This question was posed on MSE here three days ago, but hasn't gotten any answers or suggestions. I hope it's okay to ask it on MO, but if I should wait a little longer, please just let me know.

Say $M$ is a (finite) matrix with entries in your favorite field. Let $P$ be the collection of nonvanishing minors of $M$, ordered by inclusion in the natural way. The height of $P$ is the rank of $M$. What else can one say about $P$? For example, here are some naive questions:

- What is the width of $P$? (Does it have any natural meaning, like the height of $P$ being the rank of $M$?)
- Does $P$ have the Sperner property?
- What posets can be realized in this way? Does it depend on the field? What if we restrict to integer $0$-$1$ matrices?

For instance, $P$ can't be a linear order, except in the pretty trivial case that $M$ has at most one nonzero entry.

All the same questions can be asked for the poset of square submatrices of $M$ with nonzero permanent. The questions were inspired by this MO question.

maximal minorsof M’ correspond toall of the minorsof M, and hence the nonvanishing minors of M correspond to the bases of the column matroid of M’. $\endgroup$