For the (strong) Bruhat order on $S_n$, I normally label vertices by permutations in one-line notation and edges by which transposition you multiply by to go up by a cover. I am now trying to better understand the reduced word characterization of the Bruhat order. I know that $u \leq v$ in the Bruhat order if and only if any reduced word for $v$ contains a subword which is a reduced word for $u$.
What I'm wondering is the following: is there a consistent way to label the vertices by reduced words so that every edge corresponds to the removal of one letter? I see a lot of diagrams in papers where intervals in the Bruhat order are labeled in this way but I can not see a way to prove you can label the whole poset like this.