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Semigroup Forum· 2026Q2

Quadratic word equations with regular constraints and the exponent of periodicity

Volker Diekert, Silas Natterer, Alexander Thumm

Short summary

The conjecture that infinitely many solutions to word equations imply solutions with arbitrarily large periodicity holds for quadratic word equations with constraints in DLG and DRG finite semigroups.

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Abstract

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 $$\textbf{DLG}$$ DLG and its dual $$\textbf{DRG}$$ DRG , encompassing all finite groups, commutative semigroups, and $$\mathcal {J}$$ J -trivial semigroups.

The authors' abstract, as published at the source. Semigroup Forum, 2026 · DOI ↗

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Field: Computational Theory and Mathematics

Computational Theory and MathematicsComputer Science