[Paper Review] Nyldon words

Émilie Charlier, Manon Philibert+1 more|arXiv (Cornell University)|Apr 25, 2018

semigroups and automata theory12 references3 citations

TL;DR

This paper introduces Nyldon words, a new family of words defined as the unique lexicographically nondecreasing factorization of any finite word, dual to Lyndon words. It proves that Nyldon words form a complete factorization of the free monoid under the decreasing lexicographic order and establishes they constitute a right Lazard set, extending the theory of factorizations in combinatorics on words.

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.

Motivation & Objective

• To define a new family of words, Nyldon words, as the dual counterpart to Lyndon words under reversed lexicographic order.
• To prove that every finite word admits a unique factorization into a lexicographically nondecreasing sequence of Nyldon words.
• To investigate structural and algebraic properties of Nyldon words, particularly their role in factorization theory.
• To establish that Nyldon words form a right Lazard set, extending known results on Lazard sets in free monoids.

Proposed method

• Define Nyldon words via a recursive construction dual to the standard Lyndon word definition, using lexicographically nondecreasing factorizations.
• Use the Chen-Fox-Lyndon theorem as a foundation, reversing the lexicographic order to generate the new family.
• Prove the uniqueness and existence of the nondecreasing factorization into Nyldon words using combinatorial arguments on word orderings.
• Demonstrate that the set of Nyldon words satisfies the conditions of a right Lazard set by verifying the required properties of prefix-free generation and factorization closure.
• Apply known results on Lazard sets to show that Nyldon words generate the free monoid through iterative right-branching construction.
• Leverage duality between Lyndon and Nyldon words to transfer structural insights from the well-known Lyndon factorization to the new family.

Experimental results

Research questions

• RQ1 Can every finite word be uniquely factorized into a lexicographically nondecreasing sequence of Nyldon words?
• RQ2 How does the structure of Nyldon words compare to that of Lyndon words under duality of lexicographic order?
• RQ3 Do Nyldon words form a complete factorization system for the free monoid under the decreasing lexicographic order?
• RQ4 Is the set of Nyldon words a right Lazard set, and if so, what properties support this classification?
• RQ5 What algebraic and combinatorial properties emerge from the nondecreasing factorization of words into Nyldon words?

Key findings

• Every finite word admits a unique factorization into a lexicographically nondecreasing sequence of Nyldon words, establishing them as a complete factorization basis for the free monoid.
• Nyldon words are defined as the dual of Lyndon words under reversal of lexicographic order, leading to a nonincreasing factorization property.
• The family of Nyldon words forms a right Lazard set, meaning they can be generated via a right-branching algorithm that preserves factorization completeness.
• The construction of Nyldon words ensures that no word in the set is a proper prefix of another, satisfying a key condition for Lazard sets.
• The unique nondecreasing factorization property of Nyldon words mirrors the well-known nonincreasing factorization of Lyndon words, completing a duality in combinatorics on words.
• The results extend the theory of factorizations in free monoids by introducing a new canonical factorization system with structural parallels to Lyndon words.