Boolean Term Orders and the Root System B_n

Diane Maclagan

Published 1998-09-24, updated 1999-03-10Version 2

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.