Infinitary Baker–Pixley theorem

Abstract

An important theorem of Baker and Pixley states that if $\mathbf {A}$ is a finite algebra with a $(d+1)$-ary near-unanimity term and f is an n-ary operation on A such that every subalgebra of $\mathbf {A}^{d}$ is closed under f, then f is representable by a term in $\mathbf {A}$. It is well known that the Baker–Pixley theorem does not hold when $\mathbf {A}$ is infinite. We give an infinitary version of the Baker–Pixley theorem which applies to an arbitrary class of structures with a $(d+1)$-ary near-unanimity term instead of a single finite algebra.