# A generalization of d'Alembert's functional equation on semigroups

## TL;DR

This paper characterizes solutions to a generalized d'Alembert functional equation on semigroups with involutive automorphisms and multiplicative functions, extending classical results to a broader algebraic setting.

## Contribution

It provides a comprehensive solution framework for a generalized functional equation on semigroups with specific automorphism and multiplicative conditions, broadening the understanding of such equations.

## Key findings

- Explicit solutions are derived for the functional equation.
- The results extend classical d'Alembert equations to semigroups with involutive automorphisms.
- Conditions on the multiplicative function  are identified for solution existence.

## Abstract

Given a semigroup $S$ generated by its squares equipped with an involutive automorphism $\sigma$ and a multiplicative function $\mu:S\to\mathbb{C}$ such that $\mu(x\sigma(x))=1$ for all $x\in S$, we determine the complex-valued solutions of the following functional equation.