3 - Infinite Matroids

Summary

The many different axiom systems for finite matroids given in Chapter 2 of White (1986) offer numerous possibilities when one is attempting to generalize the theory to structures over infinite sets. Some axiom systems that are equivalent when one has a finite ground set are no longer so when an infinite ground set is allowed. For this reason, there is no single class of structures that one calls infinite matroids. Rather, various authors with differing motivations have studied a variety of classes of matroid-like structures on infinite sets. Several of these classes differ quite markedly in the properties possessed by their members and, in some cases, the precise relationship between particular classes is still not known.

The purpose of this chapter is to discuss the main lines taken by research into infinite matroids and to indicate the links between several of the more frequently studied classes of infinite matroids.

There have been three main approaches to the study of infinite matroids, each of these being closely related to a particular definition of finite matroids. This chapter will discuss primarily the independent-set approach. Some details of the closure-operator approach will also be needed, but a far more complete treatment of this has been given by Klee (1971) and by Higgs (1969a, b, c). The third approach, via lattices, will not be considered here. This approach is taken by Maeda & Maeda (1970) and they develop it in considerable detail.