  The intersection data of a hyperplane arrangement is described by a geometric
lattice, or equivalently a simple matroid. There is a rich interplay between
this combinatorial structure and the topology of the arrangement complement. In
this paper, we characterize the combinatorial structure underlying an abelian
arrangement (such as a toric or elliptic arrangement) by defining a class of
geometric posets and a generalization of matroids called matroid schemes. The
intersection data of an abelian arrangement is encoded in a geometric poset,
and we prove that a geometric poset is equivalent to a simple matroid scheme.
We lay foundations for the theory of matroid schemes, discussing rank, flats,
and independence. We also extend the definition of the Tutte polynomial to this
setting and prove that it satisfies a deletion-contraction recurrence.
