Completeness in oriented matroids

AbstractThe oriented matroid is a structure combining the notions of independent set and opposite element. Dependence induces a closure operator which in the vector space model is the convex hull. Weak completeness is defined as having every maximal convex set contain a maximal subspace; completeness means that every subspace is weakly complete. It is shown that all finite oriented matroids are complete, that in many infinite cases there is an easy criterion for completeness, and that in the vector space model completeness is equivalent to (Dedekind) completeness of the underlying field. A brief discussion of the axioms and basic properties of oriented matroids is also included