Note on Archimedean property 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 $X_+$.