There is no prime functional digraph: Seifert's proof revisited

Abstract

A functional digraph is a finite digraph in which each vertex has a unique out-neighbor. Considered up to isomorphism and endowed with the directed sum and product, functional digraphs form a semigroup that has recently attracted significant attention, particularly regarding its multiplicative structure. In this context, a functional digraph $X$ divides a functional digraph $A$ if there exists a functional digraph $Y$ such that $XY$ is isomorphic to $A$. The digraph $X$ is said to be prime if it is not the identity for the product, and if, for all functional digraphs $A$ and $B$, the fact that $X$ divides $AB$ implies that $X$ divides $A$ or $B$. In 2020, Antonio E. Porreca asked whether prime functional digraphs exist, and in 2023, his work led him to conjecture that they do not. However, in 2024, Barbora Hudcová discovered that this result had already been proved by Ralph Seifert in 1971, in a somewhat forgotten paper. The terminology in that work differs significantly from that used in recent studies, the framework is more general, and the non-existence of prime functional digraphs appears only as a part of broader results, relying on (overly) technical lemmas developed within this general setting. The aim of this note is to present a much more accessible version of Seifert's proof $-$ that no prime functional digraph exists $-$ by using the current language and simplifying each step as much as possible.