the plane and the three lines, but not the empty set. There are four regions, none of them bounded. If we add a line crossing the three parallels, then the intersection semilattice consists of the plane, the four lines, and the three points of intersection. There are eight regions, still none of them bounded. If we add one more line, parallel to the last, then there are 12 regions, of which two are bounded parallelograms. Typical problems about an arrangement in n-dimensional real space are to say how many regions there are, or how many faces of dimension 4, or how many bounded regions. These questions can be answered just from the intersection semilattice. For instance, two basic theorems, from Zaslavsky (1975), are that the number of regions of an affine arrangement equals (−1)npA(−1) and the number of bounded regions equals (−1)npA(1). Similarly, the number of k-dimensional faces or bounded faces can be read off as the coefficient of xn−k in (−1)n wA (−x, −1) or (−1)nwA(−x, 1). designed a fast algorithm to determine the face of an arrangement of hyperplanes containing an input point. Another question about an arrangement in real space is to decide how many regions are simplices (the n-dimensional generalization of triangles and tetrahedra). This cannot be answered based solely on the intersection semilattice. The McMullen problem asks for the smallest arrangement of a given dimension in general position in real projective space for which there does not exist a cell touched by all hyperplanes. A real linear arrangement has, besides its face semilattice, a poset of regions, a different one for each region. This poset is formed by choosing an arbitrary base region, B0, and associating with each region R the set S(R) consisting of the hyperplanes that separate R from B. The regions are partially ordered so that R1 ≥ R2 if S(R1, R) contains S(R2, R). In the special case when the hyperplanes arise from a root system, the resulting poset is the corresponding Weyl group with the weak order. In general, the poset of regions is ranked by the number of separating hyperplanes and its Möbius function has been computed . Vadim Schechtman and Alexander Varchenko introduced a matrix indexed by the regions. The matrix element for the region and is given by the product of indeterminate variables for every hyperplane H that separates these two regions. If these variables are specialized