Functions on semigroups with vanishing finite Cauchy differences.

Let $${(S,\cdot)}$$ be a semigroup, ( H, +) an abelian group and $${f: S \to H}$$ . The first and second order Cauchy differences of f are Higher order Cauchy differences C f are defined recursively. In the case of H = R, a ring where multiplication is distributive over addition, we show that functions $${f: S\to R}$$ with vanishing finite Cauchy differences are closed under multiplication. The equation C f = 0 is considered for cyclic groups, free abelian groups and other selected quotients of the free groups. [ABSTRACT FROM AUTHOR]