  In this paper we characterize those linear bijective maps on the monoid of
all $n \times n$ square matrices over an anti-negative semifield which preserve
and strongly preserve each of Green's equivalence relations $\mathcal{L},
\mathcal{R}, \mathcal{D}, \mathcal{J}$ and the corresponding three
pre-orderings $\leq_\mathcal{L}, \leq_\mathcal{R}, \leq_\mathcal{J}$. These
results apply in particular to the tropical and boolean semirings, and for
these two semirings we also obtain corresponding results for the $\mathcal{H}$
relation.
