# Single-Valued Neutro Hyper BCK-Subalgebras

Authors: M. Hamidi; F. Smarandache
Journal: Journal of Mathematics
 (2021)

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## 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.


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## Body

## 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 MBJ-neutrosophic hyper BCK-ideals in hyper BCK-algebras, an approach to BMBJ-neutrosophic hyper BCK-ideals of hyper BCK-algebras, structures on doubt neutrosophic ideals of BCK/BCI-algebras under S,T-norms, BMBJ-neutrosophic subalgebras in BCI/BCK-algebras, MBJ-neutrosophic ideals of BCK/BCI-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 hyperBCK−algebras. 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 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 X≠∅. Then a universal algebra X,ϑ,0 of type 2,0 is called a BCK-algebra if, for all, x,y,z∈X:BCI−1xϱyϱxϱzϱzϱy=0,BCI−2xϱxϱyϱy=0,BCI−3xϱx=0,BCI−4xϱy=0 and yϱx=0 imply x=y,BCK−50ϱx=0, where ϱx,y is denoted by xϱy.Definition 2.
(see [1]). Let X≠∅ and P∗X=Y|∅≠Y⊆X. Then for a map ϑ:X2⟶P∗X, a hyperalgebraic system X,ϑ,0 is called a hyper BCK-algebra if, for all, x,y,z∈X:H1xϑzϑyϑz≪xϑy,H2xϑyϑz=xϑzϑy,H3xϑX≪x,H4x≪y and y≪x imply x=y,wherex≪y is defined by 0∈xϑy, ∀A,B⊆H, A≪B⟺∀a∈A∃b∈Bs.ta≪b, AϑB=∪a∈A,b∈Baϑb, and ϑx,y is denoted by xϑy.
We will callX is a weak commutative hyper BCK-algebra if ∀x,y∈X,xϑxϑy∩yϑyϑx≠∅ [21].Theorem 1 (see [1]).
LetX,ϑ,0 be a hyper BCK-algebra. Then ∀x,y,z∈X and A,B⊆X:(i)
0ϑ0=0,0≪x,0ϑx=0, x∈xϑ0 and A≪0⇒A=0(ii)
x≪x, xϑy≪x and y≪z implies that xϑz≪xϑy(iii)
AϑB≪A, A≪A and A⊆B implies A≪BDefinition 3.
(see [22]). Let X,ϑ,0 be a hyper BCK-algebra. A fuzzy set μ:X⟶0,1 is called a fuzzy hyper BCK-subalgebra if ∀x,y∈X,∧μxϑy≥Tminμx,μy.Definition 4.
(see [5]). Let V be a universal set. A neutrosophic subset (NS) X in V is an object having the following form: X=x,TXx,IXx,FXx|x∈V, or X:V⟶0,1×0,1×0,1, which is characterized by a truth-membership function TX, an indeterminacy-membership function IX, and a falsity-membership function FX. There is no restriction on the sum of TXx,IXx, and FXx.


## 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-algebraX,ϑ,0 by X. From now on, for all, x,y∈0,1, Tminx,y=minx,y and Smaxx,y=maxx,y 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 setA=TA,IA,FA in an X is called a single-valued neutrosophic hyper BCK-subalgebra of X, if(i)
∧TAxϑy≥TminTAx,TAy(ii)
∨IAxϑy≤SmaxIAx,IAy(iii)
∨FAxϑy≤SmaxFAx,FAy
The importance of the following theorems is to determine the role and the effect of truth-membership functionTA, indeterminacy-membership function IA, and falsity-membership function FA on the element 0∈A.Theorem 2.
LetA=TA,IA,FA be a single-valued neutrosophic hyper BCK-subalgebra of X. Then(i)
TA0≥TAx(ii)
∧TAxϑ0=TAx(iii)
∧TA0ϑx=TA0Proof.
(i) Letx∈X. Since 0∈xϑx, we get that TA0≥∧TAxϑx≥TminTAx,TAx=TAx.
(ii) Letx∈X. Since x∈xϑ0, we get that TAx≥∧TAxϑ0≥TminTAx,TA0=TAx. So ∧TAxϑ0=TAx.
(iii) Immediate by Theorem1.Theorem 3.
LetA=TA,IA,FA be a single-valued neutrosophic hyper BCK-subalgebra of X. Then(i)
IA0≤IAx(ii)
∨IAxϑ0=IAx(iii)
∨IA0ϑx=IA0Proof.
(i)
Letx∈X. Since 0∈xϑx, we get that IA0≤∨IAxϑx≤SmaxIAx,IAx=IAx.
(ii)
Letx∈X. Since x∈xϑ0, we get that IAx≤∨IAxϑ0≤SmaxIAx,IA0=IAx. So ∨IAxϑ0=IAx.
(iii)
Immediate by Theorem1.Corollary 1.
LetA=TA,IA,FA be a single-valued neutrosophic hyper BCK-subalgebra of X. Then(i)
FA0≤FAx(ii)
∨FAxϑ0=FAx(iii)
∨FA0ϑx=FA0(iv)
TminTAx,IA0,FA0≤TminTA0,IAx,FAx
In the following theorem, we construct single-valued neutrosophic subset on any nonempty set.Theorem 4.
Let0∉X≠∅. Then there exist a hyperoperation “ϑ,” a single-valued neutrosophic subset A=TA,IA,FA of X′=X∪0 such that X′,ϑ,0 is a hyper BCK-algebra and A is a single-valued neutrosophic hyper BCK-subalgebra of X′.Proof.
Letx,y∈X′. Define “ϑ” on X′ by xϑy=0,ifx=0,0,x,ifx=y,x≠0,x,otherwise. Clearly, X′,ϑ,0 is a hyper BCK-algebra. Now, it is easy to see that every single-valued neutrosophic set A=TA,IA,FA that TA0=1,IA0=FA0=0 is a single-valued neutrosophic hyper BCK-subalgebra of X′.
LetSVNh = {A=TA,IA,FA|Ais a single−valued neutrosophic hyper BCK−subalgebra ofX}, whence X is a hyper BCK-algebra and X≥1.Corollary 2.
LetX≠∅. Then X can be extended to a hyper BCK-algebra that SVNh=ℝ.Proof.
LetX=x. Then X,ϑ,x is a hyper BCK-algebra such that xϑx=x. Then for a single-valued neutrosophic set, A=TA,IA,FA by TAx=IAx=FAx=α is a single-valued neutrosophic hyper BCK-subalgebra of X, where α∈0,1. If X≥2; then by Theorem 4, we can construct at least a hyper BCK-subalgebra on X. Now, ∀α∈0,1 define A=TAα,IAα,FAα by(1)TAαx=1,ifx=0,α,ifx≠0,IAαx=0,ifx=0,α,ifx≠0,FAαx=0,ifx=0,α,ifx≠0.
Obviously,A=TAα,IAα,FAα a single-valued neutrosophic hyper BCK-subalgebra of X and so SVNh=0,1.
LetX be a hyper BCK-algebra, A=TA,IA,FA a single-valued neutrosophic hyper BCK-subalgebra of X and α,β,γ∈0,1. Define TAα=x∈X|TAx≥α,IAβ=x∈X|IAx≤β, FAγ=x∈X|FAx≤γ, and Aα,β,γ=x∈X|TAx≥α,IAx≤β,FAx≤γ.
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.
LetA=TA,IA,FA be a single-valued neutrosophic hyper BCK-subalgebra of X. Then(i)
0∈Aα,β,γ=TAα∩IAβ∩FAγ(ii)
Aα,β,γ is a hyper BCK-subalgebra of X(iii)
If0≤α≤α′≤1, then TAα′⊆TAα,IAα′⊇IAα and FAα′⊇FAαProof.
(i) Clearly,Aα,β,γ=Aα∩Aβ∩Aγ and by Theorems 2 and 3, and Corollary 1, we get that 0∈Aα,β,γ.
(ii) Letx,y∈TAα. Then TminTAx,TAy≥α. Now, for any, z∈xϑy,TAz≥infTAxϑy≥TminTAx,TAy≥α. Hence, z∈TAα, and so xϑy⊆TAα. In similar a way, x,y∈IAβ∩FAγ implies that xϑy⊆IAβ∩FAγ. Then Aα,β,γ is a hyper BCK-subalgebra of X.
(iii) Immediate.Corollary 3.
LetA=TA,IA,FA be a single-valued neutrosophic hyper BCK-subalgebra of X. If 0≤α≤α′≤1, then Aα′,α,α is a hyper BCK-subalgebra of Aα,α′,α′.LetX be a hyper BCK-algebra, S be a hyper BCK-subalgebra of X and α,α′,β,β′,γ,γ′∈0,1. Define(2)TAα,α′x=α′,ifx∈S,α,ifx∉S,IAβ,β′x=β′,ifx∈S,β,ifx∉S,FAγ,γ′x=γ′,ifx∈S,γ,ifx∉S.Thus, we have the following theorem.Theorem 6.
LetX be a hyper BCK-algebra and S be a hyper BCK-subalgebra of X. Then(i)
TAα,α′ is a fuzzy hyper BCK-subalgebra of X(ii)
IAβ,β′ is a fuzzy hyper BCK-subalgebra of X(iii)
FAγ,γ′ is a fuzzy hyper BCK-subalgebra of X(iv)
A=TAα,α′,IAβ,β′,FAγ,γ′ is a single-valued neutrosophic hyper BCK-subalgebra of XProof.
(i) Letx,y∈X. If x,y∈S, since S is a hyper subalgebra of X, we get that xϑy⊆S and so(3)∧TAα,α′xϑy≥∧TAα,α′S=α′≥TminTAα,α′x,TAα,α′y.
Ifx∈Sandy∉S or x∈Sandy∈S or x∈Sandy∈S, then ∧TAα,α′xϑy∈α,α′. Thus, ∧TAα,α′xϑy≥TminTAα,α′x,TAα,α′y, and so TAα,α′ is a fuzzy hyper BCK-subalgebra of X.
(ii) and (iii) They are similar to (i).
(iv) Letx,y∈X. If x,y∈S, since S is a hyper BCK-subalgebra of X, we get that xϑy⊆S, and so ∨IAβ,β′xϑy≤∨IAβ,β′S=α′≤SmaxIAβ,β′x,IAβ,β′y. If x∈Sandy∉S or x∈Sandy∈S or x∈Sandy∈S, then ∨IAβ,β′xϑy∈β,β′. Thus, ∨TAβ,β′xϑy≤SmaxIAβ,β′x,IAβ,β′y. In a similar way, we can see that ∨FAγ,γ′xϑy≤SmaxFAγ,γ′x,FAγ,γ′y an by item (i), A=TAα,α′,IAβ,β′,FAγ,γ′ is a single-valued neutrosophic hyper BCK-subalgebra of X.
LetX be a hyper BCK-algebra and x,y∈X. Then xβy⟺∃n∈ℕ,a1,…,an∈Xnand∃u∈ϑa1,…,ansuch thatx,y⊆u. The relation β is a reflexive and symmetric relation but not transitive relation. Let Cβ be the transitive closure of β (the smallest transitive relation such that contains β). Borzooei et al. in [21], proved that for any given weak commutative hyper BCK-algebra X, Cβ is a strongly regular relation on X, and X/Cβ,ϱ,0¯ is a BCK-algebra, where CβxϱCβy=Cβxϑy and 0¯=Cβ0.
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.
LetX be a weak commutative hyper BCK-subalgebra and A=TA,IA,FA be a single-valued neutrosophic hyper BCK-subalgebra of X. Then there exists a single-valued neutrosophic set A¯=TA¯,IA¯,FA¯ of BCK-algebra X/Cβ,ϑ,0¯ that ∀x,y∈X,(i)
TA¯Cβ0≥TA¯Cβx(ii)
ifyCβx, then TA¯Cβx=TA¯Cβy(iii)
IA¯Cβ0≤IA¯Cβx(iv)
ifyCβx, then IA¯Cβx=IA¯Cβy(v)
FA¯Cβ0≤FA¯Cβx(vi)
ifyCβx, then FA¯Cβx=FA¯CβyProof.
Letx,y,t∈X. Then on X/Cβ, define TA¯Cβt=TA0,if0∈Cβx,∧tCβxTAx,otherwise,, IA¯Cβt=IA0,if0∈Cβx,∨tCβxIAx,otherwise,, and FA¯Cβt=FA0,if0∈Cβx,∨tCβxFAx,otherwise,. Using Theorems 2 and 3, we get that:
(i)TA¯Cβ0=TA0≥∧t′CβxTAt′=TA¯Cβx
(ii) SincexCβy and Cβ is transitive, we get that TA¯Cβx=∧tCβxTAt≥∧tCβyTAt=TA¯Cβy
(iii)IA¯Cβ0=IA0≤∨t′CβxIAt′=IA¯Cβx
(iv) SincexCβy and Cβ is transitive, we get that IA¯Cβx=∨tCβxIAt=∨tCβyIAt=IA¯Cβy
(v) and (vi) They are similar to (iii) and (iv), respectively.Theorem 8.
LetX be a weak commutative hyper BCK-subalgebra and A=TA,IA,FA be a single-valued neutrosophic hyper BCK-subalgebra of X. Then there exists a single-valued neutrosophic subset A¯=TA¯,IA¯,FA¯ of BCK-algebra X/Cβ,ϑ,0¯ that ∀x,y∈X:(i)
There existst∈xϑy such that TA¯Cβxϑy=TAt(ii)
There existst′∈xϑy such that IA¯Cβxϑy=IAt(iii)
There existst″∈xϑy such that FA¯Cβxϑy=FAtProof.
(i) Letx,y∈X. Applying Theorem 7,(4)TA¯CβxϱCβy=TA¯Cβxϑy=TA¯Cβm|m∈xϑy=∧sCβmm∈xϑyTAs.
Now, sincesCβm and m∈xϑy, then s∈xϑy, and so there exists t∈xϑy such that TAt=∧sCβmm∈xϑyTAs.
(ii) Letx,y∈X. Then(5)IA¯CβxϱCβy=IA¯Cβxϑy=IA¯Cβn|n∈xϑy=∨tCβnn∈xϑyIAt.
Now, sincetCβn and n∈xϑy, then t∈xϑy, and so there exists t′∈xϑy such that IAt′=∧tCβnn∈xϑyIAt.
(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.
LetX be a weak commutative hyper BCK-algebra and A=TA,IA,FA be a single-valued neutrosophic hyper BCK-subalgebra of X. Then there exists a single-valued neutrosophic BCK-subalgebra B=TB,IB,FB of X/Cβ,FB,Cβ0 that TBϑπ≤TA,IBϑπ≥IAandIBϑFB≥FA or the following diagrams are quasi commutative:(6)X⟶TA01π↓↗TBXCβ,X⟶IA01π↓↗IBXCβ,X⟶FA01π↓↗FBXCβ.Proof.
ChoiceTB=TA¯,IB=IA¯ and FB=FA¯. Then by Theorem 7, (i) ∀x∈X,(7)TBCβ0≥TBCβx,IBCβ0≤IBCβx,FBCβ0≤FBCβx.
(ii) By Theorem8, ∀x,y∈X; there exists t,t′,t″⊆xϑy that(8)TBCβxϑy=TAt,IBCβxϑy=IAt′,FBCβxϑy=FAt″.
So(9)TBCβxϱCβy=TBCβxϑy=TAt≥∧TAxϑy≥TminTAx,TAy≥TminTBCβx,TBCβy,IBCβxϱCβy=IBCβxϑy=IAt′≤∨IAxϑy≤SmaxIAx,IAy≤SmaxIBCβx,IBCβy,FBCβxϱCβy=FBCβxϑy=FAt″≤∨FAxϑy≤SmaxFAx,FAy≤SmaxFBCβx,FBCβy.
Therefore,B=TB,IB,FB is a single-valued neutrosophic BCK-subalgebra of X/Cβ, TBϑπ≤TA,IBϑπ≥IA, and IBϑπ≥FA.
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) LetX,ϱ,0 be a BCK-algebra and Y,ϑ,0 be a hyper BCK-algebra. We say that the BCK-algebra X is derived from the hyper BCK-algebra Y if X is isomorphic to a nontrivial quotient of YX≅Y/Cβ.
(ii) A single-valued neutrosophic BCK-subalgebraA=TA,IA,FA of X is called an extendable single-valued neutrosophic BCK-subalgebra, if there exist a hyper BCK-algebra Y,ϑ,0, a single-valued neutrosophic hyper BCK-subalgebra B=TB,IB,FB of Y, and n∈ℕ such that X,ϑ,A=Y,ϑ,B−n, and BCK-algebra X is derived of hyper BCK-algebra Y. If X=Y and almost everywhere TA,IA,FA=TB,IB,FB (TA,IA,FA=TB,IB,FBa.e that means x;TAx≠TBx,IAx≠IBx,FAx≠FBx=1), 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.
LetX=−1,−2,−3,−4. Then A=TA,IA,FA is a single-valued neutrosophic BCK-subalgebra of BCK-algebra X,ϑ,−1 (see Table 1).
Now, setY=0,−1,−2,−3,−4=X∪0. Then B=TB,IB,FB is a single-valued neutrosophic hyper BCK-subalgebra of Y,ϑ,0 (see Table 2).
Clearly,Y/Cβ≅X, Y=X+1, and so A=TA,IA,FA is an extendable single-valued neutrosophic BCK-subalgebra of X,ϑ,−1.
In the following theorem, we try to generate BCK-algebras based on single-valued neutrosophic hyper BCK-subalgebras.Table 1
ϱ
−
1
−
2
−
3
−
4
−
1
−
1
−
1
−
1
−
1
−
2
−
2
−
1
−
2
−
2
−
3
−
3
−
3
−
1
−
3
−
4
−
4
−
4
−
4
−
1
−
1
−
2
−
3
−
4
T
A
1
0.2
0.4
0.6
I
A
0.1
0.3
0.7
0.9
F
A
0.05
0.25
0.45
0.65Table 2
ϑ
0
−
1
−
2
−
3
−
4
e
0
0
0
0
0
−
1
−
1
0
,
−
1
0
,
−
1
e
,
−
1
0
,
−
1
−
2
−
2
−
2
0
,
−
1
−
2
−
2
−
3
−
3
−
3
−
3
0
,
−
1
−
3
−
4
−
4
−
4
−
4
−
4
0
,
−
1
0
−
1
−
2
−
3
−
4
T
B
1
1
0.2
0.4
0.6
I
B
0.1
0.1
0.3
0.7
0.9
F
B
0.05
0.05
0.25
0.45
0.65Theorem 10.
LetX,ϑ,0 be a hyper BCK-algebra, A=TA,IA,FA be a single-valued neutrosophic hyper BCK-subalgebra of X, and X¯=TAx,IAx,FAx|x∈X. If A is one to one map, then:(i)
There exists a hyperoperation “ϑ′” on X¯ such that X¯,ϑ′,TA0,IA0,FA0 is a hyper BCK-algebra(ii)
There exists a single-valued neutrosophic hyper BCK-subalgebraA¯=TA¯,IA¯,FA¯ of X¯ related to A=TA,IA,FA(iii)
There exists an operation “ϱ” related toϑ on X¯ that X¯,ϱ,TA0,IA0,FA0 is a BCK-algebraProof.
(i) Letx,y∈X. Define a hyperoperation ϑ′ on X¯, by(10)TAx,IAx,FAxϑ′TAy,IAy,FAy=TAxϑy,IAxϑy,FAxϑy.
It can be easily seen thatTAx,IAx,FAx≪′TAy,IAy,FAy⟺x≪y. It is easy to see that X¯,ϑ′,TA0,IA0,FA0 is a hyper BCK-algebra.
(ii) Letx∈X. Define A¯Ax=Ax. Clearly, A¯=TA¯,IA¯,FA¯ is a single-valued neutrosophic hyper BCK-subalgebra of X¯,ϑ′.
(iii) Assumex,y∈X. Define an operation ϱ on X¯ by(11)TAx,IAx,FAxϱTAy,IAy,FAy=TAx,IAx,FAx,ify=0,∨TAxϑy,∧IAxϑy,∧FAxϑyotherwise.
We just proveBCI-4. Let x,y∈X and(12)TAx,IAx,FAxϱTAy,IAy,FAy=TAx,IAx,FAxϱTAy,IAy,FAy=TA0,IA0,FA0.
SinceA is a one to one map, 0∈xϑy and 0∈yϑx. It follows that TAx,IAx,FAx=TAy,IAy,FAy. It is easy to see that BCI-1, BCI-2, BCI-3, and BCK-5 are valid, and so X¯,ϱ,TA0,IA0,FA0 is a BCK-algebra.Corollary 4.
LetX¯,ϑ,TA0,IA0,FA0 be a hyper BCK-algebra and A=TA,IA,FA be a single-valued neutrosophic hyper BCK-subalgebra of X¯. Then there exists a binary operation “ϱ” on X¯, such that X¯,ϱ,TA0,IA0,FA0 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.
LetX be a nonempty set, 0∉X and X′=X∪0. Then there exist a hyperoperation “ϑ” on X′, a hyperoperation “ϑ′” on X′¯, a binary operation “ϱ” on X′¯, a single-valued neutrosophic subset A=TA,IA,FA of X′, and a single-valued neutrosophic subset B=TB,IB,FB of X′¯ that:(i)
X′,ϑ,0 is a hyper BCK-algebra, and A=TA,IA,FA is a single-valued neutrosophic hyper BCK-subalgebra of X′(ii)
X′¯,ϑ′,TA0,IA0,FA0 is a hyper BCK-algebra, and A=TA,IA,FA is a single-valued neutrosophic hyper BCK-subalgebra of X′¯(iii)
X′¯,ϱ,TA0,IA0,FA0 is a BCK-algebra, and B=TB,IB,FB is a single-valued neutrosophic BCK-subalgebra of X′¯(iv)
X′=X′¯+1Proof.
LetX≥2 and b∈X be fixed. For any x,y∈X′, define a binary hyperoperation ϑ on X′ as follows:(13)xϑy=0,ifx=0,0,b,ifx=yandx≠0,b,ifx=bandy=0,0,b,ifx=bandy≠0,x,otherwise.
Now, we show thatX′,ϑ,0 is a hyper BCK-algebra. We just check that conditions (H1) and (H2) are valid.
(H1): Letx,y,z∈X′. If x=0, then xϑzϑyϑz=0ϑyϑz=0≪xϑy. If x=b, then xϑzϑyϑz⊆0,bϑyϑz⊆0,b≪xϑy. If x∉0,b, we consider the following cases:Case 1:x=y≠z. Then xϑzϑyϑz=xϑy=xϑx=0,b≪0,b=xϑy.Case 2:x=z≠y. Then xϑzϑyϑz=0,bϑyϑz=0,b≪x=xϑy.Case 3:y=z≠x. Then xϑzϑyϑz⊆xϑ0,b=0,b≪x=xϑy.Case 4:x≠y≠z. Then xϑzϑyϑz=xϑy=x≪x=xϑy.Case 5:x=y=z. Then xϑzϑyϑz=0,b≪0,b=xϑy.
(H2): Letx,y,z∈X. The proof of xϑyϑz=xϑzϑy is similar to that of (H1), and then it is easy to see that X′,ϑ,0 is a hyper BCK-algebra. Consider a single-valued neutrosophic subset A=TA,IA,FA of X′ such that TA0=TAb=1,IA0=IAb=FA0=FAb=0; by equation (2) and some modifications, we get that(14)∧TAxϑy≥TminTAx,TAy,∨IAxϑy≤SmaxIAx,IAy,∨FAxϑy≤SmaxFAx,FAy.
Hence,A=TA,IA,FA is a single-valued neutrosophic hyper BCK-subalgebra of X′,ϑ,0. Now, ∀x,y∈X; define a hyperoperation ϑ′ on X′¯ by(15)Axϑ′Ay=TAx,IAx,FAxϑ′TAy,IAy,FAy=TAxϑy,IAxϑy,FAxϑy.
Define a single-valued neutrosophic subsetB=TB,IB,FB of X′¯ by(16)BAx=Ax,orTBTAx,IBIAx,FBFAx=TAx,IAx,FAx,and an operation ϱ on X′¯ by(17)TAx,IAx,FAxϱTAy,IAy,FAy=∨TAxϑ′TAy,∧IAxϑ′IAy,∧FAxϑ′FAy.
It can be easily seen thatTAx,IAx,FAx≪′TAy,IAy,FAy⟺x≪y, X′¯,ϑ′,TA0,IA0,FA0 is a hyper BCK-algebra, A=TAx,IAx,FAx is a single-valued neutrosophic hyper BCK-subalgebra of X′¯, X′¯,ϑ,TA0,IA0,FA0 is a BCK-algebra, and B=TBx,IBx,FBx is a single-valued neutrosophic BCK-subalgebra of X′¯, and since TA0=TAb=1,IA0=IAb=FA0=FAb=0, we get that X′=X′¯+1.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 n-ary (for integer n≥1) hyperoperation ϑ:Xn⟶PY is called a neutro hyper operation if it has n-plets in Xn for which the hyperoperation is well-defined ϑa1,a2,…,an∈PY (degree of truth T), n-plets in Xn for which the hyperoperation is indeterminate (degree of indeterminacy I), and n-plets in Xn for which the hyperoperation is outer-defined ϑa1,a2,…,an∉PY (degree of falsehood F), where T,I,F∈0,1, with T,I,F≠1,0,0 that represents the n-ary (total) hyper operation and T,I,F≠0,0,1 that represents the n-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.
LetX≠∅ and P∗X=Y|∅≠Y⊆X. Then for a map ϑ:X2⟶P∗X, a hyperalgebraic system X,ϑ,0 is called a neutro hyper BCK-algebra if it satisfies in the following neutro axioms:(H1) (∃x,y,z∈X that xϑzϑyϑz≪xϑy) and (∃x′,y′,z′∈X that x′ϑz′ϑy′ϑz′≪x′ϑy′ or indeterminate)(H2) (∃x,y,z∈X that xϑyϑz=xϑzϑy) and (∃x′,y′,z′∈X that x′ϑy′ϑz′≠x′ϑz′ϑy′ or indeterminate)(H3) (∃x∈X that xϑX≪x) and (∃x′∈X that x′ϑX≪x′ or indeterminate)(H4) (∃x,y∈X that if x≪y and y≪x imply x=y) and (∃x′,y′∈X that if x′≪y′ and y′≪x′ imply x′≠y′ or indeterminate),wherea≪b is defined by 0∈aϑb, and ∀A,B⊆H, A≪B⟺∀a∈A∃b∈Bs.ta≪b
IfX,ϑ,0 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.
LetX,ϑ,0 be a neutro hyper BCK-algebra, x,y,z∈X and A,B,C⊆X. Then(i)
∃x,y∈X such that xϑy≪x(ii)
∃x,y∈X such that xϑy≪x(iii)
∃x∈X such that x≪x(iv)
∃x∈X such that x≪x(v)
∃A,B⊆X such that A≪A(vi)
∃A,B⊆X such that A≪AProof.
We prove only the item (ii), and other items are similar to it. SinceX,ϑ,0 is a neutro hyper BCK-algebra, there exists x∈Xsuch that xϑX≪X. It follows that there exist a,y∈X such that a∈xϑy and a≪x. Hence, xϑy≪x.Theorem 13.
LetX,ϑ,0 be a neutro hyper BCK-algebra, x,y,z∈X and A,B,C⊆X. Then(i)
ifA≪B, then A∪C≪B∪C(ii)
ifA≪B, then A∪C≪B∪CProof.
(i) Leta∈A be arbitrary. Since A≪B, there exists b∈B such that a≪b. Hence, for a∈A∪C, there exists b∈B∪C such that a≪b and so A∪C≪B∪C.
(ii) SinceA≪B, there exists a∈A such that for all, b∈B, we have a≪b. Hence, there exists a∈A∪C such that for all, b∈B∪C, we get that a≪b and so A∪C≪B∪C.Example 2.
(i) Every neutro BCK-algebraX,ϑ,0 is a neutro hyper BCK-algebra. Since, for all, x,y∈X, can define a hyperoperation ϑ on X by xϑy=xϱy.
(ii) Considerℕ∗=0,1,2,3,…. Define xϑy=0,xifx≤y0x,y=2,3orx,y=3,22x=y=1orx,y=0,1xotherwise. Clearly, ℕ∗,ϑ,0 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.
LetX,ϑ,0 be a hyper BCK-algebra and α∉X. For all, x,y∈X∪α, define ϑα on X∪α by xϑαy=xϑy, where, x,y∈X and whence α∈x,y, define xϑαy is indeterminate or xϑαy∈X∪α.
We show that how to construct neutro hyper BCK-algebras from BCK-algebras.Example 3.
LetX=0,1,2,3,4 and consider Table 3.
Then(i)
Ifa=0, then X,ϑ1,0 is a neutro hyper BCK-algebra and if a=1, then X\3,4,5,ϑ1,0 is a hyper BCK-algebra(ii)
X,ϑ2,0 is a neutro hyper BCK-algebra and X\4,5,ϑ2,0 is a hyper BCK-algebra(iii)
Ifs=z=0,w=3, then X,ϑ3,0 is a neutro hyper BCK-algebra, and for s=1,z=3, X\5,ϑ3,0 is a hyper BCK-algebra. If s=z=0,w=2, then X,ϑ3,0 is a neutro hyper BCK-algebra of type 4
The importance of the following theorem is to construct of neutro hyper BCK-algebra from any given nonempty set.Table 3
Neutro hyper BCK-algebras.
ϑ
1
0
1
2
3
4
5
0
0
0
0
0
2
0
1
1
0
a
2
4
3
2
2
2
0,2
0
2
0
3
3
0
1
2
4
5
4
1
4
2
1
4
3
5
0
4
0
1
4
0
ϑ
2
0
1
2
3
4
5
0
0
0
0
0
2
0
1
1
0,1
0
0,1
4
5
2
2
2
0
2
5
0
3
3
3
3
0
0
0
4
2
1
2
4
1
2
5
5
0
4
0
0
x
ϑ
3
0
1
2
3
4
5
0
0
0
0
0
0
5
1
1
0,2
1
1
s
0
2
2
0,2
0,2
0,2
0,2
3
3
3
3
3
0,2
z
0
4
4
4
4
4
0,2
1
5
2
0
2
2
2
wTheorem 15.
Let0∉X≠∅. Then there exists a hyperoperation “ϑ” on X′=X∪0 such that X′,ϑ,0 is a neutro hyper BCK-algebra.Proof.
Let0∉X≠∅. Using Theorem 4, there exist a hyperoperation “ϑ” on X′=X∪0 such that X′,ϑ,0 is a hyper BCK-algebra. Now, apply Theorem 14; there exist a hyperoperation “ϑ′” on X′=X∪0 such that X′,ϑ′,0 is a neutro hyper BCK-algebra.
LetX1,ϑ1,01 and X2,ϑ2,02 be two neutro hyper BCK-algebras. Define ϑ on X1×X2 by x,yϑx′,y′=xϑ1x′,yϑ2y′, where x,y,x′,y′∈X1×X2 and say that x,y≪x′,y′⟺01,02∈x,yϑx′,y′. The following theorem investigates the properties of partial order relation on product of Neutro hyper BCK algebras.Theorem 16.
LetX1,ϑ1,01 and X2,ϑ2,02 be two neutro hyper BCK-algebras. Then(i)
∀x,y,x′,y′∈X1×X2,x,y≪x′,y′⟺x≪1x′andy≪2y′(ii)
∀x,y,x′,y′∈X1×X2,x,y≪x′,y′⟺x≪1x′ory≪2y′(iii)
∃x,y,x′,y′∈X1×X2,01,02∈x,yϑx′,y′ϑx,y(iv)
∃x,y,x′,y′∈X1×X2,01,02∉x,yϑx′,y′ϑx,yProof.
(i)
Immediate
(ii)
Letx,y,x′,y′∈X1×X2. Then 01,02∈x,yϑx′,y′, if and only if 01,02∈xϑ1x′,yϑ2y′, if and if only 01∉xϑx′ or 02∉yϑy′, and if and only if x≪1x′ory≪2y′
(iii)
SinceX1,ϑ1,01 and X2,ϑ2,02 be two neutro hyper BCK-algebras, there exist x,y∈X1,x′,y′∈X2 such that 01∈xϑyϑx and 02∈x′ϑy′ϑx′. It follows that ∃x,y,x′,y′∈X1×X2,01,02∈x,yϑx′,y′ϑx,y
(iv)
SinceX1,ϑ1,01 and X2,ϑ2,02 be two neutro hyper BCK-algebras, there exist x,y∈X1,x′,y′∈X2 such that 01∉xϑyϑx and 02∉x′ϑy′ϑx′. It follows that ∃x,y,x′,y′∈X1×X2,01,02∈x,yϑx′,y′ϑx,y
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.
LetX1,ϑ1,01 and X2,ϑ2,02 be two neutro hyper BCK-algebras. Then X1×X2,ϑ,01,02 is a neutro hyper BCK-algebra.Proof.
We prove only the item (H4), and other items by Theorem16 are valid. Since X1,ϑ1,01 and X2,ϑ2,02 are neutro hyper BCK-algebras, there exist x1,x2,y1,y2,x1′,x2′,y1′,y2′∈X1×X2 that if x1≪1y1,y1≪1x1, then x1=y1, and if x2≪2y2,y2≪2x2, then x2=y2. Also, if x1′≪1y1′,y1′≪1x1′, then x1≠y1, and if x2′≪2y2′,y2′≪2x2′, then x2≠y2. By (i), it follows that there exist x1,x2,y1,y2,x1′,x2′,y1′,y2′∈X1×X2 that if x1,x2≪y1,y2,y1,y2≪x1,x2, we have x1,x2=y1,y2, and if x1′,x2′≪y1′,y2′,y1′,y2′≪x1′,x2′, we have x1′,x2′≠y1′,y2′.
LetX1,ϑ1,01 and X2,ϑ2,02 be hyper BCK-algebras, where X1∩X2=∅. For some x,y∈X, define a hyperoperations ϑt,ϑs as follows:(18)xϑty=xϑ1y\01,ifx,y∈X1\X2,xϑ2y,ifx,y∈X2\X1,t,ifx∈X1,y∈X2,02,ifx∈X2,y∈X1,xϑsy=xϑ1y,ifx,y∈X1\X2,xϑ2y\02,ifx,y∈X2\X1,s,ifx∈X1,y∈X2,01,ifx∈X2,y∈X1,and 01ϑt01=01ϑt02=02ϑt01=01,01ϑs02=02ϑs01=02ϑs02=02, where 02≠t∈X2,01≠s∈X1. 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.
LetX1,ϑ1,01 and X2,ϑ2,02 be hyper BCK-algebras, where X1∩X2=∅ and X=X1∪X2. Then(i)
For all,A⊆X1,A≪01,t(ii)
For all,A⊆X1,A≪02(iii)
For all,A⊆X1,A≪A, and for all, B⊆X2,B≪B(iv)
For all,A⊆X2,A≪02,s(v)
For all,A⊆X2,A≪01Proof.
(i) LetA⊆X1. Then Aϑt01=∪a∈Aaϑt01=∪a∈Aaϑ01\01. It follows that 01∉Aϑt01, so A≪01. In addition, Aϑtt=∪a∈Aaϑtt=t and 01∉tϑt01. It follows that 01∉Aϑt01, so A≪t.
(ii) LetA⊆X1. Then Aϑt02=∪a∈Aaϑt0t=t and 01∉tϑt02. It follows that 01∉Aϑt01, so A≪02. In addition, Aϑtt=∪a∈Aaϑtt=t and 01∉tϑt01. It follows that 01∉Aϑt01, so A≪t.
(iii) LetA⊆X1 and B⊆X2. Since AϑtA=∪a,a′∈Aaϑta′=∪a,a′∈Aaϑta′\01 and BϑsS=∪b,b′∈Bbϑtb′=∪b,b′∈Bbϑsb′\02, we get that 01∈AϑtA and 02∈BϑsB. Thus A≪A and B≪B.
(iv) and (v) are similar to (i) and (ii), respectively.Theorem 19.
LetX1,ϑ1,01 and X2,ϑ2,02 be hyper BCK-algebras, where X1∩X2=∅ and X=X1∪X2. Then(i)
X,ϑt,01 is a neutro hyper BCK-algebra(ii)
X,ϑs,02 is a neutro hyper BCK-algebraProof.
(i)H1:¯ For some, x,y,z∈X2\X1,xϑtzϑtyϑtz≪xϑty. Since, for x∈X1,xϑ01\01\01ϑt02=t≠02, we get that(19)xϑt01ϑt02ϑt01=xϑ01\01ϑt01=xϑ01\01\01≪02=01ϑt02.
H
2
:
¯ For some, x,y,z∈X2\X1,xϑtyϑtz=xϑtzϑty. In addition, for x∈X1,(20)xϑt02ϑt01=tϑt01=02≠t=xϑ01\01ϑt02=xϑt01ϑt02.
H
3
:
¯ For some, x∈X2X1,xϑtX=xϑX2≪X2=X. Since tϑt01=02 and ∪x∈X101ϑx01ϑt01=∪x∈X101ϑx\01\01, we get that(21)01ϑtX=01ϑtX1∪01ϑtX2=∪x∈X101ϑtx∪∪y∈X201ϑty=∪x∈X101ϑx\01∪t≪01.
H
3
:
¯ Because 01≪01 and 01∈01ϑt02 and 01∈02ϑt01, while 01≠02, we get the item H3:¯ is valid. Therefore, X,ϑt,01 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). LetX=a=China,b=Italy,c=Iran,d=Spain,e=Germany,f=USA be a set of top countries, which are in an economic network. Suppose ϑ is the relations on X, which is described in Table 4, and for x≠y,x∗y=D means that D is the set of countries that benefit from this economic partnership, whence the country x starts to country y, and for x=y, it means that the country x maintains its capital.
Clearly,X,∗,China 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.Table 4
Neutro hyper BCK-algebra of an economic network.
ϑ
a
b
c
d
e
f
a
a
a
a
a
a
f
b
b
a
,
c
b
b
a
a
c
c
a
,
c
a
,
c
a
,
c
a
,
c
d
d
d
d
d
a
,
c
a
a
e
e
e
e
e
a
,
c
b
f
c
a
c
c
c
?
?
?Example 5.
(data network). LetY=a,b,c,d,e be a set of mobile sets, which are in a data network. Suppose ϑ is the relations on Y, which is described in Table 3, and for all, x≠,x∗y=D means that D is a set of mobile sets that receive contents of messages that mobile set x starts to mobile set y, and for x=y, it means that the mobile set x retains its information. In addition, for any y∈Y,TBy,IBy,FBy are the cryptographic power, battery life, and RAM of mobile set y, respectively. Then B=TB,IB,FB is a single-valued neutrosophic hyper BCK-subalgebra of Y,ϑ,a 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 b is the best in this regard.Table 5
Single-valued neutrosophic hyper BCK-subalgebra of a data network.
ϑ
a
b
c
d
e
a
a
a
a
a
a
b
b
a
,
b
a
,
b
e
,
b
a
,
b
c
c
c
a
,
b
c
c
d
d
d
d
a
,
b
d
e
e
e
e
e
a
,
b
a
b
c
d
e
T
B
1
1
0.2
0.4
0.6
I
B
0.1
0.1
0.3
0.7
0.9
F
B
0.05
0.05
0.25
0.45
0.65


## 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). LetX=a=China,b=Italy,c=Iran,d=Spain,e=Germany,f=USA be a set of top countries, which are in an economic network. Suppose ϑ is the relations on X, which is described in Table 4, and for x≠y,x∗y=D means that D is the set of countries that benefit from this economic partnership, whence the country x starts to country y, and for x=y, it means that the country x maintains its capital.
Clearly,X,∗,China 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.Table 4
Neutro hyper BCK-algebra of an economic network.
ϑ
a
b
c
d
e
f
a
a
a
a
a
a
f
b
b
a
,
c
b
b
a
a
c
c
a
,
c
a
,
c
a
,
c
a
,
c
d
d
d
d
d
a
,
c
a
a
e
e
e
e
e
a
,
c
b
f
c
a
c
c
c
?
?
?Example 5.
(data network). LetY=a,b,c,d,e be a set of mobile sets, which are in a data network. Suppose ϑ is the relations on Y, which is described in Table 3, and for all, x≠,x∗y=D means that D is a set of mobile sets that receive contents of messages that mobile set x starts to mobile set y, and for x=y, it means that the mobile set x retains its information. In addition, for any y∈Y,TBy,IBy,FBy are the cryptographic power, battery life, and RAM of mobile set y, respectively. Then B=TB,IB,FB is a single-valued neutrosophic hyper BCK-subalgebra of Y,ϑ,a 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 b is the best in this regard.Table 5
Single-valued neutrosophic hyper BCK-subalgebra of a data network.
ϑ
a
b
c
d
e
a
a
a
a
a
a
b
b
a
,
b
a
,
b
e
,
b
a
,
b
c
c
c
a
,
b
c
c
d
d
d
d
a
,
b
d
e
e
e
e
e
a
,
b
a
b
c
d
e
T
B
1
1
0.2
0.4
0.6
I
B
0.1
0.1
0.3
0.7
0.9
F
B
0.05
0.05
0.25
0.45
0.65


## 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 anyα∈0,1 and a single-valued neutrosophic subset hyper BCK-subalgebra, A=TA,IA,FA, A=TAα,IAα,FAα is a hyper BCK-subalgebra. Through the concept of fundamental relation Cβ, 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.



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*Source: 6656739-2021-09-22.xml*