The sorting order on a Coxeter group

Let (W,S) be an arbitrary Coxeter system and let ω be an arbitrary word in the generators, called the sorting word. Each group element that occurs as a subword of ω has a canonical form---its ω-sorted word. We show that the collection of sorted words has a remarkable structure---that of a supersolvable antimatroid---and hence the subword order on these is a join-distributive lattice in which every interval is supersolvable. Moreover, this ω-sorting order is strictly between the weak and strong Bruhat orders on W.