Some Generalized Forms of Fuzzy Interval Valued Hyperideals in a

Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2014, Article ID 173267, 9 pages http://dx.doi.org/10.1155/2014/173267

Research Article Some Generalized Forms of Fuzzy Interval Valued Hyperideals in a Hyperring Hongjie Li,1 Zeyuan Li,2 and Yunqiang Yin3,4 1

Mathematics Department, Zhoukou Normal University, Zhoukou 466001, China College of Mechanical and Electrical Engineering, Harbin Institute of Technology, Harbin 150001, China 3 College of Sciences, East China Institute of Technology, Nanchang 330013, China 4 Faculty of Science, Kunming University of Science and Technology, Kunming 650093, China 2

Correspondence should be addressed to Hongjie Li; [email protected] Received 15 October 2013; Revised 20 May 2014; Accepted 22 May 2014; Published 12 June 2014 Academic Editor: Hector Pomares Copyright © 2014 Hongjie Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Some generalized forms of the hyperideals of a hyperring in the paper of Zhan et al. (2008) will be given. As a generalization of the interval valued (𝛼, 𝛽)-fuzzy hyperideals of a hyperring with 𝛼, 𝛽 ∈ {∈, 𝑞, ∈ ∧𝑞, ∈ ∨𝑞} and 𝛼 ≠ ∈ ∧𝑞, the notion of generalized interval valued (𝛼, 𝛽)-fuzzy hyperideals of a hyperring is also introduced. Special attention is concentrated on the interval valued (∈𝛾̃, ∈𝛾̃ ∨𝑞𝛿̃)-fuzzy hyperideals. As a consequence, some characterizations theorems of interval valued (∈𝛾̃, ∈𝛾̃ ∨𝑞𝛿̃)-fuzzy hyperideals will be provided.

1. Introduction The concept of hyperstructure was first introduced by Marty [1] in 1934. People later observed that hyperstructures have many applications in pure and applied sciences. A comprehensive review of hyperstructures can be found in [2, 3]. In a recent monograph of Corsini and Leoreanu [4], the authors have collected numerous applications of algebraic hyperstructures, especially those from the last decade to the following subjects: geometry, hypergraphs, binary relations, lattices, fuzzy sets and rough sets, automata, cryptography, codes, median algebras, relation algebras, artificial intelligence, and probabilities. After introducing the concept of fuzzy sets of Zadeh in 1965 (see [5]), there are many papers devoted to fuzzify the classical mathematics into fuzzy mathematics. The relationships between the fuzzy sets and algebraic hyperstructures (structures) have been considered by Corsini, Davvaz, Kehagias, Leoreanu, Yin, Zahedi, Zhan, and others. The reader is referred to the papers [6–24]. By using the notion “belongingness (∈)” and “quasicoincidence (𝑞)” of a fuzzy point with a fuzzy set introduced by Pu and Liu [25], the concept of (𝛼, 𝛽)-fuzzy subgroups, where 𝛼, 𝛽 are any two

in {∈, 𝑞, ∈ ∨ 𝑞, ∈ ∧ 𝑞} with 𝛼 ≠ ∈ ∧ 𝑞, was introduced by Bhakat and Das [26] in 1992, in which they first generalized Rosenfeld’s fuzzy subgroup [27]. The detailed study of (∈ , ∈ ∨ 𝑞)-fuzzy subgroups has been considered by Bhakat and Das in [28] and Bhakat in [29, 30]. The concept of the (∈, ∈ ∨ 𝑞)-fuzzy subhyperquasigroups of hyperquasigroups was introduced by Davvaz and Corsini [12]. By using the idea of “quasicoincidence” of a fuzzy interval value with an interval valued fuzzy set, Zhan et al. [21] introduced and investigated the notion of interval valued (𝛼, 𝛽)-fuzzy hyperideals of a hyperring. As a generalization of the concepts of “belongingness (∈)” and “quasicoincidence (𝑞)” of a fuzzy point with a fuzzy set introduced by Pu and Liu [25], Yin and Zhan [31] introduced the concept of “𝛾-belongingness (∈𝛾 )” and “𝛿quasicoincidence (𝑞𝛿 )” of a fuzzy point with a fuzzy set. Using these new concepts, Yin and Zhan [31] 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 , 𝑞𝛿 , ∈𝛾 ∧ 𝑞𝛿 , ∈𝛾̃ ∨ 𝑞𝛿̃}, 𝛼, 𝛽 ∈ {∈𝛾 , 𝑞𝛿 , ∈𝛾 ∧ 𝑞𝛿 , ∈𝛾 ∨ 𝑞𝛿 }, 𝛼 ≠ ∈𝛾 ∧ 𝑞𝛿 , and 𝛽 ≠ ∈𝛾 ∧ 𝑞𝛿 , and some related properties

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were investigated. This idea is continued and studied by Ma et al. [32], Ma and Zhan [33], Yin and Zhan [34], and so on. Following this idea, in this paper, we will give some of the generalized forms of the interval fuzzy hyperideals given by Zhan et al. in [21]. The notion of generalized interval valued (𝛼, 𝛽)-fuzzy hyperideals is introduced which is the generalization of the notion of interval valued (𝛼, 𝛽)-fuzzy hyperideals. Special attention will be paid to the interval valued (∈𝛾̃, ∈𝛾̃ ∨ 𝑞𝛿̃)-fuzzy hyperideals. It is noteworthy that the notion of an interval valued (𝛼, 𝛽)-fuzzy hyperideal is a special generalized interval valued (𝛼, 𝛽)-fuzzy hyperideal and hence many results in [21] will become easy corollaries of our results given in this paper.

2. Hyperrings We first recall that a hyperstructure is a nonempty set 𝐻 together with a mapping “∘”: 𝐻 × 𝐻 → P∗ (𝐻), where P∗ (𝐻) is the set of all nonempty subsets of 𝐻, written as (𝑥, 𝑦) 󳨃→ 𝑥 ∘ 𝑦. We note that if 𝑥 ∈ 𝐻 and 𝐴, 𝐵 are nonempty subsets of 𝐻, then by 𝐴 ∘ 𝐵 𝐴 ∘ 𝑥 and 𝑥 ∘ 𝐵, we mean that 𝐴 ∘ 𝐵 = ⋃𝑎∈𝐴,𝑏∈𝐵 𝑎 ∘ 𝑏, 𝐴 ∘ 𝑥 = 𝐴 ∘ {𝑥} and 𝑥 ∘ 𝐵 = {𝑥} ∘ 𝐵, respectively. Now, we call a hyperstructure (𝐻, ∘) a canonical hypergroup [35] if the following axioms are satisfied: (1) for every 𝑥, 𝑦, 𝑧 ∈ 𝐻, (𝑥 ∘ 𝑦) ∘ 𝑧 = 𝑥 ∘ (𝑦 ∘ 𝑧); (2) for every 𝑥, 𝑦 ∈ 𝐻, 𝑥 ∘ 𝑦 = 𝑦 ∘ 𝑥; (3) there is a unique 0 ∈ 𝐻 such that 0 ∘ 𝑥 = 𝑥, for all 𝑥 ∈ 𝐻; (4) for every 𝑥 ∈ 𝐻, there exists a unique element 𝑥󸀠 ∈ 𝐻 such that 0 ∈ 𝑥∘𝑥󸀠 (we call the element 𝑥󸀠 the opposite of 𝑥). Definition 1 (see [36]). A hyperring is an algebraic structure (𝑅, +, ⋅) which satisfies the following axioms. (1) (𝑅, +) is a canonical hypergroup (we will write −𝑥 for 𝑥󸀠 ). (2) (𝑅, ⋅) is a semigroup having zero as a bilaterally absorbing element. (3) The multiplication is distributive with respect to the hyperoperation “+.” Let (𝑅, +, ⋅) be a hyperring and let 𝐴 be a nonempty subset of 𝑅. Then 𝐴 is called a subhyperring of 𝑅 if (𝐴, +, ⋅) itself is a hyperring. In what follows, let 𝑅 denote a hyperring unless otherwise stated. 𝐴 subhyperring 𝐴 of 𝑅 is a left (right) hyperideal of 𝑅 if 𝑟𝑎 ∈ 𝐴 (𝑎𝑟 ∈ 𝐴), for all 𝑟 ∈ 𝑅 and 𝑎 ∈ 𝐴. 𝐴 subhyperring 𝐴 is called a hyperideal if 𝐴 is both left and right hyperideal.

3. Interval Valued Fuzzy Sets By an interval number 𝑎̃ [37], we mean an interval [𝑎− , 𝑎+ ], where 0 ≤ 𝑎− ≤ 𝑎+ ≤ 1. The set of all interval numbers is denoted by 𝐷[0, 1]. The interval [𝑎, 𝑎] can be simply identified

by the number 𝑎. For the interval numbers 𝑎̃𝑖 = [𝑎𝑖− , 𝑎𝑖+ ], ̃𝑏𝑖 = [𝑏𝑖− , 𝑏𝑖+ ] ∈ 𝐷[0, 1], 𝑖 ∈ 𝐼, where 𝐼 is an index set, we define rmax {̃ 𝑎𝑖 , ̃𝑏𝑖 } = [max (𝑎𝑖− , 𝑏𝑖− ) , max (𝑎𝑖+ , 𝑏𝑖+ )] , rmin {̃ 𝑎𝑖 , ̃𝑏𝑖 } = [min (𝑎𝑖− , 𝑏𝑖− ) , min (𝑎𝑖+ , 𝑏𝑖+ )] , ⋁𝑎̃𝑖 = [⋁𝑎𝑖− , ⋁𝑎𝑖+ ] , 𝑖∈𝐼

𝑖∈𝐼

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𝑖∈𝐼

⋀𝑎̃𝑖 = [⋀𝑎𝑖− , ⋀𝑎𝑖+ ] . 𝑖∈𝐼

𝑖∈𝐼

𝑖∈𝐼

For any intervals 𝑎̃ = [𝑎− , 𝑎+ ] and ̃𝑏 = [𝑏− , 𝑏+ ], where 0 ≤ 𝑎− ≤ 𝑎+ and 0 ≤ 𝑏− ≤ 𝑏+ , we define 𝑎̃ + ̃𝑏 = [𝑎− + 𝑏− , 𝑎+ + 𝑏+ ] , 𝑎̃ − ̃𝑏 = [𝑎− − 𝑏− , 𝑎+ − 𝑏+ ] , 𝑘̃ 𝑎 = [𝑘𝑎− , 𝑘𝑎+ ] ,

whenever 𝑎− ≤ 𝑏− , 𝑎+ ≤ 𝑏+ , whenever 𝑘 ≥ 0, (2)

and we put (i) 𝑎̃ ≤ ̃𝑏 ⇔ 𝑎− ≤ 𝑏− and 𝑎+ ≤ 𝑏+ , (ii) 𝑎̃ = ̃𝑏 ⇔ 𝑎− = 𝑏− and 𝑎+ = 𝑏+ , (iii) 𝑎̃ < ̃𝑏 ⇔ 𝑎̃ ≤ ̃𝑏 and 𝑎̃ ≠ ̃𝑏. Denote by 𝐷󸀠 [0, 1] the set all of interval numbers such that any two elements of 𝐷󸀠 [0, 1] are comparable. Then, it is clear that both (𝐷[0, 1], ≤, ∧ , ∨) and (𝐷[0, 1]󸀠 , ≤, ∧ , ∨) form a complete lattice with 0 = [0, 0] as their least element and 1 = [1, 1] as their greatest element. Recall that an interval valued fuzzy subset 𝐹 on 𝑋 is the set 𝐹 = {(𝑥, [𝜇𝐹− (𝑥) , 𝜇𝐹+ (𝑥)]) | 𝑥 ∈ 𝑋} ,

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where 𝜇𝐹− and 𝜇𝐹+ are two fuzzy subsets of 𝑋 such that 𝜇𝐹− (𝑥) ≤ 𝜇𝐹+ (𝑥) for all 𝑥 ∈ 𝑋. Putting 𝜇̃𝐹 (𝑥) = [𝜇𝐹− (𝑥), 𝜇𝐹+ (𝑥)], we see that 𝐹 = {(𝑥, 𝜇̃𝐹 (𝑥)) | 𝑥 ∈ 𝑋}, where 𝜇̃𝐹 : 𝑋 → 𝐷[0, 1]. Denote by IF(𝑋) the set of all interval valued fuzzy subsets of 𝑋 such that, for any 𝐹 ∈ IF(𝑋), Im𝐹 = {[𝜇𝐹− (𝑥), 𝜇𝐹+ (𝑥)] | 𝑥 ∈ 𝑋} ⊆ 𝐷󸀠 [0, 1]. If 𝐹 and 𝐺 are two interval valued fuzzy subsets of 𝑋, then we define 𝐹 ⊆ 𝐺 if and only if, for all 𝑥 ∈ 𝑋, 𝜇𝐹− (𝑥) ≤ 𝜇𝐺− (𝑥) and + 𝜇𝐹 (𝑥) ≤ 𝜇𝐺+ (𝑥), 𝐹 = 𝐺 if and only if, for all 𝑥 ∈ 𝑋, 𝜇𝐹− (𝑥) = 𝜇𝐺− (𝑥) and + 𝜇𝐹 (𝑥) = 𝜇𝐺+ (𝑥). Also, the union and intersection are defined as follows: 𝐹 ∪ 𝐺 = {(𝑥, [max {𝜇𝐹− (𝑥) , 𝜇𝐺− (𝑥)} , max {𝜇𝐹+ (𝑥) , 𝜇𝐺+ (𝑥)}]) | 𝑥 ∈ 𝑋} , 𝐹 ∩ 𝐺 = {(𝑥, [min {𝜇𝐹− (𝑥) , 𝜇𝐺− (𝑥)} , min {𝜇𝐹+ (𝑥) , 𝜇𝐺+ (𝑥)}]) | 𝑥 ∈ 𝑋} .

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An interval valued fuzzy subset 𝐹 of 𝑋 of the form 𝜇̃𝐹 (𝑦) = {

𝑟̃ ( ≠ [0, 0]) [0, 0]

if 𝑦 = 𝑥, otherwise,

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is said to be a fuzzy interval value with support 𝑥 and interval value 𝑟̃ and is denoted by 𝑈(𝑥; 𝑟̃). ̃ For a fuzzy interval Let 𝛾̃, 𝛿̃ ∈ 𝐷[0, 1] be such that 𝛾̃ < 𝛿. value 𝑈(𝑥; ̃𝑡) and an interval valued fuzzy set 𝐹, we say that (1) 𝑈(𝑥; 𝑟̃)∈𝛾̃ 𝐹 if 𝜇̃𝐹 (𝑥) ≥ 𝑟̃ > 𝛾̃; ̃ (2) 𝑈(𝑥; 𝑟̃)𝑞𝛿̃ 𝐹 if 𝜇̃𝐹 (𝑥) + 𝑟̃ > 2𝛿; (3) 𝑈(𝑥; 𝑟̃)∈𝛾̃ ∨ 𝑞𝛿̃ 𝐹 if 𝑈(𝑥; 𝑟̃)∈𝛾̃𝐹 or 𝑈(𝑥; 𝑟̃) 𝑞𝛿̃ 𝐹; (4) 𝑈(𝑥; 𝑟̃)∈𝛾̃ ∧ 𝑞𝛿̃𝐹 if 𝑈(𝑥; 𝑟̃)∈𝛾̃𝐹 and 𝑈(𝑥; 𝑟̃) 𝑞𝛿̃ 𝐹; (5) 𝑈(𝑥; 𝑟̃)𝛼 𝐹 if 𝑈(𝑥; 𝑟̃) 𝛼 𝐹 does not hold for 𝛼 ∈ {∈𝛾̃, 𝑞𝛿̃, ∈𝛾̃ ∨ 𝑞𝛿̃, ∈𝛾̃ ∧ 𝑞𝛿̃}. Next we define a new ordering relation “⊆ ∨ 𝑞(̃𝛾,𝛿) ̃ ” on IF(𝑋), which is called the interval valued fuzzy inclusion or quasicoincidence relation, as follows. ̃ For any 𝐹, 𝐺 ∈ IF(𝑋) and 𝛾̃, 𝛿̃ ∈ [0, 1] such that 𝛾̃ < 𝛿, ̃ ̃ 𝐺 we mean that 𝑈(𝑥; 𝑟 )∈ 𝐹 ⇒ 𝑈(𝑥; 𝑟 )∈ by 𝐹 ⊆ ∨ 𝑞(̃𝛾,𝛿) ̃ 𝛾̃ 𝛾̃ ∨ 𝑞𝛿̃ 𝐺 for all 𝑥 ∈ 𝑋 and 𝛾̃ < 𝑟̃ ≤ [1, 1]. And we define a relation “≈” on IF(𝑋) as follows. For any 𝐹, 𝐺 ∈ IF(𝑋), by 𝐹 ≈ 𝐺 we mean that 𝐹 ⊆ ∨ 𝑞(̃𝛾,𝛿) ̃ 𝐺 and 𝐺 ⊆ ∨ 𝑞(̃ ̃ 𝐹. 𝛾,𝛿) In the sequel, unless otherwise stated, we let 𝑅 be a hyperring, 𝛾̃, 𝛿̃ ∈ 𝐷[0, 1], such that 𝛾̃ < 𝛿̃ and denote by ⊆ ∨ 𝑞(̃𝛾,𝛿) ̃ that ⊆ ∨ 𝑞(̃ ̃ is not true. We also emphasize 𝛾,𝛿) that any interval valued fuzzy subset 𝐹 of 𝑋 must satisfy the following conditions: 𝜇̃𝐹 (𝑥) ≤ 𝛾̃ or 𝛾̃ < 𝜇̃𝐹 (𝑥) and 𝜇̃𝐹 (𝑥) ≤ 𝛿̃ or 𝛿̃ < 𝜇̃𝐹 (𝑥) for all 𝑥 ∈ 𝑋. Lemma 2. Let 𝑋 be a nonempty set and 𝐹, 𝐺 ∈ IF(𝑋). 𝜇𝐺(𝑥), 𝛾̃} ≥ Then 𝐹 ⊆ ∨ 𝑞(̃𝛾,𝛿) ̃ 𝐺 if and only if rmax {̃ ̃ rmin {̃ 𝜇 (𝑥), 𝛿} for all 𝑥 ∈ 𝑋. 𝐹

Proof. Assume that 𝐹

⊆

∨ 𝑞(̃𝛾,𝛿) ∈ 𝑋. If ̃ 𝐺. Let 𝑥 ̃ rmax{̃ 𝜇𝐺(𝑥), 𝛾̃} < rmin{̃ 𝜇𝐹 (𝑥), 𝛿}, then there exists 𝑟̃ such ̃ that is, that rmax{̃ 𝜇𝐺(𝑥), 𝛾̃} < 𝑟̃ < rmin{̃ 𝜇𝐹 (𝑥), 𝛿}; 𝑈(𝑥; 𝑟̃)∈𝛾̃𝐹 but 𝑈(𝑥; 𝑟̃) ∈𝛾̃ ∨ 𝑞𝛿̃ 𝐺, a contradiction. Hence ̃ rmax{̃ 𝜇𝐺(𝑥), 𝛾̃} ≥ rmin{̃ 𝜇𝐹 (𝑥), 𝛿}. ̃ Conversely, assume that rmax{̃ 𝜇𝐺(𝑥), 𝛾̃} ≥ rmin{̃ 𝜇𝐹 (𝑥), 𝛿} for all 𝑥 ∈ 𝑋. If 𝐹 ⊆ ∨ 𝑞(̃𝛾,𝛿) ̃ 𝐺, then there exist 𝑥 ∈ 𝑋 and 𝑟̃ > 𝛾̃ such that 𝑈(𝑥; 𝑟̃)∈𝛾̃𝐹 but 𝑈(𝑥; 𝑟̃) ∈𝛾̃ ∨ 𝑞𝛿̃ 𝐺, and so ̃ This gives that ̃ ̃, and 𝜇 ̃ ̃ < 2𝛿. 𝜇̃𝐹 (𝑥) ≥ 𝑟̃, 𝜇 𝐺 (𝑥) < 𝑟 𝐺 (𝑥) + 𝑟 ̃ a contradiction. Hence 𝐹 ⊆ rmax{̃ 𝜇𝐺(𝑥), 𝛾̃} < rmin{̃ 𝜇𝐹 (𝑥), 𝛿}, 𝐺. ∨ 𝑞(̃𝛾,𝛿) ̃

Lemma 3. Let 𝑋 be a nonempty set and 𝐹, 𝐺, 𝐻 ∈ IF(𝑋) such that 𝐹 ⊆ ∨ 𝑞(̃𝛾,𝛿) ̃ 𝐺 ⊆ ∨ 𝑞(̃ ̃ 𝐻. Then 𝐹 ⊆ ∨ 𝑞(̃ ̃ 𝐻. 𝛾,𝛿) 𝛾,𝛿) Proof. It is straightforward by Lemma 2.

Note that Lemma 2 gives that 𝐹 ≈ 𝐺 if and only if ̃ 𝛾̃} = rmax{rmin{̃ ̃ 𝛾̃} for all rmax{rmin{̃ 𝜇𝐹 (𝑥), 𝛿}, 𝜇𝐺(𝑥), 𝛿}, 𝑥 ∈ 𝑋 and 𝐹, 𝐺 ∈ IF(𝑋), and it follows from Lemmas 2 and 3 that “≈” is an equivalence relation on IF(𝑋). We now define some operations of interval valued fuzzy subsets of 𝑅. Definition 4. Let 𝐹, 𝐺 ∈ IF(𝑅). Define the sum and product of 𝐹 and 𝐺, denoted by 𝐹 ⊕ 𝐺 and 𝐹 ⊗ 𝐺, by ̃ 𝜇𝐹 (𝑦) , 𝜇 𝜇̃ 𝐹⊕𝐺 (𝑥) = ⋁ rmin {̃ 𝐺 (𝑧)} , 𝑥∈𝑦+𝑧

̃ 𝜇𝐹 (𝑦) , 𝜇 ⋁ rmin {̃ { 𝐺 (𝑧)} { { 𝑥=𝑦𝑧 { { { if there exist 𝑦, 𝑧 ∈ 𝑅 such that 𝑥 = 𝑦𝑧 𝜇̃ 𝐹⊗𝐺 (𝑥) = { { {[0, 0] { { { { otherwise,

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for all 𝑥 ∈ 𝑅. By Lemma 2 and Definition 4, we can easily deduce the following results. Lemma 5. Let 𝐹1 , 𝐹2 , 𝐺1 , 𝐺2 ∈ IF(𝑅) be such that 𝐹1 ⊆ ∨ 𝑞(̃𝛾,𝛿) ̃ 𝐹2 and 𝐺1 ⊆ ∨ 𝑞(̃ ̃ 𝐺2 . Then 𝛾,𝛿) (1) 𝐹1 ⊕ 𝐺1 ⊆ ∨ 𝑞(̃𝛾,𝛿) ̃ 𝐹2 ⊕ 𝐺2 , 𝐹1 ⊗ 𝐺1 ⊆ ∨ 𝑞(̃ ̃ 𝐹2 ⊗ 𝐺2 ; 𝛾,𝛿) (2) 𝐹1 ∩ 𝐺1 ⊆ ∨ 𝑞(̃𝛾,𝛿) ̃ 𝐹2 ∩ 𝐺2 . Lemma 6. Let 𝐹, 𝐺, 𝐻 ∈ IF(𝑅). Consider (1) 𝐹 ⊕ 𝐺 = 𝐺 ⊕ 𝐹; (2) 𝐹 ⊕ (𝐺 ⊕ 𝐻) = (𝐹 ⊕ 𝐺) ⊕ 𝐻, 𝐹 ⊗ (𝐺 ⊗ 𝐻) = (𝐹 ⊗ 𝐺) ⊗ 𝐻; (3) 𝐹⊕(𝐺∪𝐻) = 𝐹⊕𝐺∪𝐹⊕𝐻, 𝐹⊗(𝐺∪𝐻) = 𝐹⊗𝐺∪𝐹⊗𝐻; (4) 𝐹⊕(𝐺∩𝐻) ⊆ 𝐹⊕𝐺∩𝐹⊕𝐻, 𝐹⊗(𝐺∩𝐻) ⊆ 𝐹⊗𝐺∩𝐹⊗𝐻. Lemma 5 indicates that the equivalence relation “≈” is a congruence relation on (IF(𝑅), ⊕).

4. Generalized Interval Valued (𝛼,𝛽)-Fuzzy Hyperideals In this section, we define and investigate generalized interval valued (𝛼, 𝛽)-fuzzy hyperideals of a hyperring, where 𝛼 ∈ {∈𝛾̃, 𝑞𝛿̃, ∈𝛾̃ ∨ 𝑞𝛿̃} and 𝛽 ∈ {∈𝛾̃, 𝑞𝛿̃, ∈𝛾̃ ∧ 𝑞𝛿̃, ∈𝛾̃ ∨ 𝑞𝛿̃}. Definition 7. An interval valued fuzzy subset 𝐹 of 𝑅 is called an interval valued (𝛼, 𝛽)-fuzzy hyperideal of 𝑅 if, for all 𝛾̃ < 𝑟̃ ≤ [1, 1], 𝛾̃ < ̃𝑡 ≤ [1, 1], and 𝑥, 𝑦 ∈ 𝑅: (F1a) 𝑈(𝑥; 𝑟̃) 𝛼 𝐹 and 𝑈(𝑦; ̃𝑡) 𝛼 𝐹 ⇒ 𝑈(𝑧; rmin {̃𝑟, ̃𝑡}) 𝛽 𝐹 for all 𝑧 ∈ 𝑥 + 𝑦; (F2a) 𝑈(𝑥; 𝑟̃) 𝛼 𝐹 ⇒ 𝑈(−𝑥; 𝑟̃) 𝛽 𝐹 for all 𝑥 ∈ 𝑅; (F3a) 𝑈(𝑥; 𝑟̃) 𝛼 𝐹 and 𝑈(𝑦; ̃𝑡) 𝛼 𝐹 ⇒ 𝑈(𝑥𝑦; rmin {̃𝑟, ̃𝑡}) 𝛽 𝐹 for all 𝑥, 𝑦 ∈ 𝑅.

4

Journal of Applied Mathematics

An interval valued (𝛼, 𝛽)-fuzzy hyperideal with 𝛾̃ = [0, 0] and 𝛿̃ = [0.5, 0.5] is called an interval valued (𝛼, 𝛽)-fuzzy hyperideal studied in [21]. The case 𝛼 = ∈𝛾 ∧ 𝑞𝛿 must be omitted since for an interval valued fuzzy subset 𝐹 of 𝑅 such that 𝜇̃𝐹 (𝑥) ≤ 𝛿̃ for any 𝑥 ∈ 𝑅 in the case 𝑈(𝑥; 𝑟̃) ∈ ∧ 𝑞 𝐹 we have 𝜇̃𝐹 (𝑥) ≥ 𝑟̃ ̃ Thus 𝜇̃ (𝑥) + 𝜇̃ (𝑥) > 𝜇̃ (𝑥) + 𝑟̃ > 2𝛿, ̃ and 𝜇̃𝐹 (𝑥) + 𝑟̃ > 2𝛿. 𝐹 𝐹 𝐹 ̃ which implies 𝜇̃𝐹 (𝑥) > 𝛿. This means that {𝑈(𝑥; 𝑟̃) : 𝑈(𝑥; 𝑟̃) ∈ ∧ 𝑞 𝐹} = 0. As it is not difficult to see, each interval valued (𝛼, 𝛽)fuzzy hyperideal of 𝑅 is an interval valued (𝛼, ∈𝛾̃ ∨ 𝑞𝛿̃)fuzzy hyperideal of 𝑅. So, in the theory of interval valued (𝛼, 𝛽)-fuzzy hyperideals the central role is played by interval valued (𝛼, ∈𝛾̃ ∨ 𝑞𝛿̃)-fuzzy hyperideals and we only need to investigate the properties of interval valued (𝛼, ∈𝛾̃ ∨ 𝑞𝛿̃)fuzzy hyperideals. In what follows, let 𝛼 ∈ {∈𝛾̃, 𝑞𝛿̃, ∈𝛾̃ ∨ 𝑞𝛿̃} unless otherwise specified. Example 8. Let 𝑅 = {𝑎, 𝑏, 𝑐, 𝑑} be a set with a hyperoperation “+” and a binary operation “⋅” as follows: + 𝑎 𝑏 𝑐 𝑑

𝑎 𝑏 𝑐 𝑑 {𝑎} {𝑏} {𝑐} {𝑑} {𝑏} {𝑎, 𝑏} {𝑑} {𝑐} {𝑐} {𝑑} {𝑎, 𝑐} {𝑏} {𝑑} {𝑐} {𝑏} {𝑎, 𝑑} , ⋅ 𝑎 𝑏 𝑐 𝑑

𝑎 {𝑎} {𝑎} {𝑎} {𝑎}

𝑏 {𝑎} {𝑏} {𝑐} {𝑑}

𝑐 {𝑎} {𝑏} {𝑐} {𝑑}

𝑑 {𝑎} {𝑏} {𝑐} {𝑑} .

(7)

(8)

Then (𝑅, +, ⋅) is a hyperring ([21]). Define an interval valued fuzzy subset 𝐹 of 𝑅 by 𝜇̃𝐹 (𝑎) = [0.6, 0.7] ,

𝜇̃𝐹 (𝑏) = [0.2, 0.4] ,

𝜇̃𝐹 (𝑐) = [0.2, 0.4] ,

𝜇̃𝐹 (𝑑) = [0.2, 0.4] .

(9)

Then 𝐹 is an interval valued (𝛼, ∈[0.2,0.4] ∨ 𝑞[0.6,0.7] )-fuzzy hyperideal of 𝑅. Proposition 9. An interval valued (∈𝛾̃ ∨ 𝑞𝛿̃, ∈𝛾̃ ∨ 𝑞𝛿̃)-fuzzy hyperideal 𝐹 of 𝑅 is an interval valued (∈𝛾̃, ∈𝛾̃ ∨ 𝑞𝛿̃)-fuzzy hyperideal of 𝑅. Proof. This is straightforward since 𝑈(𝑥; 𝑟̃)∈𝛾̃𝐹 implies 𝑈(𝑥; 𝑟̃)∈𝛾̃ ∨ 𝑞𝛿̃ 𝐹 for all 𝑥 ∈ 𝑅 and 𝛾̃ < 𝑟̃ ≤ [1, 1]. Proposition 10. Any interval valued (∈𝛾̃, ∈𝛾̃)-fuzzy hyperideal 𝐹 of 𝑅 is an interval valued (𝑞𝛿̃, ∈𝛾̃ ∨ 𝑞𝛿̃)-fuzzy hyperideal of 𝑅. Proof. This part is straightforward. The following example shows that the converse of Propositions 9 and 10 may not be true in general.

Example 11. Consider Example 8. Define an interval valued fuzzy subset 𝐹 of 𝑅 by 𝜇̃𝐹 (𝑎) = [0.6, 0.7] ,

𝜇̃𝐹 (𝑏) = [0.7, 0.8] ,

𝜇̃𝐹 (𝑐) = [0.3, 0.5] ,

𝜇̃𝐹 (𝑑) = [0.2, 0.4] .

(10)

Then (1) it is routine to verify that 𝐹 is an (∈[0.3,0.5] ; ∈[0.3,0.5] ∨ 𝑞[0.6,0.7] )-fuzzy hyperideal of 𝑅; (2) 𝐹 is not an (∈[0.3,0.5] ; ∈[0.3,0.5] )-fuzzy hyperideal of 𝑅 since 𝑈(𝑏; [0.7, 0.8])∈[0.3,0.5] 𝐹 but 𝑎 ∈ 𝑏 + 𝑏 and 𝑈(𝑎; [0.7, 0.8]) ∈[0.7,0.8] 𝐹; (3) 𝐹 is not an (∈[0.3,0.5] ∨ 𝑞[0.6,0.7] ; ∈[0.3,0.5] ∨ 𝑞[0.6,0.7] )fuzzy hyperideal of 𝑅 since 𝑈(𝑏; [1, 1]) ∈[0.3,0.5] ∨ 𝑞[0.6,0.7] 𝐹 and 𝑈(𝑐; [1, 1])∈[0.3,0.5] ∨ 𝑞[0.6,0.7] 𝐹 but 𝑈(𝑏 + 𝑐; [1, 1]) = 𝑈(𝑑; [1, 1]) ∈[0.3,0.5] ∨ 𝑞[0.6,0.7] 𝐹. Theorem 12. Let 2𝛿̃ = [1, 1] + 𝛾̃ and 𝐹 be an interval valued (𝛼, ∈𝛾̃ ∨ 𝑞𝛿̃)-fuzzy hyperideal of 𝑅. Then the set 𝐹𝛾̂̃ is a hyperideal of 𝑅, where 𝐹𝛾̂̃ = {𝑥 ∈ 𝑅|̃ 𝜇𝐹 (𝑥) > 𝛾̃}. Proof. Assume that 𝐹 is an interval valued (𝛼, ∈𝛾̃ ∨ 𝑞𝛿̃)-fuzzy hyperideal of 𝑅. Let 𝑥, 𝑦 ∈ 𝐹𝛾̂̃ . Then 𝜇̃𝐹 (𝑥) > 𝛾̃, 𝜇̃𝐹 (𝑦) > 𝛾̃. We consider the following two cases. Case 1. If 𝛼 ∈ {∈𝛾̃, ∈𝛾̃ ∨ 𝑞𝛿̃}, then 𝑈(𝑥; 𝜇̃𝐹 (𝑥)) 𝛼 𝐹 and 𝜇𝐹 (𝑥), 𝜇̃𝐹 (𝑦)})∈𝛾̃ ∨ 𝑈(𝑦; 𝜇̃𝐹 (𝑦)) 𝛼 𝐹. By (F1a), 𝑈(𝑧; rmin{̃ 𝜇𝐹 (𝑥), 𝜇̃𝐹 (𝑦)})∈𝛾̃𝐹 𝑞𝛿̃ 𝐹 for all 𝑧 ∈ 𝑥 + 𝑦, that is, 𝑈(𝑧; rmin{̃ or 𝑈(𝑧; rmin{̃ 𝜇𝐹 (𝑥), 𝜇̃𝐹 (𝑦)}) 𝑞𝛿̃ 𝐹. It follows that 𝐹(𝑧) ≥ ̃ rmin{̃ 𝜇𝐹 (𝑥), 𝜇̃𝐹 (𝑦)} > 𝛾̃ or 𝜇̃𝐹 (𝑧) + rmin{̃ 𝜇𝐹 (𝑥), 𝜇̃𝐹 (𝑦)} > 2𝛿, and so 𝜇̃𝐹 (𝑧) ≥ rmin{̃ 𝜇𝐹 (𝑥), 𝜇̃𝐹 (𝑦)} > 𝛾̃ or 𝜇̃𝐹 (𝑧) > 2𝛿̃ − rmin{̃ 𝜇𝐹 (𝑥), 𝜇̃𝐹 (𝑦)} ≥ 2𝛿̃ − [1, 1] = 𝛾̃. Hence 𝜇̃𝐹 (𝑧) > 𝛾̃ and so 𝑧 ∈ 𝐹𝛾̂̃ . Therefore, 𝑥 + 𝑦 ⊆ 𝐹𝛾̃. By (F2a) and (F3a), 𝑈(−𝑥; 𝜇̃𝐹 (𝑥))∈𝛾̃ ∨ 𝑞𝛿̃ 𝐹 and 𝑈(𝑥𝑦; rmin{̃ 𝜇𝐹 (𝑥), 𝜇̃𝐹 (𝑦)})∈𝛾̃ ∨ 𝑞𝛿̃ 𝐹. Analogous to the above proof, we have 𝜇̃𝐹 (𝑥) > 𝛾̃ and 𝜇̃𝐹 (𝑥𝑦) > 𝛾̃; that is, −𝑥 ∈ 𝐹𝛾̃ and 𝑥𝑦 ∈ 𝐹𝛾̃. Therefore, 𝐹𝛾̂̃ is a hyperideal of 𝑅. Case 2. If 𝛼 = 𝑞𝛿̃, then 𝑈(𝑥; [1, 1]) 𝛼 𝐹 and 𝑈(𝑦; [1, 1]) 𝛼 𝐹 since 2𝛿̃ = [1, 1]+ 𝛾̃. Analogous to the proof of Case 1, we may prove that 𝐹𝛾̂̃ is a hyperideal of 𝑅. If we take 𝛾̃ = [0, 0] and 𝛿̃ = [0.5, 0.5] in Theorem 12, then we have the following corollary. Corollary 13. Let 𝐹 be an interval valued (𝛼, ∈ ∨ 𝑞)-fuzzy hyperideal of 𝑅. Then the set 𝐹[0,0] ̂ is a hyperideal of 𝑅, where 𝛼 ∈ {∈, 𝑞, ∈ ∨ 𝑞}. Theorem 14. Let 2𝛿̃ = [1, 1]+ 𝛾̃ and 𝐴 be a nonempty subset of 𝑅. Then 𝐴 is a hyperideal of 𝑅 if and only if the interval valued fuzzy subset 𝐹 of 𝑅 such that 𝜇̃𝐹 (𝑥) ≥ 𝛿̃ for all 𝑥 ∈ 𝐴 and 𝜇̃𝐹 (𝑥) ≤ 𝛾̃ otherwise is an interval valued (𝛼, ∈𝛾̃ ∨ 𝑞𝛿̃)-fuzzy hyperideal of 𝑅.

Journal of Applied Mathematics

5

Proof. Assume that 𝐴 is a hyperideal of 𝑅. Let 𝑥, 𝑦 ∈ 𝑅 and 𝛾̃ < 𝑟̃ ≤ [1, 1], 𝛾̃ < ̃𝑡 ≤ [1, 1] be such that 𝑈(𝑥; 𝑟̃) 𝛼 𝐹 and 𝑈(𝑦; ̃𝑡) 𝛼 𝐹. Then we have the following four cases. Case 1. If 𝑈(𝑥; 𝑟̃)∈𝛾̃𝐹 and 𝑈(𝑦; ̃𝑡)∈𝛾̃𝐹, then 𝜇̃𝐹 (𝑥) ≥ 𝑟̃ > 𝛾̃ ̃ that is, and 𝜇̃𝐹 (𝑦) ≥ ̃𝑡 > 𝛾̃. Thus, 𝜇̃𝐹 (𝑥) ≥ 𝛿̃ and 𝜇̃𝐹 (𝑦) ≥ 𝛿; 𝑥, 𝑦 ∈ 𝐴. Case 2. If 𝑈(𝑥; 𝑟̃) 𝑞𝛿̃ 𝐹 and 𝑈(𝑦; ̃𝑡) 𝑞𝛿̃ 𝐹, then 𝜇̃𝐹 (𝑥) + 𝑟̃ > 2𝛿̃ ̃ and so 𝜇̃ (𝑥) > 2𝛿̃ − 𝑟̃ ≥ 2𝛿̃ − [1, 1] = 𝛾̃ and 𝜇̃𝐹 (𝑦) + ̃𝑡 > 2𝛿, 𝐹 ̃ ̃ and 𝜇̃𝐹 (𝑦) > 2𝛿 − 𝑡 ≥ 2𝛿̃ − [1, 1] = 𝛾̃. It follows that 𝜇̃𝐹 (𝑥) ≥ 𝛿̃ ̃ that is, 𝑥, 𝑦 ∈ 𝐴. and 𝜇̃𝐹 (𝑦) ≥ 𝛿; Case 3. If 𝑈(𝑥; 𝑟̃)∈𝛾̃𝐹 and 𝑈(𝑦; ̃𝑡) 𝑞𝛿̃ 𝐹, then 𝜇̃𝐹 (𝑥) ≥ 𝑟̃ > 𝛾̃ ̃ Analogous to the proof of Cases 1 and 2, and 𝜇̃𝐹 (𝑦) + ̃𝑡 > 2𝛿. ̃ that is, 𝑥, 𝑦 ∈ 𝐴. 𝜇̃𝐹 (𝑥) ≥ 𝛿̃ and 𝜇̃𝐹 (𝑦) ≥ 𝛿; Case 4. If 𝑈(𝑥; 𝑟̃) 𝑞𝛿̃ 𝐹 and 𝑈(𝑦; ̃𝑡)∈𝛾̃𝐹, then 𝜇̃𝐹 (𝑥) + 𝑟̃ > 2𝛿̃ and 𝜇̃𝐹 (𝑦) ≥ ̃𝑡 > 𝛾̃. Analogous to the proof of Cases 1 and 2, ̃ that is, 𝑥, 𝑦 ∈ 𝐴. 𝜇̃𝐹 (𝑥) ≥ 𝛿̃ and 𝜇̃𝐹 (𝑦) ≥ 𝛿; Thus, in any case, 𝑥, 𝑦 ∈ 𝐴. Hence 𝑧 ∈ 𝐴 for all 𝑧 ∈ 𝑥 + 𝑦, ̃ If rmin{̃ ̃ which implies that 𝜇̃𝐹 (𝑧) ≥ 𝛿. 𝜇𝐹 (𝑥), 𝜇̃𝐹 (𝑦)} ≤ 𝛿, then 𝜇̃𝐹 (𝑧) ≥ 𝛿̃ ≥ rmin{̃ 𝜇𝐹 (𝑥), 𝜇̃𝐹 (𝑦)} > 𝛾̃; it follows 𝜇𝐹 (𝑥), 𝜇̃𝐹 (𝑦)} > that 𝑈(𝑧; rmin{̃ 𝜇𝐹 (𝑥), 𝜇̃𝐹 (𝑦)})∈𝛾̃𝐹. If rmin{̃ ̃ 𝛿, then 𝜇̃𝐹 (𝑧) + rmin{̃ 𝜇𝐹 (𝑥), 𝜇̃𝐹 (𝑦)} > 𝛿̃ + 𝛿̃ = ̃ 2𝛿; it follows that 𝑈(𝑧; rmin{̃ 𝜇𝐹 (𝑥), 𝜇̃𝐹 (𝑦)}) 𝑞𝛿̃ 𝑓. Therefore, 𝑈(𝑧; rmin{̃ 𝜇𝐹 (𝑥), 𝜇̃𝐹 (𝑦)})∈𝛾̃ ∨ 𝑞𝛿̃ 𝑓. In a similar way, we may prove that 𝑈(−𝑥; 𝜇̃𝐹 (𝑥))∈𝛾̃ ∨ 𝑞𝛿̃ 𝐹 and 𝑈(𝑥𝑦; rmin{̃ 𝜇𝐹 (𝑥), 𝜇̃𝐹 (𝑦)})∈𝛾̃ ∨ 𝑞𝛿̃ 𝐹. Therefore, 𝐹 is an interval valued (𝛼, ∈𝛾̃ ∨ 𝑞𝛿̃)-fuzzy hyperideal of 𝑅. Conversely, assume that 𝐹 is an interval valued (𝛼, ∈𝛾̃ ∨ 𝑞𝛿̃)-fuzzy hyperideal of 𝑅. It is easy to see that 𝐴 = 𝐹𝛾̂̃ . It follows from Theorem 12 that 𝐴 is a hyperideal of 𝑅. If we take 𝛾̃ = [0, 0] and 𝛿̃ = [0.5, 0.5] in Theorem 14, then we have the following corollary. Corollary 15. Let 𝐴 be a nonempty subset of 𝑅. Then 𝐴 is a hyperideal of 𝑅 if and only if the interval valued fuzzy subset 𝐹 of 𝑅 such that 𝜇̃𝐹 (𝑥) ≥ [0.5, 0.5] for all 𝑥 ∈ 𝐴 and 𝜇̃𝐹 (𝑥) = [0, 0] otherwise is an (𝛼, ∈ ∨ 𝑞)-fuzzy hyperideal of 𝑅, where 𝛼 ∈ {∈, 𝑞, ∈ ∨ 𝑞}. Theorem 16. Let 𝐹 be an (𝑞𝛿̃, ∈𝛾̃ ∨ 𝑞𝛿̃)-fuzzy hyperideal of 𝑅. Then the following conditions hold: ̃ for (F1b) rmax {⋀𝑧∈𝑥+𝑦 𝜇̃𝐹 (𝑧), 𝛾̃} ≥ rmin {̃ 𝜇𝐹 (𝑥), 𝜇̃𝐹 (𝑦), 𝛿} all 𝑥, 𝑦 ∈ 𝑅; ̃ for all 𝑥 ∈ 𝑅; 𝜇 (𝑥), 𝛿} (F2b) rmax {̃ 𝜇 (−𝑥), 𝛾̃} ≥ rmin {̃ 𝐹

𝐹

̃ for all 𝑥 ∈ (F3b) rmax {̃ 𝜇𝐹 (𝑥𝑦), 𝛾̃} ≥ rmin {̃ 𝜇𝐹 (𝑥), 𝜇̃𝐹 (𝑦), 𝛿} 𝑅. Proof. Let 𝐹 be an (𝑞𝛿̃, ∈𝛾̃ ∨ 𝑞𝛿̃)-fuzzy hyperideal of 𝑅. We only show (F1b). The other properties can be similarly proved. Assume that there exist 𝑥, 𝑦, 𝑧 ∈ 𝑅 such that 𝑧 ∈ 𝑥 + 𝑦 ̃ Then, for all 𝜇𝐹 (𝑥), 𝜇̃𝐹 (𝑦), 𝛿}. and rmax{̃ 𝜇𝐹 (𝑧), 𝛾̃} < rmin{̃

𝑟̃ ∈ 𝐷[0, 1] such that 2𝛿̃ − rmax{̃ 𝜇𝐹 (𝑧), 𝛾̃} > 𝑟̃ > 2𝛿̃ − ̃ rmin{̃ 𝜇𝐹 (𝑥), 𝜇̃𝐹 (𝑦), 𝛿}, we have 𝜇𝐹 (𝑧) , 𝛾̃} > 𝑟̃ 2𝛿̃ − 𝜇̃𝐹 (𝑧) ≥ 2𝛿̃ − rmax {̃ ̃ , > rmax {2𝛿̃ − 𝜇̃𝐹 (𝑥) , 2𝛿̃ − 𝜇̃𝐹 (𝑦) , 𝛿}

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̃ 𝜇̃ (𝑦) + 𝑟̃ > 2𝛿, ̃ 𝜇̃ (𝑧) + 𝑟̃ < and so 𝜇̃𝐹 (𝑥) + 𝑟̃ > 2𝛿, 𝐹 𝐹 ̃ ̃ 2𝛿, and 𝜇̃𝐹 (𝑧) < 𝛿 < 𝑟̃. Hence 𝑈(𝑥; 𝑟̃)𝑞𝛿̃ 𝐹, 𝑈(𝑦; 𝑟̃)𝑞𝛿̃ 𝐹 but 𝑈(𝑧; 𝑟̃) ∈𝛾̃ ∨ 𝑞𝛿̃ 𝐹, a contradiction. Hence (F1b) is valid.

5. Interval Valued (∈𝛾̃,∈𝛾̃ ∨ 𝑞𝛿̃)-Fuzzy Hyperideals In this section, using a new ordering relation on IF(𝑅), we investigate the interval valued (∈𝛾̃, ∈𝛾̃ ∨ 𝑞𝛿̃)-fuzzy hyperideals of a hyperring. In what follows, we take 𝛼 = ∈𝛾̃ and 𝛽 = ∈𝛾̃ ∨ 𝑞𝛿̃ in Definition 7. Before proceeding, we present some characterizations of interval valued (∈𝛾̃, ∈𝛾̃ ∨ 𝑞𝛿̃)-fuzzy hyperideals. Lemma 17. Let 𝐹 ∈ IF(𝐻). Then (F1a) holds if and only if one of the following conditions holds: ̃ for 𝜇𝐹 (𝑥), 𝜇̃𝐹 (𝑦), 𝛿} (F1b) rmax {⋀𝑧∈𝑥+𝑦 𝜇̃𝐹 (𝑧), 𝛾̃} ≥ rmin {̃ all 𝑥, 𝑦 ∈ 𝑅; (F1c) 𝐹 ⊕ 𝐹 ⊆ ∨ 𝑞(̃𝛾,𝛿) ̃ 𝐹. Proof. (F1a)⇒(F1b) Let 𝑥, 𝑦 ∈ 𝑅. Let, if possible, 𝑧 ∈ 𝑅 be such that 𝑧 ∈ 𝑥 + 𝑦 and rmax{̃ 𝜇𝐹 (𝑧), 𝛾̃} < ̃ Then 𝑈(𝑥; 𝑟̃)∈ ̃𝐹, 𝑈(𝑦; 𝑟̃)∈ ̃𝐹, 𝑟̃ = rmin{̃ 𝜇𝐹 (𝑥), 𝜇̃𝐹 (𝑦), 𝛿}. 𝛾 𝛾 ̃ that is, 𝑈(𝑧; 𝑟̃) ∈𝛾̃ 𝐹, and 𝜇̃𝐹 (𝑧) + 𝑟̃ < 𝑟̃ + 𝑟̃ ≤ 2𝛿; 𝑈(𝑧; 𝑟̃) ∈𝛾̃ ∨ 𝑞𝛿̃ 𝐹, a contradiction. Hence (F1b) is valid. (F1b)⇒(F1c) For 𝑈(𝑥; 𝑟̃) ∈𝛾̃ 𝐹 ⊕ 𝐹, suppose, if possible, 𝑈(𝑥; 𝑟̃) ∈𝛾̃ ∨ 𝑞𝛿̃ 𝐹. Then 𝑈(𝑥; 𝑟̃) ∈𝛾̃ 𝐹 and 𝑈(𝑥; 𝑟̃) 𝑞𝛿̃ 𝐹; that is, 𝜇̃𝐹 (𝑥) < 𝑟̃ and 𝜇̃𝐹 (𝑥) + 𝑟̃ ≤ 2𝛿̃ which implies that ̃ If 𝑥 ∈ 𝑦 + 𝑧 for some 𝑦, 𝑧 ∈ 𝑅, by (F1b), we have 𝜇̃𝐹 (𝑥) < 𝛿. ̃ ̃ it follows that 𝛿 > rmax{̃ 𝜇𝐹 (𝑥), 𝛾̃} ≥ rmin{̃ 𝜇𝐹 (𝑦), 𝜇̃𝐹 (𝑧), 𝛿}, 𝜇𝐹 (𝑦), 𝜇̃𝐹 (𝑧)}. Hence we have rmax{̃ 𝜇𝐹 (𝑥), 𝛾̃} ≥ rmin{̃ 𝜇𝐹 (𝑦) , 𝜇̃𝐹 (𝑧)} 𝑟̃ ≤ (𝐹 ⊕ 𝐹) (𝑥) = ⋁ rmin {̃ 𝑥∈𝑎+𝑏

𝜇𝐹 (𝑥) , 𝛾̃} = rmax {̃ ≤ ⋁ rmax {̃ 𝜇𝐹 (𝑥) , 𝛾̃} ;

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𝑥∈𝑎+𝑏

which contradicts 𝛾̃ < 𝑟̃ and 𝜇̃𝐹 (𝑥) < 𝑟̃. Hence (F1c) is satisfied. (F1c)⇒(F1a) Let 𝑥, 𝑦 ∈ 𝑅 and 𝛾̃ < 𝑟̃ ≤ [1, 1], 𝛾̃ < ̃𝑡 ≤ [1, 1] be such that 𝑈(𝑥; 𝑟̃) ∈𝛾̃ 𝐹 and 𝑈(𝑦; ̃𝑡) ∈𝛾̃ 𝐹. Then for any 𝑧 ∈ 𝑥 + 𝑦, we have 𝜇𝐹 (𝑎) , 𝜇̃𝐹 (𝑏)} (𝐹 ⊕ 𝐹) (𝑧) = ⋁ rmin {̃ 𝑧∈𝑎+𝑏

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≥ rmin {̃ 𝜇𝐹 (𝑥) , 𝜇̃𝐹 (𝑦)} ≥ rmin {̃𝑟, ̃𝑡} > 𝛾̃. Hence 𝑈(𝑧; rmin{̃ 𝜇𝐹 (𝑥), 𝜇̃𝐹 (𝑦)}) ∈𝛾̃ 𝐹 ⊕ 𝐹 and so (F1a) holds.

6

Journal of Applied Mathematics

Lemma 18. Let 𝐹 ∈ IF(𝑅). Then (F2a) holds if and only if one of the following conditions holds:

Corollary 24. Let 𝐹 ∈ IF(𝑅). Then

̃ for all 𝑥 ∈ 𝑅; 𝜇𝐹 (𝑥), 𝛿} (F2b) rmax {̃ 𝜇𝐹 (−𝑥), 𝛾̃} ≥ rmin {̃

(1) any interval valued (∈ ∨ 𝑞, ∈ ∨ 𝑞)-fuzzy hyperideal of 𝑅 is an interval valued (∈, ∈ ∨ 𝑞)-fuzzy hyperideal of 𝑅 [21];

̃ (F2c) 𝐹 ⊆ ∨ 𝑞(̃𝛾,𝛿) ̃ − 𝐹, where −𝐹 is defined by 𝜇 −𝐹 (𝑥) = 𝜇̃𝐹 (−𝑥) for all 𝑥 ∈ 𝑅.

(2) any interval valued (𝑞, ∈ ∨ 𝑞)-fuzzy hyperideal of 𝑅 is an interval valued (∈, ∈ ∨ 𝑞)-fuzzy hyperideal of 𝑅; (3) any interval valued (∈, ∈)-fuzzy hyperideal of 𝑅 is an interval valued (∈ ∨ 𝑞, ∈ ∨ 𝑞)-fuzzy hyperideal of 𝑅 [21].

Proof. The proof is analogous to the proof of Lemma 17. Lemma 19. Let 𝐹 ∈ IF(𝑅). Then (F3a) holds if and only if one of the following conditions holds: ̃ for all 𝑥 ∈ 𝜇𝐹 (𝑥), 𝜇̃𝐹 (𝑦), 𝛿} (F3b) rmax {̃ 𝜇𝐹 (𝑥𝑦), 𝛾̃} ≥ rmin {̃ 𝑅; (F3c) 𝐹 ⊗ 𝐹 ⊆ ∨ 𝑞(̃𝛾,𝛿) ̃ 𝐹. Proof. The proof is analogous to Lemma 17.

From Lemmas 17–19, we obtain the following results. Theorem 20. Let 𝐹 ∈ IF(𝑅). Then 𝐹 is an (∈𝛾̃, ∈𝛾̃ ∨ 𝑞𝛿̃)-fuzzy hyperideal of 𝑅 if and only if it satisfies conditions (F1b)–(F3b) or conditions (F1c)–(F3c). Remark 21. For any interval valued (∈𝛾̃, ∈𝛾̃ ∨ 𝑞𝛿̃)-fuzzy hyperideal 𝐹 of 𝑅, we can conclude that (1) 𝐹 is an interval valued fuzzy hyperideal of 𝑅 when 𝛾̃ = [0, 0] and 𝛿̃ = [1, 1] ([21], Definition 4.1); (2) 𝐹 is an interval valued (∈, ∈ ∨ 𝑞)-fuzzy hyperideal of 𝑅 when 𝛾̃ = [0, 0] and 𝛿̃ = [0.5, 0.5] ([21], Definition 4.3); (3) 𝐹 is an interval valued fuzzy hyperideal of 𝑅 with ̃ ([21], Definition 4.9). thresholds (̃ 𝛾, 𝛿) Combining Theorems 16 and 20, we have the following result. Proposition 22. Any interval valued (𝑞𝛿̃, ∈𝛾̃ ∨ 𝑞𝛿̃)-fuzzy hyperideal of 𝑅 is an interval valued (∈𝛾̃, ∈𝛾̃ ∨ 𝑞𝛿̃)-fuzzy hyperideal of 𝑅. The following example shows that the converse of Proposition 22 is not true in general. Example 23. Let 𝑅 and 𝐹 be as in Example 11. Then 𝐹 is an (∈[0.3,0.5] ; ∈[0.3,0.5] ∨ 𝑞[0.6,0.7] )-fuzzy hyperideal of 𝑅, but it is not an (𝑞[0.6,0.7] ; ∈[0.3,0.5] ∨ 𝑞[0.6,0.7] )-fuzzy hyperideal of 𝑅 since 𝑈(𝑎; [0.7, 0.8])𝑞[0.6,0.7] 𝐹 and 𝑈(𝑐; [1, 1])∈[0.3,0.5] ∨ = 𝑞[0.6,0.7] 𝐹 but 𝑈(𝑎 + 𝑐; rmin {[0.7, 0.8], [1, 1]}) 𝑈(𝑐; [0.7, 0.8]) ∈[0.3,0.5] ∨ 𝑞[0.6,0.7] 𝐹. If we take 𝛾̃ = [0, 0] and 𝛿̃ = [0.5, 0.5] in Propositions 9, 10, and 22, then we have the following corollary.

̃ 𝐹𝑟̃𝛿

For any 𝐹 ∈ IF(𝑅), we define 𝐹𝑟̃ = {𝑥 ∈ 𝑅 | 𝑈(𝑥; 𝑟̃)∈𝛾̃𝐹}, ̃

= {𝑥 ∈ 𝑅 | 𝑈(𝑥; 𝑟̃) 𝑞𝛿̃ 𝐹}, and [𝐹]𝛿𝑟̃ = {𝑥 ∈ 𝑅 | 𝑈(𝑥; 𝑟̃)∈𝛾̃ ∨ ̃

̃

𝑞𝛿̃ 𝐹} for all 𝑟̃ ∈ 𝐷󸀠 [0, 1]. It is clear that [𝐹]𝛿𝑟̃ = 𝐹𝑟̃ ∪ 𝐹𝑟̃𝛿 . The next theorem provides the relationship between the interval valued (∈𝛾̃, ∈𝛾̃ ∨ 𝑞𝛿̃)-fuzzy hyperideals of 𝑅 and the crisp hyperideals of 𝑅. Theorem 25. Let 𝐹 ∈ IF(𝑅). (1) 𝐹 is an interval valued (∈𝛾̃, ∈𝛾̃ ∨ 𝑞𝛿̃)-fuzzy hyperideal of 𝑅 if and only if 𝐹𝑟̃ (𝐹𝑟̃ ≠ 0) is an hyperideal of 𝑅 for ̃ all 𝛾̃ < 𝑟̃ ≤ 𝛿. (2) If 2𝛿 = 1 + 𝛾, then 𝐹 is an interval valued (∈𝛾̃, ∈𝛾̃ ∨ ̃

̃

𝑞𝛿̃)-fuzzy hyperideal of 𝑅 if and only if 𝐹𝑟̃𝛿 (𝐹𝑟̃𝛿 ≠ 0) is a hyperideal of 𝑅 for all 𝛿̃ < 𝑟̃ ≤ [1, 1]. (3) If 2𝛿 = 1+𝛾, then 𝐹 is an interval valued (∈𝛾̃, ∈𝛾̃ ∨ 𝑞𝛿̃)̃

̃

fuzzy hyperideal of 𝑅 if and only if [𝐹]𝛿𝑟̃ ([𝐹]𝛿𝑟̃ ≠ 0) is a hyperideal of 𝑅 for all 𝛾̃ < 𝑟̃ ≤ [1, 1].

Proof. We only show (3). Let 𝐹 be an interval valued (∈𝛾̃, ∈𝛾̃ ∨ ̃

𝑞𝛿̃)-fuzzy hyperideal of 𝑅 and 𝑥, 𝑦 ∈ [𝐹]𝛿𝑟̃ for some 𝛾̃ < 𝑟̃ ≤ [1, 1]. Then 𝑈(𝑥; 𝑟̃)∈𝛾̃ ∨ 𝑞𝛿̃ 𝐹 and 𝑈(𝑦; 𝑟̃)∈𝛾̃ ∨ 𝑞𝛿̃ 𝐹; that is, 𝜇̃𝐹 (𝑥) ≥ 𝑟̃ or 𝜇̃𝐹 (𝑥) > 2𝛿̃ − 𝑟̃ > 2𝛿̃ − [1, 1] = 𝛾̃, and 𝜇̃𝐹 (𝑦) ≥ 𝑟̃ or 𝜇̃𝐹 (𝑦) > 2𝛿̃ − 𝑟̃ > 2𝛿̃ − [1, 1] = 𝛾̃. Since 𝐹 is an interval valued (∈𝛾̃, ∈𝛾̃ ∨ 𝑞𝛿̃)-fuzzy hyperideal of 𝑅, we ̃ for all 𝑧 ∈ 𝑥 + 𝑦 since have 𝜇̃𝐹 (𝑧) ≥ rmin{̃ 𝜇𝐹 (𝑥), 𝜇̃𝐹 (𝑦), 𝛿} ̃ We consider the following cases. 𝛾̃ < rmin{̃ 𝜇𝐹 (𝑥), 𝜇̃𝐹 (𝑦), 𝛿}. ̃ Since 𝛾̃ < 𝑟̃ ≤ 𝛿, ̃ we have 2𝛿̃ − 𝑟̃ ≥ Case 1. Consider 𝛾̃ < 𝑟̃ ≤ 𝛿. ̃ = 𝑟̃ or 𝜇̃ (𝑧) ≥ rmin{̃𝑟, 2𝛿− ̃ 𝛿̃ ≥ 𝑟̃ and so 𝜇̃𝐹 (𝑧) ≥ rmin{̃𝑟, 𝑟̃, 𝛿} 𝐹 ̃ ̃ ̃ ̃ ̃ 𝑟̃, 𝛿} = 𝑟̃, or 𝜇̃𝐹 (𝑧) ≥ rmin{2𝛿−̃𝑟, 2𝛿−̃𝑟, 𝛿} = 𝑟̃ = 𝛿 ≥ 𝑟̃. Hence 𝑈(𝑧; 𝑟̃) ∈𝛾̃ 𝐹. Case 2. Consider 𝛿̃ < 𝑟̃ ≤ [1, 1]. Since 𝛿̃ < 𝑟̃ ≤ [1, 1], we have ̃ = 𝛿̃ > 2𝛿̃ − 𝑟̃ or 2𝛿̃ − 𝑟̃ < 𝛿̃ < 𝑟̃ and so 𝜇̃𝐹 (𝑧) ≥ rmin{̃𝑟, 𝑟̃, 𝛿} ̃ ̃ ̃ 𝜇̃𝐹 (𝑧) > rmin{̃𝑟, 2𝛿 − 𝑟̃, 𝛿} = 2𝛿 − 𝑟̃, or 𝜇̃𝐹 (𝑧) > rmin{2𝛿̃ − ̃ = 2𝛿̃ − 𝑟̃. Hence 𝑈(𝑧; 𝑟̃) 𝑞 ̃ 𝐹. 𝑟̃, 2𝛿̃ − 𝑟̃, 𝛿} 𝛿 ̃

Thus, in any case, 𝑈(𝑧; 𝑟̃)∈𝛾̃ ∨ 𝑞𝛿̃ 𝐹; that is, 𝑧 ∈ [𝐹]𝛿𝑟̃ for ̃

all 𝑧 ∈ 𝑥 + 𝑦. Similarly we can show that −𝑥 ∈ [𝐹]𝛿𝑟̃ and ̃ [𝐹]𝛿𝑟̃ .

̃ [𝐹]𝛿𝑟̃

𝑥𝑦 ∈ Therefore, is a hyperideal of 𝑅. Conversely, assume that the given condition holds. Let 𝑥, 𝑦 ∈ 𝑅. If there exists 𝑧 ∈ 𝑅 such that 𝑧 ∈ 𝑥 + 𝑦 and

Journal of Applied Mathematics

7

̃ then 𝑥, 𝑦 ∈ [𝐹]𝛿̃ 𝜇𝐹 (𝑥), 𝜇̃𝐹 (𝑦), 𝛿}, rmax{̃ 𝜇𝐹 (𝑧), 𝛾̃} < 𝑟̃ = rmin{̃ 𝑟̃ ̃ [𝐹]𝛿𝑟̃ ,

but 𝑧 ∉ a contradiction. Therefore, rmax{̃ 𝜇𝐹 (𝑧), 𝛾̃} ≥ ̃ for all 𝑧 ∈ 𝑥 + 𝑦. Similarly we can rmin{̃ 𝜇𝐹 (𝑥), 𝜇̃𝐹 (𝑦), 𝛿} show that conditions (F2b) and (F3b) are valid. Therefore, 𝐹 is an interval valued (∈𝛾̃, ∈𝛾̃ ∨ 𝑞𝛿̃)-fuzzy hyperideal of 𝑅 by Theorem 20.

(2) Let 𝑥 ∈ 𝑅. Then ̃} rmax {𝜇̃ 𝐹⊕𝐺 (−𝑥) , 𝛾 ̃} ̃ 𝜇𝐹 (𝑦) , 𝜇 = rmax { ⋁ rmin {̃ 𝐺 (𝑧)} , 𝛾 −𝑥∈𝑦+𝑧

= ⋁ rmin {rmax {̃ 𝜇𝐹 (𝑦) , 𝛾̃} , rmax {̃ 𝜇𝐺 (𝑧) , 𝛾̃}} −𝑥∈𝑦+𝑧

As a direct consequence of Theorem 25, we have the following result.

= ⋁ rmin {rmax {̃ 𝜇𝐹 (𝑦) , 𝛾̃} , rmax {̃ 𝜇𝐺 (𝑧) , 𝛾̃}} 𝑥∈−𝑦−𝑧

̃ 𝛿̃󸀠 ∈ 𝐷[0, 1] be such that 𝛾̃ < 𝛿, ̃ 𝛾̃󸀠 < Corollary 26. Let 𝛾̃, 𝛾̃󸀠 , 𝛿, ̃ Then every interval valued (∈ ̃, ∈ ̃ ∨ 𝛿̃󸀠 , 𝛾̃ < 𝛾̃󸀠 , and 𝛿̃󸀠 < 𝛿. 𝛾 𝛾 𝑞𝛿̃)-fuzzy hyperideal of 𝑅 is an interval valued (∈𝛾̃󸀠 , ∈𝛾̃󸀠 ∨ 𝑞𝛿̃󸀠 )fuzzy hyperideal of 𝑅.

̃ ̃ , rmin {̃ ≥ ⋁ rmin {rmin {̃ 𝜇𝐹 (−𝑦) , 𝛿} 𝜇𝐺 (−𝑧) , 𝛿}} 𝑥∈−𝑦−𝑧

̃ ̃ = rmin { ⋁ rmin {̃ 𝜇𝐹 (−𝑦) , 𝜇 𝐺 (−𝑧)} , 𝛿}

The following example shows that the converse of Corollary 26 is not true in general.

𝑥∈−𝑦−𝑧

̃ = rmin {𝜇̃ 𝐹⊕𝐺 (𝑥) , 𝛿} .

Example 27. Consider Example 8. Define interval valued fuzzy subset 𝐹 of 𝑅 by 𝜇̃𝐹 (𝑎) = [0.6, 0.7] ,

𝜇̃𝐹 (𝑏) = [0.2, 0.3] ,

𝜇̃𝐹 (𝑐) = [0.3, 0.4] ,

𝜇̃𝐹 (𝑑) = [0.3, 0.4] .

(14)

(15) (3) Let 𝑥, 𝑦 ∈ 𝑅. We have ̃ rmin {𝜇̃ 𝐹⊕𝐺 (𝑥) , 𝛿}

Then 𝐹 is an interval valued (∈[0.3,0.4] , ∈[0.3,0.4] ∨ 𝑞[0.6,0.7] )fuzzy hyperideal of 𝑅 but is not an interval valued (∈[0.2,0.3] , ∈[0.2,0.3] ∨ 𝑞[0.6,0.7] )-fuzzy hyperideal of 𝑅.

̃ ̃ 𝜇𝐹 (𝑎) , 𝜇 = rmin { ⋁ rmin {̃ 𝐺 (𝑏)} , 𝛿}

If we take 𝛾̃ = [0, 0] and 𝛿̃ = [0.5, 0.5] in Theorem 25, then we have the following corollary.

̃ , rmin {̃ ̃ 𝜇𝐹 (𝑎) , 𝛿} 𝜇𝐺 (𝑏) , 𝛿}} = ⋁ rmin {rmin {̃

Corollary 28. Let 𝐹 ∈ IF(𝑅).

𝜇𝐹 (𝑎𝑦) , 𝛾̃} , rmax {̃ 𝜇𝐺 (𝑏𝑦) , 𝛾̃}} ≤ ⋁ rmin {rmax {̃

𝑥∈𝑎+𝑏

𝑥∈𝑎+𝑏

𝑥∈𝑎+𝑏

(1) 𝐹 is an interval valued (∈, ∈ ∨ 𝑞)-fuzzy hyperideal of 𝑅 if and only if 𝐹𝑟̃ (𝐹𝑟̃ ≠ 0) is a hyperideal of 𝑅 for all [0, 0] < 𝑟̃ ≤ [0.5, 0.5].

̃ ̃} = rmax { ⋁ rmin {̃ 𝜇𝐹 (𝑎𝑦) , 𝜇 𝐺 (𝑏𝑦)} , 𝛾 𝑥∈𝑎+𝑏

(2) 𝐹 is an interval valued (∈, ∈ ∨ 𝑞)-fuzzy hyperideal of 𝑅 if and only if 𝐹𝑟̃[0.5,0.5] (𝐹𝑟̃[0.5,0.5] ≠ 0) is a hyperideal of 𝑅 for all [0.5, 0.5] < 𝑟̃ ≤ [1, 1].

̃} . ≤ rmax {𝜇̃ 𝐹⊕𝐺 (𝑥𝑦) , 𝛾 (16)

(3) 𝐹 is an interval valued (∈, ∈ ∨ 𝑞)-fuzzy hyperideal of 𝑅 if and only if [𝐹][0.5,0.5] ([𝐹][0.5,0.5] ≠ 0) is a 𝑟̃ 𝑟̃ hyperideal of 𝑅 for all [0, 0] < 𝑟̃ ≤ [1, 1].

̃} ≥ rmin In a similar way, we have rmax{𝜇̃ 𝐹⊕𝐺 (𝑥𝑦), 𝛾 ̃ Hence, rmax{𝜇̃ (𝑥𝑦), 𝛾̃} ≥ rmin{𝜇̃ (𝑥), {𝜇̃ (𝑦), 𝛿}. 𝐹⊕𝐺 𝐹⊕𝐺 𝐹⊕𝐺 ̃ 𝜇̃ (𝑦), 𝛿}.

Theorem 29. Let 𝐹 and 𝐺 be two interval valued (∈𝛾̃, ∈𝛾̃ ∨ 𝑞𝛿̃)-fuzzy hyperideals of 𝑅. Then so are 𝐹 ⊕ 𝐺 and 𝐹 ∩ 𝐺.

Theorem 30. Let IFI(𝑅) be the set of all interval valued (∈𝛾̃, ∈𝛾̃ ∨ 𝑞𝛿̃)-fuzzy hyperideals of 𝑅 with the same tip “𝑡” ̃ (i.e., 𝜇̃𝐹 (0) = 𝜇 𝐺 (0) for all 𝐹, 𝐺 ∈ IFI(𝑅)). Then (IFI(𝑅), ∩, ⊕, ⊆ ) is a modular lattice. ∨ 𝑞(̃𝛾,𝛿) ̃

Proof. We only show that 𝐹⊕𝐺 is an interval valued (∈𝛾̃, ∈𝛾̃ ∨ 𝑞𝛿̃)-fuzzy hyperideal of 𝑅. The verification is as follows. (1) By Lemmas 5, 6, and 17, (𝐹 ⊕ 𝐺) ⊕ (𝐹 ⊕ 𝐺) = (𝐹 ⊕ 𝐹) ⊕ (𝐺 ⊕ 𝐺) ⊆ ∨ 𝑞(̃𝛾,𝛿) ̃ 𝐹 ⊕ 𝐺.

𝐹⊕𝐺

Proof. Let 𝐹, 𝐺 ∈ IFI(𝑅). Clearly, 𝐹 ∩ 𝐺 is the greatest lower bound of 𝐹 and 𝐺. By Theorem 29, we have 𝐹 ⊕ 𝐺 ∈ IFI(𝑅). Now, we prove that 𝐹 ⊕ 𝐺 = 𝐹 ∨ 𝐺. Let 𝑥 ∈ 𝑅. Since 𝑥 = 0 + 𝑥

8

Journal of Applied Mathematics

̃ for and rmax{̃ 𝜇𝐹 (0), 𝛾̃} = rmax{̃ 𝜇𝐺(0), 𝛾̃} ≥ rmin{̃ 𝜇𝐺(𝑦), 𝛿} all 𝑦 ∈ 𝑅, we have

̃ , ̃ , rmin {̃ 𝜇𝐹 (𝑥) , 𝛿} 𝜇𝐺 (𝑦) , 𝛿} ≥ ⋁ rmin {rmin {̃ 𝑥∈𝑦+𝑧

̃ rmin {̃ 𝜇𝐻 (𝑧) , 𝛿}}

̃} rmax {𝜇̃ 𝐹⊕𝐺 (𝑥) , 𝛾

̃ 𝜇𝐺 (𝑦) , 𝜇̃ = rmin {rmin {̃ 𝜇𝐹 (𝑥) , ⋁ rmin {̃ 𝐻 (𝑧)}} , 𝛿}

̃ ̃} 𝜇𝐹 (𝑦) , 𝜇 = rmax { ⋁ rmin {̃ 𝐺 (𝑧)} , 𝛾

𝑥∈𝑦+𝑧

𝑥∈𝑦+𝑧

̃ ̃} ≥ rmax {rmin {̃ 𝜇𝐹 (0) , 𝜇 𝐺 (𝑥)} , 𝛾

(17)

𝜇𝐺 (𝑥) , 𝛾̃}} = rmin {rmax {̃ 𝜇𝐹 (0) , 𝛾̃} , rmax {̃ ̃ , rmin {̃ ̃ 𝜇𝐺 (𝑥) , 𝛿}} ≥ rmin {rmin {̃ 𝜇𝐺 (𝑥) , 𝛿} ̃ ; = rmin {̃ 𝜇𝐺 (𝑥) , 𝛿} it follows that 𝐹 ⊆ ∨ 𝑞(̃𝛾,𝛿) ̃ 𝐹⊕ 𝐺. Similarly, 𝐺 ⊆ ∨ 𝑞(̃ ̃ 𝐹⊕𝐺 𝛾,𝛿) and so 𝐹 ⊕ 𝐺 is an upper bound of 𝐹 and 𝐺. Now, let 𝐻 ∈ IFI(𝑅) be such that 𝐹 ⊆ ∨ 𝑞(̃𝛾,𝛿) ̃ 𝐻 and 𝐺 ⊆ ∨ 𝑞(̃ ̃ 𝐻. It is 𝛾,𝛿) easy to see 𝐹 ⊕ 𝐺 ⊆ ∨ 𝑞(̃𝛾,𝛿) ̃ 𝐻. So, 𝐹 ⊕ 𝐺 = 𝐹 ∨ 𝐺. Therefore, (IFI(𝑅), ∩, ⊕, ⊆ ∨ 𝑞(̃𝛾,𝛿) ̃ ) is a lattice. For modularity, we first let 𝐹, 𝐺, 𝐻 ∈ IFI(𝑅) be such that 𝐹 ⊆ ∨ 𝑞(̃𝛾,𝛿) ̃ 𝐻. We need to show that 𝐹∩(𝐺⊕𝐻) ≈ (𝐹∩𝐺)⊕𝐻. But, the relation (𝐹 ∩ 𝐺) ⊕ 𝐻 ⊆ ∨ 𝑞(̃𝛾,𝛿) ̃ 𝐹 ∩ (𝐺 ⊕ 𝐻) is clear. It remains to show that 𝐹 ∩ (𝐺 ⊕ 𝐻) ⊆ ∨ 𝑞(̃𝛾,𝛿) ̃ (𝐹 ∩ 𝐺) ⊕ 𝐻. Now let 𝑥 ∈ 𝑅. Then we have ̃ (𝑥) , 𝛾̃} rmax {𝜇(𝐹∩𝐺)⊕𝐻 ̃} ̃ = rmax { ⋁ rmin {𝜇̃ 𝐹∩𝐺 (𝑦) , 𝜇 𝐻 (𝑧)} , 𝛾 𝑥∈𝑦+𝑧

̃} ̃ ̃ = rmax { ⋁ rmin {̃ 𝜇𝐹 (𝑦) , 𝜇 𝐺 (𝑦) , 𝜇 𝐻 (𝑧)} , 𝛾 𝑥∈𝑦+𝑧

̃ = rmin {rmin {̃ 𝜇𝐹 (𝑥) , 𝜇̃ 𝐺⊕𝐻 (𝑥)} , 𝛿} ̃ . ̃ (𝑥) , 𝛿} = rmin {𝜇𝐹∩(𝐺⊕𝐻) (18) It follows that 𝐹 ∩ (𝐺 ⊕ 𝐻) ⊆ ∨ 𝑞(̃𝛾,𝛿) ̃ (𝐹 ∩ 𝐺) ⊕ 𝐻. Thus 𝐹 ∩ (𝐺 ⊕ 𝐻) ≈ (𝐹 ∩ 𝐺) ⊕ 𝐻 and so (IFI(𝑅), ∩, ⊕, ⊆ ∨ 𝑞(̃𝛾,𝛿) ̃ ) is a modular lattice.

6. Conclusions In this paper, we introduced and studied the generalized interval valued (𝛼, 𝛽)-fuzzy hyperideals of a hyperring, with 𝛼, 𝛽 ∈ {∈, 𝑞, ∈ ∧ 𝑞, ∈ ∨ 𝑞} and 𝛼 ≠ ∈ ∧ 𝑞, in which special attention was concentrated on the interval valued (∈𝛾̃, ∈𝛾̃ ∨ 𝑞𝛿̃)-fuzzy hyperideals. As a consequence, some characterizations theorems of interval valued (∈𝛾̃, ∈𝛾̃ ∨ 𝑞𝛿̃)fuzzy hyperideals which generalize many results obtained in [21] were provided. Our future work on this topic will focus on studying some other classes of fuzzy interval valued hyperstructure.

Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

= ⋁ rmin {rmax {̃ 𝜇𝐹 (𝑦) , 𝛾̃} , rmax {̃ 𝜇𝐺 (𝑦) , 𝛾̃} , 𝑥∈𝑦+𝑧

rmax {̃ 𝜇𝐻 (𝑧) , 𝛾̃}} ̃ , ≥ ⋁ rmin {rmin {̃ 𝜇𝐹 (𝑥) , 𝜇̃𝐹 (−𝑧) , 𝛿} 𝑥∈𝑦+𝑧

𝜇𝐻 (𝑧) , 𝛾̃}} rmax {̃ 𝜇𝐺 (𝑦) , 𝛾̃} , rmax {̃ ̃ , rmin {̃ ̃ , 𝜇𝐹 (−𝑧) , 𝛿} 𝜇𝐹 (𝑥) , 𝛿} = ⋁ rmin {rmin {̃ 𝑥∈𝑦+𝑧

rmax {̃ 𝜇𝐺 (𝑦) , 𝛾̃} , rmax {̃ 𝜇𝐻 (𝑧) , 𝛾̃}} ̃ , rmax {̃ 𝜇𝐺 (𝑦) , 𝛾̃} , 𝜇𝐹 (𝑥) , 𝛿} = ⋁ rmin {rmin {̃ 𝑥∈𝑦+𝑧

rmax {̃ 𝜇𝐻 (𝑧) , 𝛾̃}} (it follows from 𝐹 ⊆ ∨𝑞(̃𝛾,𝛿) ̃ 𝐻 that ̃ ≤ rmax {̃ rmin {̃ 𝜇𝐹 (−𝑧) , 𝛿} 𝜇𝐻 (𝑧) , 𝛾̃})

The authors are thankful to the Editor and two anonymous referees for their valuable comments and suggestions that help to improve this paper. This paper was supported in part by the National Natural Science Foundation of China (71301022) and in part by the TianYuan Special Funds of the National Natural Science Foundation of China (11226264).

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