A note on directly ordered subspaces of $\mathbb R^{\small{n}}$

Abstract

Presented is a comprehensive method of determining if a subspace of usually ordered space Rn is directly-ordered. Also, it is proven in an elementary way that if a directly-ordered vector space has a positive cone generated by its extreme vectors then the Riesz Decomposition Property implies the lattice conditions. In particular every directly-ordered subspace of Rn is a lattice-subspace if and only if it satisfies the Riesz Decomposition Property.