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

Adrien Richard

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.

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 divides a functional digraph if there exists a functional digraph such that is isomorphic to . The digraph is said to be prime if it is not the identity for the product, and if, for all functional digraphs and , the fact that divides implies that divides or . 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.

Paper Structure

This paper contains 8 sections, 6 theorems, 28 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem \ref{['thm:Seifert']}
  4. Prime functional digraphs are connected
  5. Prime functional digraphs contain a cycle of length one
  6. Seifert's proof that $\mathbb{F}_1$ has no prime functional digraphs
  7. Highlights on Seifert's original work
  8. Acknowledgments

Key Result

Theorem 1

There is no prime functional digraph.

Figures (4)

  • Figure 1: Illustration of $A$, obtained from $XP$ by adding the blue vertex $(u$) and the blue edge.
  • Figure 2: Illustration of $B$ (for $X$ as in Figure \ref{['fig:A']}).
  • Figure 3: Illustration of $Y$ (for $X$ as in Figure \ref{['fig:A']}), obtained from $PB$ (in black) by adding the blue vertices (that of $V_\chi(B)$) and the blue edges.
  • Figure 4: Illustration of $XY$ and $AB$ (for $X$ as in Figure \ref{['fig:A']}). Vertices in the same rectangle have the same out-neighbor. The drawing of $AB$ is obtained from that of $XY$ by relabeling the vertices of $XY$ with the function $\phi$.

Theorems & Definitions (11)

  • Theorem 1: seifert1971prime
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • ...and 1 more