# A characterization of directly ordered subspaces of Rn

#### Abstract

In the finite dimensional ordered vector space R^{n} , we consider the standard positive cone to be the set R^{n}+ = {*x* ∈ R^{n} : *x* ≥ 0}. Given a subspace *V* of R^{n} , we define the *positive cone* of *V* as *V*_{+} = *V* ∩ R^{n}+ . The cone *V*_{+} is said to be generating if *V* = *V*_{+} − *V*_{ +}, that is, if any vector *v* ∈ *V* can be expressed as the difference of two vectors, *v* = * x − y* where *x*, *y* ∈ * V*_{+}. Ordered vector spaces with generating cones are generally referred to as *directly ordered*. Well-known from Order Theory is that all lattices and thus lattice-subspaces are directed. However, not all directly ordered spaces are lattices, and often it is difficult to determine when a space is directed. Since directly ordered spaces enjoy a number of desirable qualities, it is useful to know when one is working in such a space. In this work, we characterize those collections of vectors in R^{n} that span directly ordered subspaces. The theory we develop naturally gives rise to a method of determining when a subspace is directed by means of a simple algorithm.^

#### Subject Area

Mathematics|Theoretical Mathematics

#### Recommended Citation

Del Valle, Jennifer J, "A characterization of directly ordered subspaces of Rn" (2011). *ETD Collection for University of Texas, El Paso*. AAI1498284.

http://digitalcommons.utep.edu/dissertations/AAI1498284