Transitive Closure Preservers for Matrices over a Class of Semirings

We introduce the transitive closure operator Γ on the set of square matrices over a class of semirings, and then define the transitive closure of a matrix. We characterize the surjective linear transformations of matrices preserving (strongly preserving) transitive closures, and show that a surjective linear transformation preserves the transitive closures if and only if it commutes with Γ, which is equivalent to preserving the kernel of Γ. We also deduce that the strong linear preservers of transitive closure on 2×2 matrices over certain inclines are surjective. In particular, we provide a complete characterization of strong linear preservers of transitive closure on n×n matrices over the binary Boolean semiring.