#### Abstract

In this article, a new idea of BMBJ-neutrosophic hyper-BCK-algebras is introduced and some of its properties are investigated. Here, BMBJ-neutrosophic hyper-BCK-ideal, BMBJ-neutrosophic weak hyper-BCK-ideal, BMBJ-neutrosophic s-weak hyper-BCK-ideal, and BMBJ-neutrosophic strong hyper-BCK-ideal are presented, and some relevant results and relations are indicated. Characterizations of BMBJ-neutrosophic (weak, s-weak, strong) hyper-BCK-ideal are considered. Conditions for a BMBJ-neutrosophic weak hyper-BCK-ideal to be a BMBJ-neutrosophic s-weak hyper-BCK-ideal are provided. Conditions for an MBJ-neutrosophic set to be a BMBJ-neutrosophic strong hyper-BCK-ideal are given.

#### 1. Introduction

A hypergroup, as a generalization of a group, was introduced by Marty  in 1934. Many authors have developed the discussion of hyperstructures (also called multialgebras), such as Corsini  and Vougiouklis . We can find well-written books for the introduction to hyperstructures, e.g., Corsini , Corsini and Leoreanu , Davvaz [5, 6], Davvaz and Cristea , and Schweigert . Another topic, which has roused the interest of several mathematicians, is that one of hyper-BCK-algebra (briefly, -BCK-algebra), introduced by Jun et al. . -BCK-algebras represent a natural extension of classical BCK-algebras. In a classical BCK-algebra, the composition of two elements is an element, while in a -BCK-algebra, the composition of two elements is a set.

As an extension of the classical notion of a set, Zadeh , in 1965, proposed fuzzy sets (briefly, FSs) as mathematical model of vagueness where elements belong to a given set to some degree that is typically a number that belongs to the unit interval . In 1986, this concept has been generalized to intuitionistic fuzzy set (briefly, IFS) theory by adding a nonmembership function by Atanassov . On the other hand, neutrosophy is an almost new branch in pure mathematics which was introduced in 1998 by Smarandache (see [12, 13]). It is an extension of the classical idea of a set and it is related to IFS theory and intuitionistic logic. Neutrosophic sets (briefly, NSs) are sets whose elements have independent degrees of truth and indeterminate and false memberships in the unit interval . The natural generalization of NS theory is the approach of MBJ-neutrosophic sets (briefly, MBJ-NSs), introduced by Takallo et al.  in 2018, when they generalized the indeterminacy membership function in a NS to an interval-valued membership function. If the interval-valued indeterminacy membership function of an MBJ-NS takes equal lower and upper values, then we go back to the NS. Thus, MBJ-NSs provide a more adequate description of uncertainty than NSs. Different extensions of FSs have been extensively implemented to algebraic structures, decision-making problems, etc. For algebraic structures (especially, BCK/BCI-algebras and semigroups), see , and for decision-making problems, see . In , Takallo et al. introduced the concept of an MBJ-N subalgebra as a generalization of a neutrosophic subalgebra in BCK/BCI-algebras and next Jun and Roh  introduced and studied the concept of an MBJ-N ideal. In B-algebras, Manokaran and Prakasam  introduced the MBJ-N subalgebra and Khalid et al.  defined and studied the MBJ-N T-ideal. As a new idea and based on MBJ-NSs, Bordbar et al.  proposed the notion of a BMBJ-N subalgebra and Takallo et al.  introduced the notions of a BMBJ-N-subalgebra and a (closed) BMBJ-N ideal in BCK/BCI-algebras. Borzooei et al.  presented a positive implicative BMBJ-N ideal in BCK-algebras.

In an algebraic hyperstructure, Jun and Xin  first studied the fuzzy -BCK-ideal of a -BCK-algebra and Bakhshi et al.  introduced fuzzy (weak, positive) implicative -BCK-ideals and then the fuzzification of -BCK-algebra started to grow up. In particular, a link between hyper -BCK-algebras and IFSs has been established by Borzooei and Jun  where they discussed intuitionistic fuzzy -BCK-ideals of -BCK-algebras. Later on, other researchers considered this field of study such as Jun  who discussed fuzzy -BCK-ideals of -BCK-algebras with multivalued membership functions. Also, an article was written by Seo et al. on -BCK-algebras and multipolar intuitionistic fuzzy -BCK-ideals (see ). As a generalization of fuzzification of hyperalgebraic structures, some researchers started working on fuzzification of hyperalgebraic structures. In fact, a link between NSs and hyperalgebraic structures was recently established and some work was done in this regard, see .

In our paper, we combine the notion of -BCK-algebras with MBJ-NSs to define some types of BMBJ-neutrosophic -BCK-ideals (briefly, BMBJ-N -BCK-ideals) of -BCK-algebras and it is organized as follows: after an introduction, Section 2 briefly reviews some preliminary results related to -BCK-algebras and MBJ-NSs that are used throughout the paper. Section 3 defines the notions of BMBJ-neutrosophic weak hyper-BCK-ideals (briefly, BMBJ-N -BCK-ideals), BMBJ-neutrosophic -weak hyper-BCK-ideals (briefly, BMBJ-N --BCK-ideals), and BMBJ-neutrosophic strong hyper-BCK-ideals (briefly, BMBJ-N -BCK-ideals) of -BCK-algebras and presents several results related to the new defined concepts. Also, we discuss BMBJ-N -BCK-ideal and BMBJ-N -BCK-ideal in relation to level cut sets. We find conditions for a BMBJ-N -BCK-ideal to be a BMBJ-N --BCK-ideal. We give conditions for an MBJ-NS to be a BMBJ-N -BCK-ideal. Finally, in Section 4, we present the conclusion and future works of the study.

#### 2. Preliminaries

In the current section, we remember some of the basic notions of -BCK-algebras and MBJ-NSs which will be very helpful in further study of the paper. Let be a -BCK-algebra in what follows, unless otherwise stated.

Let be a nonempty set and let “” be a mappingwhich is said to be hyperoperation. For any two subsets and , denote by , the set . shall use instead of , , or .

By a -BCK-algebra (see ), we mean a set with a special element 0 and a hyperoperation , for all , that satisfies the following axioms:(HI) ,(HII) ,(HIII) ,(HIV) and imply ,For all , where is defined by and is defined by such that .In a -BCK-algebra , the axiom (HIII) is equivalent to (HV), where(HV) for all .

Proposition 1 (see ). Every -BCK-algebra satisfies the following conditions, for all and for any nonempty subsets of ,(1), , ,(2), , ,(3),(4),(5), ,(6),(7),(8), , ,(9).

Definition 1. Let be a -BCK-algebra. A subset of is called as follows:A hyper-BCK-ideal (briefly, -BCK-ideal) of (see ) if(1),(2).A weak hyper-BCK-ideal (briefly, -BCK-ideal) of (see ), if it satisfies (1) and(3),A strong hyper-BCK-ideal (briefly, -BCK-ideal) of (see ), if it satisfies (1) and(4).

By an interval number , we mean an interval , where . The set of all closed interval numbers is denoted by . The interval is identified with the number .

For two interval numbers and , we define

Furthermore, we have(1),(2),(3)

Let be a nonempty set. A function is called an interval-valued fuzzy set (briefly, IVFS) over a universe .

Let stand for the set of all IVFSs . For any and , is called the degree of membership of an element to , where and are FSs over a universe which are called a lower FS and an upper FS over , respectively. For simplicity, we denote .

Let be a nonempty set. An NS over a universe (see ) is a structure of the form:where , , and are FSs over a universe , which are called a truth and an indeterminate and false membership functions, respectively.

For the sake of simplicity, we shall use the symbol for the NS

In , Takallo et al. introduced the idea of an MBJ-NS as follows.

Definition 2. Let be a nonempty set. By an MBJ-NS over a universe , we mean a structure of the following form:where and are FSs over a universe , which are called a truth and a false membership functions, respectively, and is an interval-valued fuzzy set over a universe which is called an indeterminate interval-valued membership function.
For the sake of simplicity, we shall use the symbol for the MBJ-NS,

Given an MBJ-NS over a universe , we consider the following sets:where .

#### 3. BMBJ-Neutrosophic Hyper-BCK-Ideals

Definition 3. An MBJ-NS on is called a BMBJ-N -BCK-ideal of , if it satisfies(1) (2)

Example 1. Let be a set with the hyperoperation ““, which is given by Table 1.
Then, is a -BCK-algebra. Let be an MBJ-NS over given by Table 2.
It is routine to check that is a BMBJ-N -BCK-ideal of .

Proposition 2. Let be a BMBJ-N -BCK-ideal of . Then,(i) .(ii)If satisfiesthen

Proof. The proof is obvious and is omitted.

Corollary 1. In a finite-BCK-algebra, every BMBJ-N -BCK-ideal over satisfies condition (9).

Lemma 1 (see ). Let be a subset of a -BCK-algebra . If is a -BCK-ideal of such that , then is contained in .

Theorem 1. An MBJ-NS over is a BMBJ-N -BCK-ideal of if and only if the nonempty sets , , , and are -BCK-ideals of for any .

Proof. Assume that is a BMBJ-N -BCK-ideal of . Let be such that , , , and are nonempty sets. Clearly, , , , and by Proposition 2(i). Let be such that , , , , , , , and . Then, , , , , , , , and . It follows thatwhich imply that , , , and for all , and . Hence, , , , and and soThus, , , , and , and hence, , , , and are -BCK-ideals of .
Conversely, suppose the nonempty sets , , , and are -BCK-ideals of for all . Let be such that , , , , , , , and . Then, , , , and , and so , , , and . From Lemma 1, we have , , , and , i.e., , , , and . Hence, , , , and . For any , letThen, , , , , andfor all , , , and , i.e, , , , and . Thus, , , , and , which imply from Proposition 1(6) that , , , and . Since , , , and are -BCK-ideals of , we have , , , and , which imply thatTherefore, is a BMBJ-N -BCK-ideal of .

Definition 4. An MBJ-NS over is called as follows:(1)A BMBJ-N -BCK-ideal of if it satisfies Proposition 2(i) and Definition 3(2).(2)A BMBJ-N --BCK-ideal of if it satisfies Proposition 2(i) and (2).

Theorem 2. Every BMBJ-N -BCK-ideal is a BMBJ-N -BCK-ideal.

Proof. Straightforward.

The converse of Theorem 2 is not true in general, as seen in the following example.For the converse of Theorem 3, it is not easy to find an example of a BMBJ-N -BCK-ideal which is not a BMBJ-N --BCK-ideal. So, we give the following theorem.

Example 2. Let be a -BCK-algebra as in Example 1. Let be an MBJ-NS over given by Table 3.
Then, is a BMBJ-N -BCK-ideal of . Note that ,Hence, is not a BMBJ-N -BCK-ideal of .

Theorem 3. In a -BCK-algebra, every BMBJ-N --BCK-ideal is a BMBJ-N -BCK-ideal.

Proof. Let be a BMBJ-N --BCK-ideal of over and let . Then, there exist , , , and such that , , , and by the condition (ii) of Proposition 2. Since , , , and , it follows thatTherefore, is a BMBJ-N -BCK-ideal of over .

Theorem 4. Let be a BMBJ-N -BCK-ideal of which satisfies condition (1) of Proposition 2. Then, is a BMBJ-N --BCK-ideal of .

Proof. For any , there exist , , , and such that , , , and . Then,Therefore, is a BMBJ-N --BCK-ideal of over .

Theorem 5. An MBJ-NS over is a BMBJ-N -BCK-ideal of if and only if the nonempty sets , , , and are -BCK-ideals of for all .

Proof. It is similar to the proof of Theorem 1.

The following definition presents the BMBJ-N -BCK-ideal of a -BCK-algebra . Next, we study some properties of this concept.

Definition 5. An MBJ-NS over is called a BMBJ-N -BCK-ideal of , if it satisfies(1) (2) (3) (4)

Example 3. Let be a set with the hyperoperation “,” which is given by Table 4.
Then, is a -BCK-algebra (see ). Let be an MBJ-NS over given by Table 5.
It is routine to check that is a BMBJ-N -BCK-ideal of .

Proposition 3. Every BMBJ-N -BCK-ideal over , satisfying the following conditions:(1) (2) (3)

Proof. (1) Since , we havefor all .
(2) Let be such that . Then, and thus , , , and . It follows from Definition 5 thati.e., , , , and for all with .
(3) Let be such that . Then,