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

Let alpha, beta is an element of R. The aim of this paper is to give an explicit description of the solutions f : R x R -> M-2(C) of the following parametric functional equations f(x(1)x(2) + alpha y(1)y(2), x(1)y(2) + beta x(2)y(1)) = f(x(1), y(1)) f(x(2), y(2)), that arise from number theory. Depending on the value of beta, 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. (C) 2021 Elsevier Inc. All rights reserved.

Automated summary

This paper provides an explicit description of the solutions to parametric functional equations in number theory, as well as different methods for solving these equations based on the value of beta. Solutions to a matrix multiplicative Cauchy functional equation on abelian regular semigroups are also presented, with implications for more general equations.