On Geometric Lattices and Matroids of Arbitrary Cardinality

Abstract

This paper studies in detail the collection of closed sets of a matroid of arbitrary cardinality ordered by inclusion. The relation between the collection, in particular the collection of a simple matroid, and a finite length geometric lattice is dealt with. Finally, one obtains that up to isomorphism, a finite length geometric lattice is a simple matroid, and vice versa.