Geometrical Structures of the Endomorphism Semiring of a Finite Chain: Simplices, Strings and Triangles

TL;DR

This paper investigates the geometric structures of endomorphism semirings of finite chains, revealing that these structures form simplices with specific properties, especially for small values of k.

Contribution

It introduces a geometric perspective on endomorphisms of finite chains, characterizing simplices and providing detailed descriptions for cases k=1 and 2.

Findings

Endomorphisms form a k-simplex with vertices as constant endomorphisms.

Complete description of strings (k=1) and triangles (k=2) within these simplices.

Properties of these simplices are explored for arbitrary k.

Abstract

We establish new results concerning endomorphisms of a finite chain if the cardinality of the image of such endomorphism is no more than some fixed number k. The semiring of all such endomorphisms can be seen as a k - simplex whose vertices are the constant endomorphisms. We explore the properties of these k - simplices and find some results for arbitrary k. For k = 1 and 2 we give a full description of simplices called strings and triangles, respectively.