D’Alembert $\mu$-functions on semigroups

Abstract

Let $\mathbb {K}$ be an algebraically closed field of characteristic different from 2 with identity element 1, and let S be a semigroup equipped with an anti-endomorphism $\psi :S\rightarrow S$ and a multiplicative function $\mu :S\rightarrow \mathbb {K}$ satisfying $\mu (x\psi (x))=1$ for all $x\in S$. In a previous paper, we gave the general solution $g:S\rightarrow \mathbb {K}$ of the variant of d’Alembert’s $\mu$-functional equation $(g(xy)+\mu (y)g(\psi (y)x)=2g(x)g(y),\,\,x,y\in S).$ Our main goal here is to show that any solution of d’Alembert’s $\mu$-functional equation

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

is central. This enables us to find its general solution.