Matroids on domains of functions.

Any integer-valued function with finite domain E defines, by means of an associated submodular function on 2E , a matroid M(E). The class pi of matroids so obtained is closed under restriction, contraction, and is self dual. We show it consists precisely of those transversal matroids having a presentation in which the sets of the presentation are nested. We give an excluded minor characterisation of M . We count the members of M on an n-set and exhibit explicitly those on a 6-set. We extend the above investigation, using Rado's Selection Principle, and permitting E to be infinite, to pregeometries. Finally, by examining some integer-valued functions on Er with r possibly greater than 1, we discuss some of the properties of the class of matroids so obtained.