On congruences of weak lattices

Abstract

We characterize when an equivalence relation on the base set of a weak lattice $\mathbf{L}=(L,\sqcup ,\sqcap )$ becomes a congruence on $\mathbf{L}$ provided it has convex classes. We show that an equivalence relation on L is a congruence on $\mathbf{L}$ if it satisfies the substitution property for comparable elements. Conditions under which congruence classes are convex are studied. If one fundamental operation of $\mathbf{L}$ is commutative then $\mathbf{L}$ is congruence distributive and all congruences of $\mathbf{L}$ have convex classes.