Idempotent elements of the endomorphism semiring of a finite chain

TL;DR

This paper investigates the structure of idempotent elements in the endomorphism semiring of a finite chain, revealing semiring properties of certain subsets and establishing their algebraic relationships.

Contribution

It introduces new results on the semiring structure of idempotents with fixed points and jump points, including their orders and ideal relationships.

Findings

The set of idempotents with fixed points forms a semiring.

Such semirings are ideals in larger known semirings.

Constructs an equivalence relation with unique idempotent representatives.

Abstract

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.