Work

Irreducibility of Endomorphisms of Finitely Generated Free Semigroups

Year: 2026

Type: article

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 • 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.

Source: Electronic Proceedings in Theoretical Computer Science

DOI: 10.4204/eptcs.451.5

Authors: Paul C. Bell, Eva Foster, Daniel Reidenbach

Institutions: Liverpool John Moores University, University of London, Birkbeck, University of London

Language: English

Topic: semigroups and automata theory

Subfield: Computational Theory and Mathematics

Field: Computer Science

Domain: Physical Sciences

Open Access status: diamond

OpenAlex ID: W7139146498

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