Exponential semi-polynomials and their characterization on semigroups

Exponential semi-polynomials on semigroups are natural generalizations of exponential polynomials on groups. We show that several of the standard properties of exponential polynomials on groups also hold for exponential semi-polynomials on semigroups. The main result is that for topological commutative monoids S belonging to a certain class, a function in C(S) is an exponential semi-polynomial if and only if it is contained in a finite dimensional translation invariant linear subspace. We also show that some standard results about polynomials on commutative semigroups are in fact valid on all semigroups.