Matrix homomorphism equations on a class of monoids and non Abelian groupoids

Abstract

Let α,β∈R. The aim of this paper is to give an explicit description of the solutions f:R×R→M2(C) of the following parametric functional equationsf(x1x2+αy1y2,x1y2+βx2y1)=f(x1,y1)f(x2,y2),that arise from number theory. Depending on the value of β, we present different methods for solving these equations. The solutions of a matrix multiplicative Cauchy functional equation on abelian regular semigroups are given. These results apply to more general equations.