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 . 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.
CAN, MAHİR BİLEN and RENNER, LEX E.
"Bruhat-Chevalley order on the rook monoid,"
Turkish Journal of Mathematics: Vol. 36:
4, Article 1.
Available at: https://journals.tubitak.gov.tr/math/vol36/iss4/1