Big cells and LU factorization in reductive monoids

It is well known that an invertible matrix admits a factorization as a product of a lower triangular matrix $L$ and an upper triangular matrix $U$ if and only if all the principal minors of the matrix are non-zero. The corresponding problem for singular matrices is much more subtle. We study this problem in the general setting of a reductive monoid and obtain a solution in terms of the Bruhat-Chevalley order. In the process we obtain a decomposition of the big cell $\overline B{}^-\overline B$ of a reductive monoid, where $B$ and $B^-$are opposite Borel subgroups of the unit group.