The lattice of convex sets of an oriented matroid

In this paper we further investigate the structure of the convex sets of an acyclic oriented matroid. This subject was first explored by Las Vergnas [6]. The main tool of our study is the theory of the anti-exchange closure [2]. We show that the convex closure of Las Vergnas satisfies the anti-exchange law in the case where the matroid is a geometry. From this we are able to detail the structure of the lattice of convex sets. This is done in Section II. In Section III this analysis is used to provide a new expression for the characteristic polynomial of an oriented matroid. For the elements of the theory of oriented matroids the reader is referred to the paper of Bland and Las Vergnas [ 11. We will be using their notation throughout. We restrict our attention to matroids in which every point is closed. These we call geometries. The relationship between anti-exchange closures and convexity in general has been studied by Jamison [4]. This paper is motivated by the special case where the acyclic oriented matroid comes from an acyclic directed simple graph. The convex closure in this instance is transitive closure on the edges of the graph. The extreme points are those edges whose orientation can be reversed and leave the graph acyclic. The characteristic polynomial is the chromatic polynomial of the underlying graph up to factors of A.