<abstract xmlns="http://www.w3.org/1999/xhtml"><p> A comprehensive method of determining if a subspace of usually ordered space R<italic>n </italic>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 R<italic>n </italic>is a lattice- subspace if and only if it satisfies the Riesz Decomposition Property.</p></abstract>

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.

A note on directly ordered subspaces of ℝ<sup>n</sup>

The University of Texas at El Paso 500 W. University Ave Burges Hall 11 El Paso, TX 79968-0514 U. S. A.

Department of Mathematical Sciences The University of Texas at El Paso El Paso, TX 79968 U. S. A.

Tatra Mountains Mathematical Publications

A note on directly ordered subspaces of ℝ

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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...

A note on directly ordered subspaces of ℝⁿ

Published Online: Nov 19, 2012

Page range: 101 - 113

DOI: https://doi.org/10.2478/v10127-012-0031-y

Keywords
ordered vector space, direct order, lattice-subspace, Riesz Decomposition.

This content is open access.

A note on directly ordered subspaces of ℝn