Irreducibility of Semigroup Morphisms

TL;DR

This paper investigates the concept of irreducibility in semigroup morphisms, providing characterizations and fundamental insights into when such morphisms can or cannot be decomposed into non-trivial factors.

Contribution

It introduces the notion of irreducibility for semigroup morphisms, characterizes this property, and explores key questions related to their factorization.

Findings

Characterization of irreducible semigroup morphisms

Criteria for reducibility and irreducibility

Fundamental questions on morphism factorization

Abstract

We study the notion of irreducibility of semigroup morphisms. Given an alphabet $\Sigma$, a morphism $\varphi:\Sigma^+\rightarrow\Sigma^+$ is irreducible if any factorisation $\varphi=\psi_2\circ\psi_1$ can only be satisfied if $\psi_1$ or $\psi_2$ is a trivial morphism. Otherwise, $\varphi$ is reducible. We introduce the notion of irreducibility, characterise this property and study a number of fundamental questions on the concepts under consideration.