Chain conditions on skew braces and solutions of the Yang-Baxter Equation

Abstract: Classical works of Hall and McLain show that solubility and local nilpotency play a key role in deriving finite generation in groups from maximal or minimal conditions on normal subgroups. In this work, brace-theoretical analogues of Hall's and McLain's results are analysed for skew braces satisfying the maximal or minimal condition on ideals. We also introduce finiteness and chain conditions on non-degenerate set-theoretic solutions of the Yang-Baxter equation, and their impact on associated structure and permutation skew braces of solutions is also described.

• The paper establishes finiteness and chain condition analogs in skew braces by extending Hall and McLain theorems to a noncommutative framework.
• The paper demonstrates that properties of Lyubashenko solutions, including finite generation and congruence conditions, are equivalent under central nilpotency.
• The paper provides explicit counterexamples delineating limits in multipermutational contexts, paving the way for future classifications of YBE solutions.

Chain Conditions on Skew Braces and Solutions to the Yang-Baxter Equation

Introduction

This work systematically analyzes finiteness and chain conditions (maximal or minimal conditions) on ideals and subbraces in skew braces, and extends those considerations to set-theoretic solutions of the Yang-Baxter Equation (YBE). Drawing on classical results by Hall and McLain concerning maximal and minimal chain conditions in group theory, the authors transfer and generalize these structural constraints to the non-commutative context of skew braces and the corresponding algebraic structures arising from YBE solutions. A significant portion of the analysis considers soluble, (hyper-)nilpotent, and multipermutational properties, elucidating the connections between chain conditions and finite generation in both skew braces and associated YBE solutions. Special focus is given to Lyubashenko solutions and the delicate interplay between congruence conditions and algebraic finiteness.

Preliminaries and Foundational Structures

The skew brace $(B,+,\cdot)$ gives rise to a rich algebraic framework by endowing a set $B$ with two (not necessarily compatible or commutative) group operations, subject to a left distributivity law. Two-sided skew braces enforce right distributivity, closely aligning them with Jacobson radical rings, while the $\lambda$-action and associated structures such as the star operation and the socle/center series parallel constructions in group and ring theory but yield nontrivial generalizations.

Associated to any non-degenerate set-theoretic solution $(X,r)$ of the YBE is its structure group $G(X,r)$, with a canonical embedding of $X$, and its corresponding structure and permutation skew braces. The paper details how properties of these objects both influence and reflect key attributes of the original solution. Notably, the infinitude of $G(X,r)$, and properties of the socle and central series, play a pivotal role in the transfer of chain conditions from classical group/ring theory to this context. The connection between multipermutational and nilpotent types across these algebraic objects is rigorously studied, ensuring that the behavior of chain conditions is amenable to translation between the solution and its structural algebraic superstructure.

Chain Conditions on Skew Braces

The authors provide precise brace-theoretical analogs to foundational group-theoretic results:

• Hall-type Theorems: For i-noetherian (i.e., satisfying the maximal condition on ideals) soluble braces or two-sided skew braces, finite generation follows necessarily. The proofs employ structural decompositions via abelian and derived series, alongside careful analysis of ideal generation and the effect of group operations and automorphism actions. Notably, the analogs rely on new inductive combinatorics and manipulation of the star product, tailored specifically to the non-abelian environment of skew braces.
• McLain-type Theorems: For locally multipermutational and locally centrally nilpotent skew braces (contextual analogs of locally nilpotent groups), the paper proves equivalence between i-noetherianity, finite generation, and s-noetherianity (maximal or minimal conditions on subbraces), subject to central nilpotency. Theoretical obstacles when generalizing further are exposed via explicit counterexamples showing that these results do not hold for general multipermutational types lacking central nilpotency.
• Core and Index Problems: In the context of subbraces of finite index, the core problem is considered: whether a finite-index subbrace necessarily contains a finite-index ideal. Complete affirmation is provided for two-sided (hypercentral/centrally nilpotent) cases and for structure braces of finite YBE solutions, but the general problem is left open. The structural linkage between central series factors and finite index ensures the scope of the result is broad within the two-sided framework.

Finiteness and Chain Conditions on YBE Solutions

The analysis extends to finiteness conditions in set-theoretic solutions, paralleling and refining the algebraic chain conditions established in the context of skew braces:

• Finite Generation and Locality: The authors formalize finite generation for solutions in terms of generation by subsets under closure in the YBE solution structure. Finite generation of a solution corresponds exactly to finite generation of the structure and permutation skew braces. The implications run one-way (not all finitely generated skew braces yield finitely generated solutions), elucidated by explicit counterexamples.
• Multipermutational and Nilpotency Equivalence: For classes $\mathfrak{X}$ (skew braces) and $\mathfrak{Y}$ (solutions), closed under substructures, being locally in $\mathfrak{X}$ or $B$0 is mathematically equivalent across structure solutions and braces. This yields equivalence of local (hyper)multipermutational properties between solutions and their structure/permutation skew braces.
• Chain Conditions on Subsolutions and Congruences: Maximal and minimal chain conditions are investigated for subsolutions and congruences (max-sub/min-sub, max-con/min-con). For the class of Lyubashenko solutions, a complete trichotomy emerges: finite generation, max-sub, min-sub, and having a finite number of orbits under the defining permutations are all equivalent. Moreover, maximal condition on congruences (max-con) is also equivalent to these, while the minimal condition on congruences (min-con) is strictly stronger, holding if and only if the solution is finite. The paper constructs explicit infinite, finitely generated Lyubashenko solutions failing min-con, showing the sharpness of their results.

Implications and Future Directions

The paper provides a robust foundation for the translation of chain and finiteness conditions between skew braces and solutions of the YBE, thereby supporting classification approaches to infinite-dimensional or infinite YBE solutions via algebraic structural properties. The explicit identification of cases where finiteness at the solution level forces corresponding algebraic finiteness, and vice versa, enables transfer of methods and results between disparate areas of algebra.

The structural results for two-sided skew braces further anchor the study of radical rings and noncommutative algebra, while the explicit computations and counterexamples offer clear demarcations of the limits of analogies with group-theoretic chain conditions. The focus on Lyubashenko solutions points toward a more precise understanding of the taxonomy of YBE solutions within and beyond the context of classical constructions.

Future work may address the general core-index problem in arbitrary skew braces, extend chain condition analysis to non-multipermutational classes of solutions, and harness these results in classification and construction of novel infinite YBE solutions with prescribed algebraic or combinatorial properties.

Conclusion

This paper rigorously establishes how maximal and minimal chain conditions on ideals and subbraces in skew braces both mirror and diverge from classical group-theoretic results when transferred to the YBE context. It demonstrates that appropriate analogues of Hall's and McLain's theorems hold in the soluble, multipermutational, and centrally nilpotent settings, with explicit examples demarcating the failure of such extensions in greater generality. The structural equivalence between finiteness conditions on solutions and on skew braces, especially in the setting of Lyubashenko solutions, provides a powerful new toolkit for the classification and analysis of YBE solutions and the algebraic structures they induce.