Posets Generated by Irreducible Elements

Abstract

Let J be a fixed partially ordered set (poset). Among all posets in which J is join-dense 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, C ₀ L, consisting of those sets between J and L which intersect each of the intervals I _j =[j _∨,j ^∨] (j∈J), where j _∨ and j ^∨ denote the greatest element of L less than, respectively, not greater than j. The complete lattices in C ₀ L form a closure system C _∞ L, consisting of all Dedekind–MacNeille completions of posets in C ₀ L. We describe explicitly those L for which C ₀ L, 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 C _κ L, the closure system of all posets in C ₀ L that are closed under meets of less than κ elements (where κ is any cardinal number).