# On generalized Lyndon words

## TL;DR

This paper introduces a generalized framework for Lyndon words based on position-dependent lexicographical orders, providing new characterizations and factorization properties that unify and extend classical results.

## Contribution

It defines generalized Lyndon words using position-specific total orders and offers new characterizations and factorizations, extending classical Lyndon word theory.

## Key findings

- New characterizations of Lyndon words and their factorizations.
- Extension of Lyndon word theory to generalized lexicographical orders.
- Specific results for classical and alternating lexicographical orders.

## Abstract

A generalized lexicographical order on infinite words is defined by choosing for each position a total order on the alphabet. This allows to define generalized Lyndon words. Every word in the free monoid can be factorized in a unique way as a nonincreasing factorization of generalized Lyndon words. We give new characterizations of the first and the last factor in this factorization as well as new characterization of generalized Lyndon words. We also give more specific results on two special cases: the classical one and the one arising from the alternating lexicographical order.