b-Property of sublattices in vector lattices

TL;DR

This paper investigates the $b$-property of sublattices and ideals within vector lattices, examining its behavior in various completions such as Dedekind, universal, and other topological completions.

Contribution

It provides a detailed analysis of the $b$-property in different types of completions of vector lattices, extending existing theory.

Findings

Characterization of $b$-property in Dedekind completions

Analysis of $b$-property in universal completions

Relationships between $b$-property and various lattice completions

Abstract

We study $b$-property of a sublattice (or an order ideal) $F$ of a vector lattice $E$. In particular, $b$-property of $E$ in $E^\delta$, the Dedekind completion of $E$, $b$-property of $E$ in $E^u$, the universal completion of $E$, and $b$-property of $E$ in $\hat{E}(\hat{\tau})$, the completion of $E$.