• Title, Summary, Keyword: semiring rank

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Linear operators that preserve spanning column ranks of nonnegative matrices

  • Hwang, Suk-Geun;Kim, Si-Ju;Song, Seok-Zun
    • Journal of the Korean Mathematical Society
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    • v.31 no.4
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    • pp.645-657
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    • 1994
  • If S is a semiring of nonnegative reals, which linear operators T on the space of $m \times n$ matrices over S preserve the column rank of each matrix\ulcorner Evidently if P and Q are invertible matrices whose inverses have entries in S, then $T : X \longrightarrow PXQ$ is a column rank preserving, linear operator. Beasley and Song obtained some characterizations of column rank preserving linear operators on the space of $m \times n$ matrices over $Z_+$, the semiring of nonnegative integers in [1] and over the binary Boolean algebra in [7] and [8]. In [4], Beasley, Gregory and Pullman obtained characterizations of semiring rank-1 matrices and semiring rank preserving operators over certain semirings of the nonnegative reals. We considers over certain semirings of the nonnegative reals. We consider some results in [4] in view of a certain column rank instead of semiring rank.

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Spanning column rank 1 spaces of nonnegative matrices

  • Song, Seok-Zun;Cheong, Gi-Sang;Lee, Gwang-Yeon
    • Journal of the Korean Mathematical Society
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    • v.32 no.4
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    • pp.849-856
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    • 1995
  • There are some papers on structure theorems for the spaces of matrices over certain semirings. Beasley, Gregory and Pullman [1] obtained characterizations of semiring rank 1 matrices over certain semirings of the nonnegative reals. Beasley and Pullman [2] also obtained the structure theorems of Boolean rank 1 spaces. Since the semiring rank of a matrix differs from the column rank of it in general, we consider a structure theorem for semiring rank in [1] in view of column rank.

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EXTREME PRESERVERS OF TERM RANK INEQUALITIES OVER NONBINARY BOOLEAN SEMIRING

  • Beasley, LeRoy B.;Heo, Seong-Hee;Song, Seok-Zun
    • Journal of the Korean Mathematical Society
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    • v.51 no.1
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    • pp.113-123
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    • 2014
  • The term rank of a matrix A over a semiring $\mathcal{S}$ is the least number of lines (rows or columns) needed to include all the nonzero entries in A. In this paper, we characterize linear operators that preserve the sets of matrix ordered pairs which satisfy extremal properties with respect to term rank inequalities of matrices over nonbinary Boolean semirings.

Rank-preserver of Matrices over Chain Semiring

  • Song, Seok-Zun;Kang, Kyung-Tae
    • Kyungpook Mathematical Journal
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    • v.46 no.1
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    • pp.89-96
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    • 2006
  • For a rank-1 matrix A, there is a factorization as $A=ab^t$, the product of two vectors a and b. We characterize the linear operators that preserve rank and some equivalent condition of rank-1 matrices over a chain semiring. We also obtain a linear operator T preserves the rank of rank-1 matrices if and only if it is a form (P, Q, B)-operator with appropriate permutation matrices P and Q, and a matrix B with all nonzero entries.

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LINEAR PRESERVERS OF SPANNING COLUMN RANK OF MATRIX PRODUCTS OVER SEMIRINGS

  • Song, Seok-Zun;Cheon, Gi-Sang;Jun, Young-Bae
    • Journal of the Korean Mathematical Society
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    • v.45 no.4
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    • pp.1043-1056
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    • 2008
  • The spanning column rank of an $m{\times}n$ matrix A over a semiring is the minimal number of columns that span all columns of A. We characterize linear operators that preserve the sets of matrix ordered pairs which satisfy multiplicative properties with respect to spanning column rank of matrices over semirings.

On spanning column rank of matrices over semirings

  • Song, Seok-Zun
    • Bulletin of the Korean Mathematical Society
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    • v.32 no.2
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    • pp.337-342
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    • 1995
  • A semiring is a binary system $(S, +, \times)$ such that (S, +) is an Abelian monoid (identity 0), (S,x) is a monoid (identity 1), $\times$ distributes over +, 0 $\times s s \times 0 = 0$ for all s in S, and $1 \neq 0$. Usually S denotes the system and $\times$ is denoted by juxtaposition. If $(S,\times)$ is Abelian, then S is commutative. Thus all rings are semirings. Some examples of semirings which occur in combinatorics are Boolean algebra of subsets of a finite set (with addition being union and multiplication being intersection) and the nonnegative integers (with usual arithmetic). The concepts of matrix theory are defined over a semiring as over a field. Recently a number of authors have studied various problems of semiring matrix theory. In particular, Minc [4] has written an encyclopedic work on nonnegative matrices.

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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
    • Kyungpook Mathematical Journal
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    • v.54 no.4
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    • pp.619-627
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    • 2014
  • 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 PRESERVERS OF SPANNING COLUMN RANK OF MATRIX SUMS OVER SEMIRINGS

  • Song, Seok-Zun
    • Journal of the Korean Mathematical Society
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    • v.45 no.2
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    • pp.301-312
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    • 2008
  • The spanning column rank of an $m{\times}n$ matrix A over a semiring is the minimal number of columns that span all columns of A. We characterize linear operators that preserve the sets of matrix pairs which satisfy additive properties with respect to spanning column rank of matrices over semirings.

Extreme Preservers of Zero-term Rank Sum over Fuzzy Matrices

  • Song, Seok-Zun;Na, Yeon-Jung
    • Kyungpook Mathematical Journal
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    • v.50 no.4
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    • pp.465-472
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    • 2010
  • 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.