A characterization of directly ordered subspaces of Rn

In the finite dimensional ordered vector space [special characters omitted], we consider the standard positive cone to be the set [special characters omitted] = {x ∈ [special characters omitted]: x ≥ 0}. Given a subspace V of [special characters omitted], we define the positive cone of V as V+ = V ∩ [special characters omitted]. 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 [special characters omitted] 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