A polynomial characterization of congruence classes

Abstract

Let \({\cal V}\) be a regular and permutable variety and \( {\cal A}=(A,F)\in {\cal V}\). Let \(\emptyset\neq C \subseteq A \). We get an explicit list L of polynomials such that C is a congruence class of some \( \theta \in Con\, A \) iff C is closed under all terms of L. Moreover, if \({\cal V}\) is a finite similarity type, L is finite. If also \( {\cal A \in {\cal V} \) is finite, all polynomials of L can be considered to be unary. We get a formula for the estimation of card L. The problem of deciding whether C is a congruence class of a finite algebra is in NP but for \( {\cal A \in {\cal V} \) it is in P.