Abelian Primitive Words

TL;DR

This paper explores Abelian primitive words, revealing their non-context-free nature, efficient recognition algorithms, multiple Abelian roots, and their connections to enumeration and coding theory.

Contribution

It introduces the concept of Abelian primitive words, shows their properties, and provides algorithms and theoretical insights distinct from classical primitive words.

Findings

Abelian primitive words are not context-free.

Recognition of Abelian primitive words is linear-time.

Words can have multiple Abelian roots.

Abstract

We investigate Abelian primitive words, which are words that are not Abelian powers. We show that unlike classical primitive words, the set of Abelian primitive words is not context-free. We can determine whether a word is Abelian primitive in linear time. Also different from classical primitive words, we find that a word may have more than one Abelian root. We also consider enumeration problems and the relation to the theory of codes.