The rook monoid is lexicographically shellable

The main result of this paper is that the rook monoid $R_n$ with respect to Bruhat-Chevalley-Renner ordering is a lexicographically shellable poset. Consequently, we know that for any interval $I$ in $R_n$, the order complex $\Delta(I)$ has the homotopy type of a wedge of spheres or balls.