# Generalized Lyndon Factorizations of Infinite Words

## TL;DR

This paper extends Lyndon factorization concepts to infinite words under a generalized lexicographic order, proving a unique factorization theorem and characterizing infinite Lyndon words by their prefixes.

## Contribution

It introduces a generalized Lyndon factorization for infinite words, proving its uniqueness and characterizing infinite Lyndon words through their prefixes.

## Key findings

- Every infinite word has a unique nonincreasing generalized Lyndon factorization.
- The factorization is finite if and only if the last term is finite.
- Infinite generalized Lyndon words have infinitely many Lyndon prefixes.

## Abstract

A generalized lexicographic order on words is a lexicographic order where the total order of the alphabet depends on the position of the comparison. A generalized Lyndon word is a finite word which is strictly smallest among its class of rotations with respect to a generalized lexicographic order. This notion can be extended to infinite words: an infinite generalized Lyndon word is an infinite word which is strictly smallest among its class of suffixes. We prove a conjecture of Dolce, Restivo, and Reutenauer: every infinite word has a unique nonincreasing factorization into finite and infinite generalized Lyndon words. When this factorization has finitely many terms, we characterize the last term of the factorization. Our methods also show that the infinite generalized Lyndon words are precisely the words with infinitely many generalized Lyndon prefixes.