Mal'cev conditions corresponding to identities for compatible reflexive relations

Abstract

We investigate Mal'cev conditions described by equations whose variables runs over the set of all compatible reflexive relations. Let $p \leq q$ be an equation in the language $\{\wedge, \circ,+\}$. We give a characterization of the class of all varieties which satisfy $p \leq q$ over the set of all compatible reflexive relations. The aim is to find an analogon of the Pixley-Wille algorithm for conditions expressed by equations over the set of all compatible reflexive relations, and to characterize when an equation $p \leq q$ expresses the same property when considered over the congruence lattices or over the sets of all compatible reflexive relations of algebras in a variety.