Linear isomorphisms preserving Greens relations for matrices over semirings

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 Greens 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.