Monoid algebra in GAP

Question

Given a monoid in GAP, for example

G:=FullTransformationSemigroup(3);
<full transformation monoid of degree 3>

Question: Can one build the monoid algebra of G over a field using GAP? Im just aware of this for groups.

Answer 1

Monoid algebras are examples of magma rings:

gap> G:=FullTransformationSemigroup(3);
<full transformation monoid of degree 3>
gap> FG:=FreeMagmaRing(GF(3),G);
<algebra-with-one over GF(3), with 3 generators>

Zero and one:

gap> z:=Zero(FG);
<zero> of ...
gap> e:=One(FG);
(Z(3)^0)*IdentityTransformation

Canonical embedding:

gap> emb:=Embedding(G,FG);
<mapping: Monoid( [ Transformation( [ 2, 3, 1 ] ), Transformation( [ 2, 1 ] ), 
  Transformation( [ 1, 2, 1 ] ) ] ) -> AlgebraWithOne( GF(3), ... ) >

Some manipulations with elements:

gap> a:=Random(G);
Transformation( [ 3, 2, 1 ] )
gap> b:=Random(G);
Transformation( [ 3, 3, 3 ] )
gap> x:=a^emb;
(Z(3)^0)*Transformation( [ 3, 2, 1 ] )
gap> y:=b^emb;
(Z(3)^0)*Transformation( [ 3, 3, 3 ] )
gap> x+y+x*y;
(Z(3)^0)*Transformation( [ 3, 2, 1 ] )+(Z(3))*Transformation( [ 3, 3, 3 ] )

For further details, please see the GAP manual here.