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

Jennifer Del Valle, Piotr J. Wojciechowski


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.

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DOI: https://doi.org/10.2478/tatra.v52i0.162