Is there a generalization of unimodular matrices for non-square matrices? It is well-known that unimodular matrices guarantee an integral solution for a linear program (if the constraint matrix is unimodular).
More specifically:
Let $S$ be a discrete set $S = \{ 1,\ldots,n\}$. Define $X$ to be the set of all possible triplets $(i,j,k)$ where $i,j,k \in S$ and $i < j$ and $k \neq i$ and $k \neq j$.
For each triplet define the vector $v \in \{ 0,1\}^n$ such that $v_i = 1$, $v_j = 1$ and $v_k = -1$. Define $A$ to be the matrix of the $v$s stacked on each other. It should be of size $|X| \times n$.
Am I guaranteed to get an integral solution for a linear program:
$\max c\cdot x$ such that $Ax \le \overline{1}$ and $0 \le x \le 1$
where $\overline{1}$ is a column vector of $1$s of length $|X|$?
Maybe there is some generalization for unimodular matrices that can show this for $A$.
Thanks.