Irreducibility of Semigroup Morphisms

We study the notion of irreducibility of semigroup morphisms. 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. We introduce the notion of irreducibility, characterise this property and study a number of fundamental questions on the concepts under consideration.