Distributive laws for commuting equivalence relations

In his paper (1942), Ore found necessary and sufficient conditions under which the modular and distributive laws hold in the lattice of equivalence relations on a set S. In the present paper, we consider commuting equivalence relations. It has been proved by Jónsson (1953) that the modular law holds in the lattice of commuting equivalence relations. We give some necessary and sufficient conditions for the distributive law and its dual to hold for commuting equivalence relations.