Idempotent Elements of the Endomorphism Semiring of a Finite Chain

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. 1. Introduction The idempotents play an essential role in the theory of finite semigroups and semirings. It is well known that in a finite semigroup some power of each element is an idempotent, so the idempotents can be taken to be like a generating system of the semigroup or the semiring. For deep results, using idempotents in the representation theory of finite semigroups, we refer the reader to [1, 2]. Let us briefly survey the contents of our paper. After the preliminaries, in Section 3 we show some facts about the fixed points of idempotent endomorphisms. The central result here is Theorem 9 where we prove that the set of all idempotents with fixed points , , is a semiring of order . Moreover, this semiring is an ideal of the semiring of all endomorphisms having at least as fixed points. In the next section we consider an equivalence relation on some finite semigroup such that for any follows if and only if , where and is an idempotent of . Then we consider the equivalence classes of semigroup which is, see [3], one subsemigroup of . Here we investigate the so-called jump points of the endomorphism and prove that between any two fixed points and of an endomorphism, where , there is a unique jump point. The main result of the paper is Theorem 19 where we prove that such an equivalence class is a semiring of order where is the th Catalan number. In the last section of the paper we consider idempotent endomorphisms with arbitrary fixed points but with certain jump points. Here we prove that the set of idempotent endomorphisms with identical jump points is a semiring. 2. Preliminaries We consider some basic definitions and facts concerning finite semigroups and that can be found in any of [1, 2, 4, 5]. As the terminology for semirings is not completely standardized, we say what our conventions are. An algebra with two binary operations + and？？ ？？on is called a semiring if(i) is a commutative