A note on homomorphisms between products of algebras

Abstract

Let K be a congruence distributive variety and call an algebra hereditarily directly irreducible (HDI) if every of its subalgebras is directly irreducible. It is shown that every homomorphism from a finite direct product of arbitrary algebras from K to an HDI algebra from K is essentially unary. Hence, every homomorphism from a finite direct product of algebras Ai (i∈I) from K to an arbitrary direct product of HDI algebras Cj (j∈J) from K can be expressed as a product of homomorphisms from Aσ(j) to Cj for a certain mapping σ from J to I. A homomorphism from an infinite direct product of elements of K to an HDI algebra will in general not be essentially unary, but will always factor through a suitable ultraproduct.