For example, the set of faces of a central or affine hyperplane
arrangement or the set of covectors of an oriented matroid (possibly affine) is a CW left regular band.
Let H an affine hyperplane
orthogonal to the line between x and y that passes through [x + y/2].
The affine Weyl group [??] is generated by reflections [s.sub.[beta],k] through the affine hyperplanes
The preimage of a toric arrangement A under the covering map p: [C.sup.d] [congruent to] [Hom.sub.Z]([LAMBDA], C) [right arrow] [Hom.sub.Z]([LAMBDA], [C.sup.*]) = [T.sub.[LAMBDA]], [phi] [??] exp x [phi] is a locally finite affine hyperplane arrangement on [Hom.sub.Z]([LAMBDA]; C).
Definition 11 A arrangement of hyperplanes in V is a collection B of affine hyperplanes in V.
Like the Shi arrangement, the Ish(ar)rangement begins with the [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] linear hyperplanes of the Coxeter arrangement and then adds another [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] affine hyperplanes
The set of values [delta] [member of] Z such that f vanishes identically on the affine hyperplane of equation f = [delta] is provided by Table 2.
Then [q.sub.I] - [q.sub.J] coincide on the affine hyperplanes [f.sub.ijk] = [delta] for the values of [delta] given by the third column in Table 1.
The folding operator [[phi].sub.i] is the operator which acts on an alcove walk by leaving its initial segment from [A.sub.0] to [A.sub.i-1] intact and by reflecting the remaining tail in the affine hyperplane
containing the face [F.sub.i].
As a set, [[PHI].sup.+] (G, w) consists of the following affine hyperplanes
A hyperplane arrangement A (or simply an arrangement) is a finite collection of affine hyperplanes