arxiv: 2604.20894 · v1 · submitted 2026-04-21 · 🧮 math.GR · math.RA

Recognition: unknown

Addendum/Corrigendum to "On solubility of skew left braces and solutions of the Yang-Baxter equation"

A. Ballester-Bolinches, P. Jim\'enez-Seral, R. Esteban-Romero, V. P\'erez-Calabuig

Pith reviewed 2026-05-10 01:24 UTC · model grok-4.3

classification 🧮 math.GR math.RA

keywords skew left bracesYang-Baxter equationsolubilityi-homomorphismsi-simplicityindecomposable solutionsstructure skew bracenon-degenerate solutions

0 comments

The pith

By introducing i-homomorphisms of solutions, every soluble solution to the Yang-Baxter equation has a soluble associated skew brace, restoring the equivalence in Theorem C.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper repairs an overlooked step in an earlier proof that a skew brace is soluble precisely when its associated solution of the Yang-Baxter equation is soluble. It defines a new class of maps called i-homomorphisms of solutions. These maps make it possible to characterize indecomposable solutions by i-simplicity and to ensure that the kernels generate ideals inside the structure skew brace. Solubility for solutions is then redefined as the class opposite to the i-simple ones. Under this definition the earlier equivalence holds again and, moreover, every soluble solution is shown to arise from a soluble skew brace.

Core claim

With the new definition of solubility via i-simplicity, every soluble solution has a soluble structure skew brace, and therefore a skew brace is soluble if and only if its associated solution is soluble.

What carries the argument

i-homomorphisms of solutions, a class of maps whose kernels generate ideals in the structure skew braces and for which indecomposable solutions coincide with the i-simple ones.

If this is right

Solubility of solutions is redefined as the opposite class of i-simple solutions.
Every soluble solution possesses a soluble structure skew brace.
The equivalence between soluble skew braces and soluble solutions holds under the revised definition.
New structural results about solutions follow from the analysis of i-kernels and i-simplicity.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The construction may extend to other chain conditions such as nilpotency by replacing simplicity with a suitable nilpotency notion for i-homomorphisms.
The category of solutions with i-homomorphisms could support a Jordan-Hölder type theorem that organises all finite non-degenerate solutions by their soluble series.
Verification that i-kernels always generate ideals would immediately allow the same solubility theory to be applied to infinite solutions.

Load-bearing premise

The i-homomorphisms must form a well-behaved class whose kernels generate ideals in the skew braces and whose simple objects correctly mark the indecomposable solutions.

What would settle it

An explicit non-degenerate solution that is soluble under the i-simplicity definition yet whose structure skew brace fails to be soluble, or an i-homomorphism whose kernel does not generate an ideal.

read the original abstract

In our previous work: Adv. Math. 455 (2024), no. 109880, solubility of solutions was introduced as an extension of solubility of skew braces in the classification context of non-degenerate solutions of the Yang-Baxter equation. One of our main results (Theorem C) proved that a skew brace is soluble if, and only if, its associated solution is soluble. A minor step depending on the definition of homomorphism of solutions was overlooked. In this work, proof of Theorem C is repaired by means of a new class of homomorphisms of solutions: i-homomorphisms of solutions. The importance of this new class is twofold: indecomposable solutions are characterised by means of i-simplicity of solutions, and i-kernels of i-homomorphisms generate ideals in structure skew braces of solutions. Hence, solubility of solutions is redefined as an opposite class of indecomposable solutions. The results obtained with this definition improve our previous outcomes: every soluble solution is proved to have a soluble structure skew brace, and consequently, Theorem C still holds. Several results stemming from this new analysis are outlined.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This corrigendum repairs the proof of Theorem C by introducing i-homomorphisms of solutions to restore the equivalence between soluble skew braces and soluble solutions.

read the letter

The key point is that this addendum fixes the proof of the main equivalence theorem by defining i-homomorphisms of solutions and using them to link solubility of solutions to solubility of the associated braces. They introduce these homomorphisms to characterize indecomposable solutions as i-simple ones and to ensure that their kernels produce ideals in the skew braces. With that, they can redefine solubility for solutions and show the desired implication both ways, keeping Theorem C intact while improving some supporting statements. This is a targeted and honest repair. The authors acknowledge the minor oversight in the original homomorphism definition and supply the missing piece without inflating the contribution. The main thing to check is whether the i-homomorphisms really form a class where kernels generate ideals under the brace structure. The paper asserts this, but the details of how the new maps interact with the addition, multiplication, and the solution map determine if the argument closes properly. That is the spot where a referee should look hardest. Specialists in algebraic structures related to the Yang-Baxter equation will get the most from this. It is a useful correction for anyone citing or building on the 2024 paper. The work shows clear thinking about the gap and how to close it, so it merits peer review rather than a quick desk decision. I recommend sending it to referees.

Referee Report

0 major / 2 minor

Summary. The manuscript is an addendum and corrigendum to the authors' previous paper on solubility of skew left braces and solutions of the Yang-Baxter equation. It repairs the proof of Theorem C by introducing the new class of i-homomorphisms of solutions. This allows indecomposable solutions to be characterized via i-simplicity, with the key property that i-kernels of i-homomorphisms generate ideals in the structure skew braces of solutions. Solubility of solutions is redefined as the opposite of indecomposability, yielding the improved statement that every soluble solution has a soluble structure skew brace and thereby restoring the equivalence asserted in Theorem C. Several additional results from the new analysis are outlined.

Significance. If the claims hold, the work strengthens the link between algebraic solubility in skew braces and the corresponding property for their associated solutions. The introduction of i-homomorphisms provides a well-behaved class that transfers ideal-theoretic information, enabling the redefinition of solubility and the restoration of Theorem C with a stronger implication (soluble solutions imply soluble braces). This technical advance addresses the original gap and supplies a more precise tool for classification of non-degenerate Yang-Baxter solutions.

minor comments (2)

A brief explicit comparison between the original notion of homomorphism of solutions and the new i-homomorphism would help readers see precisely where the repair occurs.
The 'several results stemming from this new analysis' mentioned in the abstract should be stated as numbered theorems or propositions with clear references for ease of citation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive assessment of our addendum/corrigendum and for recommending minor revision. The summary accurately captures the role of i-homomorphisms in repairing the proof of Theorem C and strengthening the equivalence between solubility of skew braces and their associated solutions. No specific major comments were raised in the report.

read point-by-point responses

Referee: The manuscript is an addendum and corrigendum to the authors' previous paper on solubility of skew left braces and solutions of the Yang-Baxter equation. It repairs the proof of Theorem C by introducing the new class of i-homomorphisms of solutions. This allows indecomposable solutions to be characterized via i-simplicity, with the key property that i-kernels of i-homomorphisms generate ideals in the structure skew braces of solutions. Solubility of solutions is redefined as the opposite of indecomposability, yielding the improved statement that every soluble solution has a soluble structure skew brace and thereby restoring the equivalence asserted in Theorem C. Several additional results from the new analysis are outlined.

Authors: We appreciate the referee's concise and accurate summary of the manuscript. The introduction of i-homomorphisms indeed provides the necessary tool to characterize indecomposable solutions via i-simplicity and to ensure that i-kernels generate ideals, allowing the redefinition of solubility and the restoration of the equivalence in Theorem C with the stronger implication that soluble solutions yield soluble structure skew braces. revision: no
Referee: If the claims hold, the work strengthens the link between algebraic solubility in skew braces and the corresponding property for their associated solutions. The introduction of i-homomorphisms provides a well-behaved class that transfers ideal-theoretic information, enabling the redefinition of solubility and the restoration of Theorem C with a stronger implication (soluble solutions imply soluble braces). This technical advance addresses the original gap and supplies a more precise tool for classification of non-degenerate Yang-Baxter solutions.

Authors: We agree that the new framework strengthens the connection between the two notions of solubility and offers a more precise tool for the classification of solutions. The i-homomorphisms are designed precisely to transfer the ideal-theoretic information from solutions back to the structure skew braces. revision: no

Circularity Check

0 steps flagged

No significant circularity; new definitions repair the proof without reducing claims to inputs by construction

full rationale

The corrigendum introduces i-homomorphisms and redefines solubility of solutions via i-simplicity (opposite of indecomposability) to repair the overlooked step in the prior Theorem C. The key asserted property—that i-kernels generate ideals in the associated skew left brace—is presented as following from the new class rather than assumed by definition or obtained via fitting. No step equates a derived result to its input by construction, renames a known pattern, or relies on a load-bearing self-citation whose content is unverified outside this work. The equivalence (soluble solution iff soluble brace) is re-established via the independent content of the new homomorphisms and the generation property, keeping the derivation self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claim rests on standard definitions of skew braces, solutions to the Yang-Baxter equation, and homomorphisms from the authors' previous paper, plus the new i-homomorphism class introduced here. No free parameters or invented physical entities are present.

axioms (2)

standard math Standard axioms of group theory and brace operations as defined in the prior work
Invoked throughout to define solubility and structure skew braces.
domain assumption The category of solutions with i-homomorphisms behaves appropriately for kernels to generate ideals
This is the key new assumption needed for the repaired proof and redefinition.

invented entities (1)

i-homomorphism of solutions no independent evidence
purpose: To repair the overlooked step in the original proof and to characterize indecomposable solutions via i-simplicity
New class of maps introduced in this addendum; no independent evidence outside the paper is provided.

pith-pipeline@v0.9.0 · 5522 in / 1460 out tokens · 32337 ms · 2026-05-10T01:24:35.760571+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

14 extracted references · 2 canonical work pages

[1]

Ballester-Bolinches, R

A. Ballester-Bolinches, R. Esteban-Romero, P. Jiménez-Seral, and V. Pérez-Calabuig. Soluble skew left braces and soluble solutions of the Yang-Baxter equations.Adv. Math., 455:109880, 27 pp., 2024

2024
[2]

Ballester-Bolinches, R

A. Ballester-Bolinches, R. Esteban-Romero, and V. Pérez-Calabuig. A Jordan-Hölder theorem for skew left braces and their applications to multipermutation solutions of the Yang-Baxter equation.Proc. Roy. Soc. Edinburgh Sect. A, 154(3):793–809, 2024

2024
[3]

Castelli, F

M. Castelli, F. Catino, and G. Pinto. Indecomposable involutive set- theoretic solutions of the Yang–Baxter equation.J. Pure Appl. Algebra, 223(10):4477–4493, 2019

2019
[4]

F. Cedó, E. Jespers, Ł. Kubat, A. Van Antwerpen, and C. Verwimp. On various types of nilpotency of the structure monoid and group of a set- theoretic solution of the Yang-Baxter equation.J. Pure Appl. Algebra, 227(2):107194, 2023

2023
[5]

Cedó and J

F. Cedó and J. Okniński. Constructing finite simple solutions of the Yang-Baxter equation.Adv. Math., 391:107968, 39 pp., 2021

2021
[6]

Cedó and J

F. Cedó and J. Okniński. Indecomposable solutions of the Yang-Baxter equation of square-free cardinality.Adv. Math., 430:109221, 26 p., 2023

2023
[7]

Colazzo, M

I. Colazzo, M. Ferrara, and M. Trombetti. On derived-indecomposable solutions of the Yang-Baxter equation.Publ. Mat., 69(1):171–193, 2025

2025
[8]

Simplesolutions of the Yang-Baxter equation

I.Colazzo,E.Jespers,andŁ.KubatA.VanAntwerpen. Simplesolutions of the Yang-Baxter equation. Preprint, arXiv:2312.09687, 2025

work page arXiv 2025
[9]

Ferrara, M

M. Ferrara, M. Trombetti, and C. Tsang. On cardinalities whose arith- metical properties determine the structure of solutions of the Yang- Baxter equation. Preprint: arXiv:2509.11001, 2025

work page arXiv 2025
[10]

The GAP Group.GAP – Groups, Algorithms, and Programming, Ver- sion 4.15.1, 2025.http://www.gap-system.org

2025
[11]

Gateva-Ivanova and S

T. Gateva-Ivanova and S. Majid. Set-theoretic solutions of the Yang- Baxterequation,graphsandcomputations.J. Symbolic Comput.,42(11– 12):1079–1112, 2007

2007
[12]

Jedlicka and A

P. Jedlicka and A. Pilitowska. Diagonals of solutions of the yang–baxter equation.Forum Math., 38(2):321–338, 2026. 21

2026
[13]

Jedlička, A

P. Jedlička, A. Pilitowska, and A. Zamojska-Dzienio. Indecomposable involutive solutions of the Yang-Baxter equation of multipermutation level 2 with non-abelian permutation group.Journal of Combinatorial Theory, Series A, 197:105753, 2023

2023
[14]

J.-H. Lu, M. Yan, and Y.-C. Zhu. On the set-theoretical Yang-Baxter equation.Duke Math. J., 104(1):1–18, 2000. 22

2000