Action of Pontryagin dual of semilattices grading algebras

Let A be a unitary algebra and G be a finite abelian group. Then a G-graded algebra is merely a G-algebra and viceversa because of the fact that G and its group of characters G* are isomorphic. This fact is no longer true if we substitute G with infinite or non-abelian groups. In this paper we try to obtain similar results for a special class of abelian monoids, i.e, the bounded semmilattices. Moreover, if S is such a monoid, we are going to investigate the role of S and its Pontryagin dual S* over the algebra A, in the case A is S-graded.