Transfinite Lyndon words

TL;DR

This paper extends Lyndon words to transfinite words, establishing unique factorizations and analyzing rational words, thus broadening the theoretical framework of Lyndon words to infinite contexts.

Contribution

It introduces a novel extension of Lyndon words to transfinite words and proves unique factorization results for these infinite structures.

Findings

Existence of unique Lyndon factorization for transfinite words

Special form of factorization for rational words

Factorization can be computed from rational expressions

Abstract

In this paper, we extend the notion of Lyndon word to transfinite words. We prove two main results. We first show that, given a transfinite word, there exists a unique factorization in Lyndon words that are densely non-increasing, a relaxation of the condition used in the case of finite words. In the annex, we prove that the factorization of a rational word has a special form and that it can be computed from a rational expression describing the word.