  Motivated by the geometry of certain hyperplane arrangements, Manin and
Schechtman defined for each positive integer n a hierarchy of finite partially
ordered sets B(n, k), indexed by positive integers k, called the higher Bruhat
orders. The poset B(n, 1) is naturally identified with the weak left Bruhat
order on the symmetric group S_n, each B(n, k) has a unique maximal and a
unique minimal element, and the poset B(n, k + 1) can be constructed from the
set of maximal chains in B(n, k). Elias has demonstrated a striking connection
between the posets B(n, k) for k = 2 and the diagrammatics of Bott-Samelson
bimodules in type A, providing significant motivation for the development of an
analogous theory of higher Bruhat orders in other Cartan-Killing types,
particularly for k = 2. In this paper we present a partial generalization to
type B, complete up to k = 2, prove a direct analogue of the main theorem of
Manin and Schechtman, and relate our construction to the weak Bruhat order and
reduced expression graph for Weyl groups of type B.
