Term inequalities in finite algebras

Abstract

Given an algebra $\mathbf{A}$, and terms $s(x_1,x_2,\dots x_k)$ and $t(x_1,x_2,\dots x_k)$ of the language of $\mathbf{A}$, we say that $s$ and $t$ are {\em separated} in $\mathbf{A}$ iff for all $a_1,a_2\dots a_k\in A$, $s(a_1,a_2,\dots a_k)$ and $t(a_1,a_2,\dots a_k)$ are never equal. We prove that given two terms that are separated in any algebra, there exists a finite algebra in which they are separated. As a corollary, we obtain that whenever the sentence $\sigma$ is a universally quantified conjunction of negated atomic formulas, $\sigma$ is consistent iff it has a finite model.