The sorting order on a Coxeter group

Let ( W , S ) be an arbitrary Coxeter system. For each word ω in the generators we define a partial order—called the ω-sorting order—on the set of group elements W ω ⊆ W that occur as subwords of ω. We show that the ω-sorting order is a supersolvable join-distributive lattice and that it is strictly between the weak and Bruhat orders on the group. Moreover, the ω-sorting order is a “maximal lattice” in the sense that the addition of any collection of Bruhat covers results in a nonlattice. the way we define a class of structures called supersolvable antimatroids and we show that these are equivalent to the class of supersolvable join-distributive lattices.