Bijective linear maps on semimodules spanned by Boolean matrices of fixed rank (Q5962288)

Let \(B_{I\times J}\) be the set of possibly infinite-sized Boolean matrices (that is, defined over a Boolean algebra \(\{0,1\}\)) with only finitely many nonzero cells, where rows are indexed by the set \(I\) and columns are indexed by the set \(J\). Then, given a fixed integer \(k\geq 2\), let \(B(I,J,k)\) be the Boolean subsemimodule of \(B_{I\times J}\) generated by matrices of (factor) rank-\(k\). NEWLINENEWLINEWith the help of tensor calculus, the authors classify bijective linear maps from \(B(I,J,k)\) to \(B(M,N,k)\). This is done by reduction to linear maps, again from \(B(I,J,k)\) to \(B(M,N,k)\), which preserve the weight, that is, the number of nonzero cells, and simultaneously preserve the set of Boolean matrices with weight \(k^2\) and (factor) rank-one. With this at hand, the classification of linear bijections between \(B_{I\times J}\) and \(B_{M\times N}\) which preserve (factor) rank-\(k\) is obtained. It is also shown that the bijectivity assumption cannot be relaxed.NEWLINENEWLINEThe applications to finite-sized Boolean matrices readily follow.