  A matrix is $k$-nonnegative if all its minors of size $k$ or less are
nonnegative. We give a parametrized set of generators and relations for the
semigroup of $k$-nonnegative $n\times n$ invertible matrices in two special
cases: when $k = n-1$ and when $k = n-2$, restricted to unitriangular matrices.
For these two cases, we prove that the set of $k$-nonnegative matrices can be
partitioned into cells based on their factorizations into generators,
generalizing the notion of Bruhat cells from totally nonnegative matrices. Like
Bruhat cells, these cells are homeomorphic to open balls and have a topological
structure that neatly relates closure of cells to subwords of factorizations.
In the case of $(n-2)$-nonnegative unitriangular matrices, we show the cells
form a Bruhat-like CW-complex.
