Polarity and point extensions in oriented matroids (Q1092911)

\textit{A. Bachem} and \textit{W. Kern} have recently extended the notion of polarity (relatively to an \({\mathbb{R}}\)-bilinear form) to oriented matroids [Combinatorica 6, 299-308 (1986; Zbl 0628.05014)]. We prove that the usual polarity properties of the face lattices of convex polytopes can be extended to the class of oriented matroids admitting an (oriented) polar. We give also a short proof of the principal result of the paper loc. cit. showing that there is a natural embedding of the poset of signed span of the cocircuits of a polar of an oriented matroid into the extension poset of this matroid. We remark that if M is a matroid admitting a polar, then every hyperplane can be intersected by every line. Oriented matroids satisfying this condition have an important role in oriented-matroid programming.