Regular Sublattices, O Closed Ideals and Dedekind MacNeille Completions

@misc{abela2026regular,
  title = {{Regular Sublattices, O Closed Ideals and Dedekind MacNeille Completions}},
  author = {Kevin Abela and Emmanuel Chetcuti},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.16280},
  eprint = {2609.16280},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.16280},
}

TY  - UNPB
TI  - Regular Sublattices, O Closed Ideals and Dedekind MacNeille Completions
AU  - Kevin Abela
AU  - Emmanuel Chetcuti
PY  - 2026
DA  - 2026/10/21
PB  - arXiv
DO  - 10.48550/arxiv.2609.16280
UR  - https://arxiv.org/abs/2609.16280
AB  - We study regular sublattices of lattices that satisfy the Join-Infinite Distributive Law (JID) under Dedekind MacNeille completion. Using order convergence, we identify the lattice of O-closed ideals with Crawleys completion, and compare it with the Dedekind MacNeille completion. When DM(L) satisfies (JID), this comparison yields extension results for lattice homomorphisms. In particular, if Y is a regular sublattice of a complete lattice L, and L and DM(Y) satisfy (JID), then the canonical extension lattice-embeds DM(Y) into L with join-regular image. We also give a sharp finite-dimensional obstruction: there exists an infinitely distributive regular sublattice of R3 whose Dedekind MacNeille completion is not distributive, and therefore cannot be lattice embedded in R3. In contrast, we prove that the inclusion of every join-regular sublattice of R2 with a least and a greatest element extends canonically to a lattice embedding of its MacNeille completion into R2.
ER  -