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.

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.