Solution to a problem concerning the intersection of maximal filters and maximal ideals in a distributive lattice

Abstract

It is well known that in a distributive lattice L, every filter is an intersection of prime filters, and dually. Moreover, L is relatively complemented if and only if every prime filter (ideal) is maximal. In 1947, A. Monteiro [3] posed the question of determining whether a distributive lattice, in which every filter is an intersection of maximal filters, and dually, is necessarily relatively complemented. This question also appears as Problem 39 in G. Gr/itzer [2; p. 156]. Monteiro states that J. Dieudonn6 and L. Nachbin have independently shown that neither the condition that every filter is an intersection of maximal filters, nor its dual, implies the other. We will solve this problem in the negative by constructing a non-Boolean distributive lattice L which satisfies the above conditions on the filters and ideals L. The referee was most helpful in simplifying the explanation of the construction of L. This lattice is necessarily infinite for if a finite distributive lattice is such that each of its filters is an intersection of maximal filters then it is a Boolean algebra. Indeed, this follows from the fact that if F, F1,..., F, are prime filters and F = Of="1 Fi then F=Fto for some io~ {1,..., n}.