A Lattice-theoretical Characterization of Oriented Matroids

IfPis the big face lattice of the covectors of an oriented matroid, it is well known that the zero map is a cover-preserving, order-reversing surjection onto the geometric lattice of the underlying (unoriented) matroid. In this paper we give a (necessary and) sufficient condition for such maps to come from the face lattice of an oriented matroid.