Fuzzy Multi-Hypergroups

Abstract

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.

Algebraic Hyperstructure of Multi-Fuzzy Soft Sets Related to Polygroups

“…In recent years, multi-fuzzy sets have become a subject of great interest to researchers and have been widely applied to algebraic structures. Some researchers-for instance, Onasanya and Hoskova-Mayerova [35]-studied the concept of multi-fuzzy groups, while Hoskova-Mayerova et al [36] studied fuzzy multi-hypergroups and also fuzzy multi-polygroups in [37]. Akın [38] studied the concept of multi-fuzzy soft groups as a generalization of fuzzy soft groups, and Kazancı et al [39] introduced a novel soft hyperstructure called the multi-fuzzy soft hyperstructure and investigated the notion of multi-fuzzy soft hypermodules and some of their structural properties on a hypermodule.…”

Applications of (Neutro/Anti)sophications to Semihypergroups

“…Semihypergroup, hypergroup, and fuzzy hypergroup of order 2 are enumerated in [7,11,12]. S. Hoskova-Mayerova et al used the fuzzy multisets to introduce the concept of fuzzy multi-hypergroups as a generalization of fuzzy hypergroups, defined the different operations on fuzzy multi-hypergroups, and extended the fuzzy hypergroups to fuzzy multihypergroups [13].…”

Comparison of Flight Safety Between Different Aircraft Modifications

“…);Bielawski et al (2017);Hoskova-Mayerova et al (2020). The resulting ratio is as follows: p = number of USC passengers number of passengers worldwide = 926 737 000 4 369 655 800 = 21.08%…”