 
Summary: Semimatroids and their Tutte polynomials
Federico Ardila
Abstract
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 TutteGrothendieck invariant for semi
matroids, and we give a combinatorial interpretation for its nonnegative
coefficients.
1 Introduction.
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 matroidtheoretic 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.
