Abstract

We 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.

Keywords: Matroids; Convex geometries; Base exchange property; Supermatroids

Article Outline

1. Introduction

2. Definitions and preliminaries on convex geometries

3. Matroids on convex geometries (cg-matroids)

3.1. Definition

3.2. Bases and an exchange property

3.3. Independent sets

4. Strict cg-matroids

4.1. The strict augmentation property

4.2. Rank functions

5. Concluding remarks

Acknowledgements

References