Cyclic semirings with idempotent noncommutative addition

The article discusses the structure of cyclic semirings with noncommutative addition. In the infinite case, the addition is idempotent and is either left or right. Addition of a finite cyclic semirings can be either idempotent or nonidempotent. In the finite additively idempotent cyclic semiring, addition is reduced to the addition of cyclic subsemiring with commutative addition and absorbing element for multiplication and the addition of a cycle that is a finite semifield.