# $\omega$-Lyndon words

## TL;DR

This paper introduces a new class of words called $oldsymbol{oldsymbol{ ext{ extomega}}}$-Lyndon words, generalizes Lyndon words to infinite sequences under a specific order, and proves a unique factorization theorem for infinite words.

## Contribution

It defines $ ext{ extomega}$-Lyndon words under a lexicographic-like order and establishes a unique factorization theorem for infinite words into these words.

## Key findings

- Every infinite word can be uniquely factored into a non-increasing sequence of $ ext{ extomega}$-Lyndon words.
- The paper generalizes Lyndon words to infinite sequences with a new order.
- Provides foundational results for the structure of infinite words.

## Abstract

Let $\A$ be a finite non-empty set and $\preceq $ a total order on $\A^\nats$ verifying the following lexicographic like condition: For each $n\in \nats$ and $u, v\in \A^n,$ if $u^\omega \prec v^\omega$ then $ux\prec vy$ for all $x, y \in \A^\nats.$ A word $x\in \A^\nats$ is called $\omega$-Lyndon if $x\prec y$ for each proper suffix $y$ of $x.$ A finite word $w\in \A^+$ is called $\omega$-Lyndon if $w^\omega \prec v^\omega$ for each proper suffix $v$ of $w.$ In this note we prove that every infinite word may be written uniquely as a non-increasing product of $\omega$-Lyndon words.