Mal'cev conditions corresponding to identities for compatible reflexive relations

TL;DR

This paper characterizes algebraic varieties satisfying specific Mal'cev conditions expressed by equations over compatible reflexive relations, aiming to find analogs of the Pixley-Wille algorithm for these conditions.

Contribution

It provides a characterization of varieties satisfying certain Mal'cev conditions over compatible reflexive relations and explores their relation to congruence lattices.

Findings

Characterization of varieties satisfying $p \,\leq\, q$ over compatible reflexive relations.

Development of an analog of the Pixley-Wille algorithm for these conditions.

Conditions under which equations over compatible reflexive relations correspond to properties over congruence lattices.

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.