1. Polygroup objects in regular categoriesAlessandro Linzi, 2024, original scientific article 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.
Keywords: polygroup, canonical hypergroup, multiring, Krasner hyperring, regular category, relation
Published in RUNG: 25.03.2024; Views: 1855; Downloads: 8
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8. Characteristic, C-characteristic and positive cones in hyperfieldsDawid Edmund Kędzierski, Alessandro Linzi, Hanna Stojałowska, 2023, original scientific article Abstract: We study the notions of positive cone, characteristic and C-characteristic in (Krasner) hyperfields. We show how these interact in order to produce interesting results in the theory of hyperfields. For instance, we give a criterion for deciding whether certain hyperfields cannot be obtained via Krasner's quotient construction. We prove that any positive integer (larger than 1) can be realized as the characteristic of some infinite hyperfield and an analogous result for C-characteristic. Finally, we study the (directed) graph associated to the strict partial order induced by a positive cone in a hyperfield in various examples.
Keywords: hyperfield, order, characteristic, positive cone
Published in RUNG: 13.02.2023; Views: 1879; Downloads: 7
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9. Dependence relations and grade fuzzy setAlessandro Linzi, Irina Elena Cristea, 2023, original scientific article Abstract: With the aim of developing the recent theory of dependence relations, we elaborate a procedure to measure the strength of the influence of an element on another with respect to a given dependence relation on a finite set. We call this measure the degree of influence. Its definition is based on a partial hyperoperation and a directed graph which we associate with any dependence relation. We compute the degree of influence in various examples and prove some general properties. Among these properties, we find symmetries that have the potential to be applied in the realization of effective algorithms for the computations.
Keywords: dependence relation, degree of influence, grade fuzzy set, hypercompositional structure, hyperoperation
Published in RUNG: 23.01.2023; Views: 2473; Downloads: 7
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