Irreducibility of Semigroup Morphisms
This work addresses a theoretical problem in abstract algebra, specifically for researchers in semigroup theory, and appears incremental as it introduces and studies a new notion without broad practical applications.
The paper tackles the problem of defining and characterizing irreducibility for semigroup morphisms, establishing foundational properties and addressing key questions about this concept.
We study the notion of irreducibility of semigroup morphisms. Given an alphabet $Σ$, a morphism $Ï:Σ^+\rightarrowΣ^+$ is irreducible if any factorisation $Ï=Ï_2\circÏ_1$ can only be satisfied if $Ï_1$ or $Ï_2$ is a trivial morphism. Otherwise, $Ï$ is reducible. We introduce the notion of irreducibility, characterise this property and study a number of fundamental questions on the concepts under consideration.