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Characterizations of Zero-Term Rank Preservers of Matrices over Semirings
Kang, Kyung-Tae,Song, Seok-Zun,Beasley, LeRoy B.,Encinas, Luis Hernandez Department of Mathematics 2014 Kyungpook mathematical journal Vol.54 No.4
Let $\mathcal{M}(S)$ denote the set of all $m{\times}n$ matrices over a semiring S. For $A{\in}\mathcal{M}(S)$, zero-term rank of A is the minimal number of lines (rows or columns) needed to cover all zero entries in A. In [5], the authors obtained that a linear operator on $\mathcal{M}(S)$ preserves zero-term rank if and only if it preserves zero-term ranks 0 and 1. In this paper, we obtain new characterizations of linear operators on $\mathcal{M}(S)$ that preserve zero-term rank. Consequently we obtain that a linear operator on $\mathcal{M}(S)$ preserves zero-term rank if and only if it preserves two consecutive zero-term ranks k and k + 1, where $0{\leq}k{\leq}min\{m,n\}-1$ if and only if it strongly preserves zero-term rank h, where $1{\leq}h{\leq}min\{m,n\}$.
Linear Operator Preserving Zero-term Rank of Nonnegative Integer Matrices
Han, Hee-Jeong,Kang, Kyung-Tae,Song, Seok-Zun 濟州大學校 基礎科學硏究所 2002 基礎科學硏究 Vol.15 No.2
주어진 행렬의 영항 계수는 그 행렬에 나타나는 모든 영 원소들을 덮을 수 있는 행과 열의 최소수로 정의된다. 본 논문에서는 비음의 정수들로 이루어진 반환에서 원소를 갖는 행렬들을 생각한다. 이 행렬들의 영항계수를 보존하는 선형연산자를 연구하여 그 형태를 규명하였고, 또 이 선형연산자와 필요충분조건이 되는 명제들을 찾아서 그 동치성을 증명하였다. Zero-term rank of an m x n matrix A is the minimum number of lines(rows or columns) needed to cover all the zero entries of A. In this thesis, we obtain characterizations the linear operators preserving zero-term rank on the set of m x n matrices over the nonnegative integer semiring.
Zero-term Rank Preservers of Fuzzy Matrices
Kim, Choon-Sim,Kang, Kyung-Tae,Song, Seok-Zun 濟州大學校 基礎科學硏究所 2002 基礎科學硏究 Vol.15 No.2
행렬의 영항 계수는 그 행렬에 나타나는 모든 영 원소를 덮을 수 있는 행과 열의 극소수로 정의된다. 본 논문에서는 퍼지집합에서 원소를 갖는 퍼지행렬들을 생각한다. 이 퍼지행렬들의 영항계수 보존자를 연구하여 그 형태를 규명하였고, 또 이 보존자와 필요충분조건들을 찾아서 그 동치성을 증명하였다. 곧, 모든 영항계수를 보존하은 선형연산자의 형태는 주어진 행렬의 좌우측에 순환행렬을 곱하며, 주어진 행렬의 영 아닌 원소들을 영 되게 하지 않는 행렬의 Schur 곱의 형태로 나타남을 밝혔다. Zero-term rank of a matrix is the minimum number of lines(rows or columns) needed to cover all the zero entries of the given matrix. In this thesis, we characterize the zero-term rank preservers of the m x n matrices over a fuzzy semiring.
Linear operators that preserve zero-term rank over fields and rings
Beasley, Leroy B.,Song, Seok-Zun 濟州大學校 基礎科學硏究所 2002 基礎科學硏究 Vol.15 No.2
The zero-term rank of a matrix is the maximum number of zeros in any generalized diagonal. This article characterrizes the linear operators that preserve zero-term rank of m × n matrices when the matrices have entries either in a field with at least mn +2 elements or in a ring whose characteristic is not 2.
BEASLEY, LEROY B.,SONG, SEOK-ZUN,LEE, SANG-GU 제주대학교 기초과학연구소 2002 基礎科學硏究 Vol.15 No.1
We obtain characterizations of those linear operators that preserve zero-term rank on the m×n matrices over antinegative semirings. That is, a linear operator T preserves zero-term rank if and only if it has the form T(X) = P(B_(o)X)Q, where P, Q are permutation matrices and B_(o)X is the Schur product with B whose entries are all nonzero and not zero-divisors.
Extreme Preservers of Zero-term Rank Sum over Fuzzy Matrices
Song, Seok-Zun,Na, Yeon-Jung Department of Mathematics 2010 Kyungpook mathematical journal Vol.50 No.4
In this paper, we consider two extreme sets of zero-term rank sum of fuzzy matrix pairs: $$\cal{z}_1(\cal{F})=\{(X,Y){\in}\cal{M}_{m,n}(\cal{F})^2{\mid}z(X+Y)=min\{z(X),z(Y)\}\};$$ $$\cal{z}_2(\cal{F})=\{(X,Y){\in}\cal{M}_{m,n}(\cal{F})^2{\mid}z(X+Y)=0\}$$. We characterize the linear operators that preserve these two extreme sets of zero-term rank sum of fuzzy matrix pairs.
EXTREME PRESERVERS OF FUZZY MATRIX PAIRS DERIVED FROM ZERO-TERM RANK INEQUALITIES
송석준,박은아 호남수학회 2011 호남수학학술지 Vol.33 No.3
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.
EXTREME PRESERVERS OF FUZZY MATRIX PAIRS DERIVED FROM ZERO-TERM RANK INEQUALITIES
Song, Seok-Zun,Park, Eun-A The Honam Mathematical Society 2011 호남수학학술지 Vol.33 No.3
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.