Date of Award


Degree Name

Master of Science


Mathematical Sciences


Piotr J. Wojciechowski


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.




Received from ProQuest

File Size

37 pages

File Format


Rights Holder

Jennifer Del Valle

Included in

Mathematics Commons