Linear operators that preserve graphical properties of matrices: isolation numbers

LeRoy B. Beasley; Seok-Zun Song; Young Bae Jun

Displaying similar documents to “Linear operators that preserve graphical properties of matrices: isolation numbers”

Linear operators that preserve Boolean rank of Boolean matrices

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

Czechoslovak Mathematical Journal

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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 $A = B C$ . 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...

Linear maps that strongly preserve regular matrices over the Boolean algebra

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

Czechoslovak Mathematical Journal

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The set of all $m \times n$ Boolean matrices is denoted by $𝕄_{m, n}$ . We call a matrix $A \in 𝕄_{m, n}$ regular if there is a matrix $G \in 𝕄_{n, m}$ 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 $min {m, n} \leq 2$ , then all operators on $𝕄_{m, n}$ strongly preserve regular matrices, and if $min {m, n} \geq 3$ , 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 $X \in 𝕄_{m, n}$ , or $m = n$ and $T (X) = U X^{T} V$ for all...

The primitive Boolean matrices with the second largest scrambling index by Boolean rank

Yan Ling Shao, Yubin Gao (2014)

Czechoslovak Mathematical Journal

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The scrambling index of an $n \times n$ primitive Boolean matrix $A$ is the smallest positive integer $k$ such that $A^{k} {(A^{T})}^{k} = J$ , where $A^{T}$ denotes the transpose of $A$ and $J$ denotes the $n \times n$ all ones matrix. For an $m \times n$ Boolean matrix $M$ , its Boolean rank $b (M)$ is the smallest positive integer $b$ such that $M = A B$ for some $m \times b$ Boolean matrix $A$ and $b \times n$ Boolean matrix $B$ . In 2009, M. Akelbek, S. Fital, and J. Shen gave an upper bound on the scrambling index of an $n \times n$ primitive matrix $M$ in terms of its Boolean rank $b (M)$ , and they also characterized...

On BPI Restricted to Boolean Algebras of Size Continuum

Eric Hall, Kyriakos Keremedis (2013)

Bulletin of the Polish Academy of Sciences. Mathematics

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(i) The statement P(ω) = “every partition of ℝ has size ≤ |ℝ|” is equivalent to the proposition R(ω) = “for every subspace Y of the Tychonoff product $2^{(ω)}$ the restriction |Y = Y ∩ B: B ∈ of the standard clopen base of $2^{(ω)}$ to Y has size ≤ |(ω)|”. (ii) In ZF, P(ω) does not imply “every partition of (ω) has a choice set”. (iii) Under P(ω) the following two statements are equivalent: (a) For every Boolean algebra of size ≤ |ℝ| every filter can be extended to an ultrafilter. (b) Every Boolean...