Semigroup Algebra

The semigroup algebra , where is a field and a semigroup, is formally defined in the same way as the group algebra . Similarly, a semigroup ring is a variation of the group ring , where the group is replaced by a semigroup . Usually, it is required that have an identity element so that is a unit ring and is a subring of .

The group algebra is the set of all formal expressions

(1)

where for all and for all but finitely many indices so that for sufficiently large (say, ). Hence, we can write the general element as

(2)

Assigning

(3)

defines an isomorphism of -algebras between and the polynomial ring .

More generally, if is the subsemigroup of generated by the elements , for , the semigroup algebra is isomorphic to the subalgebra of the polynomial ring generated by the monomials