An Approach to BMBJ-Neutrosophic Hyper-BCK-Ideals of Hyper-BCK-Algebras

Alsubie, Abdelaziz; Al-Masarwah, Anas; Jun, Young Bae; Ahmad, Abd Ghafur

doi:https://doi.org/10.1155/2021/5552060

Journal of Mathematics

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Research Article | Open Access

Volume 2021 | Article ID 5552060 | https://doi.org/10.1155/2021/5552060

An Approach to BMBJ-Neutrosophic Hyper-BCK-Ideals of Hyper-BCK-Algebras

Abdelaziz Alsubie,¹Anas Al-Masarwah,²Young Bae Jun,³and Abd Ghafur Ahmad²

Academic Editor: Feng Feng

Received18 Jan 2021

Revised10 Apr 2021

Accepted30 Apr 2021

Published25 May 2021

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 [1] in 1934. Many authors have developed the discussion of hyperstructures (also called multialgebras), such as Corsini [2] and Vougiouklis [3]. We can find well-written books for the introduction to hyperstructures, e.g., Corsini [2], Corsini and Leoreanu [4], Davvaz [5, 6], Davvaz and Cristea [7], and Schweigert [8]. 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. [9]. -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 [10], 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 [11]. 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. [14] 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 [15–21], and for decision-making problems, see [22–24]. In [14], 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 [25] introduced and studied the concept of an MBJ-N ideal. In B-algebras, Manokaran and Prakasam [26] introduced the MBJ-N subalgebra and Khalid et al. [27] defined and studied the MBJ-N T-ideal. As a new idea and based on MBJ-NSs, Bordbar et al. [28] proposed the notion of a BMBJ-N subalgebra and Takallo et al. [29] introduced the notions of a BMBJ-N-subalgebra and a (closed) BMBJ-N ideal in BCK/BCI-algebras. Borzooei et al. [30] presented a positive implicative BMBJ-N ideal in BCK-algebras.

In an algebraic hyperstructure, Jun and Xin [31] first studied the fuzzy -BCK-ideal of a -BCK-algebra and Bakhshi et al. [32] 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 [33] where they discussed intuitionistic fuzzy -BCK-ideals of -BCK-algebras. Later on, other researchers considered this field of study such as Jun [34] 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 [35]). 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 [36–39].

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 [9]), 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 [9]). 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 [9]) if(1),(2). A weak hyper-BCK-ideal (briefly, -BCK-ideal) of (see [9]), if it satisfies (1) and(3), A strong hyper-BCK-ideal (briefly, -BCK-ideal) of (see [40]), 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 [12]) 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 [14], 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 [41]). 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 [9]). 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,for all with .

Corollary 2. If is a BMBJ-N -BCK-ideal over , then condition (2) of Definition 3 is valid.

Proof. Note that , , , and for all with . Hence, condition (2) of Definition 3 follows from Proposition 3(2).

Theorem 6. Every BMBJ-N -BCK-ideal is a BMBJ-N -BCK-ideal.

Proof. Straightforward.

The converse of Theorem 6 is not true in general. That is, a BMBJ-N -BCK-ideal may not be a 7BMBJ-N -BCK-ideal.

Example 4. Let be a hyper-BCK-algebra as in Example 1. Let be an MBJ-NS over given by Table 6.
Then, is a BMBJ-N -BCK-ideal of , but it is not a BMBJ-N -BCK-ideal of , since

Theorem 7. Let be an MBJ-NS over . If is a BMBJ-N -BCK-ideal of , then the nonempty sets , , , and are -BCK-ideals of for all .

Proof. Assume that is an MBJ-NS over . Let be such that , , , and are nonempty sets. Then, there exist , , , and , and so , , , and . Clearly, , , , and by Proposition 3(1). Now, let be such that , , , , , , , and . Then, there exist , , , and . Hence, , , , and . It follows thatHence, , , and . Therefore, , , , and are -BCK-ideals of .

Theorem 8. Let be an MBJ-NS over which satisfies the following condition:If the nonempty sets , , , and are -BCK-ideals of for all , then is a BMBJ-N -BCK-ideal of .

Proof. Suppose the nonempty sets , , , and are -BCK-ideals of for all . Then, , , , and for some , and so , , , and . By Lemma 1, we have , , , and . Thus, for any , , and , we get , and . Hence, , and . It follows thatFor any , taking , , , and . Then, by assumption, , , , and are strong hyper-BCK-ideals of . Condition (3) implies that there exist , , , and such that , , , and . Hence,This imply that , , , and . Hence, , , , , and thus , , and . It follows thatTherefore, is a BMBJ-N -BCK-ideal of .

4. Conclusions

This paper found a new link between hyperalgebraic structures and MBJ-NSs by introducing BMBJ-N -BCK-ideal, BMBJ-N --BCK-ideal, and BMBJ-N -BCK-ideal of -BCK-algebras and studying their properties and relations. BMBJ-N -BCK-ideals and BMBJ-N -BCK-ideals in relation to level subsets have been discussed. Conditions for a BMBJ-N -BCK-ideal to be a BMBJ-N --BCK-ideal have been found. Conditions for an MBJ-NS to be a BMBJ-N -BCK-ideal have been given. The results in this paper can be considered as a generalization of the results known for (intuitionistic) fuzzy hyperideals of -BCK-algebras. In future work, various types of BMBJ-N -BCK-ideals will be defined and discussed.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest.

References

F. Marty, Sur une generalization de la notion de groupe, Huitieme congress de Mathematiciens, Scandenaves, Stockholm, 1934.
P. Corsini, Prolegomena of Hypergroup Theory, Aviani Editore, Tricesimo, Italy, 1993.
T. Vougiouklis, Hyperstructures and Their Representations, Hadronic Press, Inc., Palm Harbor, FL, USA, 1994.
P. Corsini and V. Leoreanu, Applications of Hyperstructure Theory, Kluwer, Dordrecht, The Netherlands, 2003.
B. Davvaz, Semihypergroup Theory, Academic Press, Cambridge, MA, USA, 1 edition, 2016.
B. Davvaz, Polygroup Theory and Related Systems, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, USA, 2013.
B. Davvaz and I. Cristea, Fuzzy Algebraic Hyperstructures An Introduction. Studies in Fuzziness and Soft Computing, Springer, International Publishing, Cham, Switzerland, 2015.
D. Schweigert, “Congruence relations of multialgebras,” Discrete Mathematics, vol. 53, pp. 249–253, 1985.
View at: Publisher Site | Google Scholar
Y. B. Jun, M. M. Zahedi, X. L. Xin, and R. A. Borzoei, “On hyper BCK-algebras,” Italian Journal of Pure and Applied Mathematics, vol. 8, pp. 127–136, 2000.
View at: Google Scholar
L. A. Zadeh, “Fuzzy sets,” Information and Control, vol. 8, no. 3, pp. 338–353, 1965.
View at: Publisher Site | Google Scholar
K. T. Atanassov, “Intuitionistic fuzzy sets,” Fuzzy Sets and Systems, vol. 20, no. 1, pp. 87–96, 1986.
View at: Publisher Site | Google Scholar
F. Smarandache, Unifying Field in Logics: Neutrosophic Logic. Neutrosophy, Neutrosophic Set, Nutrosophic Probability, American Reserch Press, Rehoboth, NM, USA, 1998.
F. Smarandache, “Neutrosophic set-a generalization of the intuitionistic fuzzy set,” International Journal of Pure and Applied Mathematics, vol. 24, pp. 287–297, 2005.
View at: Google Scholar
M. M. Takallo, R. A. Borzooei, and Y. B. Jun, “MBJ-neutrosophic structures and its applications in BCK/BCI-algebras,” Neutrosophic Sets System, vol. 23, pp. 72–84, 2018.
View at: Google Scholar
Y. Jun, S. Kim, and F. Smarandache, “Interval neutrosophic sets with applications in BCK/BCI-algebra,” Axioms, vol. 7, no. 2, p. 23, 2018.
View at: Publisher Site | Google Scholar
A. M. Al-Masarwah and A. G. Ahmad, “On some properties of doubt bipolar fuzzy H-ideals in BCK/BCI-algebras,” European Journal of Pure and Applied Mathematics, vol. 11, no. 3, pp. 652–670, 2018.
View at: Publisher Site | Google Scholar
A. Al-Masarwah and A. G. Ahmad, “Novel concepts of doubt bipolar fuzzy H-ideals of BCK/BCI-algebras,” International Journal of Innovative Computing, vol. 14, pp. 2025–2041, 2018.
View at: Google Scholar
A. Al-Masarwah and A. G. Ahmad, “m-Polar fuzzy ideals of BCK/BCI-algebras,” Journal of King Saud University - Science, vol. 31, no. 4, pp. 1220–1226, 2019.
View at: Publisher Site | Google Scholar
A. Al-Masarwah and A. G. Ahmad, “m-Polar (α, β) -Fuzzy Ideals in BCK/BCI-Algebras,” Symmetry, vol. 11, no. 1, p. 44, 2019.
View at: Publisher Site | Google Scholar
A. Al-Masarwah, A. G. Ahmad, G. Muhiuddin, and D. Al-Kadi, “Generalized m-polar fuzzy positive implicative ideals of BCK-algebras,” Journal of Mathematics, vol. 2021, Article ID 6610009, 10 pages, 2021.
View at: Publisher Site | Google Scholar
M. Khan, M. Gulistan, N. Shabir, and S. Muhammad, “Neutrosophic cubic (α, β)-ideals in semigroups with application -ideals in semigroups with application,” Journal of Intelligent & Fuzzy Systems, vol. 35, no. 2, pp. 2469–2483, 2018.
View at: Publisher Site | Google Scholar
M. Abu Qamar and N. Hassan, “An approach toward a Q-neutrosophic soft set and its application in decision making,” Symmetry, vol. 11, no. 2, p. 139, 2019.
View at: Publisher Site | Google Scholar
G. Muhammad, B. Ismat, and Y. Naveed, “A new approach in decision making problems under the environment of neutrosophic cubic soft matrices,” Journal of Intelligent & Fuzzy Systems, vol. 36, no. 1, pp. 295–307, 2019.
View at: Google Scholar
M. Khan, M. Gulistan, N. Yaqoob, M. Khan, and F. Smarandache, “Neutrosophic cubic Einstein geometric aggregation operators with application to multi-criteria decision making method,” Symmetry, vol. 11, no. 2, p. 247, 2019.
View at: Publisher Site | Google Scholar
Y. B. Jun and E. H. Roh, “MBJ-neutrosophic ideals of BCK/BCI-algebras,” Open Mathematics, vol. 17, no. 1, pp. 588–601, 2019.
View at: Publisher Site | Google Scholar
S. Manokaran and M. Prakasam, “On MBJ-Neutrosophic B-Subalgebra,” Neutrosophic Sets System, vol. 28, pp. 216–227, 2019.
View at: Google Scholar
M. Khalid, N. A. Khalid, and R. Iqbal, “MBJ-neutrosophic T-ideal on B-algebra,” International Journal of Neutrosophic Science, vol. 1, pp. 29–39, 2020.
View at: Google Scholar
B. Bordbar, M. M. Takallo, R. A. Borzooei, and Y. B. Jun, “BMBJ-neutrosophic subalgebras in BCI/BCK-algebras,” Neutrosophic Sets System, vol. 13, pp. 1–13, 2020.
View at: Google Scholar
M. M. Takallo, H. Bordbar, R. A. Borzooei, and Y. B. Jun, “BMBJ-neutrosophic ideals in BCK/BCI-algebras,” Neutrosophic Sets System, vol. 27, pp. 1–16, 2019.
View at: Google Scholar
R. A. Borzooei, M. M. Takallo, F. Smarandache, and Y. B. Jun, “Positive implicative BMBJ-neutrosophic ideals in BCK-algebras,” Neutrosophic Sets System, vol. 23, pp. 126–141, 2018.
View at: Google Scholar
Y. B. Jun and X. L. Xin, “Fuzzy hyper BCK-ideals of hyper BCK-algebras,” Scientiae Mathematicae Japonicae, vol. 53, pp. 353–360, 2001.
View at: Google Scholar
M. Bakhshi, M. M. Zahedi, and R. A. Borzooei, “Fuzzy (positive, weak) implicative hyper BCK-ideals. Iran,” International Journal of Fuzzy Systems, vol. 1, no. 2, pp. 63–79, 2004.
View at: Google Scholar
R. A. Borzooei and Y. B. Jun, “Intuitionistic fuzzy hyper BCK-ideals of hyper BCK-ideals. Iran,” International Journal of Fuzzy Systems, vol. 1, no. 1, pp. 65–78, 2004.
View at: Google Scholar
Y. B. Jun, “Multipolar fuzzy hyper BCK-ideals of hyper BCK-algebras,” Journal of Algebraic Hyperstructures and Logical Algebras, vol. 1, no. 1, pp. 37–47, 2020.
View at: Publisher Site | Google Scholar
Y. J. Seo, H. S. Kim, Y. B. Jun, and S. S. Ahn, “Multipolar intuitionistic fuzzy hyper BCK-ideals in hyper BCK-algebras,” Mathematics, vol. 8, no. 8, p. 1373, 2020.
View at: Publisher Site | Google Scholar
S. Khademan, M. M. Zahedi, R. A. Borzooei, and Y. B. Jun, “Neutrosophic hyper BCK-ideals,” Neutrosophic Sets System, vol. 27, pp. 201–217, 2019.
View at: Google Scholar
D. Mandal, “Neutrosophic hyperideals of -semihyperrings,” Bulletin of International Mathematical Virtual Institute, vol. 7, pp. 153–164, 2017.
View at: Google Scholar
D. Preethi, S. Rajareega, J. Vimala, G. Selvachandran, and F. Florentin Smarandache, “Single-valued neutrosophic hyperrings and hyperideals,” Neutrosophic Sets System, vol. 29, pp. 121–128, 2019.
View at: Google Scholar
A. A. A. Agboola and B. Davvaz, “Introduction to neutrosophic hypergroups,” ROMAI Journal, vol. 9, pp. 1–10, 2013.
View at: Google Scholar
Y. B. Jun, X. L. Xin, M. M. Zahedi, and E. H. Roh, “Strong hyper BCK-ideals of hyper BCK-algebras,” Japanese Journal of Mathematics, vol. 51, pp. 493–498, 2000.
View at: Google Scholar
Y. B. Jun and X. L. Xin, “Scalar elements and hyper atoms of hyper BCK-algebras,” Science China Mathematics, vol. 2, pp. 303–309, 1999.
View at: Google Scholar

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Copyright © 2021 Abdelaziz Alsubie 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.

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