• Communications of the Korean Mathematical Society
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• v.22 no.2
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• pp.195-205
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• 2007

The maximal column rank of an $m{\times}n$ matrix A over the ring of integers, is the maximal number of the columns of A that are weakly independent. We characterize the linear operators that preserve the maximal column ranks of integer matrices.

• Journal of the Korean Mathematical Society
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• v.47 no.1
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• pp.71-81
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• 2010

We characterize linear operators that preserve sets of matrix ordered pairs which satisfy extreme cases with respect to maximal column rank inequalities of matrix multiplications over semirings.

• Journal of the Korean Mathematical Society
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• v.43 no.1
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• pp.77-86
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• 2006

We consider the set of commuting pairs of matrices and their preservers over binary Boolean algebra, chain semiring and general Boolean algebra. We characterize those linear operators that preserve the set of commuting pairs of matrices over a general Boolean algebra and a chain semiring.

• Honam Mathematical Journal
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• v.29 no.3
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• pp.427-443
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• 2007

The spanning column rank of an $m{\times}n$ integer matrix A is the minimum number of the columns of A that span its column space. We compare the spanning column rank with column rank of matrices over the ring of integers. We also characterize the linear operators that preserve the spanning column rank of integer matrices.

• Honam Mathematical Journal
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• v.33 no.3
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• pp.301-310
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• 2011

In this paper, we construct the sets of fuzzy matrix pairs. These sets are naturally occurred at the extreme cases for the zero-term rank inequalities derived from the multiplication of fuzzy matrix pairs. We characterize the linear operators that preserve these extreme sets of fuzzy matrix pairs.

• Kyungpook Mathematical Journal
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• v.53 no.2
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• pp.273-283
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• 2013

In this paper, we consider the row rank inequalities derived from comparisons of the row ranks of the additions and multiplications of nonnegative integer matrices and construct the sets of nonnegative integer matrix pairs which is occurred at the extreme cases for the row rank inequalities. We characterize the linear operators that preserve these extreme sets of nonnegative integer matrix pairs.

• Journal of the Korean Mathematical Society
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• v.43 no.3
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• pp.539-552
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• 2006

For an $n{\times}n$ Boolean matrix A, A is called nilpotent if $A^m=O$ for some positive integer m. We consider the set of $n{\times}n$ nilpotent Boolean matrices and we characterize linear operators that strongly preserve nilpotent matrices over Boolean algebras.

• Journal of the Korean Mathematical Society
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• v.52 no.3
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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.

• Journal of the Korean Mathematical Society
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• v.32 no.3
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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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• v.29 no.5_6
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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.