Fuzzy Algebraic Hyperstructures

Davvaz, Bijan; Cristea, Irina

doi:10.1007/978-3-319-14762-8

Cited by 80 publications

(31 citation statements)

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“…The book by Davvaz and Leoreanu-Fotea [6] deals with the hyperring theory and applications. A more recent book [7] gives an introduction into fuzzy algebraic hyperstructures.…”

Section: Introductionmentioning

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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“…The book by Davvaz and Leoreanu-Fotea [6] deals with the hyperring theory and applications. A more recent book [7] gives an introduction into fuzzy algebraic hyperstructures.…”

Section: Introductionmentioning

confidence: 99%

Fuzzy Multi-Hypergroups

2020

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“…This study can be considered in two different directions, corresponding to the two different approaches of the theory of hypergroups associated with fuzzy sets. An overview of this theory is covered by the monograph [13] written by Davvaz and Cristea, the only one on this topic published till now. The fuzzy aspect of reducibility could be investigated in two ways: by studying the indistinguishability between the elements of a fuzzy hypergroup (i.e., a structure endowed with a fuzzy hyperoperation) or by studying the indistinguishability between the images of the elements of a crisp hypergroup through a fuzzy set.…”

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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“…The name "weak hyperstructures" is due to having some axioms of classical algebraic hyperstructures are replaced by their corresponding weak axioms in weak hyperstructures. Many researchers such as Corsini [2], Corsini and Leoreanu [3], Davvaz [4,5], Davvaz and Leoreanu-Fotea [6], Davvaz and Cristea [7] and Vougiouklis [8] wrote books related to (weak) hyperstructure theory and their applications. An overview about hyperstructure theory was published by Hoskova and Chvalina in [9].…”

Section: Introductionmentioning

confidence: 99%

Some Results on (Generalized) Fuzzy Multi-Hv-Ideals of Hv-Rings

2019

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The concept of fuzzy multiset is well established in dealing with many real life problems. It is possible to find various applications of algebraic hypercompositional structures in natural, technical and social sciences, where symmetry, or the lack of symmetry, is clearly specified and laid out. In this paper, we use fuzzy multisets to introduce the concept of fuzzy multi- H v -ideals as a generalization of fuzzy H v -ideals. Moreover, we introduce the concept of generalized fuzzy multi- H v -ideals as a generalization of generalized fuzzy H v -ideals. Finally, we investigate the properties of these new concepts and present different examples.

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Fuzzy Algebraic Hyperstructures

Cited by 80 publications

References 0 publications

Fuzzy Multi-Hypergroups

Fuzzy Multi-Hypergroups

Fuzzy Reduced Hypergroups

Some Results on (Generalized) Fuzzy Multi-Hv-Ideals of Hv-Rings

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