Word equations and the exponent of periodicity
Volker Diekert, Silas Natterer, Alexander Thumm
TL;DR
The paper investigates whether a word equation with regular constraints that has infinitely many solutions necessarily admits solutions with arbitrarily large exponent of periodicity. It introduces the notion of a semigroup class being nice for a class of equations and proves closure properties that yield broad transfer results; in particular, all finite nilpotent semigroups are nice for two-variable equations, and every finite semigroup in $\mathbf{DLG}$ is nice for all quadratic word equations. This provides positive results for the conjecture in a nontrivial regular-constraint setting and highlights deep connections between word equations and the structure of finite semigroups, including groups, commutative semigroups, and $\mathcal{J}$-trivial semigroups via the $\mathbf{DLG}$ variety. The methods rely on compression techniques and structural analysis of semigroups, leveraging Green's relations and related algebraic properties.
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.