Properties of poset of nonvanishing minors of a matrix

Say $M$ is a matrix over your favorite field. Let $P$ be the collection of nonvanishing minors of $M$, partially ordered by inclusion in the apparent way.

The vague question is: What can one say about the poset $P$?

A little more explicitly, say $M$ is $m \times n$. Let $B_m$ and $B_n$ be the Boolean lattices on the row and column indices of $M$, respectively; that is, each $B_k$ is simply the power set of $[k]=\{1,\dotsc,k\}$ partially ordered by inclusion. Then $B_m \times B_n$ is the poset (in fact lattice) of submatrices of $M$ of all sizes. Let $\Delta \subset B_m \times B_n$ be the collection of square submatrices, i.e., the same number of rows and columns. This $\Delta$ is only a subposet, not a sublattice (the intersection or "union" of two square submatrices is not necessarily square). And $P \subseteq \Delta$ is the collection of nonvanishing minors, i.e., square submatrices with nonzero determinant.

Here I am a little bit abusing terminology by letting minor refer to both a square submatrix, and its determinant. When I say that I minor [doesn't] vanish, I mean that the submatrix's determinant is [not] equal to zero.

Let's adopt the convention that the "empty" $0 \times 0$ minor of $M$ is nonvanishing (with value $1$).

If all the $r$-minors of $M$ vanish, then so do all the $(r+1)$-minors. If some $r$-minor of $M$ is nonvanishing, then it contains at least one nonvanishing $(r-1)$-minor. So every element of $P$ is "connected" to the $0 \times 0$ "empty" minor by a path in $\Delta$ (I guess, a path whose steps are covering relations). In particular $P$ is graded, the grading is as expected (it coincides with the grading of $\Delta$ by size of minor), and the height of $P$ is equal to the rank of $M$.

Are there any other interesting properties of this poset? For example, can anyone suggest answers (or references) for any of the following, or similar questions?

1. What is the width of $P$?
2. Does $P$ have the Sperner property?
3. It seems impossible for $P$ to be a total order, except trivially if $M$ has at most one nonzero entry. What other posets are realizable or nonrealizable in this way? Does it depend on the characteristic of the field?

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

• If $B_m \times B_n$ is the collection of nonvanishing minors then minors are not square. So what is a vanishing minor? Dec 16 '17 at 4:24
• @StephenMeskin Thank you, I edited to clarify that $B_m \times B_n$ is the collection of submatrices (not necessarily square) and $P$ is a subset of the square ones. Dec 16 '17 at 4:29
• Just to be sure, by "vanishing" you mean determinant $=0$. Also, I have found multi-part questions problematic because different people might answer different parts and their answers deleted as not being complete. Dec 16 '17 at 4:55
• @StephenMeskin Thank you, I edited the question to clarify that "vanishing" means "equals zero". And thank you for the comment on multi-part questions, but in this case I think it makes sense to leave these very closely related questions grouped together. If you have any information (or a reference) related to this question I would be very happy to hear it. Dec 16 '17 at 6:10
• Posted on MO mathoverflow.net/questions/288872/…. Dec 19 '17 at 21:48