Characterizations of Zero-Term Rank Preservers of Matrices over Semirings

Abstract

Let M(S) denote the set of all mn matrices over a semiring S. For A ∈ 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 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 M(S) that preserve zero-term rank. Conse- quently we obtain that a linear operator on M(S) preserves zero-term rank if and only if it preserves two consecutive zero-term ranks k and k + 1, where 0 ≤ k ≤ min{m.n} - 1 if and only if it strongly preserves zero-term rank h, where 1 ≤ h ≤ min{m,n}.