Unbounded order convergence on infinitely distributive lattices
Functional Analysis
2025-06-12 v3
Abstract
We study uO convergence on infinitely distributive lattices, extending key properties known from Riesz spaces. We show that order continuity of uO convergence characterizes infinite distributivity. We examine O-adherence and uO adherence of sublattices and ideals, proving that the uO and O closures of a sublattice coincide and form a sublattice, and that the first uO adherence of an ideal is an O closed ideal. We also analyze the Dedekind MacNeille completion of a sublattice Y within that of a lattice L, identifying conditions (A) and (B) under which the completion of Y embeds regularly in that of L. In this case, we show that the first uO adherence of Y covers its O closure.
Keywords
Cite
@article{arxiv.2405.07366,
title = {Unbounded order convergence on infinitely distributive lattices},
author = {Abela Kevin and Chetcuti Emmanuel},
journal= {arXiv preprint arXiv:2405.07366},
year = {2025}
}