Separability of distinct Boolean rank-1 matrices (Q2386807)

Let \(M_{m,n}(\mathbb{B})\) denote the set of all \(m\times n\)-matrices with entries in the Boolean algebra \(\mathbb{B}=\{0,1\}\). The Boolean rank \(b(A)\) of a nonzero Boolean matrix \(A\) is the least \(k\) for which there exist an \(m\times k\) Boolean matrix \(B\) and a \(k\times n\) Boolean matrix \(C\) with \(A= BC\). A rank-1 matrix \(C\) is called a separating matrix of two distinct rank-1 matrices \(A\) and \(B\) if the rank of \(A+ C\) is 2 but the rank of \(B+ C\) is 1 or vice versa. In this case, the matrices \(A\) and \(B\) are called separable. The author tackles the question which pairs of distinct rank-1 matrices over Boolean algebras are separable. He shows that any two distinct Boolean rank-1 matrices are separable. As a first step towards this result the author points out an error in a proof of \textit{L. B. Beasley} and \textit{N. J. Pullman} [Linear Algebra Appl. 59, 55--77 (1984; Zbl 0536.20044)]. He checks his observation by a counterexample and corrects the proof.