On homomorphisms between products of median algebras

Abstract

Homomorphisms of products of median algebras are studied with particular attention to the case when the codomain is a tree. In particular, we show that all mappings from a product \({\mathbf{A_1} \times\ldots\times {\mathbf{A}_{n}}}\) of median algebras to a median algebra \({\mathbf{B}}\) are essentially unary whenever the codomain \({\mathbf{B}}\) is a tree. In view of this result, we also characterize trees as median algebras and semilattices by relaxing the defining conditions of conservative median algebras.