Abstract
The purpose of this paper is to introduce the notation of single-valued neutrosophic hyper BCK-subalgebras and a novel concept of neutro hyper BCK-algebras as a generalization and alternative of hyper BCK-algebras, that have a larger applicable field. In order to realize the article’s goals, we construct single-valued neutrosophic hyper BCK-subalgebras and neutro hyper BCK-algebras on a given nonempty set. The result of the research is the generalization of single-valued neutrosophic BCK-subalgebras and neutro BCK-algebras to single-valued neutrosophic hyper BCK-subalgebras and neutro hyper BCK-algebras, respectively. Also, some results are obtained between extended (extendable) single-valued neutrosophic BCK-subalgebras and single-valued neutrosophic hyper BCK-subalgebras via fundamental relation. The paper includes implications for the development of single-valued neutrosophic BCK-subalgebras and neutro BCK-algebras and for modelling the uncertainty problems by single-valued neutrosophic hyper BCK-subalgebras and neutro hyper BCK-algebras. The new conception of single-valued neutrosophic hyper BCK-subalgebras and neutro hyper BCK-algebras was given for the first time in this paper. We find a method that can apply these concepts in some complex networks.
1. Introduction
The theory of logical (hyper) algebra is related to the study of certain propositional calculi and tries to solve logical problems using (hyper) algebraic methods. Jun et al. [1] has introduced a logical (hyper) algebra named hyper BCK-algebras as development of BCK-algebras, which were initiated by Imai and Iseki [2] in 1966 as a generalization of the concept of set-theoretic difference and propositional calculus. The theory of neutrosophic set as an extension of classical set and (intuitionistic) fuzzy set [3], and interval-valued (intuitionistic) fuzzy set, is introduced by Smarandache for the first time in 1998 [4] and mentioned second time in 2005 [5]. This concept handles problems involving imprecise, indeterminacy, and inconsistent data and describes an important role in the modelling of unsure hypernetworks in all sciences. Recently, due to the importance of these subjects, by combining the neutrosophic sets and (hyper) BCK-algebras, some researchers worked in more branches of neutrosophic (hyper) BCK-algebras such as -neutrosophic hyper BCK-ideals in hyper BCK-algebras, an approach to -neutrosophic hyper BCK-ideals of hyper BCK-algebras, structures on doubt neutrosophic ideals of -algebras under -norms, -neutrosophic subalgebras in -algebras, -neutrosophic ideals of -algebras, implicative neutrosophic quadruple BCK-algebras and ideals, neutrosophic hyper BCK-ideals, implicative neutrosophic quadruple BCK-algebras and ideals, bipolar-valued fuzzy soft hyper BCK ideals in hyper BCK-algebras, single-valued neutrosophic ideals in Sostak’s sense, and multipolar intuitionistic fuzzy hyper BCK-ideals in hyper BCK-algebras [6–16]. Recently, a novel concept of neutrosophy theory titled neutro (hyper) algebra as development of classical (hyper) algebra and partial (hyper) algebra is introduced by Smarandache [17].
A neutro (hyper) algebra is a system that has at least one neutro (hyper) operation or one neutro axiom (axiom that is true for some elements, indeterminate for other elements, and false for the other elements), while a partial (hyper) algebra is a (hyper) algebra that has at least one partial (hyper) operation, and all its axioms are classical (i.e., axioms true for all elements). Smarandache proved that a neutron (hyper) algebra is a generalization of a partial (hyper) algebra and showed that neutro (hyper) algebras are not partial (hyper) algebras, necessarily. Hamidi and Smarandache [18] introduced the concept of neutro BCK-subalgebras as a generalization of BCK-algebras and presented main results in neutro BCK-subalgebras as an extension of BCK-algebras structures and their applications. In addition, the concept of neutro (hyper) algebra is studied in different branches such as neutro algebra structures and neutro (hyper) graph [19, 20].
Regarding these points, one of the aims of this paper is to introduce the concept of single-valued neutrosophic hyper BCK-subalgebras and extendable single-valued neutrosophic BCK-subalgebras and generalize the notion of single-valued neutrosophic hyper BCK-subalgebras by considering the notion of single-valued neutrosophic BCK-subalgebras. Also, we want to establish the relationship between single-valued neutrosophic BCK-algebras and single-valued neutrosophic hyper . So a strongly regular relation is applied on any hyper BCK-algebras using the concept of single-valued neutrosophic hyper BCK-subalgebras, and a quotient hyper BCK-algebras can be obtained. The main aim of this study is to introduce the notation of neutro hyper BCK-algebras as a generalization of neutro BCK-algebras in regard to single-valued neutrosophic hyper BCK-subalgebras. In the study of neutro hyper BCK-algebra, despite having key mathematical tools, there are some limitations. The union of two neutro hyper BCK-algebra is not necessarily a neutro hyper BCK-algebra so the class of neutro hyper BCK-algebra is not closed under any given algebraic operation. In addition, neutro hyper BCK-algebras are different with (intuitionistic fuzzy) hyper BCK-algebras and single-valued neutrosophic hyper BCK-algebras so could not generalize the capabilities of (intuitionistic fuzzy) single-valued neutrosophic hyper BCK-algebras to neutro hyper BCK-algebras.
2. Preliminaries
Definition 1. (see [2]) Let . Then a universal algebra of type is called a BCK-algebra if, for all, : , , , and imply , , where is denoted by .
Definition 2. (see [1]). Let and . Then for a map , a hyperalgebraic system is called a hyper BCK-algebra if, for all, : , , , and imply , where is defined by , , , , and is denoted by .We will call is a weak commutative hyper BCK-algebra if [21].
Theorem 1 (see [1]). Let be a hyper BCK-algebra. Then and :(i), and (ii), and implies that (iii), and implies
Definition 3. (see [22]). Let be a hyper BCK-algebra. A fuzzy set is called a fuzzy hyper BCK-subalgebra if .
Definition 4. (see [5]). Let be a universal set. A neutrosophic subset (NS) in is an object having the following form: , or , which is characterized by a truth-membership function , an indeterminacy-membership function , and a falsity-membership function . There is no restriction on the sum of , and .
3. Single-Valued Neutrosophic Hyper BCK-Subalgebras
In this section, the concept of single-valued neutrosophic hyper BCK-subalgebras will be considered as a generalization of single-valued neutrosophic BCK-subalgebras, and some of its properties will be investigated. We will also prove that single-valued neutrosophic hyper BCK-subalgebras and single-valued neutrosophic BCK-subalgebras are related, and single-valued neutrosophic hyper BCK-subalgebras and single-valued neutrosophic BCK-subalgebras can be constructed from single-valued neutrosophic hyper BCK-subalgebras via a fundamental relation. We will define the concept of extendable single-valued neutrosophic BCK-subalgebras and will show that any infinite set is an extended single-valued neutrosophic BCK-subalgebra.
Throughout this section, we denote hyper BCK-algebra by . From now on, for all, , and are considered as triangular norm and triangular conorm, respectively. In the following definition, the notation of single-valued neutrosophic hyper BCK-subalgebra of any given nonempty is defined.
Definition 5. A single-valued neutrosophic set in an is called a single-valued neutrosophic hyper BCK-subalgebra of , if(i)(ii)(iii)The importance of the following theorems is to determine the role and the effect of truth-membership function , indeterminacy-membership function , and falsity-membership function on the element .
Theorem 2. Let be a single-valued neutrosophic hyper BCK-subalgebra of . Then(i)(ii)(iii)
Proof. (i) Let . Since , we get that .
(ii) Let . Since , we get that . So .
(iii) Immediate by Theorem 1.
Theorem 3. Let be a single-valued neutrosophic hyper BCK-subalgebra of . Then(i)(ii)(iii)
Proof. (i)Let . Since , we get that .(ii)Let . Since , we get that . So .(iii)Immediate by Theorem 1.
Corollary 1. Let be a single-valued neutrosophic hyper BCK-subalgebra of . Then(i)(ii)(iii)(iv)In the following theorem, we construct single-valued neutrosophic subset on any nonempty set.
Theorem 4. Let . Then there exist a hyperoperation “,” a single-valued neutrosophic subset of such that is a hyper BCK-algebra and is a single-valued neutrosophic hyper BCK-subalgebra of .
Proof. Let . Define “” on by . Clearly, is a hyper BCK-algebra. Now, it is easy to see that every single-valued neutrosophic set that is a single-valued neutrosophic hyper BCK-subalgebra of .
Let = {}, whence is a hyper BCK-algebra and .
Corollary 2. Let . Then can be extended to a hyper BCK-algebra that .
Proof. Let . Then is a hyper BCK-algebra such that . Then for a single-valued neutrosophic set, by is a single-valued neutrosophic hyper BCK-subalgebra of , where . If ; then by Theorem 4, we can construct at least a hyper BCK-subalgebra on . Now, define byObviously, a single-valued neutrosophic hyper BCK-subalgebra of and so .
Let be a hyper BCK-algebra, a single-valued neutrosophic hyper BCK-subalgebra of and . Define , , and .
Considering the relation between single-valued neutrosophic hyper BCK-subalgebras and (fuzzy) hyper BCK-subalgebra is the main aim of the following results via the level subsets.
Theorem 5. Let be a single-valued neutrosophic hyper BCK-subalgebra of . Then(i)(ii) is a hyper BCK-subalgebra of (iii)If , then and
Proof. (i) Clearly, and by Theorems 2 and 3, and Corollary 1, we get that .
(ii) Let . Then . Now, for any, . Hence, , and so . In similar a way, implies that . Then is a hyper BCK-subalgebra of .
(iii) Immediate.
Corollary 3. Let be a single-valued neutrosophic hyper BCK-subalgebra of . If , then is a hyper BCK-subalgebra of .
Let be a hyper BCK-algebra, be a hyper BCK-subalgebra of and . Define
Thus, we have the following theorem.
Theorem 6. Let be a hyper BCK-algebra and be a hyper BCK-subalgebra of . Then(i) is a fuzzy hyper BCK-subalgebra of (ii) is a fuzzy hyper BCK-subalgebra of (iii) is a fuzzy hyper BCK-subalgebra of (iv) is a single-valued neutrosophic hyper BCK-subalgebra of
Proof. (i) Let . If , since is a hyper subalgebra of , we get that and soIf or or , then . Thus, , and so is a fuzzy hyper BCK-subalgebra of .
(ii) and (iii) They are similar to (i).
(iv) Let . If , since is a hyper BCK-subalgebra of , we get that , and so . If or or , then . Thus, . In a similar way, we can see that an by item (i), is a single-valued neutrosophic hyper BCK-subalgebra of .
Let be a hyper BCK-algebra and . Then . The relation is a reflexive and symmetric relation but not transitive relation. Let be the of (the smallest transitive relation such that contains ). Borzooei et al. in [21], proved that for any given weak commutative hyper BCK-algebra , is a strongly regular relation on , and is a BCK-algebra, where and .
Considering the relation between single-valued neutrosophic hyper BCK-subalgebras and single-valued neutrosophic BCK-subalgebras has very important, especially in extension of single-valued neutrosophic BCK-subalgebras. So we prove the following theorems and corollaries.
Theorem 7. Let be a weak commutative hyper BCK-subalgebra and be a single-valued neutrosophic hyper BCK-subalgebra of . Then there exists a single-valued neutrosophic set of BCK-algebra that ,(i)(ii)if , then (iii)(iv)if , then (v)(vi)if , then
Proof. Let . Then on , define , , and . Using Theorems 2 and 3, we get that:
(i)
(ii) Since and is transitive, we get that
(iii)
(iv) Since and is transitive, we get that
(v) and (vi) They are similar to (iii) and (iv), respectively.
Theorem 8. Let be a weak commutative hyper BCK-subalgebra and be a single-valued neutrosophic hyper BCK-subalgebra of . Then there exists a single-valued neutrosophic subset of BCK-algebra that :(i)There exists such that (ii)There exists such that (iii)There exists such that
Proof. (i) Let . Applying Theorem 7,Now, since and , then , and so there exists such that .
(ii) Let . ThenNow, since and , then , and so there exists such that .
(iii) It is similar to item (ii).
Some categorical properties of single-valued neutrosophic BCK-subalgebras is investigated in the following theorem based on the categorical properties of single-valued neutrosophic hyper BCK-subalgebras.
Theorem 9. Let be a weak commutative hyper BCK-algebra and be a single-valued neutrosophic hyper BCK-subalgebra of . Then there exists a single-valued neutrosophic BCK-subalgebra of that or the following diagrams are quasi commutative:
Proof. Choice and . Then by Theorem 7, (i) ,(ii) By Theorem 8, ; there exists thatSoTherefore, is a single-valued neutrosophic BCK-subalgebra of , , and .
Based on the fundamental relation, we can obtain the single-valued neutrosophic BCK-subalgebras, and single-valued neutrosophic BCK-subalgebras are derived from some single-valued neutrosophic hyper BCK-subalgebras. In this regard, it is important that single-valued neutrosophic BCK-subalgebras are derived from single-valued neutrosophic hyper BCK-subalgebra with minimal order. So the concepts of (extended) extendable single-valued neutrosophic BCK-subalgebra are introduced as follows.
Definition 6. (i) Let be a BCK-algebra and be a hyper BCK-algebra. We say that the BCK-algebra is derived from the hyper BCK-algebra if is isomorphic to a nontrivial quotient of .
(ii) A single-valued neutrosophic BCK-subalgebra of is called an extendable single-valued neutrosophic BCK-subalgebra, if there exist a hyper BCK-algebra , a single-valued neutrosophic hyper BCK-subalgebra of , and such that , and BCK-algebra is derived of hyper BCK-algebra . If and almost everywhere ( that means ), we will say that it is an extended single-valued neutrosophic BCK-subalgebra.
The following example introduces an extendable single-valued neutrosophic BCK-subalgebra.
Example 1. Let . Then is a single-valued neutrosophic BCK-subalgebra of BCK-algebra (see Table 1).
Now, set . Then is a single-valued neutrosophic hyper BCK-subalgebra of (see Table 2).
Clearly, , , and so is an extendable single-valued neutrosophic BCK-subalgebra of .
In the following theorem, we try to generate BCK-algebras based on single-valued neutrosophic hyper BCK-subalgebras.
Theorem 10. Let be a hyper BCK-algebra, be a single-valued neutrosophic hyper BCK-subalgebra of , and . If is one to one map, then:(i)There exists a hyperoperation “” on such that is a hyper BCK-algebra(ii)There exists a single-valued neutrosophic hyper BCK-subalgebra of related to (iii)There exists an operation “” on that is a BCK-algebra
Proof. (i) Let . Define a hyperoperation on , byIt can be easily seen that . It is easy to see that is a hyper BCK-algebra.
(ii) Let . Define . Clearly, is a single-valued neutrosophic hyper BCK-subalgebra of .
(iii) Assume . Define an operation on byWe just prove -4. Let andSince is a one to one map, and . It follows that . It is easy to see that -1, -2, -3, and BCK-5 are valid, and so is a BCK-algebra.
Corollary 4. Let be a hyper BCK-algebra and be a single-valued neutrosophic hyper BCK-subalgebra of . Then there exists a binary operation “” on , such that is a BCK-algebra.
In the following theorem, we try to generate hyper BCK-algebras based on single-valued neutrosophic hyper BCK-subalgebras.
Theorem 11. Let be a nonempty set, and . Then there exist a hyperoperation “” on , a hyperoperation “” on , a binary operation “” on , a single-valued neutrosophic subset of , and a single-valued neutrosophic subset of that:(i) is a hyper BCK-algebra, and is a single-valued neutrosophic hyper BCK-subalgebra of (ii) is a hyper BCK-algebra, and is a single-valued neutrosophic hyper BCK-subalgebra of (iii) is a BCK-algebra, and is a single-valued neutrosophic BCK-subalgebra of (iv)
Proof. Let and be fixed. For any , define a binary hyperoperation on as follows:Now, we show that is a hyper BCK-algebra. We just check that conditions (H1) and (H2) are valid.
(H1): Let . If , then . If , then . If , we consider the following cases: Case 1: . Then . Case 2: . Then . Case 3: . Then . Case 4: . Then . Case 5: . Then .(H2): Let . The proof of is similar to that of (H1), and then it is easy to see that is a hyper BCK-algebra. Consider a single-valued neutrosophic subset of such that ; by equation (2) and some modifications, we get thatHence, is a single-valued neutrosophic hyper BCK-subalgebra of . Now, ; define a hyperoperation on byDefine a single-valued neutrosophic subset of byand an operation on byIt can be easily seen that , is a hyper BCK-algebra, is a single-valued neutrosophic hyper BCK-subalgebra of , is a BCK-algebra, and is a single-valued neutrosophic BCK-subalgebra of , and since , we get that .
Corollary 5. Each nonempty set can be constructed to an extendable single-valued neutrosophic BCK-subalgebra.
4. Neutro Hyper BCK-Algebras
Smarandache in [17] introduced the concept of neutro hyper operation. An -ary (for integer ) hyperoperation is called a neutro hyper operation if it has -plets in for which the hyperoperation is well-defined (degree of truth ), -plets in for which the hyperoperation is indeterminate (degree of indeterminacy ), and -plets in for which the hyperoperation is outer-defined (degree of falsehood ), where , with that represents the -ary (total) hyper operation and that represents the -ary anti hyper operation.
In this section, we introduce a novel concept of neutro hyper BCK-algebras as a generalization of neutro BCK-algebras and analyze their properties. The main motivation of the concept of neutro hyper BCK-algebra is a generalization of neutro BCK-algebra, which is defined as follows.
Definition 7. Let and . Then for a map , a hyperalgebraic system is called a neutro hyper BCK-algebra if it satisfies in the following neutro axioms: (H1) ( that ) and ( that or indeterminate) (H2) ( that ) and ( that or indeterminate) (H3) ( that ) and ( that or indeterminate) (H4) ( that if and imply ) and ( that if and imply or indeterminate), where is defined by , and , If is a neutro hyperalgebra and satisfies in condition (H1) to (H4), then we will call it is a neutro hyper BCK-algebra of type 4 (i.e., it satisfies 4 neutro axioms).
Investigation of partial order relation on neutro hyper BCK-algebra plays a main role in Hass diagram, so we have the following results.
Theorem 12. Let be a neutro hyper BCK-algebra, and . Then(i) such that (ii) such that (iii) such that (iv) such that (v) such that (vi) such that
Proof. We prove only the item (ii), and other items are similar to it. Since is a neutro hyper BCK-algebra, there exists such that . It follows that there exist such that and . Hence, .
Theorem 13. Let be a neutro hyper BCK-algebra, and . Then(i)if , then (ii)if , then
Proof. (i) Let be arbitrary. Since , there exists such that . Hence, for , there exists such that and so .
(ii) Since , there exists such that for all, , we have . Hence, there exists such that for all, , we get that and so .
Example 2. (i) Every neutro BCK-algebra is a neutro hyper BCK-algebra. Since, for all, , can define a hyperoperation on by .
(ii) Consider . Define . Clearly, is a neutro hyper BCK-algebra.
The following theorem shows that neutro hyper BCK-algebras are the generalization of hyper BCK-algebras.
Theorem 14. Every hyper BCK-algebra can be extended to a neutro hyper BCK-algebra.
Proof. Let be a hyper BCK-algebra and . For all, , define on by , where, and whence , define is indeterminate or .
We show that how to construct neutro hyper BCK-algebras from BCK-algebras.
Example 3. Let and consider Table 3.
Then(i)If , then is a neutro hyper BCK-algebra and if , then is a hyper BCK-algebra(ii) is a neutro hyper BCK-algebra and is a hyper BCK-algebra(iii)If , then is a neutro hyper BCK-algebra, and for , is a hyper BCK-algebra. If , then is a neutro hyper BCK-algebra of type 4The importance of the following theorem is to construct of neutro hyper BCK-algebra from any given nonempty set.
Theorem 15. Let . Then there exists a hyperoperation “” on such that is a neutro hyper BCK-algebra.
Proof. Let . Using Theorem 4, there exist a hyperoperation “” on such that is a hyper BCK-algebra. Now, apply Theorem 14; there exist a hyperoperation “” on such that is a neutro hyper BCK-algebra.
Let and be two neutro hyper BCK-algebras. Define on by , where and say that . The following theorem investigates the properties of partial order relation on product of Neutro hyper BCK algebras.
Theorem 16. Let and be two neutro hyper BCK-algebras. Then(i)(ii)(iii)(iv)
Proof. (i)Immediate(ii)Let . Then , if and only if , if and if only or , and if and only if (iii)Since and be two neutro hyper BCK-algebras, there exist such that and . It follows that (iv)Since and be two neutro hyper BCK-algebras, there exist such that and . It follows that We need to extend neutro hyper BCK-algebras to a larger class of neutro hyper BCK-algebras, so we apply the notation of product on neutro hyper BCK-algebras as follows.
Theorem 17. Let and be two neutro hyper BCK-algebras. Then is a neutro hyper BCK-algebra.
Proof. We prove only the item (H4), and other items by Theorem 16 are valid. Since and are neutro hyper BCK-algebras, there exist that if , then , and if , then . Also, if , then , and if , then . By (i), it follows that there exist that if , we have , and if , we have .
Let and be hyper BCK-algebras, where . For some define a hyperoperations as follows:and , where . Thus, we have the following theorem.
We want to extend neutro hyper BCK-algebras to a larger class of neutro hyper BCK-algebras, so we apply the notation of union on neutro hyper BCK-algebras as follows.
Theorem 18. Let and be hyper BCK-algebras, where and . Then(i)For all, (ii)For all, (iii)For all, , and for all, (iv)For all, (v)For all,
Proof. (i) Let . Then . It follows that , so . In addition, and . It follows that , so .
(ii) Let . Then and . It follows that , so . In addition, and . It follows that , so .
(iii) Let and . Since and , we get that and . Thus and .
(iv) and (v) are similar to (i) and (ii), respectively.
Theorem 19. Let and be hyper BCK-algebras, where and . Then(i) is a neutro hyper BCK-algebra(ii) is a neutro hyper BCK-algebra
Proof. (i) For some, . Since, for , we get that For some, . In addition, for For some, . Since and , we get that Because and and , while , we get the item is valid. Therefore, is a neutro hyper BCK-algebra.
(ii) It is similar to item (i).
4.1. Application of Neutro Hyper BCK-Algebras and Single-Valued Neutrosophic Hyper BCK-Subalgebras
In this subsection, we describe some applications of neutro hyper BCK-algebra and single-valued neutrosophic hyper BCK-subalgebra in some complex (hyper) networks.
Example 4. (economic network). Let be a set of top countries, which are in an economic network. Suppose is the relations on , which is described in Table 4, and for means that is the set of countries that benefit from this economic partnership, whence the country starts to country , and for , it means that the country maintains its capital.
Clearly, is a neutro hyper BCK-algebra in this model. We obtain that the USA is main source of this network; since if the USA starts to any other country, it does not benefit. In addition, if the USA starts to itself, this participation becomes indeterminate. Also, if any country starts to China, we conclude that China loss, else with USA, and if China starts to any other country, then China benefit else USA.
Example 5. (data network). Let be a set of mobile sets, which are in a data network. Suppose is the relations on , which is described in Table 3, and for all, means that is a set of mobile sets that receive contents of messages that mobile set starts to mobile set , and for , it means that the mobile set retains its information. In addition, for any are the cryptographic power, battery life, and RAM of mobile set , respectively. Then is a single-valued neutrosophic hyper BCK-subalgebra of in Table 5.
It is clear that if mobile set named “a” starts, then none of the devices receive the message, and if other devices start to name a mobile set “a”, then this device (mobile set a) cannot receive their messages; hence, it is not suitable node in this network, since furthermore to its complex cryptography, its battery life, and RAM is weak. Also, one can see that the mobile set is the best in this regard.
5. Conclusion
To conclude, the current paper has presented and analyzed the notion of single-valued neutrosophic hyper BCK-subalgebras and neutro hyper BCK-algebras and investigated some of their new useful properties. We defined the concept of the extended single-valued neutrosophic BCK-subalgebras and showed that for any and a single-valued neutrosophic subset hyper BCK-subalgebra, , is a hyper BCK-subalgebra. Through the concept of fundamental relation , we have generated the single-valued neutrosophic BCK-subalgebras from single-valued neutrosophic hyper BCK-subalgebras, so some categorical properties of single-valued neutrosophic BCK-subalgebras are investigated based on the categorical properties of single-valued neutrosophic hyper BCK-subalgebras. In addition, on any nonempty set, we have constructed at least one single-valued neutrosophic BCK-subalgebra and one extendable single-valued neutrosophic BCK-subalgebra. The concept of neutro hyper BCK-algebra as a generalization of neutro BCK-algebra is introduced in this study, and it is constructed the class of product of neutro hyper BCK-algebras and union of neutro hyper BCK-algebras via hyper BCK-algebras. In study of neutro hyper BCK-algebras, despite having key mathematical tools, there are some limitations. The union of two neutro hyper BCK-algebras is not necessarily; a neutro hyper BCK-algebras so the class of neutro hyper BCK-algebras is not closed under any given algebraic operation. In addition, neutro hyper BCK-algebras are different from single-valued neutrosophic hyper BCK-subalgebras so could not generalize the capabilities of single-valued neutrosophic hyper BCK-subalgebras to neutro hyper BCK-algebras and conversely. In final, we can apply these concepts in real world, especially in some complex (hyper) networks.
We hope that these results are helpful for further studies in single-valued neutrosophic logical algebras. In our future studies, we hope to obtain more results regarding single-valued neutrosophic (hyper) logical-subalgebras, neutro (hyper) logical-subalgebras, and their applications.
Data Availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.