Functions on semigroups with vanishing finite Cauchy differences

Abstract

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

$$\begin{aligned} C^{1}f(x, y)= & {} f(xy) - f(x) - f(y),\\ C^{2}f(x, y, z)= & {} f(xyz) - f(xy) - f(yz) - f(xz) + f(x) + f(y) + f(z).\end{aligned}$$

Higher order Cauchy differences C ^k 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 ^k f = 0 is considered for cyclic groups, free abelian groups and other selected quotients of the free groups.