whose intersection is empty. (The dimension of the empty set is defined to be −1.) This polynomial helps to solve some basic questions; see below. Another polynomial associated with A is the Whitney-number polynomial wA(x, y), defined by summed over B ⊆ C ⊆ A such that f(B) is nonempty. Being a geometric lattice or semilattice, L(A) has a characteristic polynomial, pL(A)(y), which has an extensive theory (see matroid). Thus it is good to know that pA(y) = yi pL(A)(y), where i is the smallest dimension of any flat, except that in the projective case it equals yi + 1pL(A)(y). The Whitney-number polynomial of A is similarly related to that of L(A). (The empty set is excluded from the semilattice in the affine case specifically so that these relationships will be valid.) The Orlik–Solomon algebra The intersection semilattice determines another combinatorial invariant of the arrangement, the Orlik–Solomon algebra. To define it, fix a commutative subring K of the base field and form the exterior algebra E of the vector space generated by the hyperplanes. A chain complex structure is defined on E with the usual boundary operator . The Orlik–Solomon algebra is then the quotient of E by the ideal generated by elements of the form for which have empty intersection, and by boundaries of elements of the same form for which has codimension less than p. Real arrangements In real affine space, the complement is disconnected: it is made up of separate pieces called cells or regions or chambers, each of which is either a bounded region that is a convex polytope, or an unbounded region that is a convex polyhedral region which goes off to infinity. Each flat of A is also divided into pieces by the hyperplanes that do not contain the flat; these pieces are called the faces of A. The regions are faces because the whole space is a flat. The faces of codimension 1 may be called the facets of A. The face semilattice of an arrangement is the set of all faces, ordered by inclusion. Adding an extra top element to the face semilattice gives the face lattice. In two dimensions (i.e., in the real affine plane) each region is a convex polygon (if it is bounded) or a convex polygonal region which goes off to infinity. As an example, if the arrangement consists of three parallel lines, the intersection semilattice consists of