Autoparatopisms of Quasigroups and Latin Squares
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866918407601913856 |
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| author | Mendis, Mahamendige Jayama Lalani Wanless, Ian M. |
| author_facet | Mendis, Mahamendige Jayama Lalani Wanless, Ian M. |
| contents | Paratopism is a well known action of the wreath product $\mathcal{S}_n\wr\mathcal{S}_3$ on Latin squares of order $n$. A paratopism that maps a Latin square to itself is an autoparatopism of that Latin square. Let $\mathrm{Par}(n)$ denote the set of paratopisms that are an autoparatopism of at least one Latin square of order $n$. We prove a number of general properties of autoparatopisms. Applying these results, we determine $\mathrm{Par}(n)$ for $n\le17$. We also study the proportion of all paratopisms that are in $\mathrm{Par}(n)$ as $n\rightarrow\infty$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_23866 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Autoparatopisms of Quasigroups and Latin Squares Mendis, Mahamendige Jayama Lalani Wanless, Ian M. Combinatorics 05B15 05E18 20N05 Paratopism is a well known action of the wreath product $\mathcal{S}_n\wr\mathcal{S}_3$ on Latin squares of order $n$. A paratopism that maps a Latin square to itself is an autoparatopism of that Latin square. Let $\mathrm{Par}(n)$ denote the set of paratopisms that are an autoparatopism of at least one Latin square of order $n$. We prove a number of general properties of autoparatopisms. Applying these results, we determine $\mathrm{Par}(n)$ for $n\le17$. We also study the proportion of all paratopisms that are in $\mathrm{Par}(n)$ as $n\rightarrow\infty$. |
| title | Autoparatopisms of Quasigroups and Latin Squares |
| topic | Combinatorics 05B15 05E18 20N05 |
| url | https://arxiv.org/abs/2603.23866 |