Linear operators that preserve term ranks of matrices over semirings

The term rank of a matrix A is the least number of lines (rows or columns) needed to include all the nonzero entries in A, and is a well-known upper bound for many standard and non-standard matrix ranks, and is one of the most important combinatorially. In this paper, we obtain a characterization of linear operators that preserve term ranks of matrices over antinegative semirings. That is, we show that a linear operator T on a matrix space over antinegative semirings preserves term rank if and only if T preserves any two term ranks k and l if and only if T strongly preserves any one term rank k.