Pexiderized d’Alembert functional equations on monoids

Abstract

Let S be a monoid ($=$ semigroup with identity), and let $\sigma :S \rightarrow S$ be a homomorphism such that $\sigma \circ \sigma = id$. In an earlier paper we solved the Pexiderized d’Alembert functional equation (PDFE) $f(xy) + g(\sigma (y)x) = h(x)k(y)$ for unknown $f,g,h,k:S \rightarrow {{\mathbb {C}}}$, assuming that S is either regular or generated by its squares and that one of the unknown functions is central. The present paper has two main results. The first describes the solutions of PDFE on a general monoid in terms of multiplicative functions, solutions of a special case of the sine subtraction law, and solutions of other functional equations with just one unknown function. The second main result uses the first one to give a more detailed solution of PDFE on a larger class of monoids than has been treated previously. We also find the continuous solutions on topological monoids. Examples are given to illustrate the results.