It was shown in Chapter 2 that the notion of rank function provides a cryptomorphic theory of matroids. The semimodularity property of the rank is essentially equivalent to the basis-exchange or circuit-elimination axioms and thus is central to matroid theory.

In this chapter we shall consider a more general class of functions on subsets of a finite set E that are semimodular and nondecreasing, but not necessarily nonnegative, normalized, or unit-increased.

The relationships between semimodular functions and matroids have been known and studied since the very beginning of matroid theory. Many results have been found and sometimes independently rediscovered. This chapter presents a unifying theory that attempts to explain most of these known results as derived essentially from Dilworth's fundamental theorem about the embedding of a point lattice into a geometric lattice. Dilworth's original proof was based on a construction of a rank function from a semimodular function, which is a special case of what will be more generally defined in this chapter as expansions. The main thrust of the exposition is to study the properties and applications of expansions.

GENERAL PROPERTIES OF SEMIMODULAR FUNCTIONS

In the most general setting that will be of interest we shall consider a point lattice L and classes of integer-valued functions defined on L.