Regular Sublattices, O Closed Ideals and Dedekind MacNeille Completions

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 $A\mapsto A^{{\sigma }_L}$. 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 $\text{DM}(L)$. When $\text{DM}(L)$ remains infinitely distributive, this comparison yields extension results for lattice homomorphisms and a join-regular realization of $\text{DM}(Y)$ 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.