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| Natura: | Preprint |
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2025
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| Accesso online: | https://arxiv.org/abs/2511.17082 |
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| _version_ | 1866908929500381184 |
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| author | Taranenko, Anna A. |
| author_facet | Taranenko, Anna A. |
| contents | The well-known quadrangle criterion states that a latin square is isotopic to the Cayley table of a group if and only if all quadrangles spanned by the same triple of symbols coincide on the fourth symbol. Gowers and Long (2020) reformulated this result in the following way: the Cayley tables of the most associative quasigroups have the maximum number of octahedra. In the present paper, we state the multidimensional quadrangle condition for $d$-dimensional latin hypercubes in terms of the reconstruction of submatrices of order $2$ from a bundle of $d+1$ entries and in terms of the maximal number of cuboctahedra. In particular, we show that the most associative $d$-ary quasigroups have Cayley tables such that every $2$-dimensional plane is isotopic to a latin square that is principally isotopic to the Cayley table of a group. We also estimate the number of cuboctahedra in latin squares and hypercubes from below and provide computational results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_17082 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Multidimensional quadrangle condition and cuboctahedra in latin hypercubes Taranenko, Anna A. Combinatorics 15B15, 20N05 The well-known quadrangle criterion states that a latin square is isotopic to the Cayley table of a group if and only if all quadrangles spanned by the same triple of symbols coincide on the fourth symbol. Gowers and Long (2020) reformulated this result in the following way: the Cayley tables of the most associative quasigroups have the maximum number of octahedra. In the present paper, we state the multidimensional quadrangle condition for $d$-dimensional latin hypercubes in terms of the reconstruction of submatrices of order $2$ from a bundle of $d+1$ entries and in terms of the maximal number of cuboctahedra. In particular, we show that the most associative $d$-ary quasigroups have Cayley tables such that every $2$-dimensional plane is isotopic to a latin square that is principally isotopic to the Cayley table of a group. We also estimate the number of cuboctahedra in latin squares and hypercubes from below and provide computational results. |
| title | Multidimensional quadrangle condition and cuboctahedra in latin hypercubes |
| topic | Combinatorics 15B15, 20N05 |
| url | https://arxiv.org/abs/2511.17082 |