Linear preservers for matrices over a class of semirings

Abstract

We introduce an equivalence relation ρ on matrices over a class of semirings, which is relevant to the Kleene star operator on matrices. We characterize those surjective linear transformations for matrices preserving ρ. Also, we show that a surjective linear transformation preserves Kleene stars of matrices if and only if it commutes with the Kleene star operator if and only if it preserves ρ. Finally, we illustrate that there exist non-surjective linear transformations that preserve ρ or preserve Kleene stars of matrices or commute with the Kleene star operator.