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.