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Keywords:
Boolean matrix; Boolean rank; Boolean linear operator
Boolean matrix; Boolean rank; Boolean linear operator
Summary:
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 $1<k\leq m$.
References:
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[2] Beasley, L. B., Pullman, N. J.: Boolean-rank-preserving operators and Boolean-rank-1 spaces. Linear Algebra Appl. 59 (1984), 55-77. DOI 10.1016/0024-3795(84)90158-7 | MR 0743045 | Zbl 0536.20044
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