A lattice-theoretical characterization of oriented matroids
To every finite matroid there is associated its lattice of flats \(L\), from which one can recover the matroid up to simplification, i.e. up to identifying parallel elements and removing loops. The lattices \(L\) which occur as lattices of flats of matroids can be characterized intrinsically as geometric lattices, that is, they are atomic and semimodular. Thus in some sense, matroids have a lattice theoretic characterization via their lattice of flats. As indicated by the title, this paper deals with the same question for oriented matroids. An oriented matroid can be specified by its set \({\mathcal O}\) of covectors, which are \(\{+,-,0\}\)-vectors, and these covectors form a lattice under the componentwise ordering induced by \(0,<,+,-\). One can ask whether the lattices arising in this way have some lattice-theoretic characterization. As the author points out, without any extra information this is probably too hard---the family of lattices of covectors of oriented matroids is very close to the family of face lattices of zonotopes, and hence characterizing these face lattices is nearly a large subcase of the very difficult Steinitz problem of characterizing face lattices of polytopes. The author's solution to this difficulty is to add an extra information. There is a natural cover-preserving order-reversing surjection \({\mathcal O} \rightarrow L\) from the poset of covectors of the oriented matroid to the lattice of flats of the underlying (simple) matroid, called the zero map. This zero map simply assigns to a covector the location of its \(0\)'s. The problem which he then solves is to characterize when a cover-preserving order-reversing surjection from a lattice \({\mathcal O}\) to a geometric lattice \(\L\) comes from an oriented matroid in this way. To this end, he introduces two technical conditions. The first, that the map \({\mathcal O} \rightarrow L\) be an antitonic folding, encompasses the properties one would expect for \({\mathcal O}\) to be the lattices of faces in a combinatorial manifold of the appropriate dimension, and that intervals of rank \(1\) and \(2\) in \({\mathcal O}\) bear ``locally'' the correct relation to their images in \(L\). The second condition, called being a symmetric antitonic folding, roughly corresponds to the antipodal symmetry of the lattice of covectors, which comes from reversing their \(+\) and \(-\)'s. With these definitions, the author proves (Theorem 5) that there is a \(1-1\) correspondence between reorientation classes of simple oriented matroids and symmetric antitonic foldings \({\mathcal O} \rightarrow L\). There is a certain amount of necessary technicality in the proof, but it is mostly quite reasonable and well-written.
scientific article; zbMATH DE number 1045837