  The rook monoid $R_n$ is the finite monoid whose elements are the 0-1
matrices with at most one nonzero entry in each row and column. The group of
invertible elements of $R_n$ is isomorphic to the symmetric group $S_n$. The
natural extension to $R_n$ of the Bruhat-Chevalley ordering on the symmetric
group is defined in \cite{Renner86}. In this paper, we find an efficient,
combinatorial description of the Bruhat-Chevalley ordering on $R_n$. We also
give a useful, combinatorial formula for the length function on $R_n$.
