(λ, μ) Hesitant fuzzy subalgebras of Boolean algebras

In this paper, the concept of (λ, μ)-hesitant fuzzy subalgebras is introduced in Boolean algebra. Some properties of (λ, μ)-hesitant fuzzy subalgebras are discussed. Finally, we proved that the intersection and direct product of two (λ, μ)-hesitant fuzzy subalgebras are also (λ, μ)-hesitant fuzzy subalgebras in Boolean algebra.


Introduction
After Zadeh put forward fuzzy set [1], many scholars combine fuzzy set with algebraic system and draw important conclusions. Later, Spanish scholar torra put forward the concept of hesitant fuzzy set [2]. Hesitant fuzzy number is more comprehensive than traditional fuzzy element. It contains different groups with a certain degree of hesitation. It has been applied in many mathematical models. Liu pengde et al. Generalized hesitant fuzzy sets and proposed the concepts of interval valued intuitionistic hesitant fuzzy sets [3] and interval valued hesitant fuzzy sets [4][5]. Combining algebraic structure with the concept of fuzzy set is an important method to study algebraic structure. In 1984, Liu Xuhua put forward fuzzy Boolean algebra [6]. The research ideas and methods of fuzzy sets were applied to Boolean algebra, and many meaningful conclusions were obtained. For example, sun Shaoquan proposed the concepts of fuzzy subalgebras, fuzzy ideals and fuzzy congruence relations of Boolean algebras in references [7] and [8], gave the isomorphism theorem of quotient Boolean algebras, and obtained the basic theorem of homomorphism of fuzzy subalgebras of Boolean algebras. Sun Zhongpin proposed ) ( q   ， fuzzy subalgebras and ) ( q   ， -fuzzy ideals of Boolean algebras in references [9], and discussed some properties of them in [10], Wang fengxiao studied I-V fuzzy ideals of Boolean algebras, and concluded that the intersection, homomorphic image and direct product of I-V fuzzy ideals of Boolean algebras are also I-V fuzzy ideals. In this paper, on the basis of references [11]- [14], we study Boolean algebras by using the research ideas and methods of hesitant fuzzy sets, propose the concept of ) , (   hesitant fuzzy subalgebras of Boolean algebras, obtain some equivalent characterizations, discuss the relationship between the image and the original image of hesitant fuzzy subalgebras under homomorphic mapping, give the definition of direct product under hesitant fuzzy sets, and study the correlation of ) , (   hesitant fuzzy direct product. The related results further enrich and perfect the theoretical research of fuzzy algebra and Boolean algebra.

Preliminaries
where ab is b a  ; (2) If there is no special explanation, Boolean algebra , the following condition holds: Then A is called a fuzzy subalgebra of R .
Definition2.3 [6] let a mapping ' : R R f  be called a homomorphic mapping from an algebraic system R to an algebraic system ' R . If all the algebraic operations of the algebraic system are preserved, the mapping f satisfies the following conditions: Definition2.5 [2] Let X be a reference set. Then a hesitant fuzzy set F in X is represented mathermatical as: So, we can define a set of fuzzy sets an HFS by union of their membership functions.
Let F is the hesitation fuzzy set in X, ]) is called a hesitation level set of F ,and ]) Definition2.6 [2] Let X be a reference set, Then the operations complement, union and Intersection are defined as follows: (1)

Hesitant fuzzy subalgebras of Boolean algebras
In this section, we give the definition of ) , (   hesitant fuzzy subalgebras and discuss their properties. The following general assumptions.
,the following conditions are true: , the following conditions are true: if and only if the conditions (1) and (3) in definition 3.2 hold.and: Proof. The necessity is obvious. To prove the sufficiency, we only need to prove the condition (2) in definition 3.1. In fact, for On the other hand, for ) ,   :  is the homomorphic surjective of Boolean algebra, so there must be :  is the homomorphic surjective of Boolean algebra 1 R to 2 R ,so there must be A is called the hesitant fuzzy subset of Y X  , and it is called the direct product of Ã and B . Theorem4 ,and so , , the hesitant fuzzy subsets of 1 R and 2 R are defined as follows: Similarly, we can prove that

Conclusion
On the basis of fuzzy set, hesitant fuzzy set is introduced. Hesitant fuzzy set replaces a number of [0,1] by a subset of [0,1], which changes the membership degree of interval. This expression is closer to human thinking. If we use a numerical value to describe an event, the result is too absolute, and the hesitant fuzzy set can better reflect the hesitant degree of certain elements in membership. In this paper, we combine hesitant fuzzy set with Boolean algebra, introduce the concept of ) , (   hesitant fuzzy subalgebra by using the idea and method of hesitant fuzzy set, we study its properties and equivalent characterization, and get some meaningful conclusions. These conclusions enrich Boolean algebra and hesitant fuzzy set theory.