Note on Archimedean property in ordered vector spaces

Abstract

It is shown that an ordered vector space $X$ is Archimedean if and only if $\underset{\tau \in \{\tau \},y\in L}{\inf }(x_{\tau }-y)=0$ 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 $X_{+}$.