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 contains just one idempotent is proposed. In our main result we prove that such an equivalence class is a semiring and find its order. We prove that the set of all idempotents with certain jump points is a semiring.
Idempotents yield much insight in the structure of finite semigroups and semirings. In this paper, we obtain results on (multiplicative) idempotents of the endomorphism semiring of a finite chain. We prove that the set of all idempotents with given fixed points is a semiring and we 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 contains just one idempotent is proposed. In our main result we prove that such equivalence class is a semiring and finds its order. We prove that the set of all idempotents with certain jump points is a semiring.
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