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 k-maximal if there does not exist another submonoid generated by at most k words containing M. We call a set $X\subseteq A^{⁎}$ 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 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 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 θ is defined on $A^{⁎}$. In this setting, the notions of θ-power, θ-primitive and θ-root are defined, and it is shown that any word has a unique θ-primitive root. This result can be obtained with our approach by showing that a word w is θ-primitive if and only if $\{w,\theta (w)\}$ is a primitive set.

给定自由半定体的（有限或无限）子集X${\mathrm{一个}}^{⁎}$在有限字母A上，X的秩是集合F的最小基数，使得$X\subseteq F^{⁎}$。我们说一个子幺中号所产生ķ元素${\mathrm{一个}}^{⁎}$如果不存在由最多m个包含M的k个单词生成的另一个子monoid，则k为最大。我们称一套$X\subseteq {\mathrm{一个}}^{⁎}$ 原始的，如果它是一个基础$|X|$-最大亚monoid。这个定义包含原始字词的概念-实际上，$\{w\}$当且仅当w是原始词时，它是原始集合。根据定义，对于任何集合X，都有一个原始集合Y使得$X\subseteq {ÿ}^{⁎}$。因此，我们称Y为X的原始根。作为主要结果，我们证明如果集合的等级为2，则它具有唯一的原始根。为了获得此结果，我们证明了两个2极大子monoid的交集是空单词还是一个单个原始单词生成的submonoid。

对于一个单词w，我们说集合$\{X，ÿ\}$是双根的瓦特如果瓦特可以写成的拷贝的串联X和ÿ和$\{X，ÿ\}$是一个原始集。我们证明每个原始词w最多具有一个双词根$\{X，ÿ\}$ 这样 $|X|+|ÿ|<\sqrt{|w|}$。即，单词的双根是唯一的，条件是单词相对于根的大小（长度的总和）足够长。

我们的结果也与以前的研究伪重复的方法进行了比较，在伪重复上定义了一个形态对合函数θ${\mathrm{一个}}^{⁎}$。在该设置中，的概念θ -power，θ -primitive和θ -root被定义，并且，示出了任何词具有唯一的θ -primitive根。通过我们的方法，可以证明结果表明，当且仅当单词w为θ-本原时，$\{w，\theta （w）\}$ 是一个原始集。