Let $Q$ be a word in the generators of some Coxeter group, and consider a subword $R$ (not necessarily reduced, though I might want $Q$ to be).

Define the *greedy* or *Demazure product* of $R$ as follows: while multiplying generators left-to-right, insist on only going upward in Bruhat order; any letter in $R$ that would take you downward should be skipped over.

For abstruse geometric reasons,
I want to divide the set of all $2^{|Q|}$ subwords of $Q$ into equivalence classes $\{R\}$, according to (the greedy product of $R$, the locations in $Q$ of the skipped letters). That is, for each pair $(w \in W, S\subseteq Q)$, I consider the set of $R$ with greedy product $w$ and skip locations at exactly $S$.

For example, each $(w,\emptyset)$ corresponds to the class of reduced subwords with (ordinary) product $w$, a well-studied object (e.g. in my paper with Ezra Miller on subword complexes). If $Q=121$ in $S_3$, the $2^3$ subwords break into a singleton for each $w$, except for $\{1--,--1\}$ and $\{1-\underline{1}\}$ for $w=(12)$, where the underlined $\underline{1}$ is the only skipped letter.

> Has anyone studied these equivalence classes of subwords before?
>
> If so, are standard results like the Coxeter exchange condition known for them?

EDIT: here's a larger example, hopefully to make clearer the equivalence relation.
Let $Q = 1231$, a word in the generators of $S_4$.
Then the classes are
$$ ---- $$
$$ 1---, \quad ---1 $$
$$ -2-- $$
$$ --3- $$
$$ 12-- $$
$$ 1-3-, \quad --31 $$
$$ 1--\underline{1} $$
$$ -23- $$
$$ -2-1 $$
$$ 123- $$
$$ 12-1 $$
$$ 1-3\underline{1} $$
$$ -231 $$
$$ 1231 $$
where I've underlined the skip positions.