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Keywords = canonical hypergroups

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21 pages, 1160 KiB  
Article
Generalized Fuzzy Rough Approximations on Hypergroups
by Canan Akın, Dilek Bayrak Delice and Sultan Yamak
Mathematics 2024, 12(16), 2445; https://doi.org/10.3390/math12162445 - 6 Aug 2024
Cited by 1 | Viewed by 743
Abstract
In this paper, we define the fuzzy set-valued homomorphisms of the canonical hypergroups as a generalization of fuzzy congruences and investigate some of their features. This structure is an extension of the definition of set-valued homomorphism defined for groups to hypergroups. With this [...] Read more.
In this paper, we define the fuzzy set-valued homomorphisms of the canonical hypergroups as a generalization of fuzzy congruences and investigate some of their features. This structure is an extension of the definition of set-valued homomorphism defined for groups to hypergroups. With this extension, it has become possible to study generalized fuzzy rough approximations in hyperalgebraic structures such as semihypergroups, polygroups, hyperrings, hypermodules, etc. This paper presents the generalized fuzzy rough approximations based on two-universe (I,T)-fuzzy model on canonical hypergroups. Full article
(This article belongs to the Section D2: Operations Research and Fuzzy Decision Making)
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35 pages, 11250 KiB  
Article
On the Borderline of Fields and Hyperfields
by Christos G. Massouros and Gerasimos G. Massouros
Mathematics 2023, 11(6), 1289; https://doi.org/10.3390/math11061289 - 7 Mar 2023
Cited by 6 | Viewed by 1874
Abstract
The hyperfield came into being due to a mathematical necessity that appeared during the study of the valuation theory of the fields by M. Krasner, who also defined the hyperring, which is related to the hyperfield in the same way as the ring [...] Read more.
The hyperfield came into being due to a mathematical necessity that appeared during the study of the valuation theory of the fields by M. Krasner, who also defined the hyperring, which is related to the hyperfield in the same way as the ring is related to the field. The fields and the hyperfields, as well as the rings and the hyperrings, border on each other, and it is natural that problems and open questions arise in their boundary areas. This paper presents such occasions, and more specifically, it introduces a new class of non-finite hyperfields and hyperrings that is not isomorphic to the existing ones; it also classifies finite hyperfields as quotient hyperfields or non-quotient hyperfields, and it gives answers to the question that was raised from the isomorphic problems of the hyperfields: when can the subtraction of a field F’s multiplicative subgroup G from itself generate F? Furthermore, it presents a construction of a new class of hyperfields, and with regard to the problem of the isomorphism of its members to the quotient hyperfields, it raises a new question in field theory: when can the subtraction of a field F’s multiplicative subgroup G from itself give all the elements of the field F, except the ones of its multiplicative subgroup G? Full article
(This article belongs to the Special Issue Hypergroup Theory and Algebrization of Incidence Structures)
15 pages, 352 KiB  
Article
Derived Hyperstructures from Hyperconics
by Vahid Vahedi, Morteza Jafarpour, Sarka Hoskova-Mayerova, Hossein Aghabozorgi, Violeta Leoreanu-Fotea and Svajone Bekesiene
Mathematics 2020, 8(3), 429; https://doi.org/10.3390/math8030429 - 16 Mar 2020
Cited by 9 | Viewed by 2882
Abstract
In this paper, we introduce generalized quadratic forms and hyperconics over quotient hyperfields as a generalization of the notion of conics on fields. Conic curves utilized in cryptosystems; in fact the public key cryptosystem is based on the digital signature schemes (DLP) in [...] Read more.
In this paper, we introduce generalized quadratic forms and hyperconics over quotient hyperfields as a generalization of the notion of conics on fields. Conic curves utilized in cryptosystems; in fact the public key cryptosystem is based on the digital signature schemes (DLP) in conic curve groups. We associate some hyperoperations to hyperconics and investigate their properties. At the end, a collection of canonical hypergroups connected to hyperconics is proposed. Full article
(This article belongs to the Special Issue Hypercompositional Algebra and Applications)
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