Matroids on Partially Ordered Sets

Abstract

Several generalizations of the notion of matroid have been proposed Ž . Brylawski, Edmonds, Faigle, Korte, Lovasz, Welsh, White . This work proposes yet another one, one that is motivated by two considerations. The first motivation comes from projective geometry. Matroids are the Ž natural setting for the study of arrangements of hyperplanes or, equiva. lently, sets of points in projective space. It is natural to ask whether arrangements of linear varieties of different dimensions in projective space may be ensconced into a similar axiomatic setting, one in which matroid] theoretic arguments, with circuits, rank, bases, etc., may be used. The second motivation is the replacement of a Boolean algebra of sets by the distributive lattice of filters of a finite partially ordered set. This replacement has proved fruitful in other contexts, most successfully in the replacement of algebraic varieties by schemes in algebraic geometry. Our definition of poset matroids allows the extension to this new setting of every notion of matroid theory. In fact, the extension of the notion of matroid to poset matroids sheds light on the mutual relation of the notions of matroid theory. Every family of linear varieties in projective space defines a poset matroid, and a classification of the possible special positions of a set of linear varieties is reflected in the combinatorial structure of the poset matroid thereby obtained. Two languages are available for poset matroids: the language of partially ordered sets and the language of distributive lattices. The translation of poset matroids into the language of distributive lattices leads to the definition of a combinatorial scheme. Again, the translation of matroid