Irreducibility of Semigroup Morphisms

We introduce and study the notion of irreducibility of semigroup morphisms over a finite alphabet. Given an alphabet $\Sigma$, a morphism $\varphi:\Sigma^+\rightarrow\Sigma^+$ is irreducible if any factorisation $\varphi=\psi_2\circ\psi_1$ can only be satisfied if $\psi_1$ or $\psi_2$ is a trivial morphism; otherwise, $\varphi$ is reducible. This definition provides a notion of primality in the endomorphism monoid of the free semigroup -- a natural and fundamental concept in this algebraic structure. The concept of the irreducibility of a morphism is related to the classical theory of simplifications of homomorphisms by Ehrenfeucht and Rozenberg, and the theory of indecomposable codes by Berstel, Perrin and Reutenauer, with the distinction that we primarily consider factorisations on a single alphabet, rather than through alphabets of smaller cardinality. We give a characterisation of irreducibility for morphisms where the domain and range alphabets coincide, characterise when a given morphism is a factor of another morphism, analyse the non-uniqueness of factorisation of a morphism into its irreducible components, and investigate the use of incidence matrices to give insights into the (ir-)reducibility of morphisms.