Note on Archimedean property in ordered vector spaces

It is shown that an ordered vector space $X$ is Archimedean if and only if $\inf\limits_{τ\in\{τ\}, y\in L}(x_τ-y) \ = 0$ for any bounded decreasing net $x_τ\downarrow$ in $X$, where $L$ is the collection of all lower bounds of $\{x_τ\}_τ$. 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 $X_+$.

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