Polynomial functions on subdirect products

Abstract

A congruence preserving function on a subdirect product of two finite Mal’cev algebras is polynomial if it induces polynomial functions on the subdirect factors and there are no skew congruences between the projection kernels. As a special case, if the direct product A × B of finite algebras A and B in a congruence permutable variety has no skew congruences, then the polynomial functions on A × B are exactly direct products of polynomials on A and on B. These descriptions apply in particular to classical polynomial functions on nonassociative rings. Also, for finite algebras A, B in a variety with majority term, the polynomial functions on A × B are exactly the direct products of polynomials on A and on B. However in arbitrary congruence distributive varieties the corresponding result is not true.