On the Transfer of Completeness and Projection Properties in Truncated Vector Lattices

TL;DR

This paper explores how key order and completeness properties transfer between truncated Riesz spaces and their unitizations, providing characterizations, equivalences, and counterexamples for various notions of completeness.

Contribution

It offers new characterizations and insights into the transfer of completeness properties in truncated Riesz spaces, including counterexamples highlighting the independence of these notions.

Findings

Characterizations of completeness transfer

Equivalences for various completeness notions

Counterexamples illustrating independence

Abstract

In this work we investigate the transfer of fundamental order and completeness properties between truncated Riesz spaces and their unitizations. Specifically, we provide characterizations and equivalences for several notions of completeness: the Archimedean property, relatively uniform completeness, Dedekind completeness, lateral completeness, universal completeness, and the projection property. Counterexamples are presented to illustrate the necessity of assumptions and the independence of various completeness notions.