FUZZINESS AND ROUGHNESS IN HYPERQUANTALES

Theories of fuzzy set and rough set are powerful mathematical tools for modelling various types of uncertainty. In this paper, we introduce the notions of bi-hyperideal, fuzzy bi-hyperideals of hyperquantales and their related properties is given. Furthermore we introduce the notion of generalized rough fuzzy bi-hyperideals. Moreover, we will describe the set-valued homomorphism and strong set-valued homomorphism of hyperquantales and some related properties will be study.


Introduction
The theory of rough sets was introduced by Pawlak [15,16], to deal with uncertain knowledge in information systems. The rough set theory has been emerged as another major mathematical approach for managing uncertainty that arises from inexact, noisy or incomplete information. It has turned out to be fundamentally important in artificial intelligence and cognitive sciences, especially in fields such as machine learning, knowledge acquisition, decision analysis, expert systems, pattern recognition. With the development of rough set theory, possible connections between rough sets and various algebraic systems were considered by many authors. Inspired by the construction of Pawlak rough set algebras and the investigation in algebraic properties Int. J. Anal. Appl. 18 (6) (2020) 999 of rough sets in [17,18]. As a combination of algebraic structures and partially ordered structures, the theory of quantales was initiated by Mulvey [20] to study the spectrum of C-algebras and the foundations of quantum mechanics. Wang and Zhao [22,23] proposed the concepts of ideals and prime ideals of quantales. Yang and Xu [24], considered quantales as universal sets and introduced the notions of rough (prime, semi-prime, primary) ideals and prime radicals of upper rough ideals of quantales. Wang in [21] studied prime radical theorem in quantales. The concept of fuzzy sets was introduced by Zadeh [19] in 1965. The theory of fuzzy sets has been developed fast and has many applications in many branches of sciences. Luo and Wang in [25], studied roughness and fuzziness in quantales. Davvaz et al. in [6] applied Atanassov's intuitionistic fuzzy set theory to quantales. In [29,30], Saqib and Shabir studied relationship between generalized rough sets and quantale by using fuzzy ideals of quantale.
Algebraic hyperstructures represent a natural extension of classical algebraic structures and they were originally proposed in 1934 by a French mathematician Marty [1], at the 8 th Congress of Scandinavian Mathematicians. One of the main reason which attracts researches towards hyperstructures is its unique property that in hyperstructures composition of two elements is a set, while in classical algebraic structures the composition of two elements is an element. Thus algebraic hyperstructures are natural extension of classical algebraic structures. Since then, hyperstructures are widely investigated from the theoretical point of view and for their applications to many branches of pure and applied mathematics (see [2][3][4][5]8]). Since then, there appeared many components of hyperalgebras such as hypergroups in [9], hyperrings etc in [10,11].
Konstantinidou and Mittas have introduced the concept of hyperlattices in [12], fuzzy ideal of hyperlattices have been introduced in [7]. The notion of hyperlattices is a generalization of the notion of lattices and there are some intimate connections between hyperlattices and lattices. In particular, Rasouli and Davvaz further studied the theory of hyperlattices and obtained some interesting results [13,14], which enrich the theory of hyperlattices. In [28], Estaji and Bayati studied rough Sets in terms of Hyperlattices. In [26], Khan et al. introduced the notions of hyperideals and fuzzy hyperideals of hyperquantales.
In this paper, we introduce the notions of bi-hyperideal and fuzzy bi-hyperideals of hyperquantales and give several characterizations. In addition, we will introduce the notions of generalized rough fuzzy bi-hyperideal in hyperquantales and some new properties will be obtain.

Preliminaries
A map * : S × S → P * (S) is called hyperoperation or join operation on the set S, where S is a non-empty set and P * (S) = P(S)\{∅} denotes the set of all non-empty subsets of S.
A hyperstructure is called the pair (S, * ) where * is a hyperoperation on the set S. Definition 2.1. (see [26]). A hyperquantale is a complete hyperlattice Q with an associative binary operation * satisfying x * i∈I Throughout this paper, we denote the least and greatest elements of a hyperquantale denoted by ⊥ and respectively.
Definition 2.2. (see [26]). Let Q be a hyperquantale. A non-empty subset A of Q is called a left (resp. right) hyperideal of Q if it satisfies the following conditions: (2) (∀ x, y ∈ Q) x ∈ A and y ≤ x imply y ∈ A.
(3) ∀ x ∈ Q and a ∈ A, we have x * a ⊆ A (resp. a * x ⊆ A).
A non empty subset A of Q is called a two sided hyperideal or simply a hyperideal of Q if it is both a left hyperideal and right hyperideal of Q. (1) x, y ∈ B implies x ∨ y ⊆ B.
For A ⊆ Q,we denote (A] := {a ∈ Q : a ≤ b for some b ∈ A}.

Fuzzy hyperideals of hyperquantale
Let Q be a hyperquantale. A function f from a nonempty set X to the unit interval [0, 1] is called a fuzzy subset of Q.
Let Q be a hyperquantale and f be a fuzzy subset of Q. Then for every t ∈ [0, 1] the set Definition 3.1. (see [26]). Let Q be a hyperquantale and f, g are any two fuzzy subsets of Q. We define the product f • g of f and g as follows: For two functions f and g then f ⊆ g if and only if f (x) ≤ g (x) .
Let Q be a hyperquantale and ∅ = A ⊆ Q. Then the characteristic function χ A of A is defined as: (see [26]). Let Q be a hyperquantale. A fuzzy subset f of Q is called a fuzzy subhyperquantale of Q if it satisfies the following conditions: Definition 3.3. (see [26]). Let Q be a hyperquantale. A fuzzy subset f of Q is called a fuzzy left (resp. right) hyperideal of Q if it satisfies the following conditions:

Fuzzy bi-hyperideal of hyperquantale
In this section, we introduce the notion of fuzzy bi-hyperideal of hyperquantale and investigate some related properties.
satisfies the following conditions: Let us define a fuzzy subset f : Q −→ [0, 1] as follows: Then it is easy to verify that f is a fuzzy bi-hyperideal of Q. Proof. Suppose that B is a hyperideal of Q. Let x, y ∈ Q. If x, y ∈ B then x ∨ y ⊆ B and x * y ⊆ B. Since Also for any β ∈ x * y ⊆ B, we have . Let x, y and z be any elements Hence for each α ∈ x * y * z, we have χ B (α) = 1, and so α ∈ B. Thus x * y * z ⊆ B. Thus B is a bi-hyperideal of Q.
for some x, y ∈ Q, then there exists t 0 ∈ [0, 1] such that , which implies that x, y ∈ U (f ; t) and x * y U (f ; t). It contradicts the fact that U (f ; t) is a bi-hyperideal of Q. Consequently, for all x, y ∈ Q. Next we show that If there exist x, y ∈ Q and t 0 ∈ [0, 1] such that . Then x, y ∈ U (f ; t) and β ∈ x ∨ y U (f ; t). It is again contradicts the fact that U (f ; t) is a bi-hyperideal of Q. Thus . Therefore f is a fuzzy bi-hyperideal of Q.
Conversely, suppose that f be a fuzzy bi-hyperideal of Q. Let x, y ∈ U (f ; t). Then f (x) ≥ t, f (y) ≥ t.
Since f is a fuzzy bi-hyperideal of Q, so we have Proof. Let x, y ∈ Q. Then, since each f i (i ∈ I) is a fuzzy bi-hyperideal of Q, so Thus for any α ∈ x ∨ y, f i (α) ≥ f i (x) f i (y) , and we have which implies that . Let β ∈ x * y and β∈x * y . Thus for any Thus . Now let x, y, z ∈ Q. Then for any γ ∈ x * y * z, we have Furthermore, if x ≤ y, then f (x) ≥ f (y) . Indeed: Since every f i (i ∈ I) is a fuzzy bi-hyperideal of Q, it can be obtained that Thus f = i∈I f i is a fuzzy bi-hyperideal of Q.

Homomorphism and generalized rough fuzzy bi-hyperideals of hyperquantales
Definition 5.1. (see [27]). Let X and Y be two nonempty universes. Let F be a set-valued mapping given by is the power set of Y . Then the triple (X, Y, F ) is referred to as a generalized approximation space or generalized rough set. Any set-valued function from X to P (Y ) defines a binary Obviously, if ρ is an arbitrary relation from X to Y , then a set-valued mapping F ρ : For any set A ⊆ Y , the lower and upper approximations represented by F − (A) and F + (A) respectively, are defined as We call the pair (F − (A) , F + (A)) generalized rough set, and F − , F + are termed as lower and upper generalized approximation operators, respectively.
Definition 5.2. Let (Q 1 , * 1 ) and (Q 2 , * 2 ) be two hyperquantales. A set-valued mapping F : where P * (Q 2 ) represents the collection of all nonempty subsets of Q 2 is called a set-valued homomorphism if, for all a i , a, b ∈ Q 1 (i ∈ I) , A set-valued mapping F : Q 1 −→ P * (Q 2 ) is called a strong set-valued homomorphism if we replace ⊆ by = in (1) and (2).
Let f be any fuzzy subset of Q 2 . Then for every x ∈ Q 1 , we defines Here Proof. Assume that f is a fuzzy bi-hyperideal of Q 2 , then we have Hence Again since F is a strong set-valued homomorphism, so we have F (x * 1 y) = F (x) * 2 F (y) ∀x, y ∈ Q 1 .
Thus for any β ∈ x * 1 y we have, Since β ∈ F (x) * 2 F (y), there exist a ∈ F (x) and b ∈ F (y) such that β ∈ a * 2 b. Hence Hence β∈x * 1y Again since f is a fuzzy bi-hyperideal of Q 2 , so for any γ ∈ x * 1 y * 1 z, we have Since γ ∈ F (x) * 2 F (y) * 2 F (z), there exist a ∈ F (x) and b ∈ F (y) and c ∈ F (z) such that γ ∈ a * 2 b * 2 c. Hence Hence γ∈(x * 1y * 1z) Theorem 5.2. Let F be a strong set-valued homomorphism and let f be a fuzzy bi-hyperideal of Q 2 . Then Proof. Assume that f is a fuzzy bi-hyperideal of Q 2 , then we have ∀x, y ∈ Q 2 and α ∈ x ∨ y. Also F is a strong set-valued homomorphism, so F (x ∨ y) = F (x) ∨ F (y) ∀x, y ∈ Q 1 . Therefore for any α ∈ x ∨ y Since α ∈ F (x) ∨ F (y), there exist a ∈ F (x) and b ∈ F (y) such that α ∈ a ∨ b. Hence .
Again since F is a strong set-valued homomorphism, so we have F (x * 1 y) = F (x) * 2 F (y) ∀x, y ∈ Q 1 .
Thus for any β ∈ x * 1 y we have, Since β ∈ F (x) * 2 F (y), there exist a ∈ F (x) and b ∈ F (y) such that β ∈ a * 2 b. Hence Hence β∈x * 1y Again since f is a fuzzy bi-hyperideal of Q 2 , so for any γ ∈ x * 1 y * 1 z, we have Since γ ∈ F (x) * 2 F (y) * 2 F (z), there exist a ∈ F (x) and b ∈ F (y) and c ∈ F (z) such that γ ∈ a * 2 b * 2 c. Hence Hence γ∈(x * 1y * 1z) Proposition 5.1. Let F be a strong set-valued homomorphism and let {f i } i∈I be a family of fuzzy bi- Proof. Since every f i is a fuzzy bi-hyperideals for every i ∈ I, and for every x, y ∈ Q 1 , Hence β∈x * 1y Again since F is a strong set-valued homomorphism, and f is a fuzzy bi-hyperideal of Q 2 , so for any γ ∈ x * 1 y * 1 z, we have, Hence γ∈x * 1y * 1z For the following Theorem we define the set f α where α ∈ [0, 1] as following F + (f α )), where f α = ∅, is a bi-hyperideal of Q 1 .
Proof. Assume that F − (f ) is a fuzzy bi-hyperideal of Q 1 . We need to show that F − (f α ) is a bi-hyperideal Hence Hence Conversely, assume that F − (f α ) is a bi-hyperideal of Q 1 . We shall show that F − (f ) is a fuzzy bi- and F − (f ) (y) ≥ α. So x, y ∈ F − (f α ) . Hence x ∨ y ⊆ F − (f α ) .

Conclusion
In the present paper, we introduced the notion of bi-hyperideals of hyperquantales. Furthermore we introduced the notions of fuzzy bi-hyperideals and generalized rough fuzzy bi-hyperideals of hyperquantales and their related properties is provided. Finally we discussed the strong set-valued homomorphism and setvalued homomorphism of hyperquantales and generalized rough fuzzy bi-hyperideals and shown that how they are related. In our future study of hyperquantales, we will apply the above new idea to other algebraic structures for more applications.

Acknowledgements
The second author is supported by UKM Grant: GP-2020-K006392.
Declaration: All authors agreed with the contents of the manuscript.

Conflicts of Interest:
The author(s) declare that there are no conflicts of interest regarding the publication of this paper.