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?