  A boolean term order is a total order on subsets of [n]={1,...,n} such that
\emptyset < alpha for all nonempty alpha contained in [n], and alpha < beta
implies alpha \cup gamma < beta \cup gamma for all gamma which do not intersect
alpha or beta. Boolean term orders arise in several different areas of
mathematics, including Gr\"obner basis theory for the exterior algebra, and
comparative probability.
  The main result of this paper is that boolean term orders correspond to one
element extensions of the oriented matroid M(B_n), where B_n is the root system
{e_i:1 \leq i \leq n \} \cup {e_i \pm e_j :1 \leq i < j \leq n}. This
establishes boolean term orders in the frame work of the Baues problem. We also
define a notion of coherence for a boolean term order, and a flip relation
between different term orders. Other results include examples of noncoherent
term orders, including an example exhibiting flip deficiency, and enumeration
of boolean term orders for small values of n.
