Vector lattice covers of ideals and bands in pre-Riesz spaces

TL;DR

This paper investigates how to extend ideals and bands from pre-Riesz spaces to their vector lattice covers, providing conditions and criteria for such extensions to preserve lattice properties.

Contribution

It offers new conditions and criteria for extending ideals and bands from pre-Riesz spaces to their vector lattice covers, enhancing understanding of their structure.

Findings

Conditions for smallest extension ideals in vector lattice covers

Criteria for bands and their extensions in covers

Properties of generated ideals and bands

Abstract

Pre-Riesz spaces are ordered vector spaces which can be order densely embedded into vector lattices, their so-called vector lattice covers. Given a vector lattice cover $Y$ for a pre-Riesz space $X$, we address the question how to find vector lattice covers for subspaces of $X$, such as ideals and bands. We provide conditions such that for a directed ideal $I$ in $X$ its smallest extension ideal in $Y$ is a vector lattice cover. We show a criterion for bands in $X$ and their extension bands in $Y$ as well. Moreover, we state properties of ideals and bands in $X$ which are generated by sets, and of their extensions in $Y$.