Posets Generated by Irreducible Elements

Abstract

Let J be a fixed partially ordered set (poset). Among all posets in which J is joindense and consists of all completely join-irreducible elements, there is an up to isomorphism unique greatest one, the Alexandroff completion L. Moreover, the class of all such posets has a canonical set of representatives, C0L, consisting of those sets between J and L which intersect each of the intervals Ij = [jv, jv] (j ε J), where jv and jv denote the greatest element of L less than, respectively, not greater than j. The complete lattices in C0L form a closure system C∞L, consisting of all Dedekind-MacNeille completions of posets in C0L. We describe explicitly those L for which C0L, respectively, C∞L is a (complete atomic) Boolean lattice, and similarly, those for which C ∞L is distributive (or modular). Analogous results are obtained for CκL, the closure system of all posets in C 0L that are closed under meets of less than κ elements (where κ is any cardinal number).