Computing congruence lattices of finite lattices

Abstract

An inequality between the number of coverings in the ordered set J(Con L) of join irreducible congruences on a lattice L and the size of L is given. Using this inequality it is shown that this ordered set can be computed in time O(n2 log2 n), where n = ILl. This paper is motivated by the problem of efficiently calculating and representing the congruence lattice Con L of a finite lattice L. Of course Con L can be exponential in the size of L; for example, when L is a chain of length n, Con L has 2n elements. However, since Con L is a distributive lattice, it can be recovered easily from the ordered set of its join irreducible elements J(Con L). Indeed any finite distributive lattice D is isomorphic to the lattice of order ideals of J(D) and this lattice is in turn isomorphic to the lattice of all antichains of J(D), where the antichains are ordered by A ? B, i.e., for each a E A there is a b E B with a < b. If P is an ordered set of size n which has N order ideals, then there are straightforward algorithms to find the order ideals of P which run in time 0(nN); see, for example, [5]. In [10] Medina and Nourine give an algorithm which runs in time 0(dN), where d is the maximum number of covers of any element of P. Thus we will concentrate on the problem of efficiently finding J(Con L).