A note on directly ordered subspaces of ℝⁿ

A comprehensive method of determining if a subspace of usually ordered space Rn is directly-ordered is presented here. 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.