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Article

Neutrosophic Commutative N -Ideals in BCK-Algebras

1
Department of Mathematics, Jeju National University, Jeju 63243, Korea
2
Department of Mathematics & Science, University of New Mexico, 705 Gurley Ave., Gallup, NM 87301, USA
3
Department of Mathematics Education, Gyeongsang National University, Jinju 52828, Korea
*
Author to whom correspondence should be addressed.
Information 2017, 8(4), 130; https://doi.org/10.3390/info8040130
Submission received: 16 September 2017 / Revised: 6 October 2017 / Accepted: 16 October 2017 / Published: 18 October 2017
(This article belongs to the Special Issue Neutrosophic Information Theory and Applications)

Abstract

:
The notion of a neutrosophic commutative 𝒩-algebras is introduced, and several properties are investigated. Relations between a neutrosophic N -ideal and a neutrosophic commutative N -ideal are discussed. Characterizations of a neutrosophic commutative N -ideal are considered.

1. Introduction

As a generalization of fuzzy sets, Atanassov [1] introduced the degree of nonmembership/falsehood (f) in 1986 and defined the intuitionistic fuzzy set.
Smarandache proposed the term “neutrosophic” because “neutrosophic” etymologically comes from “neutrosophy” [French neutre < Latin neuter, neutral, and Greek sophia, skill/wisdom] which means knowledge of neutral thought, and this third/neutral represents the main distinction between “fuzzy”/“intuitionistic fuzzy” logic/set and “neutrosophic” logic/set, i.e., the included middle component (Lupasco–Nicolescu’s logic in philosophy), i.e., the neutral/indeterminate/unknown part (besides the “truth”/“membership” and “falsehood”/“non-membership” components that both appear in fuzzy logic/set). Smarandache introduced the degree of indeterminacy/neutrality (i) as an independent component in 1995 (published in 1998) and defined the neutrosophic set on three components
(t, i, f) = (truth, indeterminacy, falsehood).
For more details, refer to the site http://fs.gallup.unm.edu/FlorentinSmarandache.htm.
Jun et al. [2] introduced a new function which is called negative-valued function, and constructed N -structures. Khan et al. [3] introduced the notion of neutrosophic N -structure and applied it to a semigroup. Jun et al. [4] applied the notion of neutrosophic N -structure to B C K / B C I -algebras. They introduced the notions of a neutrosophic N -subalgebra and a (closed) neutrosophic N -ideal in a B C K / B C I -algebra, and investigated related properties. They also considered characterizations of a neutrosophic N -subalgebra and a neutrosophic N -ideal, and discussed relations between a neutrosophic N -subalgebra and a neutrosophic N -ideal. They provided conditions for a neutrosophic N -ideal to be a closed neutrosophic N -ideal. B C K -algebras entered into mathematics in 1966 through the work of Imai and Iséki [5], and have been applied to many branches of mathematics, such as group theory, functional analysis, probability theory and topology. Such algebras generalize Boolean rings as well as Boolean D-posets (= M V -algebras). Also, Iséki introduced the notion of a B C I -algebra which is a generalization of a B C K -algebra (see [6]).
In this paper, we introduce the notion of a neutrosophic commutative N -ideal in B C K -algebras, and investigate several properties. We consider relations between a neutrosophic N -ideal and a neutrosophic commutative N -ideal. We discuss characterizations of a neutrosophic commutative N -ideal.

2. Preliminaries

By a BCI-algebra, we mean a system X : = ( X , , 0 ) K ( τ ) in which the following axioms hold:
(I)
( ( x y ) ( x z ) ) ( z y ) = 0 ,
(II)
( x ( x y ) ) y = 0 ,
(III)
x x = 0 ,
(IV)
x y = y x = 0 x = y
for all x , y , z X . If a BCI-algebra X satisfies 0 x = 0 for all x X , then we say that X is a BCK-algebra.
We can define a partial ordering ⪯ by
( x , y X ) ( x y x y = 0 ) .
In a BCK/BCI-algebra X, the following hold:
( x X ) ( x 0 = x ) ,
( x , y , z X ) ( ( x y ) z = ( x z ) y ) .
A B C K -algebra X is said to be commutative if it satisfies the following equality:
( x , y X ) x ( x y ) = y ( y x ) .
A subset I of a B C K / B C I -algebra X is called an ideal of X if it satisfies
0 I ,
( x , y X ) x y I , y I x I .
A subset I of a B C K -algebra X is called a commutative ideal of X if it satisfies (4) and
( x , y , z X ) ( x y ) z I , z I x ( y ( y x ) ) I .
Lemma 1.
An ideal I is commutative if and only if the following assertion is valid.
( x , y X ) x y I x ( y ( y x ) ) I .
We refer the reader to the books [7,8] for further information regarding BCK/BCI-algebras.
For any family { a i i Λ } of real numbers, we define
{ a i i Λ } : = max { a i i Λ } if Λ is finite , sup { a i i Λ } otherwise .
{ a i i Λ } : = min { a i i Λ } if Λ is finite , inf { a i i Λ } otherwise .
Denote by F ( X , [ 1 , 0 ] ) the collection of functions from a set X to [ 1 , 0 ] . We say that an element of F ( X , [ 1 , 0 ] ) is a negative-valued function from X to [ 1 , 0 ] (briefly, N -function on X). By an N -structure, we mean an ordered pair ( X , f ) of X and an N -function f on X (see [2]). A neutrosophic N -structure over a nonempty universe of discourse X (see [3]) is defined to be the structure
X N : = X ( T N , I N , F N ) = x ( T N ( x ) , I N ( x ) , F N ( x ) ) x X
where T N , I N and F N are N -functions on X which are called the negative truth membership function, the negative indeterminacy membership function and the negative falsity membership function, respectively, on X.
Note that every neutrosophic N -structure X N over X satisfies the condition:
( x X ) 3 T N ( x ) + I N ( x ) + F N ( x ) 0 .

3. Neutrosophic Commutative 𝒩-Ideals

In what follows, let X denote a B C K -algebra unless otherwise specified.
Definition 1 
([4]). A neutrosophic N -structure X N over X is called a neutrosophic N -ideal of X if the following assertion is valid.
( x , y X ) T N ( 0 ) T N ( x ) { T N ( x y ) , T N ( y ) } I N ( 0 ) I N ( x ) { I N ( x y ) , I N ( y ) } F N ( 0 ) F N ( x ) { F N ( x y ) , F N ( y ) } .
Definition 2.
A neutrosophic N -structure X N over X is called a neutrosophic commutative N -ideal of X if the following assertions are valid.
( x X ) T N ( 0 ) T N ( x ) , I N ( 0 ) I N ( x ) , F N ( 0 ) F N ( x ) ,
( x , y , z X ) T N ( x ( y ( y x ) ) ) { T N ( ( x y ) z ) , T N ( z ) } I N ( x ( y ( y x ) ) ) { I N ( ( x y ) z ) , I N ( z ) } F N ( x ( y ( y x ) ) ) { F N ( ( x y ) z ) , F N ( z ) } .
Example 1.
Consider a B C K -algebra X = { 0 , 1 , 2 , 3 , 4 } with the Cayley table which is given in Table 1.
The neutrosophic N -structure
X N = 0 ( 0.8 , 0.2 , 0.9 ) , 1 ( 0.3 , 0.9 , 0.5 ) , 2 ( 0.7 , 0.7 , 0.4 ) , 3 ( 0.3 , 0.6 , 0.7 ) , 4 ( 0.5 , 0.3 , 0.1 )
over X is a neutrosophic commutative N -ideal of X.
Theorem 1.
Every neutrosophic commutative N -ideal is a neutrosophic N -ideal.
Proof. 
Let X N be a neutrosophic commutative N -ideal of X. For every x , z X , we have
T N ( x ) = T N ( x ( 0 ( 0 x ) ) ) { T N ( ( x 0 ) z ) , T N ( z ) } = { T N ( x z ) , T N ( z ) } , I N ( x ) = I N ( x ( 0 ( 0 x ) ) ) { I N ( ( x 0 ) z ) , I N ( z ) } = { I N ( x z ) , I N ( z ) } , F N ( x ) = F N ( x ( 0 ( 0 x ) ) ) { F N ( ( x 0 ) z ) , F N ( z ) } = { F N ( x z ) , F N ( z ) }
by putting y = 0 in (11) and using (1). Therefore, X N is a neutrosophic commutative N -ideal of X.  ☐
The converse of Theorem 1 is not true in general as seen in the following example.
Example 2.
Consider a B C K -algebra X = { 0 , 1 , 2 , 3 , 4 } with the Cayley table which is given in Table 2.
The neutrosophic N -structure
X N = 0 ( 0.8 , 0.1 , 0.7 ) , 1 ( 0.7 , 0.6 , 0.6 ) , 2 ( 0.6 , 0.2 , 0.4 ) , 3 ( 0.3 , 0.8 , 0.4 ) , 4 ( 0.3 , 0.8 , 0.4 )
over X is a neutrosophic N -ideal of X. But it is not a neutrosophic commutative N -ideal of X since F N ( 2 ( 3 ( 3 2 ) ) = F N ( 2 ) = 0.4 0.7 = { F N ( ( 2 3 ) 0 ) , F N ( 0 ) } .
We consider characterizations of a neutrosophic commutative N -ideal.
Theorem 2.
Let X N be a neutrosophic N -ideal of X. Then, X N is a neutrosophic commutative N -ideal of X if and only if the following assertion is valid.
( x , y X ) T N ( x ( y ( y x ) ) ) T N ( x y ) , I N ( x ( y ( y x ) ) ) I N ( x y ) , F N ( x ( y ( y x ) ) ) F N ( x y ) .
Proof. 
Assume that X N is a neutrosophic commutative N -ideal of X. The assertion (12) is by taking z = 0 in (11) and using (1) and (10).
Conversely, suppose that a neutrosophic N -ideal X N of X satisfies the condition (12). Then,
( x , y X ) T N ( x y ) { T N ( ( x y ) z ) , T N ( z ) } I N ( x y ) { I N ( ( x y ) z ) , I N ( z ) } F N ( x y ) { F N ( ( x y ) z ) , F N ( z ) } .
It follows that the condition (11) is induced by (12) and (13). Therefore, X N is a neutrosophic commutative N -ideal of X. ☐
Lemma 2 
([4]). For any neutrosophic N -ideal X N of X, we have
( x , y , z X ) x y z T N ( x ) { T N ( y ) , T N ( z ) } I N ( x ) { I N ( y ) , I N ( z ) } F N ( x ) { F N ( y ) , F N ( z ) } .
Theorem 3.
In a commutative B C K -algebra, every neutrosophic N -ideal is a neutrosophic commutative N -ideal.
Proof. 
Let X N be a neutrosophic N -ideal of a commutative B C K -algebra X. For any x , y , z X , we have
( ( x ( y ( y x ) ) ) ( ( x y ) z ) ) z = ( ( x ( y ( y x ) ) ) z ) ( ( x y ) z ) ( x ( y ( y x ) ) ) ( x y ) = ( x ( x y ) ) ( y ( y x ) ) = 0 ,
that is, ( x ( y ( y x ) ) ) ( ( x y ) z ) z . It follows from Lemma 2 that
T N ( x ( y ( y x ) ) ) { T N ( ( x y ) z ) , T N ( z ) } , I N ( x ( y ( y x ) ) ) { I N ( ( x y ) z ) , I N ( z ) } , F N ( x ( y ( y x ) ) ) { F N ( ( x y ) z ) , F N ( z ) } .
Therefore, X N is a neutrosophic commutative N -ideal of X. ☐
Let X N be a neutrosophic N -structure over X and let α , β , γ [ 1 , 0 ] be such that 3 α + β + γ 0 . Consider the following sets.
T N α : = { x X T N ( x ) α } , I N β : = { x X I N ( x ) β } , F N γ : = { x X F N ( x ) γ } .
The set
X N ( α , β , γ ) : = { x X T N ( x ) α , I N ( x ) β , F N ( x ) γ }
is called the ( α , β , γ ) -level set of X N . It is clear that
X N ( α , β , γ ) = T N α I N β F N γ .
Theorem 4.
If X N is a neutrosophic N -ideal of X, then T N α , I N β and F N γ are commutative ideals of X for all α , β , γ [ 1 , 0 ] with 3 α + β + γ 0 whenever they are nonempty.
We call T N α , I N β and F N γ level commutative ideals of X N .
Proof. 
Assume that T N α , I N β and F N γ are nonempty for all α , β , γ [ 1 , 0 ] with 3 α + β + γ 0 . Then, x T N α , y I N β and z F N γ for some x , y , z X . Thus, T N ( 0 ) T N ( x ) α , I N ( 0 ) I N ( y ) β , and F N ( 0 ) F N ( z ) γ , that is, 0 T N α I N β F N γ . Let ( x y ) z T N α and z T N α . Then, T N ( ( x y ) z ) α and T N ( z ) α , which imply that
T N ( x ( y ( y x ) ) ) { T N ( ( x y ) z ) , T N ( z ) } α ,
that is, x ( y ( y x ) ) T N α . If ( a b ) c I N β and c I N β , then I N ( ( a b ) c ) β and I N ( c ) β .
Thus
I N ( a ( b ( b c ) ) ) { I N ( ( a b ) c ) , I N ( c ) } β ,
and so a ( b ( b c ) ) I N β . Finally, suppose that ( u v ) w F N γ and w F N γ . Then, F N ( ( u v ) w ) γ and F N ( w ) γ . Thus,
F N ( u ( v ( v w ) ) ) { F N ( ( u v ) w ) , F N ( w ) } γ ,
that is, u ( v ( v w ) ) F N γ . Therefore, T N α , I N β and F N γ are commutative ideals of X.  ☐
Corollary 1.
Let X N be a neutrosophic N -structure over X and let α , β , γ [ 1 , 0 ] be such that 3 α + β + γ 0 . If X N is a neutrosophic commutative N -ideal of X, then the nonempty ( α , β , γ ) -level set of X N is a commutative ideal of X.
Proof. 
Straightforward. ☐
Lemma 3 
([4]). Let X N be a neutrosophic N -structure over X and assume that T N α , I N β and F N γ are ideals of X for all α , β , γ [ 1 , 0 ] with 3 α + β + γ 0 . Then X N is a neutrosophic N -ideal of X.
Theorem 5.
Let X N be a neutrosophic N -structure over X and assume that T N α , I N β and F N γ are commutative ideals of X for all α , β , γ [ 1 , 0 ] with 3 α + β + γ 0 . Then, X N is a neutrosophic commutative N -ideal of X.
Proof. 
If T N α , I N β and F N γ are commutative ideals of X, then they are ideals of X. Hence, X N is a neutrosophic N -ideal of X by Lemma 3. Let x , y X and α , β , γ [ 1 , 0 ] with 3 α + β + γ 0 such that T N ( x y ) = α , I N ( x y ) = β and F N ( x y ) = γ . Then, x y T N α I N β F N γ . Since T N α I N β F N γ is a commutative ideal of X, it follows from Lemma 1 that x ( y ( y x ) ) T N α I N β F N γ .
Hence
T N ( x ( y ( y x ) ) ) α = T N ( x y ) , I N ( x ( y ( y x ) ) ) β = I N ( x y ) , F N ( x ( y ( y x ) ) ) γ = F N ( x y ) .
Therefore, X N is a neutrosophic commutative N -ideal of X by Theorem 2. ☐
Theorem 6.
Let f : X X be an injective mapping. Given a neutrosophic N -structure X N over X, the following are equivalent.
(1)
X N is a neutrosophic commutative N -ideal of X, satisfying the following condition.
( x X ) T N ( f ( x ) ) = T N ( x ) I N ( f ( x ) ) = I N ( x ) F N ( f ( x ) ) = F N ( x ) .
(2)
T N α , I N β and F N γ are commutative ideals of X N , satisfying the following condition.
f ( T N α ) = T N α , f ( I N β ) = I N β , f ( F N γ ) = F N γ .
Proof. 
Let X N be a neutrosophic commutative N -ideal of X, satisfying the condition (15). Then, T N α , I N β and F N γ are commutative ideals of X N by Theorem 4. Let α Im ( T N ) , β Im ( I N ) , γ Im ( F N ) and x T N α I N β F N γ . Then T N ( f ( x ) ) = T N ( x ) α , I N ( f ( x ) ) = I N ( x ) β and F N ( f ( x ) ) = F N ( x ) γ . Thus, f ( x ) T N α I N β F N γ , which shows that f ( T N α ) T N α , f ( I N β ) I N β and f ( F N γ ) F N γ . Let y X be such that f ( y ) = x . Then, T N ( y ) = T N ( f ( y ) ) = T N ( x ) α , I N ( y ) = I N ( f ( y ) ) = I N ( x ) β and F N ( y ) = F N ( f ( y ) ) = F N ( x ) γ , which imply that y T N α I N β F N γ . Thus, x = f ( y ) f ( T N α ) f ( I N β ) f ( F N γ ) , and so T N α f ( T N α ) , I N β f ( I N β ) and F N γ f ( F N γ ) . Therefore (16) is valid.
Conversely, assume that T N α , I N β and F N γ are commutative ideals of X N , satisfying the condition (16). Then, X N is a neutrosophic commutative N -ideal of X by Theorem 5. Let x , y , z X be such that T N ( x ) = α , I N ( y ) = β and F N ( z ) = γ . Note that
T N ( x ) = α x T N α and x T N α ˜ for all α > α ˜ , I N ( y ) = β y I N β and y I N β ˜ for all β < β ˜ , F N ( z ) = γ z F N γ and z F N γ ˜ for all γ > γ ˜ .
It follows from (16) that f ( x ) T N α , f ( y ) I N β and f ( z ) F N γ . Hence, T N ( f ( x ) ) α , I N ( f ( y ) ) β and F N ( f ( z ) ) γ . Let α ˜ = T N ( f ( x ) ) , β ˜ = I N ( f ( y ) ) and γ ˜ = F N ( f ( z ) ) . If α > α ˜ , then f ( x ) T N α ˜ = f T N α ˜ , and thus x T N α ˜ since f is one to one. This is a contradiction. Hence, T N ( f ( x ) ) = α = T N ( x ) . If β < β ˜ , then f ( y ) I N β ˜ = f I N β ˜ which implies from the injectivity of f that y I N β ˜ , a contradiction. Hence, I N ( f ( x ) ) = β = I N ( x ) . If γ > γ ˜ , then f ( z ) F N γ ˜ = f F N γ ˜ . Since f is one to one, we have z F N γ ˜ which is a contradiction. Thus, F N ( f ( x ) ) = γ = F N ( x ) . This completes the proof.  ☐
For any elements ω t , ω i , ω f X , we consider sets:
X N ω t : = x X T N ( x ) T N ( ω t ) , X N ω i : = x X I N ( x ) I N ( ω i ) , X N ω f : = x X F N ( x ) F N ( ω f ) .
Obviously, ω t X N ω t , ω i X N ω i and ω f X N ω f .
Lemma 4 (
([4]). Let ω t , ω i and ω f be any elements of X. If X N is a neutrosophic N -ideal of X, then X N ω t , X N ω i and X N ω f are ideals of X.
Theorem 7.
Let ω t , ω i and ω f be any elements of X. If X N is a neutrosophic commutative N -ideal of X, then X N ω t , X N ω i and X N ω f are commutative ideals of X.
Proof. 
If X N is a neutrosophic commutative N -ideal of X, then it is a neutrosophic N -ideal of X and so X N ω t , X N ω i and X N ω f are ideals of X by Lemma 4. Let x y X N ω t X N ω i X N ω f for any x , y X . Then, T N ( x y ) T N ( ω t ) , I N ( x y ) T N ( ω i ) and F N ( x y ) F N ( ω f ) . It follows from Theorem 2 that
T N ( x ( y ( y x ) ) ) T N ( x y ) T N ( ω t ) , I N ( x ( y ( y x ) ) ) I N ( x y ) I N ( ω i ) , F N ( x ( y ( y x ) ) ) F N ( x y ) F N ( ω f ) .
Hence, x ( y ( y x ) ) X N ω t X N ω i X N ω f , and therefore X N ω t , X N ω i and X N ω f are commutative ideals of X by Lemma 1. ☐
Theorem 8.
Any commutative ideal of X can be realized as level commutative ideals of some neutrosophic commutative N -ideal of X.
Proof. 
Let A be a commutative ideal of X and let X N be a neutrosophic N -structure over X in which
T N : X [ 1 , 0 ] , x α if x A , 0 otherwise , I N : X [ 1 , 0 ] , x β if x A , 1 otherwise , F N : X [ 1 , 0 ] , x γ if x A , 0 otherwise
where α , γ [ 1 , 0 ) and β ( 1 , 0 ] . Division into the following cases will verify that X N is a neutrosophic commutative N -ideal of X.
If ( x y ) z A and z A , then x ( y ( y x ) A . Thus,
T N ( ( x y ) z ) = T N ( z ) = T N ( x ( y ( y x ) ) ) = α , I N ( ( x y ) z ) = I N ( z ) = I N ( x ( y ( y x ) ) ) = β , F N ( ( x y ) z ) = F N ( z ) = F N ( x ( y ( y x ) ) ) = γ ,
and so (11) is clearly verified.
If ( x y ) z A and z A , then T N ( ( x y ) z ) = T N ( z ) = 0 , I N ( ( x y ) z ) = I N ( z ) = 1 and F N ( ( x y ) z ) = F N ( z ) = 0 . Hence
T N ( x ( y ( y x ) ) ) { T N ( ( x y ) z ) , T N ( z ) } , I N ( x ( y ( y x ) ) ) { I N ( ( x y ) z ) , I N ( z ) } , F N ( x ( y ( y x ) ) ) { F N ( ( x y ) z ) , F N ( z ) } .
If ( x y ) z A and z A , then T N ( ( x y ) z ) = α , T N ( z ) = 0 , I N ( ( x y ) z ) = β , I N ( z ) = 1 , F N ( ( x y ) z ) = γ and F N ( z ) = 0 . Therefore,
T N ( x ( y ( y x ) ) ) { T N ( ( x y ) z ) , T N ( z ) } , I N ( x ( y ( y x ) ) ) { I N ( ( x y ) z ) , I N ( z ) } , F N ( x ( y ( y x ) ) ) { F N ( ( x y ) z ) , F N ( z ) } .
Similarly, if ( x y ) z A and z A , then (11) is verified. Therefore, X N is a neutrosophic commutative N -ideal of X. Obviously, T N α = A , I N β = A and F N γ = A . This completes the proof.  ☐

4. Conclusions

In order to deal with the negative meaning of information, Jun et al. [2] have introduced a new function which is called negative-valued function, and constructed N -structures. The concept of neutrosophic set (NS) has been developed by Smarandache in [9,10] as a more general platform which extends the concepts of the classic set and fuzzy set, intuitionistic fuzzy set and interval valued intuitionistic fuzzy set. In this article, we have introduced the notion of a neutrosophic commutative N -ideal in B C K -algebras, and investigated several properties. We have considered relations between a neutrosophic N -ideal and a neutrosophic commutative N -ideal. We have discussed characterizations of a neutrosophic commutative N -ideal.

Acknowledgments

The first author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (No. 2016R1D1A1B02006812). The authors wish to thank the anonymous reviewers for their valuable suggestions.

Author Contributions

Y.B. Jun initiated the main idea of the work and wrote the paper. S.Z. Song and Y.B. Jun conceived and designed the new definitions and results. F. Smarandache and S.Z. Song performed finding examples and checking contents. All authors have read and approved the final manuscript for submission.

Conflicts of Interest

The authors declare no conflict of interest.

References

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  10. Smarandache, F. Neutrosophic set-a generalization of the intuitionistic fuzzy set. Int. J. Pure Appl. Math. 2005, 24, 287–297. [Google Scholar]
Table 1. Cayley table for the binary operation “*”.
Table 1. Cayley table for the binary operation “*”.
*01234
000000
110111
222022
333303
444440
Table 2. Cayley table for the binary operation “*”
Table 2. Cayley table for the binary operation “*”
*01234
000000
110100
222000
333300
444430

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Song, S.-Z.; Smarandache, F.; Jun, Y.B. Neutrosophic Commutative N -Ideals in BCK-Algebras. Information 2017, 8, 130. https://doi.org/10.3390/info8040130

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Song S-Z, Smarandache F, Jun YB. Neutrosophic Commutative N -Ideals in BCK-Algebras. Information. 2017; 8(4):130. https://doi.org/10.3390/info8040130

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Song, Seok-Zun, Florentin Smarandache, and Young Bae Jun. 2017. "Neutrosophic Commutative N -Ideals in BCK-Algebras" Information 8, no. 4: 130. https://doi.org/10.3390/info8040130

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