Mal’cev conditions corresponding to identities for compatible reflexive relations

We investigate Mal’cev conditions described by those equations whose variables runs over the set of all compatible reflexive relations. Let p≤ q be an equation for the language { ∧ , ∘ , + }. We give a characterization of the class of all varieties which satisfy p≤ 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≤ 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.