Completeness in oriented matroids (Q1105612)

An oriented matroid E is said to be weakly complete if every maximal convex set contains a maximal flat. Moreover, E is complete if every subspace, including E, is weakly complete. And finally, E is locally complete if every subspace of rank 2 is weakly complete. In this paper under review, the author investigates the concept of completeness mentioned above, and proves the following three main results: (1) if E is a finite oriented matroid, then E is complete; (2) a full oriented matroid E is complete if and only if it is locally complete; and (3) the vector space \(V^ n(K)\) over an ordered field K is complete if and only if the field K is complete in the sense of Dedekind. A brief discussion concerning the result (2) is included at the end of the paper, as well.