A property of the hyperplanes of a matroid and an extension of Dilworth's theorem

Abstract

In this paper, I show that in any combinatorial geometry, that is a matroid without loops or parallel edges, there exist at least as many hyperplanes as elements. This is equivalent to there being at least as many copoints as points in a geometric lattice, and hence extends Dilworth’s theorem [4] which proves equality only for modular lattices. Since proving this result I have discovered that both Baster-field and Kelly [l] and Greene [5] have previously established it. However my proof is totally different as it is incorporated in settling the case of equality. For I shall also show that the only geometries which have precisely as many hyperplanes as elements are the union of disjoint connected matroids, each of which is either a projective geometry or is a geometry of rank two or less. This is not true of matroids in general. The equivalent result for matroids is that there are at least as many hyperplanes as closed sets of rank one. In [3], Dembowski and Wagner essentially prove that if the blocks of a symmetric block design form the hyperplanes of a matroid, then this matroid is a projective geometry. This result may be deduced as a corollary of the main result of this paper. Lastly I show that the conditions of the DembowskiWagner theorem may be relaxed, inasmuch as I prove that if the hyperplanes of a combinatorial geometry are such that any two intersect in h elements, then it is either a projective geometry or has a single base or is the union of a single element and a geometry of rank two.