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Linear maps that strongly preserve regular matrices over the Boolean algebra

Kyung-Tae Kang, Seok-Zun Song (2011)

Czechoslovak Mathematical Journal

The set of all m × n Boolean matrices is denoted by 𝕄 m , n . We call a matrix A 𝕄 m , n regular if there is a matrix G 𝕄 n , m such that A G A = A . In this paper, we study the problem of characterizing linear operators on 𝕄 m , n that strongly preserve regular matrices. Consequently, we obtain that if min { m , n } 2 , then all operators on 𝕄 m , n strongly preserve regular matrices, and if min { m , n } 3 , then an operator T on 𝕄 m , n strongly preserves regular matrices if and only if there are invertible matrices U and V such that T ( X ) = U X V for all X 𝕄 m , n , or m = n and T ( X ) = U X T V for all X 𝕄 n .

Linear operators that preserve Boolean rank of Boolean matrices

LeRoy B. Beasley, Seok-Zun Song (2013)

Czechoslovak Mathematical Journal

The Boolean rank of a nonzero m × n Boolean matrix A is the minimum number k such that there exist an m × k Boolean matrix B and a k × n Boolean matrix C such that A = B C . 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...

Linear operators that preserve graphical properties of matrices: isolation numbers

LeRoy B. Beasley, Seok-Zun Song, Young Bae Jun (2014)

Czechoslovak Mathematical Journal

Let A be a Boolean { 0 , 1 } matrix. The isolation number of A is the maximum number of ones in A such that no two are in any row or any column (that is they are independent), and no two are in a 2 × 2 submatrix of all ones. The isolation number of A is a lower bound on the Boolean rank of A . A linear operator on the set of m × n Boolean matrices is a mapping which is additive and maps the zero matrix, O , to itself. A mapping strongly preserves a set, S , if it maps the set S into the set S and the complement of...

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