Overview
This article examines word equations within free semigroups, specifically addressing a conjecture concerning the relationship between the existence of infinitely many solutions and solutions exhibiting an arbitrarily large exponent of periodicity. The investigation extends to the broader framework of word equations constrained by regular expressions. Positive results are established for a specific class of these equations.
Research Context
The central conjecture under investigation posits that if a word equation possesses infinitely many solutions, it must also have solutions with an arbitrarily large exponent of periodicity. The study situates this conjecture within the scope of word equations. It particularly considers the impact of regular constraints, which restrict the possible solutions to those belonging to a specified regular language.
The research focuses on quadratic word equations. These are equations where each variable appears at most twice. The constraints considered are those defined within finite semigroups from the variety $\mathbf{DLG}$ and its left-right dual, $\mathbf{DRG}$. These varieties of semigroups are relevant because they encompass several well-known algebraic structures.
Findings
The study establishes new positive results regarding the conjecture mentioned. Specifically, the conjecture holds true for all quadratic word equations that incorporate constraints operating within finite semigroups belonging to the $\mathbf{DLG}$ variety. Furthermore, these findings also extend to quadratic word equations with constraints in finite semigroups from $\mathbf{DRG}$, which is the left-right dual of $\mathbf{DLG}$.
The scope of the finite semigroups covered by these results is notable. The varieties $\mathbf{DLG}$ and $\mathbf{DRG}$ explicitly encompass several important types of semigroups, including all finite groups. This means the established results apply to quadratic word equations where constraints are defined over the elements of any finite group. Additionally, the findings cover all commutative semigroups, indicating applicability across a wide range of algebraic structures where the order of elements in multiplication does not affect the result. The study also confirms the conjecture's validity for $\mathcal{J}$-trivial semigroups.
No other findings or methodologies are explicitly discussed in the source document.