A characterization of directly ordered subspaces of Rn

Jennifer J Del Valle, University of Texas at El Paso


In the finite dimensional ordered vector space Rn , we consider the standard positive cone to be the set Rn+ = {x ∈ Rn : x ≥ 0}. Given a subspace V of Rn , we define the positive cone of V as V+ = V ∩ Rn+ . The cone V+ is said to be generating if V = V+V +, that is, if any vector vV 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 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.^

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.