All matrices are real. The operator $\max$ on matrices returns the largest value in each row. We are interested in characterizing the set of matrices $D$ of size $n \times m$, $m < n$ such that we have the following.
$\forall G \in \mathbb{R}^{m \times l}\; \text{assuming} \;(\max D G) \in \text{Range}(D) \; \text{we have} \; D \max G = \max D G$
We know that one valid choice for $D$ are matrices containing a subset of rows of some permutation matrix. But I am interested in a complete characterization, or barring that, at least a richer class of valid choices of $D$.
Added by Alex Ravsky. It is possible that there is a characterization of the set of the matrices $D$ like each row of $D$ contains at most one non-zero element, which is also positive. I hope to wrote my ideas when I’ll have time for that. Therefore I voted to undelete this question.