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Article

Neutrosophic N-Structures Applied to BCK/BCI-Algebras

by
Young Bae Jun
1,
Florentin Smarandache
2 and
Hashem Bordbar
3,*
1
Department of Mathematics Education, Gyeongsang National University, Jinju 52828, Korea
2
Mathematics & Science Department, University of New Mexico, 705 Gurley Ave., Gallup, NM 87301, USA
3
Department of Mathematics, Shiraz University, Shiraz 7616914111, Iran
*
Author to whom correspondence should be addressed.
Information 2017, 8(4), 128; https://doi.org/10.3390/info8040128
Submission received: 12 September 2017 / Accepted: 6 October 2017 / Published: 16 October 2017
(This article belongs to the Special Issue Neutrosophic Information Theory and Applications)

Abstract

:
Neutrosophic N -structures with applications in B C K / B C I -algebras is discussed. The notions of a neutrosophic N -subalgebra and a (closed) neutrosophic N -ideal in a B C K / B C I -algebra are introduced, and several related properties are investigated. Characterizations of a neutrosophic N -subalgebra and a neutrosophic N -ideal are considered, and relations between a neutrosophic N -subalgebra and a neutrosophic N -ideal are stated. Conditions for a neutrosophic N -ideal to be a closed neutrosophic N -ideal are provided.

1. Introduction

B C K -algebras entered into mathematics in 1966 through the work of Imai and Iséki [1], and they 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). Additionally, Iséki introduced the notion of a B C I -algebra, which is a generalization of a B C K -algebra (see [2]).
A (crisp) set A in a universe X can be defined in the form of its characteristic function μ A : X { 0 , 1 } yielding the value 1 for elements belonging to the set A and the value 0 for elements excluded from the set A . So far, most of the generalizations of the crisp set have been conducted on the unit interval [ 0 , 1 ] , and they are consistent with the asymmetry observation. In other words, the generalization of the crisp set to fuzzy sets relied on spreading positive information that fit the crisp point { 1 } into the interval [ 0 , 1 ] . Because no negative meaning of information is suggested, we now feel a need to deal with negative information. To do so, we also feel a need to supply a mathematical tool. To attain such an object, Jun et al. [3] introduced a new function, called a negative-valued function, and constructed N -structures. Zadeh [4] introduced the degree of membership/truth (t) in 1965 and defined the fuzzy set. As a generalization of fuzzy sets, Atanassov [5] introduced the degree of nonmembership/falsehood (f) in 1986 and defined the intuitionistic fuzzy 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 following site:
In this paper, we discuss a neutrosophic N -structure with an application to B C K / B C I -algebras. We introduce the notions of a neutrosophic N -subalgebra and a (closed) neutrosophic N -ideal in a B C K / B C I -algebra, and investigate related properties. We consider characterizations of a neutrosophic N -subalgebra and a neutrosophic N -ideal. We discuss relations between a neutrosophic N -subalgebra and a neutrosophic N -ideal. We provide conditions for a neutrosophic N -ideal to be a closed neutrosophic N -ideal.

2. Preliminaries

We let K ( τ ) be the class of all algebras with type τ = ( 2 , 0 ) . A BCI-algebra refers to a system X : = ( X , , θ ) K ( τ ) in which the following axioms hold:
(I)
( ( x y ) ( x z ) ) ( z y ) = θ ,
(II)
( x ( x y ) ) y = θ ,
(III)
x x = θ ,
(IV)
x y = y x = θ x = y .
for all x , y , z X . If a BCI-algebra X satisfies θ x = θ 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 = θ )
In a BCK/BCI-algebra X, the following hold:
( x X ) ( x θ = x )
( x , y , z X ) ( ( x y ) z = ( x z ) y )
A non-empty subset S of a B C K / B C I -algebra X is called a subalgebra of X if x y S for all x , y S .
A subset I of a B C K / B C I -algebra X is called an ideal of X if it satisfies the following:
(I1)
0 I ,
(I2)
( x , y X ) ( x y I , y I x I ) .
We refer the reader to the books [6,7] 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
We 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). An N -structure refers to an ordered pair ( X , f ) of X and an N -function f on X (see [3]). In what follows, we let X denote the nonempty universe of discourse unless otherwise specified.
A neutrosophic N -structure over X (see [8]) 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.
We 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. Application in BCK / BCI -Algebras

In this section, we take a B C K / B C I -algebra X as the universe of discourse unless otherwise specified.
Definition 1.
A neutrosophic N -structure X N over X is called a neutrosophic N -subalgebra of X if the following condition is valid:
( x , y X ) T N ( x y ) { T N ( x ) , T N ( y ) } I N ( x y ) { I N ( x ) , I N ( y ) } F N ( x y ) { F N ( x ) , F N ( y ) }
Example 1.
Consider a B C K -algebra X = { θ , a , b , c } with the following Cayley table.
* θ a b c
θθθθθ
aaθθa
bbaθb
ccccθ
The neutrosophic N -structure
X N = θ ( 0.7 , 0.2 , 0.6 ) , a ( 0.5 , 0.3 , 0.4 ) , b ( 0.5 , 0.3 , 0.4 ) , c ( 0.3 , 0.8 , 0.5 )
over X is a neutrosophic N -subalgebra 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 . Note that
X N ( α , β , γ ) = T N α I N β F N γ
Theorem 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 N -subalgebra of X, then the nonempty ( α , β , γ ) -level set of X N is a subalgebra of X.
Proof. 
Let α , β , γ [ 1 , 0 ] be such that 3 α + β + γ 0 and X N ( α , β , γ ) . If x , y X N ( α , β , γ ) , then T N ( x ) α , I N ( x ) β , F N ( x ) γ , T N ( y ) α , I N ( y ) β and F N ( y ) γ . It follows from Equation (4) that
T N ( x y ) { T N ( x ) , T N ( y ) } α ,
I N ( x y ) { I N ( x ) , I N ( y ) } β , and
F N ( x y ) { F N ( x ) , F N ( y ) } γ .
Hence, x y X N ( α , β , γ ) , and therefore X N ( α , β , γ ) is a subalgebra of X. ☐
Theorem 2.
Let X N be a neutrosophic N -structure over X and assume that T N α , I N β and F N γ are subalgebras of X for all α , β , γ [ 1 , 0 ] with 3 α + β + γ 0 . Then X N is a neutrosophic N -subalgebra of X.
Proof. 
Assume that there exist a , b X such that T N ( a b ) > { T N ( a ) , T N ( b ) } . Then T N ( a b ) > t α { T N ( a ) , T N ( b ) } for some t α [ 1 , 0 ) . Hence a , b T N t α but a b T N t α , which is a contradiction. Thus
T N ( x y ) { T N ( x ) , T N ( y ) }
for all x , y X . If I N ( a b ) < { I N ( a ) , I N ( b ) } for some a , b X , then
I N ( a b ) < t β < { I N ( a ) , I N ( b ) }
where t β : = 1 2 I N ( a b ) + { I N ( a ) , I N ( b ) } . Thus a , b I N t β and a b I N t β , which is a contradiction. Therefore
I N ( x y ) { I N ( x ) , I N ( y ) }
for all x , y X . Now, suppose that there exist a , b X and t γ [ 1 , 0 ) such that
F N ( a b ) > t γ { F N ( a ) , F N ( b ) }
Then a , b F N t γ and a b F N t γ , which is a contradiction. Hence
F N ( x y ) { F N ( x ) , F N ( y ) }
for all x , y X . Therefore X N is a neutrosophic N -subalgebra of X. ☐
Because [ 1 , 0 ] is a completely distributive lattice with respect to the usual ordering, we have the following theorem.
Theorem 3.
If { X N i i N } is a family of neutrosophic N -subalgebras of X, then { X N i i N } , forms a complete distributive lattice.
Proposition 1.
If a neutrosophic N -structure X N over X is a neutrosophic N -subalgebra of X, then T N ( θ ) T N ( x ) , I N ( θ ) I N ( x ) and F N ( θ ) F N ( x ) for all x X .
Proof. 
Straightforward. ☐
Theorem 4.
Let X N be a neutrosophic N -subalgebra of X. If there exists a sequence { a n } in X such that lim n T N ( a n ) = 1 , lim n I N ( a n ) = 0 and lim n F N ( a n ) = 1 , then T N ( θ ) = 1 , I N ( θ ) = 0 and F N ( θ ) = 1 .
Proof. 
By Proposition 1, we have T N ( θ ) T N ( x ) , I N ( θ ) I N ( x ) and F N ( θ ) F N ( x ) for all x X . Hence T N ( θ ) T N ( a n ) , I N ( a n ) I N ( θ ) and F N ( θ ) F N ( a n ) for every positive integer n. It follows that
1 T N ( θ ) lim n T N ( a n ) = 1 0 I N ( θ ) lim n I N ( a n ) = 0 1 F N ( θ ) lim n F N ( a n ) = 1
Hence T N ( θ ) = 1 , I N ( θ ) = 0 and F N ( θ ) = 1 . ☐
Proposition 2.
If every neutrosophic N -subalgebra X N of X satisfies:
T N ( x y ) T N ( y ) , I N ( x y ) I N ( y ) , F N ( x y ) F N ( y )
for all x , y X , then X N is constant.
Proof. 
Using Equations (1) and (5), we have T N ( x ) = T N ( x θ ) T N ( θ ) , I N ( x ) = I N ( x θ ) I N ( θ ) and F N ( x ) = F N ( x θ ) F N ( θ ) for all x X . It follows from Proposition 1 that T N ( x ) = T N ( θ ) , I N ( x ) = I N ( θ ) and F N ( x ) = F N ( θ ) for all x X . Therefore X N is constant. ☐
Definition 2.
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 ( θ ) T N ( x ) { T N ( x y ) , T N ( y ) } I N ( θ ) I N ( x ) { I N ( x y ) , I N ( y ) } F N ( θ ) F N ( x ) { F N ( x y ) , F N ( y ) }
Example 2.
The neutrosophic N -structure X N over X in Example 1 is a neutrosophic N -ideal of X.
Example 3.
Consider a B C I -algebra X : = Y × Z where ( Y , , θ ) is a B C I -algebra and ( Z , , 0 ) is the adjoint B C I -algebra of the additive group ( Z , + , 0 ) of integers (see [6]). Let X N be a neutrosophic N -structure over X given by
X N = x ( α , 0 , γ ) x Y × ( N { 0 } ) x ( 0 , β , 0 ) x Y × ( N { 0 } )
where α , γ [ 1 , 0 ) and β ( 1 , 0 ] . Then X N is a neutrosophic N -ideal of X.
Proposition 3.
Every neutrosophic N -ideal X N of X satisfies the following assertions:
( x , y X ) x y T N ( x ) T N ( y ) , I N ( x ) I N ( y ) , F N ( x ) F N ( y )
Proof. 
Let x , y X be such that x y . Then x y = θ , and so
T N ( x ) { T N ( x y ) , T N ( y ) } = { T N ( θ ) , T N ( y ) } = T N ( y )
I N ( x ) { I N ( x y ) , I N ( y ) } = { I N ( θ ) , I N ( y ) } = I N ( y )
F N ( x ) { F N ( x y ) , F N ( y ) } = { F N ( θ ) , F N ( y ) } = F N ( y )
This completes the proof. ☐
Proposition 4.
Let X N be a neutrosophic N -ideal of X. Then
(1)
T N ( x y ) T N ( ( x y ) y ) T N ( ( x z ) ( y z ) ) T N ( ( x y ) z )
(2)
I N ( x y ) I N ( ( x y ) y ) I N ( ( x z ) ( y z ) ) I N ( ( x y ) z )
(3)
F N ( x y ) F N ( ( x y ) y ) F N ( ( x z ) ( y z ) ) F N ( ( x y ) z )
for all x , y , z X .
Proof. 
Note that
( ( x ( y z ) ) z ) z ( x y ) z
for all x , y , z X . Assume that T N ( x y ) T N ( ( x y ) y ) , I N ( x y ) I N ( ( x y ) y ) and F N ( x y ) F N ( ( x y ) y ) for all x , y X . It follows from Equation (2) and Proposition 3 that
T N ( ( x z ) ( y z ) ) = T N ( ( x ( y z ) ) z ) T N ( ( ( x ( y z ) ) z ) z ) T N ( ( x y ) z )
I N ( ( x z ) ( y z ) ) = I N ( ( x ( y z ) ) z ) I N ( ( ( x ( y z ) ) z ) z ) I N ( ( x y ) z )
and
F N ( ( x z ) ( y z ) ) = F N ( ( x ( y z ) ) z ) F N ( ( ( x ( y z ) ) z ) z ) F N ( ( x y ) z )
for all x , y X .
Conversely, suppose
T N ( ( x z ) ( y z ) ) T N ( ( x y ) z ) I N ( ( x z ) ( y z ) ) I N ( ( x y ) z ) F N ( ( x z ) ( y z ) ) F N ( ( x y ) z )
for all x , y , z X . If we substitute z for y in Equation (9), then
T N ( x z ) = T N ( ( x z ) θ ) = T N ( ( x z ) ( z z ) ) T N ( ( x z ) z ) I N ( x z ) = I N ( ( x z ) θ ) = I N ( ( x z ) ( z z ) ) I N ( ( x z ) z ) F N ( x z ) = F N ( ( x z ) θ ) = F N ( ( x z ) ( z z ) ) F N ( ( x z ) z )
for all x , z X by using (III) and Equation (1). ☐
Theorem 5.
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 N -ideal of X, then the nonempty ( α , β , γ ) -level set of X N is an ideal of X.
Proof. 
Assume that X N ( α , β , γ ) for α , β , γ [ 1 , 0 ] with 3 α + β + γ 0 . Clearly, θ X N ( α , β , γ ) . Let x , y X be such that x y X N ( α , β , γ ) and y X N ( α , β , γ ) . Then T N ( x y ) α , I N ( x y ) β , F N ( x y ) γ , T N ( y ) α , I N ( y ) β and F N ( y ) γ . It follows from Equation (6) that
T N ( x ) { T N ( x y ) , T N ( y ) } α I N ( x ) { I N ( x y ) , I N ( y ) } β F N ( x ) { F N ( x y ) , F N ( y ) } γ
so that x X N ( α , β , γ ) . Therefore X N ( α , β , γ ) is an ideal of X. ☐
Theorem 6.
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.
Proof. 
If there exist a , b , c X such that T N ( θ ) > T N ( a ) , I N ( θ ) < I N ( b ) and F N ( θ ) > F N ( c ) , respectively, then T N ( θ ) > a t T N ( a ) , I N ( θ ) < b i I N ( b ) and F N ( θ ) > c f F N ( c ) for some a t , c f [ 1 , 0 ) and b i ( 1 , 0 ] . Then θ T N a t , θ I N b i and θ F N c f . This is a contradiction. Hence, T N ( θ ) T N ( x ) , I N ( θ ) I N ( x ) and F N ( θ ) F N ( x ) for all x X . Assume that there exist a t , b t , a i , b i , a f , b f X such that T N ( a t ) > { T N ( a t b t ) , T N ( b t ) } , I N ( a i ) < { I N ( a i b i ) , I N ( b i ) } and F N ( a f ) > { F N ( a f b f ) , F N ( b f ) } . Then there exist s t , s f [ 1 , 0 ) and s i ( 1 , 0 ] such that
T N ( a t ) > s t { T N ( a t b t ) , T N ( b t ) } I N ( a i ) < s i { I N ( a i b i ) , I N ( b i ) } F N ( a f ) > s f { F N ( a f b f ) , F N ( b f ) }
It follows that a t b t T N s t , b t T N s t , a i b i I N s i , b i I N s i , a f b f F N s f and b f F N s f . However, a t T N s t , a i I N s i and a f F N s f . This is a contradiction, and so
T N ( x ) { T N ( x y ) , T N ( y ) } I N ( x ) { I N ( x y ) , I N ( y ) } F N ( x ) { F N ( x y ) , F N ( y ) }
for all x , y X . Therefore X N is a neutrosophic N -ideal of X. ☐
Proposition 5.
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 ) }
Proof. 
Let x , y , z X be such that x y z . Then ( x y ) z = θ , and so
T N ( x y ) { T N ( ( x y ) z ) , T N ( z ) } = { T N ( θ ) , T N ( z ) } = T N ( z ) I N ( x y ) { I N ( ( x y ) z ) , I N ( z ) } = { I N ( θ ) , I N ( z ) } = I N ( z ) F N ( x y ) { F N ( ( x y ) z ) , F N ( z ) } = { F N ( θ ) , F N ( z ) } = F N ( z )
It follows that
T N ( x ) { T N ( x y ) , T N ( y ) } { T N ( y ) , T N ( z ) } I N ( x ) { I N ( x y ) , I N ( y ) } { I N ( y ) , I N ( z ) } F N ( x ) { F N ( x y ) , F N ( y ) } { F N ( y ) , F N ( z ) }
This completes the proof. ☐
Theorem 7.
In a B C K -algebra, every neutrosophic N -ideal is a neutrosophic N -subalgebra.
Proof. 
Let X N be a neutrosophic N -ideal of a B C K -algebra X. For any x , y X , we have
T N ( x y ) { T N ( ( x y ) x ) , T N ( x ) } = { T N ( ( x x ) y ) , T N ( x ) } = { T N ( θ y ) , T N ( x ) } = { T N ( θ ) , T N ( x ) } { T N ( x ) , T N ( y ) }
I N ( x y ) { I N ( ( x y ) x ) , I N ( x ) } = { I N ( ( x x ) y ) , I N ( x ) } = { I N ( θ y ) , I N ( x ) } = { I N ( θ ) , I N ( x ) } { I N ( y ) , I N ( x ) }
and
F N ( x y ) { F N ( ( x y ) x ) , F N ( x ) } = { F N ( ( x x ) y ) , F N ( x ) } = { F N ( θ y ) , F N ( x ) } = { F N ( θ ) , F N ( x ) } { F N ( x ) , F N ( y ) }
Hence X N is a neutrosophic N -subalgebra of a B C K -algebra X. ☐
The converse of Theorem 7 may not be true in general, as seen in the following example.
Example 4.
Consider a B C K -algebra X = { θ , 1 , 2 , 3 , 4 } with the following Cayley table.
* θ 1234
θθθθθθ
11θθθθ
221θ1θ
3333θθ
44443θ
Let X N be a neutrosophic N -structure over X, which is given as follows:
X N = { θ ( 0.8 , 0 , 1 ) , 1 ( 0.8 , 0.2 , 0.9 ) , 2 ( 0.2 , 0.6 , 0.5 ) , 3 ( 0.7 , 0.4 , 0.7 ) , 4 ( 0.4 , 0.8 , 0.3 ) }
Then X N is a neutrosophic N -subalgebra of X, but it is not a neutrosophic N -ideal of X as T N ( 2 ) = 0.2 > 0.7 = { T N ( 2 3 ) , T N ( 3 ) } , I N ( 4 ) = 0.8 < 0.4 = { I N ( 4 3 ) , I N ( 3 ) } , or F N ( 4 ) = 0.3 > 0.7 = { F N ( 4 3 ) , F N ( 3 ) } .
Theorem 7 is not valid in a B C I -algebra; that is, if X is a B C I -algebra, then there is a neutrosophic N -ideal that is not a neutrosophic N -subalgebra, as seen in the following example.
Example 5.
Consider the neutrosophic N -ideal X N of X in Example 3. If we take x : = ( θ , 0 ) and y : = ( θ , 1 ) in Y × ( N { 0 } ) , then x y = ( θ , 0 ) ( θ , 1 ) = ( θ , 1 ) Y × ( N { 0 } ) . Hence
T N ( x y ) = 0 > α = { T N ( x ) , T N ( y ) } I N ( x y ) = β < 0 = { I N ( x ) , I N ( y ) } or F N ( x y ) = 0 > γ = { F N ( x ) , F N ( y ) }
Therefore X N is not a neutrosophic N -subalgebra of X.
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 )
Clearly, ω t X N ω t , ω i X N ω i and ω f X N ω f .
Theorem 8.
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.
Proof. 
Clearly, θ X N ω t , θ X N ω i and θ X N ω f . Let x , y X be such that x y X N ω t X N ω i X N ω f and y X N ω t X N ω i X N ω f . Then
T N ( x y ) T N ( ω t ) , T N ( y ) T N ( ω t ) I N ( x y ) I N ( ω i ) , I N ( y ) I N ( ω i ) F N ( x y ) F N ( ω f ) , F N ( y ) F N ( ω f )
It follows from Equation (6) that
T N ( x ) { T N ( x y ) , T N ( y ) } T N ( ω t ) I N ( x ) { I N ( x y ) , I N ( y ) } I N ( ω i ) F N ( x ) { F N ( x y ) , F N ( y ) } F N ( ω f )
Hence 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 ideals of X. ☐
Theorem 9.
Let ω t , ω i , ω f X and let X N be a neutrosophic N -structure over X. Then
(1)
If X N ω t , X N ω i and X N ω f are ideals of X, then the following assertion is valid:
( x , y , z X ) T N ( x ) { T N ( y z ) , T N ( z ) } T N ( x ) T N ( y ) I N ( x ) { I N ( y z ) , I N ( z ) } I N ( x ) I N ( y ) F N ( x ) { F N ( y z ) , F N ( z ) } F N ( x ) F N ( y )
(2)
If X N satisfies Equation (11) and
( x X ) T N ( θ ) T N ( x ) , I N ( θ ) I N ( x ) , F N ( θ ) F N ( x )
then X N ω t , X N ω i and X N ω f are ideals of X for all ω t Im ( T N ) , ω i Im ( I N ) and ω f Im ( F N ) .
Proof. 
(1) Assume that X N ω t , X N ω i and X N ω f are ideals of X for ω t , ω i , ω f X . Let x , y , z X be such that T N ( x ) { T N ( y z ) , T N ( z ) } , I N ( x ) { I N ( y z ) , I N ( z ) } and F N ( x ) { F N ( y z ) , F N ( z ) } . Then y z X N ω t X N ω i X N ω f and z X N ω t X N ω i X N ω f , where ω t = ω i = ω f = x . It follows from (I2) that y X N ω t X N ω i X N ω f for ω t = ω i = ω f = x . Hence T N ( y ) T N ( ω t ) = T N ( x ) , I N ( y ) I N ( ω i ) = I N ( x ) and F N ( y ) F N ( ω f ) = F N ( x ) .
(2) Let ω t Im ( T N ) , ω i Im ( I N ) and ω f Im ( F N ) and suppose that X N satisfies Equations (11) and (12). Clearly, θ X N ω t X N ω i X N ω f by Equation (12). Let x , y X be such that x y X N ω t X N ω i X N ω f and y X N ω t X N ω i X N ω f . Then
T N ( x y ) T N ( ω t ) , T N ( y ) T N ( ω t ) I N ( x y ) I N ( ω i ) , I N ( y ) I N ( ω i ) F N ( x y ) F N ( ω f ) , F N ( y ) F N ( ω f )
which implies that { T N ( x y ) , T N ( y ) } T N ( ω t ) , { I N ( x y ) , I N ( y ) } I N ( ω i ) , and { F N ( x y ) , F N ( y ) } F N ( ω f ) . It follows from Equation (11) that T N ( ω t ) T N ( x ) , I N ( ω i ) I N ( x ) and F N ( ω f ) F N ( x ) . Thus, 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 ideals of X.☐
Definition 3.
A neutrosophic N -ideal X N of X is said to be closed if it is a neutrosophic N -subalgebra of X.
Example 6.
Consider a B C I -algebra X = { θ , 1 , a , b , c } with the following Cayley table.
* θ 1 a b c
θθθabc
11θabc
aaaθcb
bbbcθa
cccbaθ
Let X N be a neutrosophic N -structure over X which is given as follows:
X N = { θ ( 0.9 , 0.3 , 0.8 ) , 1 ( 0.7 , 0.4 , 0.7 ) , a ( 0.6 , 0.8 , 0.3 ) , b ( 0.2 , 0.6 , 0.3 ) , c ( 0.2 , 0.8 , 0.5 ) }
Then X N is a closed neutrosophic N -ideal of X.
Theorem 10.
Let X be a B C I -algebra, For any α 1 , α 2 , γ 1 , γ 2 [ 1 , 0 ) and β 1 , β 2 ( 1 , 0 ] with α 1 < α 2 , γ 1 < γ 2 and β 1 > β 2 , let X N : = X ( T N , I N , F N ) be a neutrosophic N -structure over X given as follows:
T N : X [ 1 , 0 ] , x α 1 if x X + α 2 otherwise I N : X [ 1 , 0 ] , x β 1 if x X + β 2 otherwise F N : X [ 1 , 0 ] , x γ 1 if x X + γ 2 otherwise
where X + = { x X θ x } . Then X N is a closed neutrosophic N -ideal of X.
Proof. 
Because θ X + , we have T N ( θ ) = α 1 T N ( x ) , I N ( θ ) = β 1 I N ( x ) and F N ( θ ) = γ 1 F N ( x ) for all x X . Let x , y X . If x X + , then
T N ( x ) = α 1 { T N ( x y ) , T N ( y ) } I N ( x ) = β 1 { I N ( x y ) , I N ( y ) } F N ( x ) = γ 1 { F N ( x y ) , F N ( y ) }
Suppose that x X + . If x y X + then y X + , and if y X + then x y X + . In either case, we have
T N ( x ) = α 2 = { T N ( x y ) , T N ( y ) } I N ( x ) = β 2 = { I N ( x y ) , I N ( y ) } F N ( x ) = γ 2 = { F N ( x y ) , F N ( y ) }
For any x , y X , if any one of x and y does not belong to X + , then
T N ( x y ) α 2 = { T N ( x ) , T N ( y ) } I N ( x y ) β 2 = { I N ( x ) , I N ( y ) } F N ( x y ) γ 2 = { F N ( x ) , F N ( y ) }
If x , y X + , then x y X + . Hence
T N ( x y ) = α 1 = { T N ( x ) , T N ( y ) } I N ( x y ) = β 1 = { I N ( x ) , I N ( y ) } F N ( x y ) = γ 1 = { F N ( x ) , F N ( y ) }
Therefore X N is a closed neutrosophic N -ideal of X.  ☐
Proposition 6.
Every closed neutrosophic N -ideal X N of a B C I -algebra X satisfies the following condition:
( x X ) T N ( θ x ) T N ( x ) , I N ( θ x ) I N ( x ) , F N ( θ x ) F N ( x )
Proof. 
Straightforward. ☐
We provide conditions for a neutrosophic N -ideal to be closed.
Theorem 11.
Let X be a B C I -algebra. If X N is a neutrosophic N -ideal of X that satisfies the condition of Equation (13), then X N is a neutrosophic N -subalgebra and hence is a closed neutrosophic N -ideal of X.
Proof. 
Note that ( x y ) x θ y for all x , y X . Using Equations (10) and (13), we have
T N ( x y ) { T N ( x ) , T N ( θ y ) } { T N ( x ) , T N ( y ) } I N ( x y ) { I N ( x ) , I N ( θ y ) } { I N ( x ) , I N ( y ) } F N ( x y ) { F N ( x ) , F N ( θ y ) } { F N ( x ) , F N ( y ) }
Hence X N is a neutrosophic N -subalgebra and is therefore a closed neutrosophic N -ideal of X. ☐

Author Contributions

In this paper, Y. B. Jun conceived and designed the main idea and wrote the paper, H. Bordbar performed the idea, checking contents and finding examples, F. Smarandache analyzed the data and checking language.

Conflicts of Interest

The authors declare no conflict of interest.

References

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Jun, Y.B.; Smarandache, F.; Bordbar, H. Neutrosophic N-Structures Applied to BCK/BCI-Algebras. Information 2017, 8, 128. https://doi.org/10.3390/info8040128

AMA Style

Jun YB, Smarandache F, Bordbar H. Neutrosophic N-Structures Applied to BCK/BCI-Algebras. Information. 2017; 8(4):128. https://doi.org/10.3390/info8040128

Chicago/Turabian Style

Jun, Young Bae, Florentin Smarandache, and Hashem Bordbar. 2017. "Neutrosophic N-Structures Applied to BCK/BCI-Algebras" Information 8, no. 4: 128. https://doi.org/10.3390/info8040128

APA Style

Jun, Y. B., Smarandache, F., & Bordbar, H. (2017). Neutrosophic N-Structures Applied to BCK/BCI-Algebras. Information, 8(4), 128. https://doi.org/10.3390/info8040128

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