Polygroup objects in regular categories

Alessandro Linzi; Alessandro Linzi

doi:10.3934/math.2024552

AIMS Mathematics

2024, Volume 9, Issue 5: 11247-11277. doi: 10.3934/math.2024552

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Polygroup objects in regular categories

Alessandro Linzi ^, ,

Centre for Information Technologies and Applied Mathematics, University of Nova Gorica, 5000 Nova Gorica, Slovenia

Received: 27 October 2023 Revised: 15 March 2024 Accepted: 18 March 2024 Published: 22 March 2024
MSC : 16Y20, 20N20

We express the fundamental properties of commutative polygroups (also known as canonical hypergroups) in category-theoretic terms, over the category $ \mathbf{Set} $ formed by sets and functions. For this, we employ regularity as well as the monoidal structure induced on the category $ {\mathbf{Rel}} $ of sets and relations by cartesian products. We highlight how our approach can be generalised to any regular category. In addition, we consider the theory of partial multirings and find fully faithful functors between certain slice or coslice categories of the category of partial multirings and other categories formed by well-known mathematical structures and their morphisms.
- polygroup,
- canonical hypergroup,
- multiring,
- Krasner hyperring,
- regular category,
- relation
Citation: Alessandro Linzi. Polygroup objects in regular categories[J]. AIMS Mathematics, 2024, 9(5): 11247-11277. doi: 10.3934/math.2024552

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Abstract

We express the fundamental properties of commutative polygroups (also known as canonical hypergroups) in category-theoretic terms, over the category $ \mathbf{Set} $ formed by sets and functions. For this, we employ regularity as well as the monoidal structure induced on the category $ {\mathbf{Rel}} $ of sets and relations by cartesian products. We highlight how our approach can be generalised to any regular category. In addition, we consider the theory of partial multirings and find fully faithful functors between certain slice or coslice categories of the category of partial multirings and other categories formed by well-known mathematical structures and their morphisms.

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