Term inequalities in finite algebras
David Hobby · arXiv (Cornell University) · 2016
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 $σ$ is a universally quantified conjunction of negated atomic formulas, $σ$ is consistent iff it has a finite model.