On cardinalities whose arithmetical properties determine the structure of solutions of the Yang--Baxter equation

2025/09/13 by Maria Ferrara, Ferrara, Maria, Marco Trombetti +3
Computer Science · Mathematics · #16T25 #20F16 #81R50 #Advanced Topics in Algebra #FOS: Mathematics #Graph theory and applications #Group Theory (math.GR) #Matrix Theory and Algorithms #Rings and Algebras (math.RA)

paper · pdf · doi:10.48550/arxiv.2509.11001

openalex publication_date 2025/09/13 · openalex created_date 2025/10/12 · openalex updated_date 2026/07/28

Abstract

The aim of this paper is to provide purely arithmetical characterisations of those natural numbers n for which every non-degenerate set-theoretic solution of cardinality n of the Yang--Baxter equation arising from a skew brace (sb-solution for short) satisfies some relevant properties, such as being a flip or being involutive. For example, it turns out that every sb-solution of cardinality n has finite multipermutation level if and only if its prime factorisation n= p₁^α₁ … p_t^α_t is cube-free, namely α_i≤ 2 for every i, and p_i does not divide p_j^α_j-1 for i≠ j. Two novel constructions of skew braces will play a central role in our proofs. We shall also introduce the notion of supersoluble solution and show how this concept is related to that of supersoluble skew brace. In doing so, we have spotted an irreparable mistake in the proof of Theorem C [Ballester-Bolinches et al., Adv. Math. 455 (2024)], which characterizes soluble solutions in terms of soluble skew braces.

Citations

• Finite skew braces of square-free order and supersolubility
• Soluble skew left braces and soluble solutions of the Yang-Baxter equation
• On the Structure of Some One-generator Braces
• Indecomposable solutions of the Yang-Baxter equation of square-free cardinality
• On derived-indecomposable solutions of the Yang--Baxter equation
• On Insoluble Transitive Subgroups in the Holomorph of a Finite Soluble Group
• Skew left braces and the Yang-Baxter equation
• Symmetric skew braces and brace systems
• Central nilpotency of skew braces
• Opposite skew left braces and applications
• Connections between properties of the additive and the multiplicative groups of a two-sided skew brace
• On the characterization of the numbers n such that any group of order n has a given property P
• Solubility Criteria for Hopf-Galois Structures
• Classification of braces of order p³

Related

• Studying solutions of the Yang-Baxter equation through skew braces, with an application to indecomposable involutive solutions with abelian permutation group
• A note on set-theoretic solutions of the Yang-Baxter equation
• Soluble skew left braces and soluble solutions of the Yang-Baxter equation
• Simple solutions of the Yang-Baxter equation of cardinality pⁿ
• Simple solutions of the Yang-Baxter equation