On Hypergroups with a β-Class of Finite Height

Salvo, Mario De; Fasino, Dario; Freni, Domenico; Faro, Giovanni Lo

doi:10.3390/sym12010168

Cited by 7 publications

(10 citation statements)

References 21 publications

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“…Other relevant results on 1-hypergroups can be found, e.g., in [13][14][15][16]. In this section, we will study the main properties of hypergroups whose heart is isomorphic to a group G, which we call G-hypergroups.…”

Section: G-hypergroupsmentioning

confidence: 99%

“…hypergroups can be built by means of the construction shown in Example 2 of [15], which we recall hereafter. Let Aut(H) be the automorphism group of a hypergroup (H, •).…”

Section: G-hypergroupsmentioning

confidence: 99%

“…In this case, that element is also the identity of the hypergroup. In [15,16], the authors characterized 1-hypergroups in terms of the height of their heart and provided a classification of the 1-hypergroups with |H| ≤ 6 based on the partition of H induced by β. By means of this technique, the authors were able to enumerate all 1-hypergroups of size up to 6 and construct explicitly all non-isomorphic 1-hypergroups of size up to 5.…”

Section: Introductionmentioning

confidence: 99%

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G-Hypergroups: Hypergroups with a Group-Isomorphic Heart

et al. 2022

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Hypergroups can be subdivided into two large classes: those whose heart coincide with the entire hypergroup and those in which the heart is a proper sub-hypergroup. The latter class includes the family of 1-hypergroups, whose heart reduces to a singleton, and therefore is the trivial group. However, very little is known about hypergroups that are neither 1-hypergroups nor belong to the first class. The goal of this work is to take a first step in classifying G-hypergroups, that is, hypergroups whose heart is a nontrivial group. We introduce their main properties, with an emphasis on G-hypergroups whose the heart is a torsion group. We analyze the main properties of the stabilizers of group actions of the heart, which play an important role in the construction of multiplicative tables of G-hypergroups. Based on these results, we characterize the G-hypergroups that are of type U on the right or cogroups on the right. Finally, we present the hyperproduct tables of all G-hypergroups of size not larger than 5, apart of isomorphisms.

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Section: G-hypergroupsmentioning

confidence: 99%

“…hypergroups can be built by means of the construction shown in Example 2 of [15], which we recall hereafter. Let Aut(H) be the automorphism group of a hypergroup (H, •).…”

Section: G-hypergroupsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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G-Hypergroups: Hypergroups with a Group-Isomorphic Heart

et al. 2022

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“…For more examples and details about hypergroups, we refer to the books [3][4][5][6]26,27] and to the papers [20,21,[28][29][30][31][32].…”

Section: Hypergroupsmentioning

confidence: 99%

Fuzzy Multi-Hypergroups

2020

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A fuzzy multiset is a generalization of a fuzzy set. This paper aims to combine the innovative notion of fuzzy multisets and hypergroups. In particular, we use fuzzy multisets to introduce the concept of fuzzy multi-hypergroups as a generalization of fuzzy hypergroups. Different operations on fuzzy multi-hypergroups are defined and discussed and some results known for fuzzy hypergroups are generalized to fuzzy multi-hypergroups. 5 × 5 lower triangular matrix (a ij ) with a 11 = a 22 = −5, a 33 = 1, a 44 = a 55 = 0 has the multiset {−5, −5, 1, 0, 0}) and roots of a polynomial (e.g., the polynomial (x + 2) 2 (x − 1)x 3 over the field of complex numbers has the multiset {−2, −2, 1, 0, 0, 0}) can be considered as multisets.As an extension of the classical notion of sets, Zadeh [10] introduced a concept similar to that of a set but whose elements have degrees of membership and he called it fuzzy set. In classical sets, the membership function can take only two values: 0 and 1. It takes 0 if the element belongs to the set, and 1 if the element does not belong to the set. In fuzzy sets, there is a gradual assessment of the membership of elements in a set which is assigned a number between 0 and 1 (both included). Several applications for fuzzy sets appear in real life. We refer to the papers [11][12][13]. Yager, in [8], generalized the fuzzy set by introducing the concept of fuzzy multiset (fuzzy bag) and he discussed a calculus for them in [14]. An element of a fuzzy multiset can occur more than once with possibly the same or different membership values. If every element of a fuzzy multiset can occur at most once, we go back to fuzzy sets [15].In [16], Onasanya and Hoskova-Mayerova defined multi-fuzzy groups and in [17,18], the authors defined fuzzy multi-polygroups and fuzzy multi-H v -ideals and studied their properties. Moreover, Davvaz in [19] discussed various properties of fuzzy hypergroups. Our paper generalizes the work in [16,17,19] to combine hypergroups and fuzzy multisets. More precisely, it is concerned with fuzzy multi-hypergroups and constructed as follows: After the Introduction, Section 2 presents some preliminary definitions and results related to fuzzy multisets and hypergroups that are used throughout the paper. Section 3 introduces, for the first time, fuzzy multi-hypergroups as a generalization of fuzzy hypergroups and studies its properties. Finally, Section 4 defines some operations (e.g., intersection, selection, product, etc.) on fuzzy multi-hypergroups and discusses them.

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“…All the equivalences having this property, i.e., they are the smallest equivalence relations defined on a hypercompositional structure such that the corresponding quotient (modulo that relation) is a classical structure with the same behaviour, are called fundamental relations, while the associated quotients are called fundamental structures. The study of the fundamental relations represents an important topic of research in Hypercompositional Algebra also nowadays [5][6][7]. But this is not the unique case when the name "fundamental" is given to an equivalence defined on a hyperstructure.…”

Section: Introductionmentioning

confidence: 99%

Fuzzy Reduced Hypergroups

Kankaraš

Cristea

2020

Mathematics

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The fuzzyfication of hypercompositional structures has developed in several directions. In this note we follow one direction and extend the classical concept of reducibility in hypergroups to the fuzzy case. In particular we define and study the fuzzy reduced hypergroups. New fundamental relations are defined on a crisp hypergroup endowed with a fuzzy set, that lead to the concept of fuzzy reduced hypergroup. This is a hypergroup in which the equivalence class of any element, with respect to a determined fuzzy set, is a singleton. The most well known fuzzy set considered on a hypergroup is the grade fuzzy set, used for the study of the fuzzy grade of a hypergroup. Based on this, in the second part of the paper, we study the fuzzy reducibility of some particular classes of crisp hypergroups with respect to the grade fuzzy set.

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On Hypergroups with a β-Class of Finite Height

Cited by 7 publications

References 21 publications

G-Hypergroups: Hypergroups with a Group-Isomorphic Heart

G-Hypergroups: Hypergroups with a Group-Isomorphic Heart

Fuzzy Multi-Hypergroups

Fuzzy Reduced Hypergroups

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