On cardinalities whose arithmetical properties determine the structure of solutions of the Yang--Baxter equation
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| Format: | Preprint |
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2025
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| author | Ferrara, Maria Trombetti, Marco Tsang, Cindy |
| author_facet | Ferrara, Maria Trombetti, Marco Tsang, Cindy |
| contents | 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 $n= p_1^{α_1} \ldots p_t^{α_t}$ is cube-free, namely $α_i\leq 2$ for every $i$, and $p_i$ does not divide $p_j^{α_j}-1$ for $i\neq j$. 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_11001 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On cardinalities whose arithmetical properties determine the structure of solutions of the Yang--Baxter equation Ferrara, Maria Trombetti, Marco Tsang, Cindy Group Theory Rings and Algebras 16T25, 20F16, 81R50 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 $n= p_1^{α_1} \ldots p_t^{α_t}$ is cube-free, namely $α_i\leq 2$ for every $i$, and $p_i$ does not divide $p_j^{α_j}-1$ for $i\neq j$. 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. |
| title | On cardinalities whose arithmetical properties determine the structure of solutions of the Yang--Baxter equation |
| topic | Group Theory Rings and Algebras 16T25, 20F16, 81R50 |
| url | https://arxiv.org/abs/2509.11001 |