Characterizations of Equivalence Lattices

2.1 Introduction

In Section 1.1.2 we introduced, for a set A, the complete lattice Eqv A of all equivalence relations on A and, in Section 1.1.3, equivalence lattices and 0-1 equivalence lattices as sublattices of Eqv A. The purpose of this chapter is to collect results which give either necessary or sufficient conditions for arithmeticity or distributivity of equivalence lattices. If it is clear from the context that an equivalence lattice contains 0_A and 1_A we shall sometimes neglect calling it a 0-1 equivalence lattice. Also note that while distributivity is a property of abstract lattices, arithmeticity for an equivalence lattice means that the lattice is distributive and consists of permuting equivalence relations. (If we apply the adjective arithmetical to an abstract lattice, this would reasonably be interpreted to imply that the lattice is arithmetical as an algebra, i.e., is congruence distributive and congruence permutable. In the present context of equivalence lattices this possible ambiguity should not arise.)