Boolean matrix; Boolean rank; Boolean linear operator
The Boolean rank of a nonzero $m\times n$ Boolean matrix $A$ is the minimum number $k$ such that there exist an $m\times k$ Boolean matrix $B$ and a $k\times n$ Boolean matrix $C$ such that $A=BC$. In the previous research L. B. Beasley and N. J. Pullman obtained that a linear operator preserves Boolean rank if and only if it preserves Boolean ranks $1$ and $2$. In this paper we extend this characterizations of linear operators that preserve the Boolean ranks of Boolean matrices. That is, we obtain that a linear operator preserves Boolean rank if and only if it preserves Boolean ranks $1$ and $k$ for some $1<k\leq m$.
 Kang, K.-T., Song, S.-Z., Heo, S.-H., Jun, Y.-B.: Linear preserves of regular matrices over general Boolean algebras
. Bull. Malays. Math. Sci. Soc. 34 (2011), 113-125. MR 2783783
 Kim, K. H.: Boolean Matrix Theory and Applications
. Pure and Applied Mathematics 70 Marcel Dekker, New York (1982). MR 0655414
| Zbl 0495.15003