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, Philibert, and Stipulanti. 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”, and show how to generate these sets easily.

Grinberg 将 Nyldon 词定义为不能分解为按字典顺序排列的较小 Nyldon 词序列的那些词。他的灵感来自林登的话，除了将“非减少”替换为“非增加”外，其他定义方式相同。Charlier、Philibert 和 Stipulanti 证明，与 Lyndon 词一样，任何词都有一个独特的非递减因式分解为 Nyldon 词。他们还表明 Nyldon 词形成了一个正确的 Lazard 集，等效地，一个正确的 Hall 集。在本文中，我们提供了与霍尔集合论相关的 Nyldon 词的唯一因式分解的新证明，并解决了 Charlier、Philibert 和 Stipulanti 的几个问题。特别地，我们证明了固定长度的 Nyldon 词形成一个循环码，我们证明了将词的幂分解为 Nyldon 词的结果，我们研究了生成 Nyldon 词的 Lazard 程序。我们展示了这些结果可以推广到一类新的霍尔集，其中 Nyldon 词就是一个例子，我们将其命名为“类 Nyldon 集”，并展示了如何轻松生成这些集。