A variant of d’Alembert’s functional equation on semigroups with an anti-endomorphism

Abstract

Let S be a semigroup, let M be a monoid with neutral element e, and let $\mathbb {K}$ be an algebraically closed field of characteristic $\ne 2$ with identity element 1. Inspired by Stetkær’s procedure [19] we describe, in terms of multiplicative functions and characters of 2-dimensional representations of S, the solutions $g:S\rightarrow \mathbb {K}$ of the functional equation

$$\begin{aligned} g(xy)+\mu (y)g(\psi (y)x)=2g(x)g(y),\quad x,y\in S, \end{aligned}$$

where $\psi :S\rightarrow S$ is an anti-endomorphism that need not be involutive and $\mu :S\rightarrow \mathbb {K}$ is a multiplicative function such that $\mu (x\psi (x))=1$ for all $x\in S$. This enables us to find the solutions $g:M\rightarrow \mathbb {K}$ of the new functional equation

$$\begin{aligned} g(x\sigma (y))+g(\psi (y)x) =2g(x)g(y),\quad x,y\in M, \end{aligned}$$

where $\sigma :M\rightarrow M$ is an involutive endomorphism.