Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
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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
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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 $1{\leq}h{\leq}min\{m,n\}$.

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Acknowledgement

Supported by : National Research Foundation of Korea

References

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