Computing Congruence Lattices 3

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(n 2 log 2 n), where n = jLj. This paper is motivated by the problem of eeciently calculating and representing the congruence lattice Con L of a nite 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 2 n 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 nite 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 order by A B, i.e., for each a 2 A there is a b 2 B with a b. If P is an ordered set of size n which has N order ideals, then there are straightforward algorithms to nd the order ideals of P which run in time O(nN); see, for example, 5]. In 10] Mediana and Nourine give an algorithm which runs in time O(dN), where d is the maximum number of covers of any element of P. Thus we will concentrate on the problem of eeciently nding J(Con L). Throughout this paper L denotes a nite lattice. J(L) denotes the set of nonzero join irreducible elements and M(L) the set of nonunit meet ir-reducible elements. These sets are ordered by the induced order from L. If a 2 J(L) then it has a unique lower cover in L which we denote by a and similarly if q 2 M(L) then q is the unique upper cover of q. The cover relation is denote by a b; Cg (x; y) is the smallest congruence identifying x and y. Throughout the paper we let n = jLj m = j J(Con L)j