Irreducibility of Endomorphisms of Finitely Generated Free Semigroups

2026-10-05

AI Summary

This study investigates the irreducibility of endomorphisms of finitely generated free semigroups, specifically addressing when an endomorphism cannot be non-trivially decomposed into a composition of other endomorphisms. Methodologically, by employing algebraic combinatorics, endomorphism monoid theory, and incidence matrix analysis, this work introduces the first formal definitions of irreducibility and primality within this algebraic structure. The primary contributions lie in establishing characteristic criteria for determining the reducibility and irreducibility of rank-preserving endomorphisms, revealing patterns of non-unique factorization, and thoroughly analyzing decomposition properties through incidence matrices. Ultimately, these findings provide a novel perspective for understanding the algebraic structure of endomorphism monoids.

Abstract

We introduce and investigate the irreducibility of endomorphisms of finitely generated free semigroups, i.e., we investigate when an endomorphism $φ: Σ^+ -> Σ^+$, where $Σ$ is any alphabet, can be nontrivially expressed as a composition $φ= ψ_2 \circ ψ_1$ of endomorphisms $ψ_1, ψ_2: Σ^+ -> Σ^+$. We, hence, study a notion of primality in the endomorphism monoid of the free semigroup---a natural and fundamental concept in this algebraic structure. We establish that irreducibility is a nontrivial property for the class of so-called rank-preserving endomorphisms, and we provide a characteristic condition separating the reducible and irreducible endomorphisms. We also characterise when an endomorphism is a factor of another endomorphism, analyse the non-uniqueness of factorisations of a rank-preserving endomorphism into its irreducible components, and investigate the use of incidence matrices to give insights into the (ir-)reducibility of rank-preserving endomorphisms.

Problem

• free semigroups
• endomorphisms
• irreducibility
• factorisation
• rank-preserving

Innovation

• free semigroups
• endomorphisms
• irreducibility
• rank-preserving
• incidence matrices