Generalized Polynomials on Semigroups

This article has two main parts. In the first part we show that some of the basic theory of generalized polynomials on commutative semi-groups can be extended to all semigroups. In the second part we show that if a sub-semigroup S of a group G generates G in the sense that G = S · S⁻¹, then a generalized polynomial on S with values in an Abelian group H can be extended to a generalized polynomial on G into H. Finally there is a short discussion of the extendability of exponential functions and generalized exponential polynomials.