Geometries on partially ordered sets

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Abstract

Geometries on finite partially ordered sets extend the concept of matroids on finite sets to partially ordered sets. Geometries are defined in terms of closure operators on partially ordered sets. The lattice of closed sets is semimodular, and every finite semimodular lattice is isomorphic to the lattice of closed sets of some geometry. A distinction between geometries and pregeometries is made, and deletion and contraction are discussed as constructions on pregeometries. Strong geometries and strong semimodular lattices are introduced, and strong geometries are characterized as those geometries for which every contraction results in a geometry. Cryptomorphic descriptions of pregeometries are given in terms of flats, hyperplanes, rank function, independent, and B-independent sets. It is shown that among the geometries precisely the strong geometries possess the Kuroš-Ore exchange property, and a general marriage theorem for pregeometries is proved. To each pregeometry on a partially ordered set a dual on the dual partial order is associated as an operation of period 2. Deletion and contraction are seen to commute under duality, and it is shown that the dual of a geometry is, in general, only a pregeometry.

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