Lattices of compatible relations satisfying a set of formulas

Abstract

The aim of the paper is to give a framework for the generation of algebraic lattices of compatible relations connected with algebras. These compatible relations satisfy a set of formulas (which can include symmetry and/or transitivity). A particular case are lattices elements of which are all relations reflexive on subalgebras satisfying the same set of formulas, e.g., all tolerances, all quasi-orders, all congruences etc., on all subalgebras of the given algebra (the subalgebra lattice being isomorphic with the ideal generated by the diagonal relation). Using the extension and intersection properties (a generalization of the CEP and the CIP), we give conditions under which these lattices satisfy a lattice identity, such as modularity and distributivity.