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Etude des morphismes pr{é}servant les mots primitifs

Francis Wlazinski

TL;DR

This work investigates when morphisms preserve primitive structure and powers of words. It leverages word equations, conjugacy properties, and code-based characterizations to show that primitive morphisms are injective and that primitivity can be characterized via the image code; it proves a key result: if an injective morphism sends a primitive word to a power of a primitive word, then only powers of that primitive word map to powers. For uniform and binary morphisms, it establishes that primitivity is equivalent to 2-primitivity, and it derives nuanced power-free results: uniform morphisms without a $k$-th power are primitive for $k\,\ge\,5$, and related implications for $k\ge3$. Finally, it shows that square-free morphisms are necessarily primitive, linking square-free behavior directly to primitivity with a concise, self-contained argument.

Abstract

This article provides a reminder of some properties of primitive words and the morphisms that preserve them. Their proofs, which I have more or less revised, are included. This makes the article almost self-contained. I also contribute by giving some properties of primitive words, but especially by showing that a morphism without powers k($\ge$ 5) is primitive and that a uniform morphism without powers k($\ge$ 2) is primitive.

Etude des morphismes pr{é}servant les mots primitifs

TL;DR

This work investigates when morphisms preserve primitive structure and powers of words. It leverages word equations, conjugacy properties, and code-based characterizations to show that primitive morphisms are injective and that primitivity can be characterized via the image code; it proves a key result: if an injective morphism sends a primitive word to a power of a primitive word, then only powers of that primitive word map to powers. For uniform and binary morphisms, it establishes that primitivity is equivalent to 2-primitivity, and it derives nuanced power-free results: uniform morphisms without a -th power are primitive for , and related implications for . Finally, it shows that square-free morphisms are necessarily primitive, linking square-free behavior directly to primitivity with a concise, self-contained argument.

Abstract

This article provides a reminder of some properties of primitive words and the morphisms that preserve them. Their proofs, which I have more or less revised, are included. This makes the article almost self-contained. I also contribute by giving some properties of primitive words, but especially by showing that a morphism without powers k( 5) is primitive and that a uniform morphism without powers k( 2) is primitive.
Paper Structure (5 sections, 17 theorems)

This paper contains 5 sections, 17 theorems.

Table of Contents

  1. Préliminaires
  2. Quelques équations
  3. Caractérisation des morphismes primitifs
  4. Morphismes primitifs et puissance $k \geq 2$
  5. Morphismes sans carré

Key Result

Proposition 1.1

Lot1983Lot2002 Si une puissance d'un mot non vide $u$ et une puissance d'un mot non vide $v$ ont un préfixe commun de longueur supérieure ou égale à $|u|+|v|-pgcd(|u|,|v|)$ alors $u$ et $v$ sont les puissances d'un même mot primitif. Autrement dit, $u$ et $v$ ont même racine primitive. De plus, si $

Theorems & Definitions (17)

  • Proposition 1.1: Fine & Wilf
  • Proposition 1.7
  • Proposition 1.13
  • Proposition 1.14
  • Proposition 1.18
  • Proposition 1.19
  • Proposition 2.7
  • Proposition 2.9
  • Proposition 2.10
  • Proposition 3.1
  • ...and 7 more