Economical presentations of matroids by functions

Matroid theory arose as an attempt to generalize and unify concepts in such relatively distinct areas of mathematics as linear algebra, graph theory, finite geometry and lattice theory. Consequently, there are many equivalent ways of viewing matroids. Our main interest in this thesis is the description of matroids by integer-valued functions on the subsets of their ground-sets. Early papers in this direction by Dilworth [14] and Ingleton [21] contain in implicit form the following characterization of a matroid by its rank function.