A short proof of the Baker–Pixley theorem for classes

Abstract

An important theorem by Baker and Pixley states that if A is a finite algebra with a (d +1)-ary near-unanimity term and f : A^n → A is any function, then f is a term function of A if f preserves all subuniverses of A^d. This result was generalized recently for (classes of) not necessarily finite algebras using sheaf-theoretic tools. In this note we give a short model-theoretic proof of this generalization. Also, we apply the theorem to obtain a characterization of dual discriminator varieties, and to give necessary and sufficient conditions for a variety with a near-unanimity term to be congruence permutable.