Note on Archimedean property in ordered vector spaces

TL;DR

This paper characterizes the Archimedean and almost Archimedean properties of ordered vector spaces using infimum conditions and linear extensions of additive mappings, providing new theoretical insights.

Contribution

It introduces a novel characterization of Archimedean and almost Archimedean properties in ordered vector spaces through infimum conditions and linear extension criteria.

Findings

Characterization of Archimedean property via infimum of bounded decreasing nets.

Characterization of almost Archimedean property through linear extension of additive mappings.

Provides theoretical criteria for identifying Archimedean properties in ordered vector spaces.

Abstract

It is shown that an ordered vector space $X$ is Archimedean if and only if $\inf\limits_{\tau\in\{\tau\}, y\in L}(x_\tau -y) \ = 0$ for any bounded decreasing net $x_\tau\downarrow$ in $X$, where $L$ is the collection of all lower bounds of $\{x_\tau\}_{\tau}$. We give also a characterization of the almost Archimedean property of $X$ in terms of existence of a linear extension of an additive mapping $T:Y_+\to X_+$ of the positive cone $Y_+$ of an ordered vector space $Y$ into $X_+$.