50.1 Creating Finitely Presented Semigroups and Monoids

creates a finitely presented semigroup given by the presentation ágens \midrels ñ where gens are the generators of the free semigroup F. Note that relations are entered as pairs of words in the generators of the free semigroup.

gap> f:=FreeSemigroup(3);;
gap> s:=GeneratorsOfSemigroup(f);;
gap> f/[ [s[1]*s[2]*s[1],s[1]] , [s[2]^4,s[1]] ];
<fp semigroup on the generators [ s1, s2, s3 ]>

creates a finitely presented semigroup given by the monoid presentation ágens \midrels ñ where gens are the generators of the free monoid F. Note that relations are entered as pairs of words in both the identity and the generators of the free monoid.

gap> f := FreeMonoid( 3 );
<free monoid on the generators [ m1, m2, m3 ]>
gap> x := GeneratorsOfMonoid( f );
[ m1, m2, m3 ]
gap> e:= Identity ( f );
<identity ...>
gap> m := f/[ [x[1]^3,e] , [x[1]*x[2],x[2] ]];
<fp semigroup on the generators [ <identity ...>, m1, m2, m3 ]>

One may also call the following functions to construct finitely presented semigroups.

F is a free semigroup (resp. monoid) and rels is a list of pairs of elements of F. Returns the fp semigroup which is the quotient of F by the least congruence on F generated by the pairs in rels.

In case F is a free monoid, this function constructs a new free semigroup with one more generator e (representing the identity) and adds new relations to rels of the form e*g = g, g*e = g for all the generators g of the original monoid then constructs the fp semigroup in the same way as for the free semigroup case.

Finally, if one has a finitely presented group, to find an isomorphic finitely presented semigroup, for example, to apply the Knuth-Bendix procedure, use:

Returns an isomorphism from S to an fp semigroup

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