Given a (finite or infinite) subset $X$ of the free monoid $A^*$ over a finite alphabet $A$, the rank of $X$ is the minimal cardinality of a set $F$ such that $X \subseteq F^*$. We say that a submonoid $M$ generated by $k$ elements of $A^*$ is {\em $k$-maximal} if there does not exist another submonoid generated by at most $k$ words containing $M$. We call a set $X \subseteq A^*$ {\em primitive} if it is the basis of a $|X|$-maximal submonoid. This definition encompasses the notion of primitive word -- in fact, $\{w\}$ is a primitive set if and only if $w$ is a primitive word. By definition, for any set $X$, there exists a primitive set $Y$ such that $X \subseteq Y^*$. We therefore call $Y$ a {\em primitive root} of $X$. As a main result, we prove that if a set has rank $2$, then it has a unique primitive root. To obtain this result, we prove that the intersection of two $2$-maximal submonoids is either the empty word or a submonoid generated by one single primitive word. For a single word $w$, we say that the set $\{x,y\}$ is a {\em bi-root} of $w$ if $w$ can be written as a concatenation of copies of $x$ and $y$ and $\{x,y\}$ is a primitive set. We prove that every primitive word $w$ has at most one bi-root $\{x,y\}$ such that $|x|+|y|<\sqrt{|w|}$. That is, the bi-root of a word is unique provided the word is sufficiently long with respect to the size (sum of lengths) of the root. Our results are also compared to previous approaches that investigate pseudo-repetitions, where a morphic involutive function $\theta$ is defined on $A^*$. In this setting, the notions of $\theta$-power, $\theta$-primitive and $\theta$-root are defined, and it is shown that any word has a unique $\theta$-primitive root. This result can be obtained with our approach by showing that a word $w$ is $\theta$-primitive if and only if $\{w, \theta(w)\}$ is a primitive set.

中文翻译：

---

原始词组

给定在有限字母表 $A$ 上的自由幺半群 $A^*$ 的（有限或无限）子集 $X$，$X$ 的秩是集合 $F$ 的最小基数，使得 $X \subseteq F^*$。如果不存在由至多 $k$ 个包含 $M$ 的单词生成的另一个子单体，我们说由 $A^*$ 的 $k$ 个元素生成的子单体 $M$ 是 {\em $k$-maximal}。如果集合 $X\subseteq A^*$ {\emprimitive} 是 $|X|$-最大次幺半群的基，我们称它为集合 $X\subseteq 这个定义包含了原始词的概念——事实上，$\{w\}$ 是一个原始集当且仅当 $w$ 是一个原始词。根据定义，对于任何集合 $X$，都存在一个原始集合 $Y$，使得 $X \subseteq Y^*$。因此，我们称 $Y$ 为 $X$ 的 {\em 原始根}。作为主要结果，我们证明如果一个集合的秩为 $2$，则它具有唯一的原始根。为了得到这个结果，我们证明了两个 $2$-maximal submonoids 的交集要么是空词，要么是由一个原始词生成的 submonoid。对于单个单词 $w$，如果 $w$ 可以写成 $x 的副本的串联，我们说集合 $\{x,y\}$ 是 $w$ 的 {\em bi-root} $ 和 $y$ 和 $\{x,y\}$ 是原始集。我们证明每个原始词 $w$ 至多有一个双根 $\{x,y\}$ 使得 $|x|+|y|<\sqrt{|w|}$。也就是说，一个词的双根是唯一的，前提是该词相对于词根的大小（长度之和）足够长。我们的结果还与之前研究伪重复的方法进行了比较，其中在 $A^*$ 上定义了形态对合函数 $\theta$。在此设置中，定义了 $\theta$-power、$\theta$-primitive 和 $\theta$-root 的概念，并且证明了任何单词都有唯一的 $\theta$-原始词根。这个结果可以用我们的方法通过证明一个词 $w$ 是 $\theta$-primitive 当且仅当 $\{w, \theta(w)\}$ 是一个原始集来获得。