Relative uniform convergence and Archimedean property in pre-ordered vector spaces

TL;DR

This paper establishes a method to construct an Archimedeanization of a pre-ordered vector space using quotient spaces and closure in the ru-topology, advancing the understanding of ordered vector space structures.

Contribution

It introduces a new approach to Archimedeanization via quotient spaces and closure in the ru-topology for pre-ordered vector spaces.

Findings

The quotient space (X/A,[W]) serves as an Archimedeanization of X.

The set W is the closure of the positive wedge in ru-topology.

A is the intersection of W and its negative in the space.

Abstract

It is proved that, for a pre-ordered vector space $X$, the quotient space $(X/A,[W])$ is an Archimedeanization of $X$, where $W$ is the closure of the positive wedge $X_+$ in ru-topology, $A=W\cap(-W)$, and $[W]$ is the quotient set of $W$ in $X/A$.