Boolean Term Orders and the Root System B_n

Abstract

A Boolean term order is a total order on subsets of [n] ={1,..., n} such that ∅ ≺ α for all α $\subseteq$ [n], α ≠ ∅, and α ≺ β ⇒ α ∪ γ ≺ β ∪ γ for all γ with γ ∩(α ∪ β) = ∅. Boolean term orders arise in several different areas of mathematics, including Gröbner 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 ≤ i ≤ n} ∪{e_i ± e_j : 1 ≤ i < j ≤ n}. This establishes Boolean term orders in the framework of the Baues problem, in the sense of (Reiner, 1998). 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.