Preservers of term ranks of symmetric matrices

The term rank of an n×n matrix A is the least number of lines (rows or columns) needed to include all the nonzero entries in A. In this paper, we study linear operators that preserve term ranks of n×n symmetric matrices with entries in a commutative antinegative semiring § and with all diagonal entries zero. Consequently, we show that a linear operator T on symmetric matrices with zero diagonal preserves term rank if and only if T preserves term ranks 2 and if and only if T preserves term ranks 3 and Other characterizations of term-rank preservers are also given.