Polynomial functions over commutative semi-groups

Let S be a commutative semigroup. We consider the semigroup P(S) with respect to composition of all transformations p: S S of the form x a,x xⁿ or x axⁿ (a S; n N) and the semigroup P(S) containing only elements of the last two forms. Since all polynomials over S have the form a, xⁿ or ax these transformations are the so-called polynomial functions over S. We investigate the relationship between the structures of S and(S) resp. P(S) — a criterion on the commutativity of S has been shown by means of polynomial functions in 2.