#### Date of Award

2011-01-01

#### Degree Name

Master of Science

#### Department

Mathematical Sciences

#### Advisor(s)

Piotr J. Wojciechowski

#### Abstract

In the finite dimensional ordered vector space **R**^{n}, we consider the standard positive cone to be the set **R**^{n}_{+} = {x : x ≥ 0} for an element x in **R**^{n}. Given a subspace *V* of **R**^{n}, we define the *positive cone* of *V* to be the intersection of *V* with **R**^{n}_{+}. The cone *V _{+}* is said to be generating if

*V =*

*V*−_{+}*V*, that is, if any vector_{+}*v*in*V*can be expressed as the difference of two vectors,*v = x − y*where*x*and*y*are elements of*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.

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#### Language

en

#### Provenance

Received from ProQuest

#### Copyright Date

2011

#### File Size

37 pages

#### File Format

application/pdf

#### Rights Holder

Jennifer Del Valle

#### Recommended Citation

Del Valle, Jennifer, "A Characterization of Directly Ordered Subspaces of R^n" (2011). *Open Access Theses & Dissertations*. 2270.

https://digitalcommons.utep.edu/open_etd/2270

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