Regular Sublattices, O Closed Ideals and Dedekind MacNeille Completions

Abstract

We study the behaviour of regular sublattices of infinitely distributive lattices under completion. Using order convergence, we associate with an infinitely distributive lattice \(L\) the complete lattice \(\mathfrak I_L\) of \(O\)-closed order ideals and describe the corresponding closure operator \(A\mapsto A^{σ_L}\). We show that every lattice homomorphism induces a canonical map between the corresponding lattices of \(O\)-closed ideals, and compare \(\mathfrak I_L\) with the Dedekind--MacNeille completion \(\DM(L)\). When \(\DM(L)\) remains infinitely distributive, this comparison yields extension results for lattice homomorphisms and a join-regular realization of \(\DM(Y)\) inside an ambient complete lattice. We also determine a sharp finite-dimensional obstruction. We construct an infinitely distributive regular sublattice \(L\subseteq\mathbb R^3\) whose Dedekind--MacNeille completion is not even modular, and therefore cannot be realized as a sublattice of \(\mathbb R^3\). In contrast, we prove that the Dedekind--MacNeille completion of every bounded sublattice of a product of two chains is distributive. Thus dimension three is the least dimension in which a bounded sublattice of a finite product of real chains can have a non-distributive Dedekind--MacNeille completion.

14 pages