Characterizations of Zero-Term Rank Preservers of Matrices over Semirings

• Journal title : Kyungpook mathematical journal
• Volume 54, Issue 4,  2014, pp.619-627
• Publisher : Department of Mathematics, Kyungpook National University
• DOI : 10.5666/KMJ.2014.54.4.619
Title & Authors
Characterizations of Zero-Term Rank Preservers of Matrices over Semirings
Kang, Kyung-Tae; Song, Seok-Zun; Beasley, LeRoy B.; Encinas, Luis Hernandez;

Abstract
Let $\small{\mathcal{M}(S)}$ denote the set of all $\small{m{\times}n}$ matrices over a semiring S. For $\small{A{\in}\mathcal{M}(S)}$, 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 $\small{\mathcal{M}(S)}$ 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 $\small{\mathcal{M}(S)}$ that preserve zero-term rank. Consequently we obtain that a linear operator on $\small{\mathcal{M}(S)}$ preserves zero-term rank if and only if it preserves two consecutive zero-term ranks k and k + 1, where $\small{0{\leq}k{\leq}min\{m,n\}-1}$ if and only if it strongly preserves zero-term rank h, where $\small{1{\leq}h{\leq}min\{m,n\}}$.
Keywords
Semiring;zero-term rank;linear operator;(strongly) preserve;
Language
English
Cited by
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