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PreprintCC BY 4.0arXiv

Irreducibility of Endomorphisms of Finitely Generated Free Semigroups

  • Paul C. Bell
  • Eva Foster
  • Daniel Reidenbach
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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= ψ2 ∘ ψ1 of endomorphisms ψ1, ψ2: 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.

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Cite

BibTeXRIS
BibTeX
@misc{bell2026irreducibility,
  title = {{Irreducibility of Endomorphisms of Finitely Generated Free Semigroups}},
  author = {Paul C. Bell and Eva Foster and Daniel Reidenbach},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.07122},
  eprint = {2610.07122},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.07122},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Irreducibility of Endomorphisms of Finitely Generated Free Semigroups
AU  - Paul C. Bell
AU  - Eva Foster
AU  - Daniel Reidenbach
PY  - 2026
DA  - 2026/10/05
PB  - arXiv
DO  - 10.48550/arxiv.2610.07122
UR  - https://arxiv.org/abs/2610.07122
AB  - 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.
ER  -