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The set of all Boolean matrices is denoted by . We call a matrix regular if there is a matrix such that . In this paper, we study the problem of characterizing linear operators on that strongly preserve regular matrices. Consequently, we obtain that if , then all operators on strongly preserve regular matrices, and if , then an operator on strongly preserves regular matrices if and only if there are invertible matrices and such that for all , or and for all .
The Boolean rank of a nonzero Boolean matrix is the minimum number such that there exist an Boolean matrix and a Boolean matrix such that . 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 and . 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...
Let be a Boolean matrix. The isolation number of is the maximum number of ones in such that no two are in any row or any column (that is they are independent), and no two are in a submatrix of all ones. The isolation number of is a lower bound on the Boolean rank of . A linear operator on the set of Boolean matrices is a mapping which is additive and maps the zero matrix, , to itself. A mapping strongly preserves a set, , if it maps the set into the set and the complement of...
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