Nyldon words

TL;DR

This paper introduces Nyldon words, a new family of words derived by reversing the lexicographic order in Lyndon word factorizations, and proves their unique factorization property and algebraic structure.

Contribution

It defines Nyldon words via reversed lexicographic order and proves they form a unique factorization and a right Lazard set, expanding Lyndon word theory.

Findings

Unique factorization of words into Nyldon words

Nyldon words form a right Lazard set

Nyldon words are a new algebraic family

Abstract

The Chen-Fox-Lyndon theorem states that every finite word over a fixed alphabet can be uniquely factorized as a lexicographically nonincreasing sequence of Lyndon words. This theorem can be used to define the family of Lyndon words in a recursive way. If the lexicographic order is reversed in this definition, we obtain a new family of words, which are called the Nyldon words. In this paper, we show that every finite word can be uniquely factorized into a lexicographically nondecreasing sequence of Nyldon words. Otherwise stated, Nyldon words form a complete factorization of the free monoid with respect to the decreasing lexicographic order. Then we investigate this new family of words. In particular, we show that Nyldon words form a right Lazard set.