A New Kind of Fuzzy 𝑛-ary Hypergroups in the Framework of Soft Set Theory

Abstract

Maji et al. introduced the concept of fuzzy soft sets as a generalization of the standard soft sets and presented an application of fuzzy soft sets in a decision making problem. The aim of this paper is to apply the concept of fuzzy soft sets to 𝑛-ary hypergroup theory. The concepts of (∈_𝛾,∈_𝛾∨𝑞_𝛿)-fuzzy soft (invertible) 𝑛-ary subhypergroups over a commutative 𝑛-ary hypergroup are introduced and some related properties and characterizations are obtained. The homomorphism properties of (∈_𝛾,∈_𝛾∨𝑞_𝛿)-fuzzy soft (invertible) 𝑛-ary subhypergroups are also derived.

1. Introduction

To solve complicated problems in economics, engineering, and environment, we cannot successfully use classical methods because of various uncertainties typical for those problems. There are three theories: theory of probability, theory of fuzzy sets, and the interval mathematics, which we can consider as mathematical tools for dealing with uncertainties. However, all of these have their advantages as well as inherent limitations in dealing with uncertainties. One major problem shared by those theories is their incompatibility with the parameterization tools. To overcome these difficulties, Molodtsov [1] introduced the concept of soft set as a new mathematical tool for dealing with uncertainties that is free from the difficulties that have troubled the usual theoretical approaches. Molodtsov pointed out several directions for the applications of soft sets. This theory has proven useful in many different fields such as decision making [2–6], data analysis [7, 8], and forecasting [9].

Up to the present, research on soft sets has been very active and many important results have been achieved in the theoretical aspect. Maji et al. [10] introduced several algebraic operations in soft set theory and published a detailed theoretical study on soft sets. Ali et al. [11] further presented and investigated some new algebraic operations for soft sets. Qin and Hong [12] further investigated the algebraic structure of soft sets and established the soft quotient algebra. Maji et al. [13] and Majumdar and Samanta [14] extended (classical) soft sets to fuzzy soft sets, respectively. Maji et al. [15] extended (classical) soft sets to intuitionistic fuzzy soft sets, which were further discussed by Maji et al. [16] and Jiang et al. [17]. Aktaş and Çağman [18] compared soft sets to the related concepts of fuzzy sets and rough sets. They also defined the notion of soft groups and derived some related properties. Aygünoğlu and Aygün [19] discussed the applications of fuzzy soft sets to group theory and investigated (normal) fuzzy soft groups. Feng et al. [20] investigated soft semirings by using the soft set theory. Jun [21] introduced and investigated the notion of soft BCK/BCI-algebras. Jun and Park [22] and Jun et al. [23] discussed the applications of soft sets in ideal theory of BCK/BCI-algebras and in 𝑑-algebras, respectively. Acar et al. [24] introduced and studied soft rings. Zhan and Jun [25] characterized the (implicative, positive implicative, and fantastic) filteristic soft 𝐵𝐿-algebras based on ∈-soft sets and 𝑞-soft sets.

On the other hand, the fuzzy sets and hyperstructures introduced by Zadeh and Marty, respectively, are now extensively applied to many disciplines. The study of fuzzy hyperstructures is an interesting research topic of fuzzy sets. There is a considerable amount of work on the connections between fuzzy sets and hyperstructures. The reader is referred to [26–38]. As a generalization of the concepts of “belongingness (∈)” and “quasi-coincidence (𝑞)” of a fuzzy point with a fuzzy set introduced by Pu and Liu [39], Yin and Zhan [40] introduced the concepts of “𝛾-belongingness (∈_𝛾)” and “𝛿-quasi-coincidence (𝑞_𝛿)” of a fuzzy point with a fuzzy set. Using these new concepts, Yin and Zhan [40] introduced the concepts of (𝛼,𝛽)-fuzzy (implicative, positive implicative, and fantastic) filters and (𝛽,𝛼)-fuzzy (implicative, positive implicative, and fantastic) filters of 𝐵𝐿-algebras, where 𝛼,𝛽∈{∈_𝛾,𝑞_𝛿,∈_𝛾∧𝑞_𝛿,∈_𝛾∨𝑞_𝛿}, 𝛼,𝛽∈{∈_𝛾,𝑞_𝛿,∈_𝛾∧𝑞_𝛿,∈_𝛾∨𝑞_𝛿}, 𝛼≠∈_𝛾∧𝑞_𝛿, and 𝛽≠∈_𝛾∧𝑞_𝛿, and some related properties were investigated. Special attention is paid to (∈_𝛾,∈_𝛾∨𝑞_𝛿)-fuzzy (implicative, positive implicative, and fantastic) filters and (∈_𝛾,∈_𝛾∨𝑞_𝛿)-fuzzy (implicative, positive implicative, and fantastic) filters which are generalizations of (∈,∈∨𝑞)-fuzzy (implicative, positive implicative, and fantastic) filters and (∈,∈∨𝑞)-fuzzy (implicative, positive implicative, and fantastic) filters studied by Ma et al. [41, 42]. This idea is continued and studied by Ma et al. [43], Yin and Zhan [44, 45], Yin et al. [46], and so on. The purpose of this paper is to deal with the algebraic structure of 𝑛-ary hypergroups by applying fuzzy soft set theory. We introduce the concepts of (∈_𝛾,∈_𝛾∨𝑞_𝛿)-fuzzy soft (invertible) 𝑛-ary subhypergroups over a commutative 𝑛-ary hypergroup and focus on investigating their characterization and algebraic properties. We also discuss the homomorphism properties of (∈_𝛾,∈_𝛾∨𝑞_𝛿)-fuzzy soft (invertible) 𝑛-ary subhypergroups.

2. Preliminaries

In this section, we recall some basic notions and results about 𝑛-ary hypergroups, fuzzy sets, and fuzzy soft sets (see [13, 28, 30, 47]).

2.1. n-Ary Hypergroups

We will be concerned primarily with a basic nonempty set 𝐻. Denote by 𝐻^𝑛 the cartesian product 𝐻×⋯×𝐻, where 𝐻 appears 𝑛 times. An element of 𝐻^𝑛 will be denoted by (𝑥₁,…,𝑥_𝑛), where 𝑥_𝑖∈𝐻 for all 1≤𝑖≤𝑛. In general, a mapping 𝑓∶𝐻^𝑛→𝒫^∗(𝐻) is called an 𝑛-ary hyperoperation and 𝑛 is called the arity of the hyperoperation 𝑓. Let 𝑓 be an 𝑛-ary hyperoperation on 𝐻 and 𝐴₁,…,𝐴_𝑛 subsets in 𝐻. Define 𝑓𝐴₁,…,𝐴_𝑛=∪𝑓𝑥₁,…,𝑥_𝑛∣𝑥_𝑖∈𝐴_𝑖,1≤𝑖≤𝑛. (1) In the sequel, we will denote the sequence 𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑗 by 𝑥^𝑗_𝑖. For 𝑗<𝑖, 𝑥^𝑗_𝑖=∅. Thus, 𝑓𝑥₁,…,𝑥_𝑖,𝑦_𝑖+1,…,𝑦_𝑗,𝑧_𝑗+1,…,𝑧_𝑛 (2) will be written as 𝑓(𝑥^𝑖₁,𝑦^𝑗_𝑖+1,𝑧^𝑛_𝑗+1). Also 𝑓(𝑎^𝑖₁,𝑥∗) means 𝑓(𝑎^𝑖₁,𝑥,…,𝑥_𝑛−𝑖) for 𝑎^𝑖₁,𝑥∈𝐻 and 1≤𝑖≤𝑛.

𝐻 with an 𝑛-ary hyperoperation 𝑓∶𝐻^𝑛→𝒫^∗(𝐻) is called an 𝑛-ary hypergroupoid and will be denoted by (𝐻,𝑓). An 𝑛-ary hypergroupoid (𝐻,𝑓) is said to be an 𝑛-ary semihypergroup if the following associative axiom holds: 𝑓𝑥₁^𝑖−1,𝑓𝑥_𝑖^{𝑛+𝑖−1},𝑥^2𝑛−1_𝑛+𝑖=𝑓𝑥₁^𝑗−1,𝑓𝑥_𝑗^{𝑛+𝑗−1},𝑥^2𝑛−1_𝑛+𝑗 (3) for all 𝑖,𝑗∈{1,…,𝑛} and 𝑥₁^2𝑛−1∈𝐻. An 𝑛-ary semihypergroup is said to be an 𝑛-ary hypergroup if, for all 𝑦,𝑦^𝑛₁∈𝐻 and fixed 𝑖∈{1,…,𝑛}, there exists 𝑥∈𝐻 such that 𝑦∈𝑓(𝑦₁^𝑖−1,𝑥,𝑦^𝑛_𝑖+1). An 𝑛-ary hypergroup is said to be commutative if, for all 𝑥^𝑛₁∈𝐻 and any permutation 𝜌 of {1,…,𝑛}, we have 𝑓(𝑥^𝑛₁)=𝑓(𝑥_𝜌1,…,𝑥_𝜌𝑛). An element 𝑒 of 𝐻 is called an identity of 𝐻 if, for any 𝑥∈𝐻 and 𝑖∈{1,…,𝑛}, we have 𝑥∈𝑓(𝑒,…,𝑒_𝑖−1,𝑥,𝑒,…,𝑒_𝑛−𝑖).

Definition 1. Let (𝐻,𝑓) be an 𝑛-ary hypergroup and 𝐾 a nonempty subset of 𝐻. One says that 𝐾 is an 𝑛-ary subhypergroup of 𝐻 if the following conditions hold:(i)𝐾is closed under the 𝑛-ary hyperoperation 𝑓;(ii)for all 𝑦,𝑦^𝑛₁∈𝐾 and fixed 𝑖∈{1,…,𝑛}, there exists 𝑥∈𝐾 such that 𝑦∈𝑓(𝑦₁^𝑖−1,𝑥,𝑦^𝑛_𝑖+1).

In what follows, (𝐻,𝑓) denotes a commutative 𝑛-ary hypergroup with an identity 𝑒 unless otherwise stated.

An 𝑛-ary subhypergroup 𝐾 of 𝐻 is said to be invertible if, for any 𝑥,𝑦∈𝐻, 𝑥∈𝑓(𝑦,𝐾,…,𝐾_𝑛−1) implies 𝑦∈𝑓(𝑥,𝐾,…,𝐾_𝑛−1).

2.2. Fuzzy Sets

Let 𝑋 be a nonempty set. A fuzzy subset 𝜇 of 𝑋 is defined as a mapping from 𝑋 into [0,1], where [0,1] is the usual interval of real numbers. We denote by ℱ(𝑋) the set of all fuzzy subsets of 𝑋. For 𝜇,𝜈∈ℱ(𝑋), by 𝜇⊆𝜈, we mean 𝜇(𝑥)≤𝜈(𝑥) for all 𝑥∈𝑋. And the union and intersection of 𝜇 and 𝜈, denoted by 𝜇∪𝜈 and 𝜇∩𝜈, are defined as the fuzzy subsets of 𝑋 by (𝜇∪𝜈)(𝑥)=max{𝜇(𝑥),𝜈(𝑥)} and (𝜇∩𝜈)(𝑥)=min{𝜇(𝑥),𝜈(𝑥)} for all 𝑥∈𝑋.

For 𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑗∈𝑋 and 𝜇∈ℱ(𝑋), the sequence 𝜇(𝑥_𝑖),𝜇(𝑥_𝑖+1),…,𝜇(𝑥_𝑗) is denoted by 𝜇^𝑥𝑗_𝑥𝑖, and 𝜇^𝑥𝑗_𝑥𝑖=0 if 𝑗<𝑖. Moreover, for 𝜇_𝑖,𝜇_𝑖+1,…,𝜇_𝑗∈ℱ(𝑋), the sequence 𝜇_𝑖,𝜇_𝑖+1,…,𝜇_𝑗 is denoted by 𝜇^𝑗_𝑖, and 𝜇^𝑗_𝑖 denotes the zero fuzzy set if 𝑗<𝑖.

For any 𝑌⊆𝑋 and 𝑟∈(0,1], define a fuzzy subset 𝑟_𝑌 of 𝑋 by 𝑟_𝑌(𝑥)=𝑟if0𝑥∈𝑌,otherwise, (4) for all 𝑥∈𝑋. In particular, when 𝑟=1, 𝑟_𝑌 is said to be the characteristic function of 𝑌, denoted by 𝜒_𝑌, and when𝑋={𝑥}, 𝑟_𝑋 is said to be a fuzzy point with support 𝑥 and value 𝑟 and is denoted by 𝑟_𝑥.

In what follows, let 𝛾,𝛿∈[0,1] be such that 𝛾<𝛿. For a fuzzy point 𝑟_𝑥 and a fuzzy subset 𝜇 of 𝑋, we say that(1)𝑟_𝑥∈_𝛾𝜇 if 𝜇(𝑥)≥𝑟>𝛾;(2)𝑟_𝑥𝑞_𝛿𝜇 if 𝜇(𝑥)+𝑟>2𝛿;(3)𝑟_𝑥∈_𝛾∨𝑞_𝛿𝜇 if 𝑟_𝑥∈_𝛾𝜇 or 𝑟_𝑥𝑞_𝛿𝜇 [40].

Let us now introduce a new relation on ℱ(𝑋), denoted by “⊆∨𝑞_(𝛾,𝛿),” as follows.

For any 𝜇,𝜈∈ℱ(𝑋), by 𝜇⊆∨𝑞_(𝛾,𝛿)𝜈, we mean that 𝑟_𝑥∈_𝛾𝜇 implies 𝑟_𝑥∈_𝛾∨𝑞_𝛿𝜈 for all 𝑥∈𝑋 and 𝑟∈(𝛾,1]. Moreover, 𝜇 and 𝜈 are said to be (𝛾,𝛿)-equal, denoted by 𝜇=_(𝛾,𝛿)𝜈, if 𝜇⊆∨𝑞_(𝛾,𝛿)𝜈 and 𝜈⊆∨𝑞_(𝛾,𝛿)𝜇.

In the sequel, unless otherwise stated, 𝛼 means that 𝛼 does not hold, where 𝛼∈{∈_𝛾,𝑞_𝛿,∈_𝛾∨𝑞_𝛿,⊆∨𝑞_(𝛾,𝛿)}.

Lemma 2 (see [40]). Let 𝜇,𝜈∈ℱ(𝑋). Then 𝜇⊆∨𝑞_(𝛾,𝛿)𝜈 if and only if max{𝜈(𝑥),𝛾}≥min{𝜇(𝑥),𝛿} for all 𝑥∈𝑋.

Lemma 3 (see [40]). Let 𝜇,𝜈,𝜔∈ℱ(𝑋). If 𝜇⊆∨𝑞_(𝛾,𝛿)𝜈 and  𝜈⊆∨𝑞_(𝛾,𝛿)𝜔, then 𝜇⊆∨𝑞_(𝛾,𝛿)𝜔.

Proof. It is straightforward by Lemma 2

Lemmas 2 and 3 give that “=_(𝛾,𝛿)” is an equivalence relation on ℱ(𝑋). It is also worth noticing that 𝜇=_(𝛾,𝛿)𝜈 if and only if max{min{𝜇(𝑥),𝛿},𝛾}=max{min{𝜈(𝑥),𝛿},𝛾} for all 𝑥∈𝑋 by Lemma 2

2.3. Fuzzy Soft Sets

Let 𝑈 be an initial universe set and 𝐸 the set of all possible parameters under consideration with respect to 𝑈. As a generalization of soft set introduced by Molodtsov [1], Maji et al. [13] defined fuzzy soft set in the following way.

Definition 4. A pair ⟨𝐹,𝐴⟩ is called a fuzzy soft set over 𝑈, where 𝐴⊆𝐸 and 𝐹 is a mapping given by 𝐹∶𝐴→ℱ(𝑈).

In general, for every 𝜀∈𝐴, 𝐹(𝜀) is a fuzzy set of 𝑈 and it is called fuzzy value set of parameter 𝜀. The set of all fuzzy soft sets over 𝑈 with parameters from 𝐸 is called fuzzy soft classes, and it is denoted by ℱ𝒮(𝑈,𝐸). A fuzzy soft set ⟨𝐹,𝐴⟩ in ℱ𝒮(𝑈,𝐸) is said to have sup-property if, for any 𝑉∈𝒫^∗(𝑈) and 𝜀∈𝐴, there exists 𝑥₀∈𝑉 such that 𝐹(𝜀)(𝑥₀)=sup_𝑥∈𝑉𝐹(𝜀)(𝑥).

Let 𝐴⊆𝐸 and 𝜇∈ℱ(𝑈). Define ⟨𝜇,𝐴⟩to be a fuzzy soft set by 𝜇(𝜀)=𝜇 for all 𝜀∈𝐴. For any fuzzy point 𝑟_𝑥 in 𝑈, define (𝑟_𝑥,𝐴)to be a fuzzy soft set by 𝑟_𝑥(𝜀)=𝑟_𝑥 for all 𝜀∈𝐴.

Definition 5 (see [6]). Let ⟨𝐹,𝐴⟩ and ⟨𝐺,𝐵⟩ be two fuzzy soft sets over𝑈. One says that ⟨𝐹,𝐴⟩ is a fuzzy soft subset of ⟨𝐺,𝐵⟩ and writes ⟨𝐹,𝐴⟩⋐⟨𝐺,𝐵⟩ if(i)𝐴⊆𝐵;(ii)for any 𝜀∈𝐴, 𝐹(𝜀)⊆𝐺(𝜀).
⟨𝐹,𝐴⟩ and ⟨𝐺,𝐵⟩ are said to be fuzzy soft equal and write ⟨𝐹,𝐴⟩=⟨𝐺,𝐵⟩ if ⟨𝐹,𝐴⟩⋐⟨𝐺,𝐵⟩ and ⟨𝐺,𝐵⟩⋐⟨𝐹,𝐴⟩.

Let us now introduce some new concepts on fuzzy soft sets analogous to the concepts introduced by Ali et al. [11].

Definition 6. The extended intersection of two fuzzy soft sets ⟨𝐹,𝐴⟩ and ⟨𝐺,𝐵⟩ over 𝑈 is a fuzzy soft set denoted by ⟨𝐻,𝐶⟩, where 𝐶=𝐴∪𝐵 and 𝐻(𝜀)=𝐹(𝜀)if𝐺(𝜀)𝜀∈𝐴−𝐵,if𝐹(𝜀)(𝜀)𝜀∈𝐵−𝐴,∩𝐺if𝜀∈𝐴∩𝐵, (5) for all 𝜀∈𝐶. This is denoted by ⟨𝐻,𝐶⟩=⟨𝐹,𝐴⟩∩⟨𝐺,𝐵⟩.

Definition 7. Let ⟨𝐹,𝐴⟩ and ⟨𝐺,𝐵⟩ be two fuzzy soft sets over 𝑈 such that 𝐴∩𝐵≠∅. The restricted intersection of ⟨𝐹,𝐴⟩ and ⟨𝐺,𝐵⟩ is defined to be the fuzzy soft set ⟨𝐻,𝐶⟩, where 𝐶=𝐴∩𝐵 and 𝐻(𝜀)=𝐹(𝜀)∩𝐺(𝜀) for all 𝜀∈𝐶. This is denoted by ⟨𝐻,𝐶⟩=⟨𝐹,𝐴⟩⋒⟨𝐺,𝐵⟩.

Definition 8. Let 𝑉⊆𝑈 and 𝑟∈(𝛾,1]. A fuzzy soft set ⟨𝐹,𝐴⟩ over 𝑉 is said to be a relative whole (𝛾,𝛿)-fuzzy soft set (with respect to universe set 𝑉, parameter set 𝐴, and value 𝑟), denoted by Σ(𝑉,𝐴,𝑟), if 𝐹(𝜀)=𝑟_𝑉 for all 𝜀∈𝐴.

Definition 9. Let ⟨𝐹,𝐴⟩ and ⟨𝐺,𝐵⟩ be two fuzzy soft sets over 𝑈. One says that ⟨𝐹,𝐴⟩ is (𝛾,𝛿)-fuzzy soft subset of ⟨𝐺,𝐵⟩and writes ⟨𝐹,𝐴⟩⋐_(𝛾,𝛿)⟨𝐺,𝐵⟩ if(i)𝐴⊆𝐵;(ii)for any 𝜀∈𝐴, 𝐹(𝜀)⊆∨𝑞_(𝛾,𝛿)𝐺(𝜀).
⟨𝐹,𝐴⟩ and ⟨𝐺,𝐵⟩ are said to be (𝛾,𝛿)-fuzzy soft equal and write ⟨𝐹,𝐴⟩≍_(𝛾,𝛿)⟨𝐺,𝐵⟩ if ⟨𝐹,𝐴⟩⋐_(𝛾,𝛿)⟨𝐺,𝐵⟩ and ⟨𝐺,𝐵⟩⋐_(𝛾,𝛿)⟨𝐹,𝐴⟩.

Clearly, ⟨𝐹,𝐴⟩⋐⟨𝐺,𝐵⟩ implies ⟨𝐹,𝐴⟩⊆∨𝑞_(𝛾,𝛿)⟨𝐺,𝐵⟩ by Lemma 2 and Definition 9.

Lemma 10. Let ⟨𝐹,𝐴⟩, ⟨𝐺,𝐵⟩, and ⟨𝐻,𝐶⟩ be fuzzy soft sets over 𝑈. If ⟨𝐹,𝐴⟩⋐_(𝛾,𝛿)⟨𝐺,𝐵⟩ and ⟨𝐺,𝐵⟩⋐_(𝛾,𝛿)⟨𝐻,𝐶⟩, then ⟨𝐹,𝐴⟩⋐_(𝛾,𝛿)⟨𝐹,𝐶⟩.

Proof. It is straightforward by Lemma 3 and Definition 9.

Lemmas 2 and 10 and Definition 9 give that “≍_(𝛾,𝛿)” is an equivalence relation on ℱ𝒮(𝑈,𝐸).

3. A New Operation on Fuzzy Soft Sets over an 𝑛-Ary Hypergroupoid

We first introduce a fuzzy 𝑛-ary hyperoperation on an 𝑛-ary hypergroupoid as follows.

Definition 11. Let (𝐻,𝑓) be an 𝑛-ary hypergroupoid and 𝜇₁,…,𝜇_𝑛∈ℱ(𝐻). Define a fuzzy 𝑛-ary hyperoperation 𝓕∶ℱ(𝐻)×⋯×ℱ(𝐻)_𝑛→ℱ(𝐻) by 𝓕𝜇₁,…,𝜇_𝑛(𝑥)=sup_{𝑦𝑥∈𝑓𝑛1}min𝜇₁𝑦₁,…,𝜇_𝑛𝑦_𝑛 (6) for all 𝑥∈𝐻. In particular, for any 𝑥^𝑛₁∈𝐻 and 𝑖∈{1,…,𝑛}, define 𝓕𝜇₁,…,𝜇_𝑖,𝑥^𝑛_𝑖+1(𝑥)=sup_{𝑥∈𝑓(𝑦𝑖1,𝑥𝑛𝑖+1)}min𝜇₁𝑦₁,…,𝜇_𝑖𝑦_𝑖 (7) for all 𝑥∈𝐻.

Clearly, for any 𝑥^𝑛₁∈𝐻 and 1≤𝑖≤𝑛, 𝓕(𝜇₁,…,𝜇_𝑖,𝑥^𝑛_𝑖+1)=𝓕(𝜇₁,…,𝜇_𝑖,1_𝑥𝑖+1,…,1_𝑥𝑛). Note that, for any 𝑟^𝑛₁∈(0,1] and 𝑥^𝑛₁∈𝐻, by Definition 11, 𝓕(𝑟_1𝑥1,𝑟_2𝑥2,…,𝑟_𝑛𝑥𝑛)=min{𝑟^𝑛₁}_𝑓(𝑥^𝑛₁₎.

Definition 12. Let (𝐻,𝑓) be an 𝑛-ary hypergroupoid and ⟨𝐹_𝑖,𝐴_𝑖⟩∈ℱ𝒮(𝐻,𝐸), where 𝑖∈𝐼 and 𝐼 is an index set, such that ⋂_𝑖∈𝐼𝐴_𝑖≠∅. One defines a fuzzy soft set over 𝐻, denoted by ⟨𝔽(𝐹₁,…,𝐹_𝑛),𝐶⟩, where 𝐶=⋂_𝑖∈𝐼𝐴_𝑖 and 𝔽𝐹₁,…,𝐹_𝑛(𝜀)=𝓕𝐹₁(𝜀),…,𝐹_𝑛(𝜀), (8) for all 𝜀∈𝐶. This is denoted by ⟨𝔽(𝐹₁,…,𝐹_𝑛),𝐶⟩=𝔉(⟨𝐹₁,𝐴₁⟩,…,⟨𝐹_𝑛,𝐴_𝑛⟩).

In the following, we will denote the sequence ⟨𝐹_𝑖,𝐴_𝑖⟩,⟨𝐹_𝑖+1,𝐴_𝑖+1⟩,…,⟨𝐹_𝑗,𝐴_𝑗⟩ by ⟨𝐹,𝐴⟩^𝑗_𝑖. For 𝑗<𝑖, ⟨𝐹,𝐴⟩^𝑗_𝑖 is the empty set. The following elementary facts follow easily from the definition.

Lemma 13. Let (𝐻,𝑓) be an 𝑛-ary hypergroupoid and ⟨𝐹,𝐴⟩₁^2𝑛−1,⟨𝐺,𝐵⟩^𝑛₁∈ℱ𝒮(𝐻,𝐸). Then(1)if (𝐻,𝑓) is an 𝑛-ary semihypergroup and ⋂^2𝑛_𝑖=1𝐴_𝑖≠∅, then 𝔉⟨𝐹,𝐴⟩₁^𝑖−1,𝔉⟨𝐹,𝐴⟩_𝑖^{𝑛+𝑖−1},⟨𝐹,𝐴⟩^2𝑛−1_𝑛+𝑖=𝔉⟨𝐹,𝐴⟩₁^𝑗−1,𝔉⟨𝐹,𝐴⟩_𝑗^{𝑛+𝑗−1},⟨𝐹,𝐴⟩^2𝑛−1_𝑛+𝑗 (9) for all 1≤𝑖,𝑗≤𝑛;(2)if (𝐻,𝑓) is commutative and ⋂^𝑛_𝑖=1𝐴_𝑖≠∅, then the value of 𝔉(⟨𝐹,𝐴⟩^𝑛₁) does not depend on the permutation of ⟨𝐹₁,𝐴₁⟩,…,⟨𝐹_𝑛,𝐴_𝑛⟩;(3)if ⟨𝐹_𝑖,𝐴_𝑖⟩⋐_(𝛾,𝛿)⟨𝐺_𝑖,𝐵_𝑖⟩ for all 1≤𝑖≤𝑛 and ⋂^𝑛_𝑖=1𝐴_𝑖≠∅, then 𝔉(⟨𝐹,𝐴⟩^𝑛₁)⋐_(𝛾,𝛿)𝔉(⟨𝐺,𝐵⟩^𝑛₁);(4)if ⟨𝐹₁,𝐴₁⟩⋐_(𝛾,𝛿)⟨𝐺₁,𝐴₁⟩ and⟨𝐹₂,𝐴₂⟩⋐_(𝛾,𝛿)⟨𝐺₂,𝐴₂⟩, then ⟨𝐹₁,𝐴₁⟩∩⟨𝐹₂,𝐴₂⟩⋐_(𝛾,𝛿)⟨𝐺₁,𝐴₁⟩∩⟨𝐺₂,𝐴₂⟩ and ⟨𝐹₁,𝐴₁⟩⋒⟨𝐹₂,𝐴₂⟩⋐_(𝛾,𝛿)⟨𝐺₁,𝐴₁⟩⋒⟨𝐺₂,𝐴₂⟩ if 𝐴₁∩𝐴₂≠∅.

4. (∈_𝛾,∈_𝛾∨𝑞_𝛿)-Fuzzy Soft (Invertible) 𝑛-Ary Subhypergroups over a Commutative 𝑛-Ary Hypergroup

In this section, we will introduce and investigate the concept of (∈_𝛾,∈_𝛾∨𝑞_𝛿)-fuzzy soft 𝑛-ary subhypergroups over a commutative 𝑛-ary hypergroup. Let us start by giving the following concept.

Definition 14. Let ⟨𝐹,𝐴⟩∈ℱ𝒮(𝐻,𝐸). Then ⟨𝐹,𝐴⟩ is called (∈_𝛾,∈_𝛾∨𝑞_𝛿)-fuzzy soft 𝑛-ary subhypergroup over 𝐻 if it satisfies the following conditions:(F1a)for all 𝑥∈𝐻 and 𝜀∈𝐴, max{𝐹(𝜀)(𝑒),𝛾}≥min{𝐹(𝜀)(𝑥),𝛿};(F2a)for all 𝑥^𝑛₁∈𝐻 and 𝜀∈𝐴, max{inf_{𝑥∈𝑓(𝑥}^𝑛₁₎𝐹(𝜀)(𝑥),𝛾}≥min{𝐹(𝜀)^𝑥𝑛_𝑥1,𝛿};(F3a)for all 𝑦,𝑦₁^𝑛−1∈𝐻 and 𝜀∈𝐴, there exists 𝑥∈𝐻 such that 𝑦∈𝑓𝑥,𝑦₁^𝑛−1,max𝐹(𝜀)(𝑥),𝛾≥min𝐹(𝜀)^𝑦𝑛−1_𝑦1,𝐹(𝜀)𝑦,𝛿. (10)An (∈_𝛾,∈_𝛾∨𝑞_𝛿)-fuzzy 𝑛-ary subhypergroup ⟨𝐹,𝐴⟩ over 𝐻 is said to be invertible if it satisfies the following:(F4a)for all 𝑥,𝑦∈𝐻 and 𝜀∈𝐴, maxmin𝓕𝑦,𝐹(𝜀),…,𝐹_𝑛−1(𝜀)(𝑥)𝓕(𝜀)(𝜀),𝛿,𝛾=maxmin𝑥,𝐹,…,𝐹_𝑛−1𝑦,𝛿,𝛾. (11)

Example 15. Let 𝑍₈ be the residue class of (𝑍,+) modulo 8 with an 𝑛-ary hyperoperation 𝑓 as follows: 𝑓∶𝑍^𝑛₈⟶𝒫^∗𝑍₈,𝑓𝑥^𝑛₁=𝑚₁𝑥₁+⋯+𝑚_𝑛𝑥_𝑛∣𝑚₁,…,𝑚_𝑛∈𝑍. (12) Then, (𝑍₈,𝑓) is a commutative 𝑛-ary hypergroup. Let 𝐸=(0.2,0.6] and 𝑟≥0.6. Define a fuzzy soft set ⟨𝐹,𝐴⟩ over 𝑍₈ by 𝐹(𝜀𝑥)=𝑟)(if𝜀𝑥=0,if𝑥∈{2,4},0.2otherwise, (13) for all 𝜀∈𝐸 and 𝑥∈𝑍₈. Then ⟨𝐹,𝐸⟩ is an (∈_0.2,∈_0.2∨𝑞_0.6)-fuzzy soft invertible 𝑛-ary subhypergroup over 𝑍₈.

Next let us first provide some auxiliary lemmas as follows.

Lemma 16. Let ⟨𝐹,𝐴⟩∈ℱ𝒮(𝐻,𝐸). Then (F2a) holds if and only if one of the following conditions holds:(F2b)𝔉(⟨𝐹,𝐴⟩,…,⟨𝐹,𝐴⟩_𝑛)⋐_(𝛾,𝛿)⟨𝐹,𝐴⟩;(F2c)for all 𝑥^𝑛₁∈𝐻 and 𝜀∈𝐴, 𝓕(𝐹(𝜀)(𝑥₁)_𝑥1,…,𝐹(𝜀)(𝑥_𝑛)_𝑥𝑛)⊆∨𝑞_(𝛾,𝛿)𝐹(𝜀);(F2d)for all 𝜀∈𝐴 and 𝑟_𝑖𝑥𝑖∈_𝛾𝐹(𝜀)(1≤𝑖≤𝑛), 𝓕(𝑟_1𝑥1,…,𝑟_𝑛𝑥𝑛)⊆∨𝑞_(𝛾,𝛿)𝐹(𝜀).

Proof. (F2a)⇒(F2b) Assume that (F2a) holds. Then, for any 𝑥∈𝐻 and 𝜀∈𝐴, we have min𝓕𝐹(𝜀),…,𝐹_𝑛(𝜀)(𝑥),𝛿=minsup_{𝑥∈𝑓(𝑦𝑛1)}min𝐹(𝜀)^𝑦𝑛_𝑦1,𝛿=sup_{𝑦𝑥∈𝑓𝑛1}min𝐹(𝜀)^𝑦𝑛_𝑦1,𝛿≤max𝐹(𝜀)(𝑥),𝛾 (14) and so 𝓕(𝐹(𝜀),…,𝐹(𝜀)_𝑛)⊆∨𝑞_(𝛾,𝛿)𝐹(𝜀) by Lemma 2. Therefore, 𝔉(⟨𝐹,𝐴⟩,…,⟨𝐹,𝐴⟩_𝑛)⋐_(𝛾,𝛿)⟨𝐹,𝐴⟩.
(F2b)⇒(F2c) Let 𝑥^𝑛₁∈𝐻 and 𝜀∈𝐴. Since 𝐹(𝜀)(𝑥_𝑖)_𝑥𝑖∈_𝛾𝐹(𝜀), for all 1≤𝑖≤𝑛, we have 𝓕𝐹(𝜀)𝑥_1𝑥1,…,𝐹(𝜀)𝑥_𝑛𝑥𝑛⊆𝓕𝐹(𝜀),…,𝐹_𝑛(𝜀)⊆∨𝑞_(𝛾,𝛿)𝐹(𝜀), (15) and so 𝓕(𝐹(𝜀)(𝑥₁)_𝑥1,…,𝐹(𝜀)(𝑥_𝑛)_𝑥𝑛)⊆∨𝑞_(𝛾,𝛿)𝐹(𝜀).
(F2c)⇔(F2d) This is straightforward.
(F2d)⇒(F2a) Let 𝜀∈𝐴, 𝑥∈𝐻, and 𝑥^𝑛₁∈𝐻 be such that 𝑥∈𝑓(𝑥^𝑛₁). Then, by Lemma 2 and (F2d), we have 𝐹(𝜀𝑥)𝓕𝐹(𝜀)𝑥max)(,𝛾≥min_1𝑥1,…,𝐹(𝜀)𝑥_𝑛𝑥𝑛(𝑥)𝐹(𝜀),𝛿=minmin^𝑥𝑛_{𝑥1𝑓(𝑥𝑛1)}(𝑥)𝐹(𝜀),𝛿=min^𝑥𝑛_𝑥1,𝛿. (16) This implies max{inf_{𝑥∈𝑓(𝑥}^𝑛₁₎𝐹(𝜀)(𝑥),𝛾}≥min{𝐹(𝜀)^𝑥𝑛_𝑥1,𝛿} for all 𝑥^𝑛₁∈𝐻 and 𝜀∈𝐴. Hence, (F2a) holds.

Lemma 17. Let ⟨𝐹,𝐴⟩∈ℱ𝒮(𝐻,𝐸). Then, one has the following.(1)(F3a) holds if and only if the following condition holds: (F3b) for all 𝑦,𝑦₁^𝑛−1∈𝐻 and 𝜀∈𝐴, there exists 𝑥∈𝐻 such that min𝐹(𝜀)^𝑦𝑛−1_𝑦1,𝐹(𝜀)𝑦_𝑦⊆∨𝑞_(𝛾,𝛿)𝓕𝐹(𝜀)(𝑥)_𝑥,𝐹(𝜀)𝑦_1𝑦1,…,𝐹(𝜀)𝑦_{𝑛−1𝑦𝑛−1}. (17)(2)If (F3a) holds, then the following condition holds:(F3c) for all 𝑦₁^𝑛−1∈𝐻 and 𝜀∈𝐴, 𝐹(𝜀)∩min𝐹(𝜀)^𝑦𝑛−1_𝑦1𝐻⊆∨𝑞_(𝛾,𝛿)𝓕𝐹(𝜀),𝐹(𝜀)𝑦_1𝑦1,…,𝐹(𝜀)𝑦_{𝑛−1𝑦𝑛−1}. (18)Moreover, (F3c) implies (F3a) if ⟨𝐹,𝐴⟩ has the sup-property or 𝐻 is finite.

Proof. (F3a)⇒(F3b) Let 𝑦,𝑦₁^𝑛−1∈𝐻 and 𝜀∈𝐴. By (F3a), there exists 𝑥∈𝐻 such that 𝑦∈𝑓𝑥,𝑦₁^𝑛−1,max𝐹(𝜀)(𝑥),𝛾≥min𝐹(𝜀)^𝑦𝑛−1_𝑦1,𝐹(𝜀)𝑦,𝛿. (19) This implies min{max{𝐹(𝜀)^𝑦𝑛−1_𝑦1,𝛾},max{𝐹(𝜀)(𝑥),𝛾}}≥min{𝐹(𝜀)^𝑦𝑛−1_𝑦1,𝐹(𝜀)(𝑦),𝛿}, and so max𝓕𝐹(𝜀)(𝑥)_𝑥,𝐹(𝜀)𝑦_1𝑦1,…,𝐹(𝜀)𝑦_{𝑛−1𝑦𝑛−1}𝑦𝐹(𝜀),𝛾=maxmin^𝑦𝑛−1_𝑦1,𝐹(𝜀)(𝑥)_{𝑓(𝑥,𝑦1𝑛−1)}𝑦𝐹(𝜀),𝛾≥min^𝑦𝑛−1_𝑦1(𝜀)𝑦𝐹(𝜀),𝐹,𝛿=minmin^𝑦𝑛−1_𝑦1,𝐹(𝜀)𝑦_𝑦𝑦,𝛿. (20) It follows that min{𝐹(𝜀)^𝑦𝑛−1_𝑦1,𝐹(𝜀)(𝑦)}_𝑦⊆∨𝑞_(𝛾,𝛿)𝓕(𝐹(𝜀)(𝑥)_𝑥, 𝐹(𝜀)(𝑦₁)_𝑦1,…,𝐹(𝜀)(𝑦_𝑛−1)_𝑦𝑛−1) and so (F3b) holds.
(F3b)⇒(F3a) Let 𝑦,𝑦₁^𝑛−1∈𝐻. By (F3b), there exists 𝑥∈𝐻 such that min𝐹(𝜀)^𝑦𝑛−1_𝑦1,𝐹(𝜀)(𝑦)_𝑦⊆∨𝑞_(𝛾,𝛿)𝓕𝐹(𝜀)(𝑥)_𝑥,𝐹(𝜀)𝑦_1𝑦1,…,𝐹(𝜀)𝑦_{𝑛−1𝑦𝑛−1}. (21) It follows from Lemma 2 that maxmin𝐹(𝜀)^𝑦𝑛−1_𝑦1,𝐹(𝜀)(𝑥)_{𝑓(𝑥,𝑦1𝑛−1)}𝑦𝓕𝐹(𝜀𝑥),𝛾=max)(_𝑥,𝐹(𝜀)𝑦_1𝑦1,…,𝐹(𝜀)𝑦_{𝑛−1𝑦𝑛−1}×𝑦𝐹(𝜀),𝛾≥minmin^𝑦𝑛−1_𝑦1,𝐹(𝜀)𝑦_𝑦𝑦𝐹(𝜀),𝛿=min^𝑦𝑛−1_𝑦1,𝐹(𝜀)𝑦,𝛿. (22) This implies 𝑦∈𝑓(𝑥,𝑦₁^𝑛−1) and max{min{𝐹(𝜀)^𝑦𝑛−1_𝑦1,𝐹(𝜀)(𝑥)},𝛾}≥min{𝐹(𝜀)^𝑦𝑛−1_𝑦1,𝐹(𝜀)(𝑦),𝛿}. Hence, max𝐹(𝜀)(𝑥),𝛾≥maxmin𝐹(𝜀)^𝑦𝑛−1_𝑦1(𝜀𝑥𝐹(𝜀),𝐹)(),𝛾≥min^𝑦𝑛−1_𝑦1,𝐹(𝜀)𝑦,𝛿 (23) and so (F3a) holds.
(F3a)⇒(F3c) This is straightforward by (F3b).
In the following, assume that ⟨𝐹,𝐴⟩ has the sup-property or 𝐻 is finite. We prove (F3c)⇒(F3a). Let 𝑦₁^𝑛−1∈𝐻. If, for any 𝑥∈𝐻 such that 𝑦∈𝑓(𝑥,𝑦₁^𝑛−1), we have 𝐹(𝜀𝑥)𝐹(𝜀𝑥)𝐹(𝜀))(≤max)(,𝛾<min^𝑦𝑛−1_𝑦1,𝐹(𝜀)𝑦,𝛿≤𝐹(𝜀)^𝑦𝑛−1_𝑦1, (24) then since⟨𝐹,𝐴⟩ has the sup-property or 𝐻 is finite, we have max𝓕𝐹(𝜀),𝐹(𝜀)𝑦_1𝑦1,…,𝐹(𝜀)𝑦_{𝑛−1𝑦𝑛−1}𝑦,𝛾=maxsup_{𝑦∈𝑓𝑥,𝑦1𝑛−1}min𝐹(𝜀)(𝑥),𝐹(𝜀)^𝑦𝑛−1_𝑦1,𝛾=maxsup_{𝑦∈𝑓𝑥,𝑦1𝑛−1}𝐹(𝜀)(𝑥),𝛾=sup_{𝑦∈𝑓𝑥,𝑦1𝑛−1}𝐹(𝜀𝑥)𝐹(𝜀)max)(,𝛾<min^𝑦𝑛−1_𝑦1(𝜀)𝑦𝐹(𝜀)𝐹(𝜀),𝐹,𝛿=min∩min^𝑦𝑛−1_𝑦1𝐻𝑦,𝛿, (25) which contradicts 𝐹(𝜀)∩min{𝐹(𝜀)^𝑦𝑛−1_𝑦1}_𝐻⊆∨𝑞_(𝛾,𝛿)𝓕(𝐹(𝜀), 𝐹(𝜀)(𝑦₁)_𝑦1,…,𝐹(𝜀)(𝑦_𝑛−1)_𝑦𝑛−1) by Lemma 2. Hence (F3a) holds.

Lemma 18. Let ⟨𝐹,𝐴⟩∈ℱ𝒮(𝐻,𝐸). Then (F4a) holds if and only if the following condition holds:
(F4c) for all 𝑥,𝑦∈𝐻, 𝜀∈𝐴, and 𝑟∈(𝛾,1], 𝑟_𝑥∈_𝛾𝓕(𝑟_𝑦,𝐹(𝜀),…,𝐹(𝜀)_𝑛−1) implies 𝑟_𝑦∈_𝛾∨𝑞_𝛿𝓕(𝑟_𝑥,𝐹(𝜀),…,𝐹(𝜀)_𝑛−1).

Proof. (F4a)⇒(F4c) Let 𝑥,𝑦∈𝐻 and 𝑟∈(𝛾,1]. If 𝑟_𝑦∈_𝛾𝓕(𝑟_𝑥,𝐹(𝜀),…,𝐹(𝜀)_𝑛−1) but 𝑟_𝑥∈_𝛾∨𝑞_𝛿 𝓕(𝑟_𝑦,𝐹(𝜀),…,𝐹(𝜀)_𝑛−1). Then 𝓕𝑥,𝐹(𝜀),…,𝐹_𝑛−1(𝜀)𝑦≥𝓕𝑟_𝑥,𝐹(𝜀),…,𝐹_𝑛−1(𝜀)≥𝑟>𝛾, (26) but 𝓕(𝑟_𝑦,𝐹(𝜀),…,𝐹(𝜀)_𝑛−1)(𝑥)<𝑟 and 𝓕(𝑟_𝑦,𝐹(𝜀),…,𝐹(𝜀)_𝑛−1)(𝑥)+𝑟≤2𝛿. Then min𝓕𝑦,𝐹(𝜀),…,𝐹_𝑛−1(𝜀)(𝑥)𝑟,𝑟=𝓕_𝑦,𝐹(𝜀),…,𝐹_𝑛−1(𝜀)(𝑥)<𝑟 (27) gives 𝐹(𝑦,𝜇,…,𝜇_𝑛−1)(𝑥)<𝑟 and so 2𝓕(𝑦,𝐹(𝜀),…,𝐹(𝜀)_𝑛−1)(𝑥) <min{𝓕(𝑦,𝐹(𝜀),…,𝐹(𝜀)_𝑛−1)(𝑥),𝑟}+𝑟=𝓕(𝑟_𝑦,𝐹(𝜀),…,𝐹(𝜀)_𝑛−1)(𝑥)+𝑟≤2𝛿; that is, 𝓕(𝑦, 𝐹(𝜀),…,𝐹(𝜀)_𝑛−1)(𝑥)<𝛿. Thus we have max{𝓕(𝑦,𝐹(𝜀),…,𝐹(𝜀)_𝑛−1)(𝑥),𝛾}<min{𝓕(𝑥,𝐹(𝜀),…,𝐹(𝜀)_𝑛−1)(𝑦),𝛿}, a contradiction. Hence (F4c) holds.
(F4c)⇒(F4a) Let 𝑥,𝑦∈𝐻. If there exists 𝑟 such that max{𝓕(𝑦,𝐹(𝜀),…,𝐹(𝜀)_𝑛−1)(𝑥),𝛾}<𝑟<min{𝓕(𝑥,𝐹(𝜀),…,𝐹(𝜀)_𝑛−1)(𝑦),𝛿}, then 𝓕𝑟_𝑦,𝐹(𝜀),…,𝐹_𝑛−1(𝜀)(𝑥)=min𝓕𝑦,𝐹(𝜀),…,𝐹_𝑛−1(𝜀)𝑥)𝓕(𝜀)(𝜀)(,𝑟<𝑟≤min𝑥,𝐹,…,𝐹_𝑛−1𝑦𝑟,𝑟=𝓕_𝑥,𝐹(𝜀),…,𝐹_𝑛−1(𝜀)𝑦 (28) and 𝛾<𝑟<𝛿. This implies that 𝑟_𝑦∈_𝛾𝓕(𝑟_𝑥,𝐹(𝜀),…,𝐹(𝜀)_𝑛−1) but 𝑟_𝑥∈_𝛾∨𝑞_𝛿𝓕(𝑟_𝑦,𝐹(𝜀),…,𝐹(𝜀)_𝑛−1), a contradiction. Hence max{𝓕(𝑦,𝐹(𝜀),…,𝐹(𝜀)_𝑛−1)(𝑥),𝛾}≥min{𝓕(𝑥, 𝐹(𝜀),…,𝐹(𝜀)_𝑛−1)(𝑦),𝛿} and so max{min{𝓕(𝑦, 𝐹(𝜀),…,𝐹(𝜀)_𝑛−1)(𝑥),𝛿},𝛾}≥max{min{𝓕(𝑥, 𝐹(𝜀),…,𝐹(𝜀)_𝑛−1)(𝑦),𝛿},𝛾}. In a similar way, we have max{min{𝓕(𝑥,𝐹(𝜀),…,𝐹(𝜀)_𝑛−1)(𝑦),𝛿},𝛾}≥max{min{𝓕(𝑦,𝐹(𝜀),…,𝐹(𝜀)_𝑛−1)(𝑥),𝛿},𝛾}. Thus (F4a) holds.

For any fuzzy soft set ⟨𝐹,𝐴⟩ over 𝐻, 𝜀∈𝐴, and𝑟∈(𝛾,1], denote𝐹(𝜀)_𝑟={𝑥∈𝐻∣𝑥_𝑟∈_𝛾𝐹(𝜀)}, ⟨𝐹(𝜀)⟩_𝑟={𝑥∈𝐻∣𝑥_𝑟𝑞_𝛿𝐹(𝜀)}, and [𝐹(𝜀)]_𝑟={𝑥∈𝐻∣𝑥_𝑟∈_𝛾∨𝑞_𝛿𝐹(𝜀)}. The next theorem presents the relationships between (∈_𝛾,∈_𝛾∨𝑞_𝛿)-fuzzy soft (invertible) 𝑛-ary subhypergroups over 𝐻 and crisp (invertible) 𝑛-ary subhypergroups of 𝐻.

Theorem 19. Let ⟨𝐹,𝐴⟩∈ℱ𝒮(𝐻,𝐸). Then,(1)⟨𝐹,𝐴⟩ is an (∈_𝛾,∈_𝛾∨𝑞_𝛿)-fuzzy soft (invertible) 𝑛-ary subhypergroup over 𝐻 if and only if nonempty subset 𝐹(𝜀)_𝑟 is an (invertible) 𝑛-ary subhypergroup of 𝐻 for all 𝜀∈𝐴 and 𝑟∈(𝛾,𝛿];(2)if 2𝛿=1+𝛾, then ⟨𝐹,𝐴⟩ is an (∈_𝛾,∈_𝛾∨𝑞_𝛿)-fuzzy soft (invertible) 𝑛-ary subhypergroup over 𝐻 if and only if nonempty subset ⟨𝐹(𝜀)⟩_𝑟 is an (invertible) 𝑛-ary subhypergroup of 𝐻 for all 𝜀∈𝐴 and 𝑟∈(𝛿,1];(3)⟨𝐹,𝐴⟩ is an (∈_𝛾,∈_𝛾∨𝑞_𝛿)-fuzzy soft (invertible) 𝑛-ary subhypergroup over 𝐻 if and only if nonempty subset [𝐹(𝜀)]_𝑟 is an (invertible) 𝑛-ary subhypergroup of 𝐻 for all 𝜀∈𝐴 and 𝑟∈(𝛾,min{2𝛿−𝛾,1}].

Proof. We only prove (2) and (3). (1) can be easily proved.
(2) Assume that 2𝛿=1+𝛾. Let ⟨𝐹,𝐴⟩ be an (∈_𝛾,∈_𝛾∨𝑞_𝛿)-fuzzy soft 𝑛-ary subhypergroup over 𝐻 and assume that ⟨𝐹(𝜀)⟩_𝑟≠∅ for some 𝜀∈𝐴 and 𝑟∈(𝛿,1]. Let 𝑥^𝑛₁∈⟨𝐹(𝜀)⟩_𝑟. Then 𝑟_𝑥1𝑞_𝛿𝐹(𝜀),…,𝑟_𝑥𝑛𝑞_𝛿𝐹(𝜀); that is, 𝐹(𝜀)(𝑥₁)+𝑟>2𝛿,…,𝐹(𝜀)(𝑥_𝑛)+𝑟>2𝛿. Since ⟨𝐹,𝐴⟩ is an (∈_𝛾,∈_𝛾∨𝑞_𝛿)-fuzzy 𝑛-ary subhypergroup over 𝐻, we have max{𝐹(𝜀)(𝑥),𝛾}≥min{𝐹(𝜀)^𝑥𝑛_𝑥1,𝛿} for all 𝑥∈𝑓(𝑥^𝑛₁). Hence, by 𝑟>𝛿, 𝐹(𝜀𝑥)𝐹(𝜀𝑥)𝐹(𝜀)max)(+𝑟,𝛾+𝑟=max)(,𝛾+𝑟≥min^𝑥𝑛_𝑥1𝐹(𝜀)𝑥,𝛿+𝑟=min₁+𝑟,…,𝐹(𝜀)𝑥_𝑛+𝑟,𝛿+𝑟>2𝛿. (29) From 𝑟≤1=2𝛿−𝛾, that is, 𝑟+𝛾≤2𝛿, we have 𝐹(𝜀)(𝑥)+𝑟>2𝛿 and so 𝑥∈⟨𝐹(𝜀)⟩_𝑟. Similarly we can show that 𝑒∈⟨𝐹(𝜀)⟩_𝑟 and, for all 𝑦,𝑦₁^𝑛−1∈⟨𝐹(𝜀)⟩_𝑟, there exists 𝑥∈⟨𝐹(𝜀)⟩_𝑟 such that 𝑦∈𝑓(𝑥,𝑦₁^𝑛−1). Therefore, ⟨𝐹(𝜀)⟩_𝑟 is an 𝑛-ary hypergroup of 𝐻.
Conversely, assume that the given condition holds. Let 𝑥^𝑛₁∈𝐻 and 𝜀∈𝐴. If there exists 𝑥∈𝑓(𝑥^𝑛₁) such that max{𝐹(𝜀)(𝑥),𝛾}<min{𝐹(𝜀)^𝑥𝑛_𝑥1,𝛿}, take 𝑟=2𝛿−max{𝐹(𝜀)(𝑥),𝛾}. Then 𝑟∈(𝛿,1], 𝐹(𝜀)(𝑥)≤2𝛿−𝑟, 𝐹(𝜀)(𝑥₁)>max{𝐺(𝜀)(𝑥),𝛾}=2𝛿−𝑟,…,𝐹(𝜀)(𝑥_𝑛)>max{𝐺(𝜀)(𝑥),𝛾}=2𝛿−𝑟, that is, 𝑥^𝑛₁∈⟨𝐹(𝜀)⟩_𝑟, but𝑥∉⟨𝐹(𝜀)⟩_𝑟, a contradiction. Hence max{𝐹(𝜀)(𝑥),𝛾}≥min{𝐹(𝜀)^𝑥𝑛_𝑥1,𝛿} and so max{inf_{𝑥∈𝑓(𝑥}^𝑛₁₎𝐹(𝜀)(𝑥),𝛾}≥min{𝐹(𝜀)^𝑥𝑛_𝑥1,𝛿}. Similarly we can show that conditions (F1a) and (F3a) hold. Therefore, ⟨𝐹,𝐴⟩ is an (∈_𝛾,∈_𝛾∨𝑞_𝛿)-fuzzy soft 𝑛-ary subhypergroup over 𝐻.
(3) Let ⟨𝐹,𝐴⟩ be an (∈_𝛾,∈_𝛾∨𝑞_𝛿)-fuzzy soft 𝑛-ary subhypergroup over 𝐻 and assume that [𝐹(𝜀)]_𝑟≠∅ for some 𝜀∈𝐴 and 𝑟∈(𝛾,min{2𝛿−𝛾,1}]. Let 𝑥^𝑛₁∈[𝐹(𝜀)]_𝑟. Then 𝑟_𝑥1∈_𝛾∨𝑞_𝛿𝐹(𝜀),…,𝑟_𝑥𝑛∈_𝛾∨𝑞_𝛿𝐹(𝜀); that is, 𝐹(𝜀)(𝑥₁)≥𝑟>𝛾 or 𝐹(𝜀)(𝑥₁)>2𝛿−𝑟≥2𝛿−(2𝛿−𝛾)=𝛾,…, 𝐹(𝜀)(𝑥_𝑛)≥𝑟>𝛾 or 𝐹(𝜀)(𝑥_𝑛)>2𝛿−𝑟≥2𝛿−(2𝛿−𝛾)=𝛾. Since ⟨𝐹,𝐴⟩ is an (∈_𝛾,∈_𝛾∨𝑞_𝛿)-fuzzy 𝑛-ary subhypergroup over 𝐻, we have max{𝐹(𝜀)(𝑥),𝛾}≥min{𝐹(𝜀)^𝑥𝑛_𝑥1,𝛿} for all 𝑥∈𝑓(𝑥^𝑛₁) and so 𝐹(𝜀)(𝑥)≥min{𝐹(𝜀)^𝑥𝑛_𝑥1,𝛿} since 𝛾<min{𝐹(𝜀)^𝑥𝑛_𝑥1,𝛿} in any case. Now we consider the following cases.
Case  1 (𝑟∈(𝛾,𝛿]). In this case, 2𝛿−𝑟≥𝛿≥𝑟. (1)If 𝐹(𝜀)(𝑥₁)≥𝑟 or ⋯ or 𝐹(𝜀)(𝑥_𝑛)≥𝑟, then 𝐹(𝜀)(𝑥)≥min{𝐹(𝜀)^𝑥𝑛_𝑥1,𝛿}≥𝑟. Hence 𝑧_𝑟∈_𝛾𝐹(𝜀).(2)If 𝐹(𝜀)(𝑥_𝑖)+𝑟>2𝛿, for all 1≤𝑖≤𝑛, then 𝐹(𝜀)(𝑥)≥min{𝐹(𝜀)^𝑥𝑛_𝑥1,𝛿}=𝛿≥𝑟. Hence 𝑧_𝑟∈_𝛾𝐹(𝜀).
Case  2 (𝑟∈(𝛿,min{2𝛿−𝛾,1}]). In this case, 𝑟>𝛿>2𝛿−𝑟. (1)If 𝐹(𝜀)(𝑥_𝑖)≥𝑟, for all 1≤𝑖≤𝑛, then 𝐹(𝜀)(𝑥)≥min{𝐹(𝜀)^𝑥𝑛_𝑥1,𝛿}=𝛿>2𝛿−𝑟. Hence 𝑧_𝑟𝑞_𝛿𝐹(𝜀).(2)If 𝐹(𝜀)(𝑥₁)+𝑟>2𝛿 or ⋯ or 𝐹(𝜀)(𝑥_𝑛)+𝑟>2𝛿, then 𝐹(𝜀)(𝑥)≥min{𝐹(𝜀)^𝑥𝑛_𝑥1,𝛿}>2𝛿−𝑟. Hence 𝑧_𝑟𝑞_𝛿𝐹(𝜀).
Thus, in any case, 𝑥_𝑟∈_𝛾∨𝑞_𝛿𝐹(𝜀); that is, 𝑥∈[𝐹(𝜀)]_𝑟. Similarly, we can show that 𝑒∈[𝐹(𝜀)]_𝑟 and, for all 𝑦,𝑦₁^𝑛−1∈[𝐹(𝜀)]_𝑟, there exists 𝑥∈[𝐹(𝜀)]_𝑟 such that 𝑦∈𝑓(𝑥,𝑦₁^𝑛−1). Therefore, [𝐹(𝜀)]_𝑟 is an 𝑛-ary hypergroup of 𝐻.
Conversely, assume that the given condition holds. Let 𝑥^𝑛₁∈𝐻 and 𝜀∈𝐴. If there exists 𝑥∈𝑓(𝑥^𝑛₁) such that max{𝐹(𝜀)(𝑥),𝛾}<𝑟=min{𝐹(𝜀)^𝑥𝑛_𝑥1,𝛿}, then 𝐹(𝜀)(𝑥₁)≥𝑟>𝛾,…,𝐹(𝜀)(𝑥_𝑛)≥𝑟>𝛾,𝐹(𝜀)(𝑥)<𝑟, and 𝐹(𝜀)(𝑥)+𝑟<2𝑟≤2𝛿, that is, (𝑥₁)_𝑟∈_𝛾𝐹(𝜀),…,(𝑥_𝑛)_𝑟∈_𝛾𝐹(𝜀), but 𝑥_𝑟∈_𝛾∨𝑞_𝛿𝐹(𝜀), that is, 𝑥^𝑛₁∈[𝐹(𝜀)]_𝑟, but 𝑥∉[𝐹(𝜀)]_𝑟, a contradiction. Hence max{𝐹(𝜀)(𝑥),𝛾}≥min{𝐹(𝜀)^𝑥𝑛_𝑥1,𝛿} and so max{inf_{𝑥∈𝑓(𝑥}^𝑛₁₎𝐹(𝜀)(𝑥),𝛾}≥min{𝐹(𝜀)^𝑥𝑛_𝑥1,𝛿}. Similarly, we can show that conditions (F1a) and (F3a) hold. Therefore, ⟨𝐹,𝐴⟩ is an (∈_𝛾,∈_𝛾∨𝑞_𝛿)-fuzzy soft 𝑛-ary subhypergroup over 𝐻.

As a direct consequence of Theorem 19, we have the following results.

Corollary 20. Let 𝛾,𝛾^′,𝛿,𝛿^′∈[0,1] be such that 𝛾<𝛿, 𝛾^′<𝛿^′, 𝛾<𝛾^′, and𝛿^′<𝛿. Then every (∈_𝛾,∈_𝛾∨𝑞_𝛿)-fuzzy (invertible) 𝑛-ary soft subhypergroup over 𝐻 is an (∈_𝛾′,∈_𝛾′∨𝑞_𝛿′)-fuzzy soft (invertible)