Categories of hypermagmas, hypergroups, and related hyperstructures
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2023
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| _version_ | 1866916737581056000 |
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| author | Nakamura, So Reyes, Manuel L. |
| author_facet | Nakamura, So Reyes, Manuel L. |
| contents | In order to diagnose the cause of some defects in the category of canonical hypergroups, we investigate several categories of hyperstructures that generalize hypergroups. By allowing hyperoperations with possibly empty products, one obtains categories with desirable features such as completeness and cocompleteness, free functors, regularity, and closed monoidal structures. We show by counterexamples that such constructions cannot be carried out within the category of canonical hypergroups. This suggests that (commutative) unital, reversible hypermagmas -- which we call mosaics -- form a worthwhile generalization of (canonical) hypergroups from the categorical perspective. Notably, mosaics contain pointed simple matroids as a subcategory, and projective geometries as a full subcategory. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2304_09273 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Categories of hypermagmas, hypergroups, and related hyperstructures Nakamura, So Reyes, Manuel L. Category Theory Combinatorics Group Theory Rings and Algebras Primary 18D15, 20N20, Secondary 05B35, 51A05 In order to diagnose the cause of some defects in the category of canonical hypergroups, we investigate several categories of hyperstructures that generalize hypergroups. By allowing hyperoperations with possibly empty products, one obtains categories with desirable features such as completeness and cocompleteness, free functors, regularity, and closed monoidal structures. We show by counterexamples that such constructions cannot be carried out within the category of canonical hypergroups. This suggests that (commutative) unital, reversible hypermagmas -- which we call mosaics -- form a worthwhile generalization of (canonical) hypergroups from the categorical perspective. Notably, mosaics contain pointed simple matroids as a subcategory, and projective geometries as a full subcategory. |
| title | Categories of hypermagmas, hypergroups, and related hyperstructures |
| topic | Category Theory Combinatorics Group Theory Rings and Algebras Primary 18D15, 20N20, Secondary 05B35, 51A05 |
| url | https://arxiv.org/abs/2304.09273 |