Order version of the Riesz Theorem

Abstract

\begin{abstract}
This paper offers an order-theoretic generalization of the classical Riesz theorem within the setting of Riesz spaces, employing the notion of order
compactness as a key conceptual tool ( see $ \left[2 \right]$ ). The main result showing that a Riesz space endowed with a strictly positive linear functional $\varphi$ must be finitedimensional if its unit sphere is $\sigma$-order compact. Moreover, if $\varphi$ is order continuous and a box $[-a,a]$ ($ 0 < a$ ) is $\sigma-$order compact then $E$ is finitedimensional.
\end{abstract}