Unbounded order convergence on infinitely distributive lattices

TL;DR

This paper explores unbounded order convergence in infinitely distributive lattices, extending properties from Riesz spaces and characterizing order continuity through infinite distributivity.

Contribution

It introduces new characterizations of order continuity via uO convergence and analyzes the structure of closures and adherence of sublattices and ideals.

Findings

uO convergence characterizes infinite distributivity

uO and O closures of sublattices coincide and form a sublattice

the first uO adherence of an ideal is an O closed ideal

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.