Matrix homomorphism equations on a class of monoids and non abelian groupoids (Q2041750)

The authors present an explicit description of the solutions \(f : \mathbb{R}^2 \to M_2(\mathbb{C})\) of the parametric functional equation \[f(x_1x_2 + \alpha y_1y_2, x_1y_2 + \beta x_2y_1) = f(x_1, y_1)f(x_2, y_2),\] where \(\alpha, \beta\in \mathbb{R}\). They explore the solutions of a matrix multiplicative Cauchy functional equation on abelian regular semigroups. As an application, they characterize the solution \(f : \mathbb{R}^2 \to M_2(\mathbb{C})\) of matrix functional equation \[f(x_1x_2 + \alpha y_1y_2, x_1y_2 + \beta x_2y_1) = f(x_1, y_1) + f(x_2, y_2) + \lambda f(x_1, y_1)f(x_2, y_2),\] where \(\lambda\in \mathbb{C}\).