> or <K>false</K>.</Returns>
 <Description>
 This property is <K>true</K> if the object <A>S</A> is a subsemigroup of an
 fp monoid, and <K>false</K> otherwise. This property is just a synonym for
 <Ref Prop="IsSemigroup" BookName="ref"/> and
 <C>IsElementOfFpMonoidCollection</C>.

 <Example><![CDATA[
gap> F := FreeSemigroup("a", "b");
<free semigroup on the generators [ a, b ]>
gap> AssignGeneratorVariables(F);
gap> R := [[a ^ 3, a], [b ^ 2, b], [(a * b) ^ 2, a]];
[ [ a^3, a ], [ b^2, b ], [ (a*b)^2, a ] ]
gap> S := F / R;
<fp semigroup with 2 generators and 3 relations of length 14>
gap> IsSubsemigroupOfFpMonoid(S);
false
gap> map := EmbeddingFpMonoid(S);
<fp semigroup with 2 generators and 3 relations of length 14> ->
<fp monoid with 2 generators and 3 relations of length 14>
gap> IsSubsemigroupOfFpMonoid(Image(map));
true
gap> IsSubsemigroupOfFpMonoid(Range(map));
true]]></Example>
 </Description>
</ManSection>
<#/GAPDoc>

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