Diagonal subalgebras and local polynomial functions of median algebras

Abstract

A well-known result of Baker and Pixley states that the local polynomial functions of an algebra endowed with a ternary majority term function are precisely the functions preserving all diagonal subalgebras of the square. In the particular case of a median algebra one can be more specific since the collection of all diagonal subalgebras can be generated from various small subcollections by means of intersection, directed union, product, and inversion. This immediately leads to convenient descriptions of the local polynomial functions of median algebras. In consequence, one can easily characterize the locally affine complete median algebras.