MBJ-neutrosophic hyper BCK-ideals in hyper BCK-algebras

In 2018, Takallo et al. introduced the concept of an MBJ-neutrosophic structure, which is a generalization of a neutrosophic structure, and applied it to a BCK/BCI-algebra. The aim of this study is to apply the notion of an MBJ-neutrosophic structure to a hyper BCK-algebra. The notions of the MBJ-neutrosophic hyper BCK-ideal, the MBJ-neutrosophic weak hyper BCK-ideal, the MBJ-neutrosophic s-weak hyper BCK-ideal and the MBJ-neutrosophic strong hyper BCK-ideal are introduced herein, and their relations and properties are investigated. These notions are discussed in connection with the MBJ-neutrosophic level cut sets.


Introduction
BCK-algebras entered into pure mathematics in 1966 through the work of Imai and Iséki [1], and were applied to various mathematical fields, such as functional analysis, group theory, topology and probability theory, etc. The hyperstructure theory was born in 1934 when Marty introduced hypergroups. In this side, he explored and applied their properties to groups and other algebraic structures [2]. Algebraic hyperstructures reflect a natural generalization of classical algebraic structures. In an algebraic hyperstructure, the composition of two elements is a set, while in a classical algebraic structure, the composition of two elements is an element. As an extension of a BCK-algebra, Jun et al. [3] introduced an algebraic hyperstructure called a hyper BCK-algebra. They studied hyper BCK-ideals in hyper BCK-algebras. Saeid and Zahedi [4] studied quotient hyper BCK-algebras and in [5] Saeid et al. introduced weak implicative and implicative hyper K-ideals of hyper K-algebras. After that, many books and several articles have been published on hyper BCK-algebras and other hyper algebraic structures.
Zadeh [6] introduced fuzzy set theory in 1965 and in 1986 this concept has been generalized to intuitionistic fuzzy set theory by adding a non-membership function by Atanassov [7]. As a generalization of the classical set and (intuitionistic) fuzzy set theory, Smarandache [8,9] launched a significant topic, that deals with indeterminacy, called neutrosophic set theory. In [10], Takallo et al. presented the notion of an MBJ-neutrosophic set as generalization of a neutrosophic set and they applied it to BCK/BCI-algebras. In an MBJ-neutrosophic set, the indeterminacy membership function is generalized to interval valued membership function. Next, Jun and Roh [11] introduced and studied the concept of an MBJ-neutrosophic ideal in BCK/BCI-algebras. In B-algebras, Manokaran and Prakasam [12] introduced the MBJ-neutrosophic subalgebra and Khalid et al. [13] defined and studied the MBJ-neutrosophic T-ideal. The notions of (intuitionistic) fuzzy sets, neutrosophic sets and other extensions of fuzzy sets have been applied to algebraic structures, decision making problems, etc. For algebraic structures, see [14][15][16][17][18][19][20][21][22] and for decision making problems, see [23,24]. In an algebraic hyperstructure, Jun and Xin [25] discussed the topic of fuzzy set theory of hyper BCK-ideals in hyper BCK-algebras and in [26] Bakhshi et al. studied fuzzy (positive, weak) implicative hyper BCK-ideals. In 2004, Borzooei and Jun [27] studied the intuitionistic fuzzy set theory of hyper BCK-ideals in hyper BCK-algebras. In addition, Khademan et al. [28] studied neutrosophic set theory of hyper BCK-ideals in hyper BCK-algebras.
As no studies have been reported so far to generalize the above mentioned concepts, so the aim of this present article is: (1) To apply the notion of an MBJ-neutrosophic structure to a hyper BCK-algebra.
To do so, the rest of the article is structured as follows: In Section 2, we review some elementary notions. In Section 3, we introduce the notions of the MBJ-neutrosophic hyper BCK-ideal, the MBJneutrosophic weak hyper BCK-ideal, the MBJ-neutrosophic s-weak hyper BCK-ideal and the MBJneutrosophic strong hyper BCK-ideal and investigate several properties. We discuss MBJ-neutrosophic (weak, strong) hyper BCK-ideal in relation to MBJ-neutrosophic level cut sets. Finally, in Section 4, we present the conclusion and future works of the study.

Preliminaries
In the current section, we remember some of the basic notions of hyper BCK-algebras which will be very helpful in further study of the paper. Let H be a hyper BCK-algebra in what follows, unless otherwise stated.
Let H be a non-empty set and let " " be a mapping which is said to be hyperoperation. For any two subsets K and F , denote by K F , the set { τ | ∈ K, τ ∈ F }. We shall use By a hyper BCK-algebra H (see [3]), we mean a non-empty set H with a special element 0 and a hyperoperation , for all , τ, η ∈ H, that satisfies the following axioms: τ and τ imply = τ, for all , τ, η ∈ H, where τ is defined by 0 ∈ τ and for any K, F ⊆ H, K F is defined by ∀r ∈ K, ∃t ∈ F such that r t.
In a hyper BCK-algebra H the axiom (HIII) is equivalent to the following axiom: Proposition 2.1.
[3] Every hyper BCK-algebra H satisfies the following conditions, for all , τ, η ∈ H and for any non-empty subsets K, F , G of H, Definition 2.2. Let (H, ) be a hyper BCK-algebra. A subset K of H is called: τ K, τ ∈ K ⇒ ∈ K, ∀ , τ ∈ H.

By an intervalũ we mean an intervalũ
The set of all closed intervals I is denoted by [I]. The interval [u, u] is identified with the number u.
For Let H be a nonempty set. A neutrosophic set over a universe H (see [9]) is a structure of the form: where D T , D I and D F are fuzzy sets over a universe H, which are called a truth, an indeterminate and a false membership functions, respectively. For the sake of simplicity, we shall use the symbol D = (D T , D I , D F ) for the neutrosophic set In [10], Takallo et el. introduced the idea of an MBJ-neutrosophic set as follows: Let H be a nonempty set. By an MBJ-neutrosophic set over a universe H, we mean a structure of the form: where M D and J D are fuzzy sets over a universe H, which are called a truth and a false membership functions, respectively, andB D is an interval-valued fuzzy set over a universe H which is called an indeterminate interval-valued membership function. Given an MBJ-neutrosophic set D = (M D ,B D , J D ) over a universe H, we consider the following sets:
Then, H is a hyper BCK-algebra (see [3]). Let D = (M D ,B D , J D ) be an MBJ-neutrosophic set over H given by Table 2.
τ}. It follows from Definition 3.1(2) that This completes the proof.   Proof.
and sup{J D (z) | z ∈ a b} ≤ γ, and so  For any , τ, u, v, a, b ∈ H .
The converse of Theorem 3.7 is not true in general, as seen in the following example.  Table 3.  Since and Question1. Is the converse of Theorem 3.8 true? It is not easy to find an example of an MBJ-neutrosophic weak hyper BCK-ideal which is not an MBJ-neutrosophic s-weak hyper BCK-ideal. However, we give the following theorem. Proof . For any , τ, u, v, a, b ∈ H Proof. It is similar to the proof of Theorem 3.5.
The following definition presents the concept of an MBJ-neutrosophic strong hyper BCK-ideal of a hyper BCK-algebra H. Next, we study some properties of this concept.
The converse of Theorem 3.14 is not true in general. That is, an MBJ-neutrosophic hyper BCK-ideal may not be an MBJ-neutrosophic strong hyper BCK-ideal.  Table 6.   (1).

Conclusions
In this paper, we have applied the MBJ-neutrosophic set to hyper BCK-algebra. We have presented the concepts of the MBJ-neutrosophic hyper BCK-ideal, the MBJ-neutrosophic weak hyper BCK-ideal, the MBJ-neutrosophic s-weak hyper BCK-ideal and the MBJ-neutrosophic strong hyper BCK-ideal, and have discussed related properties and their relations. We have investigated MBJ-neutrosophic (weak, s-weak, strong) hyper BCK-ideals in relation to level cut sets. In the future work, we will use the concept and results in this paper to study other hyper algebraic structures, for instance, hyper BCI-algebra, hyper hoop, hyper MV-algebra and hyper B-algebra.