Weak Z-Property of the Absolute Order on Groups Generated by Sets Closed Under Taking Inverses

The Bruhat order is a well-studied partial order on Coxeter groups and Schubert varieties. Deodhar provided several characterizations of the Bruhat order, including the so-called “ Z -property.” Another partial order on Coxeter groups that is relevant in combinatorics is the absolute order, which extends the notion of the classical noncrossing partitions to Coxeter groups. In this note, we prove a result that implies that the absolute order satisfies a weaker version of the Z -property.