to be all value q, then this is called the q-matrix (over the Euclidean domain ) for the arrangement and much information is contained in its Smith normal form. Complex arrangements In complex affine space (which is hard to visualize because even the complex affine plane has four real dimensions), the complement is connected (all one piece) with holes where the hyperplanes were removed. A typical problem about an arrangement in complex space is to describe the holes. The basic theorem about complex arrangements is that the cohomology of the complement M(A) is completely determined by the intersection semilattice. To be precise, the cohomology ring of M(A) (with integer coefficients) is isomorphic to the Orlik–Solomon algebra on Z. The isomorphism can be described explicitly and gives a presentation of the cohomology in terms of generators and relations, where generators are represented (in the de Rham cohomology) as logarithmic differential forms with any linear form defining the generic hyperplane of the arrangement. Technicalities Sometimes it is convenient to allow the degenerate hyperplane, which is the whole space S, to belong to an arrangement. If A contains the degenerate hyperplane, then it has no regions because the complement is empty. However, it still has flats, an intersection semilattice, and faces. The preceding discussion assumes the degenerate hyperplane is not in the arrangement. Sometimes one wants to allow repeated hyperplanes in the arrangement. We did not consider this possibility in the preceding discussion, but it makes no material difference. See also Supersolvable arrangement Oriented matroid References . . . . Discrete geometry Combinatorics Oriented matroids