Matroids on convex geometries (cg-matroids)

AbstractWe consider matroidal structures on convex geometries, which we call cg-matroids. The concept of a cg-matroid is closely related to but different from that of a supermatroid introduced by Dunstan, Ingleton, and Welsh in 1972. Distributive supermatroids or poset matroids are supermatroids defined on distributive lattices or sets of order ideals of posets. The class of cg-matroids includes distributive supermatroids (or poset matroids). We also introduce the concept of a strict cg-matroid, which turns out to be exactly a cg-matroid that is also a supermatroid. We show characterizations of cg-matroids and strict cg-matroids by means of the exchange property for bases and the augmentation property for independent sets. We also examine submodularity structures of strict cg-matroids

Poset matching—a distributive analog of independent matching

AbstractGiven poset matroids (or distributive supermatroids) on two finite posets and an ordered binary relation which associates with every element of each poset an (order) ideal of other poset, a poset matching matches independent ideals with independent ideals. The general theory for this distributive analog of independent matching is investigated and analogs of the independent matching algorithms and theorems are obtained. The class of transversal and matching poset matroids is discussed. An interpretation of the Dilworth completion of a poset matroid is given in terms of matchings