A Characterization of Directly Ordered Subspaces of R^n

Author

Jennifer Del Valle, University of Texas at El PasoFollow

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 Rⁿ, we consider the standard positive cone to be the set Rⁿ₊ = {x : x ≥ 0} for an element x in Rⁿ. Given a subspace V of Rⁿ, we define the positive cone of V to be the intersection of V with Rⁿ₊. 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 Rⁿ 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.