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 $θ$ 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 $\{w, θ(w)\}$ is a primitive set.
Completely Reachable Automata, Primitive Groups and the State Complexity of the Set of Synchronizing Words
我们给出了一个新的特征的本原置换群的概念，完全可达自动机。此外，我们引入同步最大置换群绑定到一组同步字的某些相关自动机的状态复杂性，并表明它们之间包含的$2$ -齐次和原始组。最后，我们定义了$k$ -可达群，类似于同步群，并受到我们对本原置换群的刻画的启发。但结果表明，一个$k$ -可达置换群的度为$n$且有$6 \le k \le n - 6$，它要么是交错群，要么是对称群.
Sync-Maximal Permutation Groups Equal Primitive Permutation Groups
一个给定的$n$状态自动机的一组同步字形成了一个可被具有$2^n - n$状态的自动机识别的正则语言。一组同步字的识别自动机的大小与同步和同步字的长度有关的计算问题有关。因此，研究具有此性质的同步自动机极值是很自然的，即，使得用于同步字组的最小确定性自动机具有$2^n - n$状态。同步极大置换群在[{ \sc S. Hoffmann}，Completely Reachable Automata，Primitive Groups and the State Complexity of the Set of Synchronizing Words，LATA 2021]通过规定与群和非置换相关联的自动机具有该极值性质。该定义类似于同步基团，类似于在上述工作中获得的双折射率的表征。与其他类别的群体的确切关系被认为是一个悬而未决的问题。在这里，我们通过证明同步极大群正是本原群来解决这个公开问题。我们的结果给出了本原群的一个新的刻画。最后，我们探索了一个替代的和更强的定义比同步最大化。
Sets Represented as the Length-n Factors of a Word
在本文中，我们考虑以下问题：有多少不同的子集Sigma^n可以出现作为一个有限字的所有长度为n的因素的集合？如果一个子集是可表示的，我们需要用多长的词来表示它？有多少个这样的子集由长度为t的单词表示？对于第一个问题，我们给出了二元情形下的形式alpha^（2^n）的上界和下界。对于第二个问题，我们给出了一个弱上界和一些实验数据。对于第三个问题，在n <= t < 2n的情况下，我们给出了一个封闭形式的公式。这些问题的数学变体以前被称为“最短公共超弦”。
