TITLE: permutations and involutions in binary arrays
QUESTION [7 upvotes]: This question arises from my research in finite regular semigroups but I've shown it is equivalent to a purely combinatorial question about binary arrays.
Let M be a finite m x n table with binary entries.  We shall call a specific position (i,j)  in M  a member. We identify M  with its set of members.  Each member then has an entry, which is either 0 or 1. 
Two members (i,j)  and (k,l)  are related if the entries at (i,l)  and (k,j)  are both 1, in which case we write (i,j) R (k,l).
A permutation f on M is called a (permutation) matching  if (i,j) R f(i,j)  for all (i,j)  in M. 
My question is, if M  has a permutation matching, is it necessarily the case that M  has a matching that is an involution?

REPLY [2 votes]: Not an answer, but a nice sufficient condition for existence of an involution when $m=n$:
If $m=n$ and $\operatorname{perm}(M)>0$, then $M$ has a matching that is an involution.
Proof. Since $\operatorname{perm}(M)>0$, there exists a permutation $\pi$ such that $$M_{1,\pi(1)}=M_{2,\pi(2)}=\dots=M_{n,\pi(n)}=1.$$ 
Then
$f((i,j)):=(\pi^{-1}(j),\pi(i))$ is a matching and an involution.