# New Results on Nyldon Words and Nyldon-like Sets

## TL;DR

This paper introduces new theoretical results on Nyldon words, including their unique factorizations, their structure as a circular code, and generalizations to Nyldon-like sets, advancing the understanding of Lyndon-like word sets.

## Contribution

It provides a new proof of Nyldon word factorizations, establishes Nyldon words as a circular code, and generalizes results to Nyldon-like sets, expanding the theoretical framework.

## Key findings

- Nyldon words of fixed length form a circular code
- Powers of words can be factorized into Nyldon words
- Results extend to Nyldon-like sets

## Abstract

Grinberg defined Nyldon words as those words which cannot be factorized into a sequence of lexicographically nondecreasing smaller Nyldon words. He was inspired by Lyndon words, defined the same way except with "nondecreasing" replaced by "nonincreasing." Charlier, Philibert, and Stipulanti proved that, like Lyndon words, any word has a unique nondecreasing factorization into Nyldon words. They also show that the Nyldon words form a right Lazard set, and equivalently, a right Hall set. In this paper, we provide a new proof of unique factorization into Nyldon words related to Hall set theory and resolve several questions of Charlier et al. In particular, we prove that Nyldon words of a fixed length form a circular code, we prove a result on factorizing powers of words into Nyldon words, and we investigate the Lazard procedure for generating Nyldon words. We show that these results generalize to a new class of Hall sets, of which Nyldon words are an example, that we name "Nyldon-like sets."