Anders Björner: Continuous matroids

Abstract

This will be a review of some work done in the 1980s in collaboration with L. Lovasz. Our main concern at that time was to provide conditions that make it possible to pass to the limit of a class of finite matroids. The characteristic property of a continuous matroid is the existence of a rank function taking as values the full real unit interval. Known examples include Lebesgue measure on the unit interval (the limit of finite Boolean lattices), and normalized dimension of hyperfinite von Neumann geometries (limit of finite projective geometries). In both these cases the lattice property of modularity plays a crucial role. A more general concept, pseudomodularity, makes possible the construction of e.g. continuous field extensions (algebraic matroids) and continuous partition lattices (graphic matroids).