Nilpotent and idempotent elements of subsemirings of the endomorphism semiring of an infinite chain

Idempotent and nilpotent elements are important in the structure of semigroups and semirings. In this article, we obtain some results of nilpotent endomorphisms of an infinite chain which applies to (multiplicatively) idempotents of the endomorphism semiring of a finite chain as well. We prove that the set of all idempotents with certain fixed points is a semiring and find its order. We further show that this semiring is an ideal in a well known semiring. The construction of an equivalence relation such that any equivalence class contain just one idempotent and roots of the idempotent is proposed. In our main result we prove that such equivalence class is a semiring and find his order.