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.
IsTransformationSemigroup( obj ) P
IsTransformationMonoid( obj ) 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.
DegreeOfTransformationSemigroup( S ) A
The number of points the semigroup acts on.
IsomorphismTransformationSemigroup( S ) A
HomomorphismTransformationSemigroup( S, r ) O
IsomorphismTransformationSemigroup is a generic attribute which is a transformation semigroup isomorphic to S (if such can be computed). In the case of an fp- semigroup, a todd-coxeter will be attempted. For a semigroup of endomorphisms of a finite domain of n elements, it will be to a semigroup of transformations of {1, ¼, n}. Otherwise, it will be the right regular representation on S or S1 if S has no MultiplicativeNeutralElement.
HomomorphismTransformationSemigroup finds a representation of S as transformations of the set of equivalence classes of the right congruence r.
GAP 4 manual