Characterizations of Zero-Term Rank Preservers of Matrices over Semirings

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