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.