50.3 Preimages in the Free Semigroup

returns the underlying free semigroup for the finitely presented semigroup T, ie, the free semigroup over which T is defined as a quotient (this is the free semigroup generated by the free generators provided by FreeGeneratorsOfFpSemigroup(T)).

returns the underlying free generators corresponding to the generators of the finitely presented semigroup T.

returns the relations of the finitely presented semigroup S as pairs of words in the free generators provided by FreeGeneratorsOfFpSemigroup(S).

gap> f := FreeSemigroup( "a" , "b" );;
gap> a := GeneratorsOfSemigroup( f )[ 1 ];;
gap> b := GeneratorsOfSemigroup( f )[ 2 ];;
gap> s := f / [ [ a^3 , a ] , [ b^3 , b ] , [ a*b , b*a ] ];             
<fp semigroup on the generators [ a, b ]>
gap> Size( s );
8
gap> FreeSemigroupOfFpSemigroup( s ) = f;
true
gap> FreeGeneratorsOfFpSemigroup( s );
[ a, b ]
gap> RelationsOfFpSemigroup( s );
[ [ a^3, a ], [ b^3, b ], [ a*b, b*a ] ]

Elements of a finitely presented semigroup are not words, but are represented using a word from the free semigroup as representative.

If elm is an element of a finitely presented semigroup, it returns the word from the free semigroup that is used as a representative for elm.

gap> w := GeneratorsOfSemigroup(s)[1] * GeneratorsOfSemigroup(s)[2];
a*b
gap> IsWord (w );
false
gap> ue := UnderlyingElement( w ); 
a*b
gap> IsWord( ue );
true

If fam is the elements family of a finitely presented semigroup and word is a word in the free generators underlying this finitely presented semigroup, this operation creates the element with the representative word in the free semigroup.

gap> ge:=ElementOfFpSemigroup( FamilyObj( GeneratorsOfSemigroup(s)[1]), a*b ); 
a*b
gap> ge in f;
false
gap> ge in s;
true

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