A note on homomorphisms between products of algebras

摘要

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 A(i) (i is an element of I) from K to an arbitrary direct product of HDI algebras C-j (j is an element of J) from K can be expressed as a product of homomorphisms from A(sigma)(j) to C-j for a certain mapping sigma from J to I is an element of 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.