[taken from preprint] We study the higher Bruhat orders B(n,k) of Manin & Schechtman [MaS] and (1) characterize them in terms of inversion sets, (2) identify them with the posets ZY(Cn+1' r ,n+l) of uniform extensions of the alternating oriented matroids Cn ' r for r := n—k (that is, with the extensions of a cyclic hyperplane arrangement by a new oriented pseudoplane), (3) show that B(n, k) is a lattice for k = 1 and for r < 3, but not in general, (4) show that B(n, k) is ordered by inclusion of inversion sets for k — 1 and for r < 4. However, B(8,3) is not ordered by inclusion. This implies that the partial order Bc (n, k) defined by inclusion of inversion sets differs from B(n, k) in general. We show that the proper part of Bc (n, k) is homotopy equivalent to Sr ~ 2 . Consequently, - £(n , k) ~ Sr ~ 2 for Jfe = 1 and for r < 4. In contrast to this, we find that the uniform extension poset of an affine hyperplane arrangement is in general not graded and not a lattice even for r = 3, and that the proper part is not always homotopy equivalent to Sr( - M '~2 .
Ziegler, G. M. (1993). Higher bruhat orders and cyclic hyperplane arrangements. Topology, 32(2), 259–279. https://doi.org/10.1016/0040-9383(93)90019-R