Summary: Semimatroids and their Tutte polynomials
We define and study semimatroids, a class of objects which abstracts
the dependence properties of an affine hyperplane arrangement. We show
that geometric semilattices are precisely the posets of flats of semima-
troids. We define and investigate the Tutte polynomial of a semimatroid.
We prove that it is the universal Tutte-Grothendieck invariant for semi-
matroids, and we give a combinatorial interpretation for its non-negative
The goal of this paper is to define and study a class of objects called semima-
troids. A semimatroid can be thought of as a matroid-theoretic abstraction of
the dependence properties of an affine hyperplane arrangement. Many proper-
ties of hyperplane arrangements are really facts about their underlying matroidal
structure. Therefore, the study of such properties can be carried out much more
naturally and elegantly in the setting of semimatroids.
The paper is organized as follows. In Section 2 we define semimatroids,
and show how we can think of a hyperplane arrangement as a semimatroid.
The following sections provide different ways of thinking about semimatroids.