Regular Sublattices, O Closed Ideals and Dedekind MacNeille Completions

Kevin Abela; Emmanuel Chetcuti

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arXiv · 2609.16280

Regular Sublattices, O Closed Ideals and Dedekind MacNeille Completions

Kevin Abela·Emmanuel Chetcuti

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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.

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Kevin Abela, Emmanuel Chetcuti. 2026-09-14. Regular Sublattices, O Closed Ideals and Dedekind MacNeille Completions. https://arxiv.org/abs/2609.16280

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