Irreducibility of Endomorphisms of Finitely Generated Free Semigroups

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.