On Relative Principal Congruences in Term Quasivarieties

Abstract

Let ${\mathcal {K}}$ be a quasivariety. We say that ${\mathcal {K}}$ is a term quasivariety if there exist an operation of arity zero e and a family of binary terms $\{t_i\}_{i\in I}$ such that for every $A \in {\mathcal {K}}$, $\theta$ a ${\mathcal {K}}$-congruence of A and $a,b\in A$ the following condition is satisfied: $(a,b)\in \theta$ if and only if $(t_{i}(a,b),e) \in \theta$ for every $i\in I$. In this paper we study term quasivarieties. For every $A\in {\mathcal {K}}$ and $a,b\in A$ we present a description for the smallest ${\mathcal {K}}$-congruence containing the pair (a, b). We apply this result in order to characterize ${\mathcal {K}}$-compatible functions on A (i.e., functions which preserve all the ${\mathcal {K}}$-congruences of A) and we give two applications of this property: (1) we give necessary conditions on ${\mathcal {K}}$ for which for every $A \in {\mathcal {K}}$ the ${\mathcal {K}}$-compatible functions on A coincides with a polynomial over finite subsets of A; (2) we give a method to build up ${\mathcal {K}}$-compatible functions.