Primitive Sets of Words

TL;DR

This paper introduces the concept of primitive sets of words, explores their properties, and proves uniqueness results for primitive roots and bi-roots, extending classical notions of primitive words and connecting to pseudo-repetitions.

Contribution

It generalizes primitive words to primitive sets, proves the uniqueness of primitive roots for sets of rank 2, and establishes bi-root uniqueness for long words, linking to pseudo-repetition concepts.

Findings

Unique primitive root for sets of rank 2.

Intersection of two 2-maximal submonoids is trivial or generated by one primitive word.

At most one bi-root for sufficiently long primitive words.

Abstract

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.