Complementation in the lattice of equivalence relations

Abstract

The aim of this paper is to study those pairs of complementary equivalence relations on a fixed set which are maximal as families of mutually complementary equivalence relations. The existence of such pairs on uncountable sets was proved by Steprāns and Watson (1995). They conjectured that such pairs do not exist in the finite and in the countable case. Here we disprove this conjecture by proving that they exist in huge quantity in both cases. We study in detail the case when: (a) one of the equivalence relations in the pair has precisely two equivalence classes; (b) one of the equivalence relations has at most three equivalence classes; (c) in one of the equivalence relations all but one equivalence classes are singletons. In cases (a) and (c) we describe all pairs of complementary equivalence relations having this extremal property. In the case (a) the non-extremal pairs are related to Turan graphs.