Bruhat-Chevalley order on the rook monoid

The rook monoid Rn 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 Rn is isomorphic to the symmetric group Sn . The natural extension to Rn of the Bruhat-Chevalley ordering on the symmetric group is defined in [1]. In this paper, we find an efficient, combinatorial description of the Bruhat-Chevalley ordering on Rn . We also give a useful, combinatorial formula for the length function on Rn .