  Idempotents yield much insight in the structure of finite semigroups and
semirings. In this article, we obtain some results on (multiplicatively)
idempotents of the endomorphism semiring of a finite chain. 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 is proposed. In our main result we prove that such
equivalence class is a semiring and find his order. We prove that the set of
all idempotents with certain jump points is a semiring.
