Derivations of some classes of additively idempotent semirings

This article investigates a new class of matrix semirings over an arbitrary (not necessarily commutative) ai-semiring. Using the considered left(right) semicentral idempotents, we define an LR-semiring. For suitably chosen basis of the LR-semiring we prove that it is isomorphic to a matrix semiring. The main results in the article are associated with the derivations of the LR-matrix semirings. We prove that the endomorphism semiring of a finite chain with n elements is isomorphic to a semiring of (0,1)-matrices.