Several identities for the characteristic polynomial of a combinatorial geometry

In this paper we explore a research problem of Greene: to find inequalities for the Möbius function which become equalities in the presence of modularity. We replace these inequalities with identities and give combinatorial interpretations for the difference

Connectivity of submodular functions

The notion of connectivity for submodular functions was introduced by Cunningham. This paper relates the connectivity of such a function f to that of certain submodular functions which are derived from f. In particular, we prove a generalisation of the well-known matroid result that, for every element x of a connected matroid M, either the deletion or contraction of x from M is connected. © 1992