Word equations and the exponent of periodicity

TL;DR

This paper investigates the relationship between the existence of infinitely many solutions to word equations and the presence of solutions with arbitrarily large periodicity, providing new results for specific classes of equations.

Contribution

It proves the conjecture holds for quadratic word equations with constraints in certain finite semigroups, expanding understanding in algebraic word equation solutions.

Findings

The conjecture is valid for quadratic equations in finite semigroups from $ extbf{DLG}$ and $ extbf{DRG}$.

Results encompass all finite groups, commutative semigroups, and $ extbf{J}$-trivial semigroups.

New positive results support the conjecture in broader algebraic contexts.

Abstract

In this article, we study word equations in free semigroups and the conjecture that the existence of infinitely many solutions entails the existence of solutions with arbitrarily large exponent of periodicity. We examine this question in the broader framework of word equations with regular constraints and establish new positive results: the conjecture holds for all quadratic word equations with constraints in finite semigroups from the variety $\mathbf{DLG}$ and its left-right dual $\mathbf{DRG}$, encompassing, in particular, all finite groups, commutative semigroups, and $\mathcal{J}$-trivial semigroups.