• 한국수학교육학회지시리즈B:순수및응용수학
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• 제21권2호
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• pp.121-128
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• 2014

The Boolean rank of a nonzero m $m{\times}n$ Boolean matrix A is the least integer k such that there are an $m{\times}k$ Boolean matrix B and a $k{\times}n$ Boolean matrix C with A = BC. In 1984, Beasley and Pullman showed that a linear operator preserves the Boolean rank of any Boolean matrix if and only if it preserves Boolean ranks 1 and 2. In this paper, we extend this characterization of linear operators that preserve the Boolean ranks of Boolean matrices. We show that a linear operator preserves all Boolean ranks if and only if it preserves two consecutive Boolean ranks if and only if it strongly preserves a Boolean rank k with $1{\leq}k{\leq}min\{m,n\}$.

• 대한수학회지
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• 제52권3호
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• pp.625-636
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• 2015

The Boolean rank of a nonzero $m{\times}n$ Boolean matrix A is the least integer k such that there are an $m{\times}k$ Boolean matrix B and a $k{\times}n$ Boolean matrix C with A = BC. We investigate the structure of linear transformations T : $\mathbb{M}_{m,n}{\rightarrow}\mathbb{M}_{p,q}$ which preserve Boolean rank. We also show that if a linear transformation preserves the set of Boolean rank 1 matrices and the set of Boolean rank k matrices for any k, $2{\leq}k{\leq}$ min{m, n} (or if T strongly preserves the set of Boolean rank 1 matrices), then T preserves all Boolean ranks.

• 대한수학회지
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• 제54권4호
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• pp.1317-1329
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• 2017

A Boolean rank one matrix can be factored as $\text{uv}^t$ for vectors u and v of appropriate orders. The perimeter of this Boolean rank one matrix is the number of nonzero entries in u plus the number of nonzero entries in v. A Boolean matrix of Boolean rank k is the sum of k Boolean rank one matrices, a rank one decomposition. The perimeter of a Boolean matrix A of Boolean rank k is the minimum over all Boolean rank one decompositions of A of the sums of perimeters of the Boolean rank one matrices. The arctic rank of a Boolean matrix is one half the perimeter. In this article we characterize the linear operators that preserve the symmetric arctic rank of symmetric Boolean matrices.

• Journal of applied mathematics & informatics
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• 제26권3_4호
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• pp.643-652
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• 2008

We construct the sets of Boolean matrix pairs, which are naturally occurred at the extreme cases for the Boolean rank inequalities relative to the sums and difference of two Boolean matrices or compared between their Boolean ranks and their real ranks. For these sets, we consider the linear operators that preserve them. We characterize those linear operators as T(X) = PXQ or $T(X)\;=\;PX^tQ$ with appropriate invertible Boolean matrices P and Q.

• Korean Journal of Mathematics
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• 제12권1호
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• pp.49-54
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• 2004

In this paper we introduce the notion of quotient Boolean algebra and study the relation between the ideals of Boolean algebra ${\wp}(S)$ and the ideals of quotient Boolean algebra ${\wp}(S)/I$.

• 대한수학회지
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• 제32권3호
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• pp.531-540
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• 1995

There is much literature on the study of matrices over a finite Boolean algebra. But many results in Boolean matrix theory are stated only for binary Boolean matrices. This is due in part to a semiring isomorphism between the matrices over the Boolean algebra of subsets of a k element set and the k tuples of binary Boolean matrices. This isomorphism allows many questions concerning matrices over an arbitrary finite Boolean algebra to be answered using the binary Boolean case. However there are interesting results about the general (i.e. nonbinary) Boolean matrices that have not been mentioned and they differ somwhat from the binary case.

• Journal of applied mathematics & informatics
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• 제29권5_6호
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• pp.1525-1532
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• 2011

The $m{\times}n$ Boolean matrix A is said to be of Boolean rank r if there exist $m{\times}r$ Boolean matrix B and $r{\times}n$ Boolean matrix C such that A = BC and r is the smallest positive integer that such a factorization exists. We consider the the sets of matrix ordered pairs which satisfy extremal properties with respect to Boolean rank inequalities of matrices over nonbinary Boolean algebra. We characterize linear operators that preserve these sets of matrix ordered pairs as the form of $T(X)=PXP^T$ with some permutation matrix P.

• 대한수학회보
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• 제35권3호
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• pp.605-613
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• 1998

We study the permanent set of the prime Boolean matrices in the semigroup of Boolean matrices. We define a class $M_n$ of prime matrices, and find all the possible permanents of the elements in $M_n$.

• 대한수학회보
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• 제36권1호
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• pp.131-138
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• 1999

We consider the Boolean linear operators that preserve Boolean rank and obtain some characterizations of the linear operators which extend the results in [1].

• 한국콘텐츠학회논문지
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• 제6권2호
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• pp.27-33
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• 2006

불리언 행렬은 다양한 분야에 응용되어 유용하게 사용되고 있으며 불리언 행렬에 대한 많은 연구가 수행되었다 대부분의 연구에서는 불리언 행렬의 곱셈을 다루고 있으나 모두 두 불리언 행렬 사이의 곱셈에 관심을 두고 있으며 다수의 n$\times$m 불리언 행렬과 모든 m$\times$k불리언 행렬 사이의 곱셈은 극히 소수의 연구에서 보이고 있다. 본 논문은 기존에 제시된 두 불리언 행렬의 최적 곱셈 알고리즘이 모든 불리언행렬에 대한 곱셈을 해야 하는 경우 부적합함을 보이고 n$\times$m 불리언 행렬과 모든 m$\times$k 불리언 행렬의 곱셈을 효율적으로 계산할 수 있는 이론을 정립한 후 이를 적용한 불리언 행렬 곱셈의 실행결과에 대하여 논한다.