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Characterizations of Zero-Term Rank Preservers of Matrices over Semirings

  • Kang, Kyung-Tae (Department of Mathematics, Jeju National University) ;
  • Song, Seok-Zun (Department of Mathematics, Jeju National University) ;
  • Beasley, LeRoy B. (Department of Mathematics and Statistics, Utah State University) ;
  • Encinas, Luis Hernandez (Department of Information Processing and Cryptography, Institute of Physical and Information Technologies, Spanish National Research Council)
  • Received : 2013.10.20
  • Accepted : 2014.04.09
  • Published : 2014.12.23

Abstract

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 $1{\leq}h{\leq}min\{m,n\}$.

Keywords

Acknowledgement

Supported by : National Research Foundation of Korea

References

  1. L. B. Beasley and N. J. Pullman, Boolean rank preserving operators and Boolean rank-1 spaces, Linear Algebra Appl., 59(1984), 55-77. https://doi.org/10.1016/0024-3795(84)90158-7
  2. L. B. Beasley and N. J. Pullman, Semiring rank versus column rank, Linear Algebra Appl., 101(1988), 33-48. https://doi.org/10.1016/0024-3795(88)90141-3
  3. L. B. Beasley and N. J. Pullman, Linear operators that preserve term rank-1, Proc. Roy. Irish Acad., 91(1990), 71-78.
  4. L. B. Beasley, S. Z. Song and S. G. Lee, Linear operators that preserve zero-term rank of Boolean matrices, J. Korean Math. Soc., 36(6)(1999), 1181-1190.
  5. L. B. Beasley, S. Z. Song and S. G. Lee, Zero-term rank preservers, Linear and Multilinear Algebra, 48(2001), 313-318. https://doi.org/10.1080/03081080108818677
  6. K. T. Kang, S. Z. Song and L. B. Beasley, Linear preservers of term ranks of matrices over semirings, Linear Algebra Appl., 436(2012), 1850-1862. https://doi.org/10.1016/j.laa.2011.08.046