  A family C of circuits of a matroid M is a linear class if, given a modular
pair of circuits in C}, any circuit contained in the union of the pair is also
in C. The pair (M,C) can be seen as a matroidal generalization of a biased
graph. We introduce and study an Orlik-Solomon type algebra determined by
(M,C). If C is the set of all circuits of M this algebra is the Orlik-Solomon
algebra of M.
