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Axioms 2018, 7(2), 41; doi:10.3390/axioms7020041

Article
Neutrosophic Quadruple BCK/BCI-Algebras
1
Department of Mathematics Education, Gyeongsang National University, Jinju 52828, Korea
2
Department of Mathematics, Jeju National University, Jeju 63243, Korea
3
Mathematics & Science Department, University of New Mexico, 705 Gurley Ave., Gallup, NM 87301, USA
4
Department of Mathematics, Shahid Beheshti University, Tehran 1983963113, Iran
*
Author to whom correspondence should be addressed.
Received: 20 April 2018 / Accepted: 18 May 2018 / Published: 18 June 2018

Abstract

:
The notion of a neutrosophic quadruple B C K / B C I -number is considered, and a neutrosophic quadruple B C K / B C I -algebra, which consists of neutrosophic quadruple B C K / B C I -numbers, is constructed. Several properties are investigated, and a (positive implicative) ideal in a neutrosophic quadruple B C K -algebra and a closed ideal in a neutrosophic quadruple B C I -algebra are studied. Given subsets A and B of a B C K / B C I -algebra, the set N Q ( A , B ) , which consists of neutrosophic quadruple B C K / B C I -numbers with a condition, is established. Conditions for the set N Q ( A , B ) to be a (positive implicative) ideal of a neutrosophic quadruple B C K -algebra are provided, and conditions for the set N Q ( A , B ) to be a (closed) ideal of a neutrosophic quadruple B C I -algebra are given. An example to show that the set { 0 ˜ } is not a positive implicative ideal in a neutrosophic quadruple B C K -algebra is provided, and conditions for the set { 0 ˜ } to be a positive implicative ideal in a neutrosophic quadruple B C K -algebra are then discussed.
Keywords:
neutrosophic quadruple BCK/BCI-number; neutrosophic quadruple BCK/BCI-algebra; neutrosophic quadruple subalgebra; (positive implicative) neutrosophic quadruple ideal
MSC:
06F35; 03G25; 08A72

1. Introduction

The notion of a neutrosophic set was developed by Smarandache [1,2,3] and is a more general platform that extends the notions of classic sets, (intuitionistic) fuzzy sets, and interval valued (intuitionistic) fuzzy sets. Neutrosophic set theory is applied to a different field (see [4,5,6,7,8]). Neutrosophic algebraic structures in B C K / B C I -algebras are discussed in [9,10,11,12,13,14,15,16]. Neutrosophic quadruple algebraic structures and hyperstructures are discussed in [17,18].
In this paper, we will use neutrosophic quadruple numbers based on a set and construct neutrosophic quadruple B C K / B C I -algebras. We investigate several properties and consider ideals and positive implicative ideals in neutrosophic quadruple B C K -algebra, and closed ideals in neutrosophic quadruple B C I -algebra. Given subsets A and B of a neutrosophic quadruple B C K / B C I -algebra, we consider sets N Q ( A , B ) , which consist of neutrosophic quadruple B C K / B C I -numbers with a condition. We provide conditions for the set N Q ( A , B ) to be a (positive implicative) ideal of a neutrosophic quadruple B C K -algebra and for the set N Q ( A , B ) to be a (closed) ideal of a neutrosophic quadruple B C I -algebra. We give an example to show that the set { 0 ˜ } is not a positive implicative ideal in a neutrosophic quadruple B C K -algebra, and we then consider conditions for the set { 0 ˜ } to be a positive implicative ideal in a neutrosophic quadruple B C K -algebra.

2. Preliminaries

A B C K / B C I -algebra is an important class of logical algebras introduced by Iséki (see [19,20]).
By a B C I -algebra, we mean a set X with a special element 0 and a binary operation ∗ that satisfies the following conditions:
(I)
( x , y , z X ) ( ( ( x y ) ( x z ) ) ( z y ) = 0 ) ;
(II)
( x , y X ) ( ( x ( x y ) ) y = 0 ) ;
(III)
( x X ) ( x x = 0 ) ;
(IV)
( x , y X ) ( x y = 0 , y x = 0 x = y ) .
If a B C I -algebra X satisfies the identity
(V)
( x X ) ( 0 x = 0 ) ,
then X is called a B C K -algebra. Any B C K / B C I -algebra X satisfies the following conditions:
( x X ) x 0 = x
( x , y , z X ) x y x z y z , z y z x
( x , y , z X ) ( x y ) z = ( x z ) y
( x , y , z X ) ( x z ) ( y z ) x y
where x y if and only if x y = 0 . Any B C I -algebra X satisfies the following conditions (see [21]):
( x , y X ) ( x ( x ( x y ) ) = x y ) ,
( x , y X ) ( 0 ( x y ) = ( 0 x ) ( 0 y ) ) .
A B C K -algebra X is said to be positive implicative if the following assertion is valid.
( x , y , z X ) ( x z ) ( y z ) = ( x y ) z .
A nonempty 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
0 I ,
( x X ) y I x y I x I .
A subset I of a B C I -algebra X is called a closed ideal (see [21]) of X if it is an ideal of X which satisfies
( x X ) ( x I 0 x I ) .
A subset I of a B C K -algebra X is called a positive implicative ideal (see [22]) of X if it satisfies (8) and
( x , y , z X ) ( ( x y ) z I , y z I x z I .
Observe that every positive implicative ideal is an ideal, but the converse is not true (see [22]). Note also that a B C K -algebra X is positive implicative if and only if every ideal of X is positive implicative (see [22]).
We refer the reader to the books [21,22] for further information regarding B C K / B C I -algebras, and to the site “http://fs.gallup.unm.edu/neutrosophy.htm” for further information regarding neutrosophic set theory.

3. Neutrosophic Quadruple BCK/BCI-Algebras

We consider neutrosophic quadruple numbers based on a set instead of real or complex numbers.
Definition 1.
Let X be a set. A neutrosophic quadruple X-number is an ordered quadruple ( a , x T , y I , z F ) where a , x , y , z X and T , I , F have their usual neutrosophic logic meanings.
The set of all neutrosophic quadruple X-numbers is denoted by N Q ( X ) , that is,
N Q ( X ) : = { ( a , x T , y I , z F ) a , x , y , z X } ,
and it is called the neutrosophic quadruple set based on X. If X is a B C K / B C I -algebra, a neutrosophic quadruple X-number is called a neutrosophic quadruple B C K / B C I -number and we say that N Q ( X ) is the neutrosophic quadruple B C K / B C I -set.
Let X be a B C K / B C I -algebra. We define a binary operation ⊙ on N Q ( X ) by
( a , x T , y I , z F ) ( b , u T , v I , w F ) = ( a b , ( x u ) T , ( y v ) I , ( z w ) F )
for all ( a , x T , y I , z F ) , ( b , u T , v I , w F ) N Q ( X ) . Given a 1 , a 2 , a 3 , a 4 X , the neutrosophic quadruple B C K / B C I -number ( a 1 , a 2 T , a 3 I , a 4 F ) is denoted by a ˜ , that is,
a ˜ = ( a 1 , a 2 T , a 3 I , a 4 F ) ,
and the zero neutrosophic quadruple B C K / B C I -number ( 0 , 0 T , 0 I , 0 F ) is denoted by 0 ˜ , that is,
0 ˜ = ( 0 , 0 T , 0 I , 0 F ) .
We define an order relation “≪” and the equality “=” on N Q ( X ) as follows:
x ˜ y ˜ x i y i for i = 1 , 2 , 3 , 4 x ˜ = y ˜ x i = y i for i = 1 , 2 , 3 , 4
for all x ˜ , y ˜ N Q ( X ) . It is easy to verify that “≪” is an equivalence relation on N Q ( X ) .
Theorem 1.
If X is a B C K / B C I -algebra, then ( N Q ( X ) ; , 0 ˜ ) is a B C K / B C I -algebra.
Proof. 
Let X be a B C I -algebra. For any x ˜ , y ˜ , z ˜ N Q ( X ) , we have
( x ˜ y ˜ ) ( x ˜ z ˜ ) = ( x 1 y 1 , ( x 2 y 2 ) T , ( x 3 y 3 ) I , ( x 4 y 4 ) F ) ( x 1 z 1 , ( x 2 z 2 ) T , ( x 3 z 3 ) I , ( x 4 z 4 ) F ) = ( ( x 1 y 1 ) ( x 1 z 1 ) , ( ( x 2 y 2 ) ( x 2 z 2 ) ) T , ( ( x 3 y 3 ) ( x 3 z 3 ) ) I , ( ( x 4 y 4 ) ( x 4 z 4 ) ) T ) ( z 1 y 1 , ( z 2 y 2 ) T , ( z 3 y 3 ) I , ( z 4 y 4 ) F ) = z ˜ y ˜
x ˜ ( x ˜ y ˜ ) = ( x 1 , x 2 T , x 3 I , x 4 F ) ( x 1 y 1 , ( x 2 y 2 ) T , ( x 3 y 3 ) I , ( x 4 y 4 ) F ) = ( x 1 ( x 1 y 1 ) , ( x 2 ( x 2 y 2 ) ) T , ( x 3 ( x 3 y 3 ) ) I , ( x 4 ( x 4 y 4 ) ) F ) ( y 1 , y 2 T , y 3 I , y 4 F ) = y ˜
x ˜ x ˜ = ( x 1 , x 2 T , x 3 I , x 4 F ) ( x 1 , x 2 T , x 3 I , x 4 F ) = ( x 1 x 1 , ( x 2 x 2 ) T , ( x 3 x 3 ) I , ( x 4 x 4 ) F ) = ( 0 , 0 T , 0 I , 0 F ) = 0 ˜ .
Assume that x ˜ y ˜ = 0 ˜ and y ˜ x ˜ = 0 ˜ . Then
( x 1 y 1 , ( x 2 y 2 ) T , ( x 3 y 3 ) I , ( x 4 y 4 ) F ) = ( 0 , 0 T , 0 I , 0 F )
and
( y 1 x 1 , ( y 2 x 2 ) T , ( y 3 x 3 ) I , ( y 4 x 4 ) F ) = ( 0 , 0 T , 0 I , 0 F ) .
It follows that x 1 y 1 = 0 = y 1 x 1 , x 2 y 2 = 0 = y 2 x 2 , x 3 y 3 = 0 = y 3 x 3 and x 4 y 4 = 0 = y 4 x 4 . Hence, x 1 = y 1 , x 2 = y 2 , x 3 = y 3 , and x 4 = y 4 , which implies that
x ˜ = ( x 1 , x 2 T , x 3 I , x 4 F ) = ( y 1 , y 2 T , y 3 I , y 4 F ) = y ˜ .
Therefore, we know that ( N Q ( X ) ; , 0 ˜ ) is a B C I -algebra. We call it the neutrosophic quadruple B C I -algebra. Moreover, if X is a B C K -algebra, then we have
0 ˜ x ˜ = ( 0 x 1 , ( 0 x 2 ) T , ( 0 x 3 ) I , ( 0 x 4 ) F ) = ( 0 , 0 T , 0 I , 0 F ) = 0 ˜ .
Hence, ( N Q ( X ) ; , 0 ˜ ) is a B C K -algebra. We call it the neutrosophic quadruple B C K -algebra. ☐
Example 1.
If X = { 0 , a } , then the neutrosophic quadruple set N Q ( X ) is given as follows:
N Q ( X ) = { 0 ˜ , 1 ˜ , 2 ˜ , 3 ˜ , 4 ˜ , 5 ˜ , 6 ˜ , 7 ˜ , 8 ˜ , 9 ˜ , 10 ˜ , 11 ˜ , 12 ˜ , 13 ˜ , 14 ˜ , 15 ˜ }
where 0 ˜ = ( 0 , 0 T , 0 I , 0 F ) , 1 ˜ = ( 0 , 0 T , 0 I , a F ) , 2 ˜ = ( 0 , 0 T , a I , 0 F ) , 3 ˜ = ( 0 , 0 T , a I , a F ) , 4 ˜ = ( 0 , a T , 0 I , 0 F ) , 5 ˜ = ( 0 , a T , 0 I , a F ) , 6 ˜ = ( 0 , a T , a I , 0 F ) , 7 ˜ = ( 0 , a T , a I , a F ) , 8 ˜ = ( a , 0 T , 0 I , 0 F ) , 9 ˜ = ( a , 0 T , 0 I , a F ) , 10 ˜ = ( a , 0 T , a I , 0 F ) , 11 ˜ = ( a , 0 T , a I , a F ) , 12 ˜ = ( a , a T , 0 I , 0 F ) , 13 ˜ = ( a , a T , 0 I , a F ) , 14 ˜ = ( a , a T , a I , 0 F ) , and 15 ˜ = ( a , a T , a I , a F ) .
Consider a B C K -algebra X = { 0 , a } with the binary operation ∗, which is given in Table 1.
Then ( N Q ( X ) , , 0 ˜ ) is a B C K -algebra in which the operation ⊙ is given by Table 2.
Theorem 2.
The neutrosophic quadruple set N Q ( X ) based on a positive implicative B C K -algebra X is a positive implicative B C K -algebra.
Proof. 
Let X be a positive implicative B C K -algebra. Then X is a B C K -algebra, so ( N Q ( X ) ; , 0 ˜ ) is a B C K -algebra by Theorem 1. Let x ˜ , y ˜ , z ˜ N Q ( X ) . Then
( x i z i ) ( y i z i ) = ( x i y i ) z i
for all i = 1 , 2 , 3 , 4 since x i , y i , z i X and X is a positive implicative B C K -algebra. Hence, ( x ˜ z ˜ ) ( y ˜ z ˜ ) = ( x ˜ y ˜ ) z ˜ ; therefore, N Q ( X ) based on a positive implicative B C K -algebra X is a positive implicative B C K -algebra. ☐
Proposition 1.
The neutrosophic quadruple set N Q ( X ) based on a positive implicative B C K -algebra X satisfies the following assertions.
( x ˜ , y ˜ , z ˜ N Q ( X ) ) ( x ˜ y ˜ z ˜ x ˜ z ˜ y ˜ z ˜ )
( x ˜ , y ˜ N Q ( X ) ) ( x ˜ y ˜ y ˜ x ˜ y ˜ ) .
Proof. 
Let x ˜ , y ˜ , z ˜ N Q ( X ) . If x ˜ y ˜ z ˜ , then
0 ˜ = ( x ˜ y ˜ ) z ˜ = ( x ˜ z ˜ ) ( y ˜ z ˜ ) ,
so x ˜ z ˜ y ˜ z ˜ . Assume that x ˜ y ˜ y ˜ . Using Equation (12) implies that
x ˜ y ˜ y ˜ y ˜ = 0 ˜ ,
so x ˜ y ˜ = 0 ˜ , i.e., x ˜ y ˜ . ☐
Let X be a B C K / B C I -algebra. Given a , b X and subsets A and B of X, consider the sets
N Q ( a , B ) : = { ( a , a T , y I , z F ) N Q ( X ) y , z B }
N Q ( A , b ) : = { ( a , x T , b I , b F ) N Q ( X ) a , x A }
N Q ( A , B ) : = { ( a , x T , y I , z F ) N Q ( X ) a , x A ; y , z B }
N Q ( A , B ) : = a A N Q ( a , B )
N Q ( A , B ) : = b B N Q ( A , b )
and
N Q ( A B ) : = N Q ( A , 0 ) N Q ( 0 , B ) .
The set N Q ( A , A ) is denoted by N Q ( A ) .
Proposition 2.
Let X be a B C K / B C I -algebra. Given a , b X and subsets A and B of X, we have
(1)
N Q ( A , B ) and N Q ( A , B ) are subsets of N Q ( A , B ) .
(1)
If 0 A B then N Q ( A B ) is a subset of N Q ( A , B ) .
Proof. 
Straightforward. ☐
Let X be a B C K / B C I -algebra. Given a , b X and subalgebras A and B of X, N Q ( a , B ) and N Q ( A , b ) may not be subalgebras of N Q ( X ) since
( a , a T , x 3 I , x 4 F ) ( a , a T , u 3 I , v 4 F ) = ( 0 , 0 T , ( x 3 u 3 ) I , ( x 4 v 4 ) F ) N Q ( a , B )
and
( x 1 , x 2 T , b I , b F ) ( u 1 , u 2 T , b I , b F ) = ( x 1 u 1 , ( x 2 u 2 ) T , 0 I , 0 F ) N Q ( A , b )
for ( a , a T , x 3 I , x 4 F ) N Q ( a , B ) , ( a , a T , u 3 I , v 4 F ) N Q ( a , B ) , ( x 1 , x 2 T , b I , b F ) N Q ( A , b ) , and ( u 1 , u 2 T , b I , b F ) N Q ( A , b ) .
Theorem 3.
If A and B are subalgebras of a B C K / B C I -algebra X, then the set N Q ( A , B ) is a subalgebra of N Q ( X ) , which is called a neutrosophic quadruple subalgebra.
Proof. 
Assume that A and B are subalgebras of a B C K / B C I -algebra X. Let x ˜ = ( x 1 , x 2 T , x 3 I , x 4 F ) and y ˜ = ( y 1 , y 2 T , y 3 I , y 4 F ) be elements of N Q ( A , B ) . Then x 1 , x 2 , y 1 , y 2 A and x 3 , x 4 , y 3 , y 4 B , which implies that x 1 y 1 A , x 2 y 2 A , x 3 y 3 B , and x 4 y 4 B . Hence,
x ˜ y ˜ = ( x 1 y 1 , ( x 2 y 2 ) T , ( x 3 y 3 ) I , ( x 4 y 4 ) F ) N Q ( A , B ) ,
so N Q ( A , B ) is a subalgebra of N Q ( X ) . ☐
Theorem 4.
If A and B are ideals of a B C K / B C I -algebra X, then the set N Q ( A , B ) is an ideal of N Q ( X ) , which is called a neutrosophic quadruple ideal.
Proof. 
Assume that A and B are ideals of a B C K / B C I -algebra X. Obviously, 0 ˜ N Q ( A , B ) . Let x ˜ = ( x 1 , x 2 T , x 3 I , x 4 F ) and y ˜ = ( y 1 , y 2 T , y 3 I , y 4 F ) be elements of N Q ( X ) such that x ˜ y ˜ N Q ( A , B ) and y ˜ N Q ( A , B ) . Then
x ˜ y ˜ = ( x 1 y 1 , ( x 2 y 2 ) T , ( x 3 y 3 ) I , ( x 4 y 4 ) F ) N Q ( A , B ) ,
so x 1 y 1 A , x 2 y 2 A , x 3 y 3 B and x 4 y 4 B . Since y ˜ N Q ( A , B ) , we have y 1 , y 2 A and y 3 , y 4 B . Since A and B are ideals of X, it follows that x 1 , x 2 A and x 3 , x 4 B . Hence, x ˜ = ( x 1 , x 2 T , x 3 I , x 4 F ) N Q ( A , B ) , so N Q ( A , B ) is an ideal of N Q ( X ) . ☐
Since every ideal is a subalgebra in a B C K -algebra, we have the following corollary.
Corollary 1.
If A and B are ideals of a B C K -algebra X, then the set N Q ( A , B ) is a subalgebra of N Q ( X ) .
The following example shows that Corollary 1 is not true in a B C I -algebra.
Example 2.
Consider a B C I -algebra ( Z , , 0 ) . If we take A = N and B = Z , then N Q ( A , B ) is an ideal of N Q ( Z ) . However, it is not a subalgebra of N Q ( Z ) since
( 2 , 3 T , 5 I , 6 F ) ( 3 , 5 T , 6 I , 7 F ) = ( 1 , 2 T , 11 I , 13 F ) N Q ( A , B )
for ( 2 , 3 T , 5 I , 6 F ) , ( 3 , 5 T , 6 I , 7 F ) N Q ( A , B ) .
Theorem 5.
If A and B are closed ideals of a B C I -algebra X, then the set N Q ( A , B ) is a closed ideal of N Q ( X ) .
Proof. 
If A and B are closed ideals of a B C I -algebra X, then the set N Q ( A , B ) is an ideal of N Q ( X ) by Theorem 4. Let x ˜ = ( x 1 , x 2 T , x 3 I , x 4 F ) N Q ( A , B ) . Then
0 ˜ x ˜ = ( 0 x 1 , ( 0 x 2 ) T , ( 0 x 3 ) I , ( 0 x 4 ) F ) N Q ( A , B )
since 0 x 1 , 0 x 2 A and 0 x 3 , 0 x 4 B . Therefore, N Q ( A , B ) is a closed ideal of N Q ( X ) . ☐
Since every closed ideal of a B C I -algebra X is a subalgebra of X, we have the following corollary.
Corollary 2.
If A and B are closed ideals of a B C I -algebra X, then the set N Q ( A , B ) is a subalgebra of N Q ( X ) .
In the following example, we know that there exist ideals A and B in a B C I -algebra X such that N Q ( A , B ) is not a closed ideal of N Q ( X ) .
Example 3.
Consider B C I -algebras ( Y , , 0 ) and ( Z , , 0 ) . Then X = Y × Z is a B C I -algebra (see [21]). Let A = Y × N and B = { 0 } × N . Then A and B are ideals of X, so N Q ( A , B ) is an ideal of N Q ( X ) by Theorem 4. Let ( ( 0 , 0 ) , ( 0 , 1 ) T , ( 0 , 2 ) I , ( 0 , 3 ) F ) N Q ( A , B ) . Then
( ( 0 , 0 ) , ( 0 , 0 ) T , ( 0 , 0 ) I , ( 0 , 0 ) F ) ( ( 0 , 0 ) , ( 0 , 1 ) T , ( 0 , 2 ) I , ( 0 , 3 ) F ) = ( ( 0 , 0 ) , ( 0 , 1 ) T , ( 0 , 2 ) I , ( 0 , 3 ) F ) N Q ( A , B ) .
Hence, N Q ( A , B ) is not a closed ideal of N Q ( X ) .
We provide conditions wherethe set N Q ( A , B ) is a closed ideal of N Q ( X ) .
Theorem 6.
Let A and B be ideals of a B C I -algebra X and let
Γ : = { a ˜ N Q ( X ) ( x ˜ N Q ( X ) ) ( x ˜ a ˜ x ˜ = a ˜ ) } .
Assume that, if Γ N Q ( A , B ) , then | Γ | < . Then N Q ( A , B ) is a closed ideal of N Q ( X ) .
Proof. 
If A and B are ideals of X, then N Q ( A , B ) is an ideal of N Q ( X ) by Theorem 4. Let a ˜ = ( a 1 , a 2 T , a 3 I , a 4 F ) N Q ( A , B ) . For any n N , denote n ( a ˜ ) : = 0 ˜ ( 0 ˜ a ˜ ) n . Then n ( a ˜ ) Γ and
n ( a ˜ ) = ( 0 ( 0 a 1 ) n , ( 0 ( 0 a 2 ) n ) T , ( 0 ( 0 a 3 ) n ) I , ( 0 ( 0 a 4 ) n ) F ) = ( 0 ( 0 a 1 n ) , ( 0 ( 0 a 2 n ) ) T , ( 0 ( 0 a 3 n ) ) I , ( 0 ( 0 a 4 n ) ) F ) = 0 ˜ ( 0 ˜ a ˜ n ) .
Hence,
n ( a ˜ ) a ˜ n = ( 0 ˜ ( 0 ˜ a ˜ n ) ) a ˜ n = ( 0 ˜ a ˜ n ) ( 0 ˜ a ˜ n ) = 0 ˜ N Q ( A , B ) ,
so n ( a ˜ ) N Q ( A , B ) , since a ˜ N Q ( A , B ) , and N Q ( A , B ) is an ideal of N Q ( X ) . Since | Γ | < , it follows that k N such that n ( a ˜ ) = ( n + k ) ( a ˜ ) , that is, n ( a ˜ ) = n ( a ˜ ) ( 0 ˜ a ˜ ) k , and thus
k ( a ˜ ) = 0 ˜ ( 0 ˜ a ˜ ) k = ( n ( a ˜ ) ( 0 ˜ a ˜ ) k ) n ( a ˜ ) = n ( a ˜ ) n ( a ˜ ) = 0 ˜ ,
i.e., ( k 1 ) ( a ˜ ) ( 0 ˜ a ˜ ) = 0 ˜ . Since 0 ˜ a ˜ Γ , it follows that 0 ˜ a ˜ = ( k 1 ) ( a ˜ ) N Q ( A , B ) . Therefore, N Q ( A , B ) is a closed ideal of N Q ( X ) . ☐
Theorem 7.
Given two elements a and b in a B C I -algebra X, let
A a : = { x X a x = a } and B b : = { x X b x = b } .
Then N Q ( A a , B b ) is a closed ideal of N Q ( X ) .
Proof. 
Since a 0 = a and b 0 = b , we have 0 A a B b . Thus, 0 ˜ N Q ( A a , B b ) . If x A a and y B b , then
0 x = ( a x ) a = a a = 0 and 0 y = ( b y ) b = b b = 0 .
Let x , y , c , d X be such that x , y x A a and c , d c B b . Then
( a y ) a = 0 y = ( 0 y ) 0 = ( 0 y ) ( 0 x ) = 0 ( y x ) = 0
and
( b d ) b = 0 d = ( 0 d ) 0 = ( 0 d ) ( 0 c ) = 0 ( d c ) = 0 ,
that is, a y a and b d b . On the other hand,
a = a ( y x ) = ( a x ) ( y x ) a y
and
b = b ( d c ) = ( b c ) ( d c ) b d .
Thus, a y = a and b d = b , i.e., y A a and d B b . Hence, A a and B b are ideals of X, and N Q ( A a , B b ) is therefore an ideal of N Q ( X ) by Theorem 4. Let x ˜ = ( x 1 , x 2 T , x 3 I , x 4 F ) N Q ( A a , B b ) . Then x 1 , x 2 A a , and x 3 , x 4 B b . It follows from Equation (15) that 0 x 1 = 0 A a , 0 x 2 = 0 A a , 0 x 3 = 0 B b , and 0 x 4 = 0 B b . Hence,
0 ˜ x ˜ = ( 0 x 1 , ( 0 x 2 ) T , ( 0 x 3 ) I , ( 0 x 4 ) F ) N Q ( A a , B b ) .
Therefore, N Q ( A a , B b ) is a closed ideal of N Q ( X ) . ☐
Proposition 3.
Let A and B be ideals of a B C K -algebra X. Then
N Q ( A ) N Q ( B ) = { 0 ˜ } ( x ˜ N Q ( A ) ) ( y ˜ N Q ( B ) ) ( x ˜ y ˜ = x ˜ ) .
Proof. 
Note that N Q ( A ) and N Q ( B ) are ideals of N Q ( X ) . Assume that N Q ( A ) N Q ( B ) = { 0 ˜ } . Let
x ˜ = ( x 1 , x 2 T , x 3 I , x 4 F ) N Q ( A ) and y ˜ = ( y 1 , y 2 T , y 3 I , y 4 F ) N Q ( B ) .
Since x ˜ ( x ˜ y ˜ ) x ˜ and x ˜ ( x ˜ y ˜ ) y ˜ , it follows that x ˜ ( x ˜ y ˜ ) N Q ( A ) N Q ( B ) = { 0 ˜ } . Obviously, ( x ˜ y ˜ ) x ˜ { 0 ˜ } . Hence, x ˜ y ˜ = x ˜ .
Conversely, suppose that x ˜ y ˜ = x ˜ for all x ˜ N Q ( A ) and y ˜ N Q ( B ) . If z ˜ N Q ( A ) N Q ( B ) , then z ˜ N Q ( A ) and z ˜ N Q ( B ) , which is implied from the hypothesis that z ˜ = z ˜ z ˜ = 0 ˜ . Hence N Q ( A ) N Q ( B ) = { 0 ˜ } . ☐
Theorem 8.
Let A and B be subsets of a B C K -algebra X such that
( a , b A B ) ( K ( a , b ) A B )
where K ( a , b ) : = { x X x a b } . Then the set N Q ( A , B ) is an ideal of N Q ( X ) .
Proof. 
If x A B , then 0 K ( x , x ) since 0 x x . Hence, 0 A B by Equation (17), so it is clear that 0 ˜ N Q ( A , B ) . Let x ˜ = ( x 1 , x 2 T , x 3 I , x 4 F ) and y ˜ = ( y 1 , y 2 T , y 3 I , y 4 F ) be elements of N Q ( X ) such that x ˜ y ˜ N Q ( A , B ) and y ˜ N Q ( A , B ) . Then
x ˜ y ˜ = ( x 1 y 1 , ( x 2 y 2 ) T , ( x 3 y 3 ) I , ( x 4 y 4 ) F ) N Q ( A , B ) ,
so x 1 y 1 A , x 2 y 2 A , x 3 y 3 B , and x 4 y 4 B . Using (II), we have x 1 K ( x 1 y 1 , y 1 ) A , x 2 K ( x 2 y 2 , y 2 ) A , x 3 K ( x 3 y 3 , y 3 ) B , and x 4 K ( x 4 y 4 , y 4 ) B . This implies that x ˜ = ( x 1 , x 2 T , x 3 I , x 4 F ) N Q ( A , B ) . Therefore, N Q ( A , B ) is an ideal of N Q ( X ) . ☐
Corollary 3.
Let A and B be subsets of a B C K -algebra X such that
( a , x , y X ) ( x , y A B , ( a x ) y = 0 a A B ) .
Then the set N Q ( A , B ) is an ideal of N Q ( X ) .
Theorem 9.
Let A and B be nonempty subsets of a B C K -algebra X such that
( a , x , y X ) ( x , y A ( or B ) , a x y a A ( or B ) ) .
Then the set N Q ( A , B ) is an ideal of N Q ( X ) .
Proof. 
Assume that the condition expressed by Equation (19) is valid for nonempty subsets A and B of X. Since 0 x x for any x A ( or B ) , we have 0 A ( or B ) by Equation (19). Hence, it is clear that 0 ˜ N Q ( A , B ) . Let x ˜ = ( x 1 , x 2 T , x 3 I , x 4 F ) and y ˜ = ( y 1 , y 2 T , y 3 I , y 4 F ) be elements of N Q ( X ) such that x ˜ y ˜ N Q ( A , B ) and y ˜ N Q ( A , B ) . Then
x ˜ y ˜ = ( x 1 y 1 , ( x 2 y 2 ) T , ( x 3 y 3 ) I , ( x 4 y 4 ) F ) N Q ( A , B ) ,
so x 1 y 1 A , x 2 y 2 A , x 3 y 3 B , and x 4 y 4 B . Note that x i ( x i y i ) y i for i = 1 , 2 , 3 , 4 . It follows from Equation (19) that x 1 , x 2 A and x 3 , x 4 B . Hence,
x ˜ = ( x 1 , x 2 T , x 3 I , x 4 F ) N Q ( A , B ) ;
therefore, N Q ( A , B ) is an ideal of N Q ( X ) . ☐
Theorem 10.
If A and B are positive implicative ideals of a B C K -algebra X, then the set N Q ( A , B ) is a positive implicative ideal of N Q ( X ) , which is called a positive implicative neutrosophic quadruple ideal.
Proof. 
Assume that A and B are positive implicative ideals of a B C K -algebra X. Obviously, 0 ˜ N Q ( A , B ) . Let x ˜ = ( x 1 , x 2 T , x 3 I , x 4 F ) , y ˜ = ( y 1 , y 2 T , y 3 I , y 4 F ) , and z ˜ = ( z 1 , z 2 T , z 3 I , z 4 F ) be elements of N Q ( X ) such that ( x ˜ y ˜ ) z ˜ N Q ( A , B ) and y ˜ z ˜ N Q ( A , B ) . Then
( x ˜ y ˜ ) z ˜ = ( ( x 1 y 1 ) z 1 , ( ( x 2 y 2 ) z 2 ) T , ( ( x 3 y 3 ) z 3 ) I , ( ( x 4 y 4 ) z 4 ) F ) N Q ( A , B ) ,
and
y ˜ z ˜ = ( y 1 z 1 , ( y 2 z 2 ) T , ( y 3 z 3 ) I , ( y 4 z 4 ) F ) N Q ( A , B ) ,
so ( x 1 y 1 ) z 1 A , ( x 2 y 2 ) z 2 A , ( x 3 y 3 ) z 3 B , ( x 4 y 4 ) z 4 B , y 1 z 1 A , y 2 z 2 A , y 3 z 3 B , and y 4 z 4 B . Since A and B are positive implicative ideals of X, it follows that x 1 z 1 , x 2 z 2 A and x 3 z 3 , x 4 z 4 B . Hence,
x ˜ z ˜ = ( x 1 z 1 , ( x 2 z 2 ) T , ( x 3 z 3 ) I , ( x 4 z 4 ) F ) N Q ( A , B ) ,
so N Q ( A , B ) is a positive implicative ideal of N Q ( X ) . ☐
Theorem 11.
Let A and B be ideals of a B C K -algebra X such that
( x , y , z X ) ( ( x y ) z A ( or B ) ( x z ) ( y z ) A ( or B ) ) .
Then N Q ( A , B ) is a positive implicative ideal of N Q ( X ) .
Proof. 
Since A and B are ideals of X, it follows from Theorem 4 that N Q ( A , B ) is an ideal of N Q ( X ) . Let x ˜ = ( x 1 , x 2 T , x 3 I , x 4 F ) , y ˜ = ( y 1 , y 2 T , y 3 I , y 4 F ) , and z ˜ = ( z 1 , z 2 T , z 3 I , z 4 F ) be elements of N Q ( X ) such that ( x ˜ y ˜ ) z ˜ N Q ( A , B ) and y ˜ z ˜ N Q ( A , B ) . Then
( x ˜ y ˜ ) z ˜ = ( ( x 1 y 1 ) z 1 , ( ( x 2 y 2 ) z 2 ) T , ( ( x 3 y 3 ) z 3 ) I , ( ( x 4 y 4 ) z 4 ) F ) N Q ( A , B ) ,
and
y ˜ z ˜ = ( y 1 z 1 , ( y 2 z 2 ) T , ( y 3 z 3 ) I , ( y 4 z 4 ) F ) N Q ( A , B ) ,
so ( x 1 y 1 ) z 1 A , ( x 2 y 2 ) z 2 A , ( x 3 y 3 ) z 3 B , ( x 4 y 4 ) z 4 B , y 1 z 1 A , y 2 z 2 A , y 3 z 3 B , and y 4 z 4 B . It follows from Equation (20) that ( x 1 z 1 ) ( y 1 z 1 ) A , ( x 2 z 2 ) ( y 2 z 2 ) A , ( x 3 z 3 ) ( y 3 z 3 ) B , and ( x 4 z 4 ) ( y 4 z 4 ) B . Since A and B are ideals of X, we get x 1 z 1 A , x 2 z 2 A , x 3 z 3 B , and x 4 z 4 B . Hence,
x ˜ z ˜ = ( x 1 z 1 , ( x 2 z 2 ) T , ( x 3 z 3 ) I , ( x 4 z 4 ) F ) N Q ( A , B ) .
Therefore, N Q ( A , B ) is a positive implicative ideal of N Q ( X ) . ☐
Corollary 4.
Let A and B be ideals of a B C K -algebra X such that
( x , y X ) ( ( x y ) y A ( or B ) x y A ( or B ) ) .
Then N Q ( A , B ) is a positive implicative ideal of N Q ( X ) .
Proof. 
If the condition expressed in Equation (21) is valid, then the condition expressed in Equation (20) is true. Hence, N Q ( A , B ) is a positive implicative ideal of N Q ( X ) by Theorem 11. ☐
Theorem 12.
Let A and B be subsets of a B C K -algebra X such that 0 A B and
( ( x y ) y ) z A ( or B ) , z A ( or B ) x y A ( or B )
for all x , y , z X . Then N Q ( A , B ) is a positive implicative ideal of N Q ( X ) .
Proof. 
Since 0 A B , it is clear that 0 ˜ N Q ( A , B ) . We first show that
( x , y X ) ( x y A ( or B ) , y A ( or B ) x A ( or B ) ) .
Let x , y X be such that x y A (or B) and y A (or B). Then
( ( x 0 ) 0 ) y = x y A ( or B )
by Equation (1), which, based on Equations (1) and (22), implies that x = x 0 A (or B). Let x ˜ = ( x 1 , x 2 T , x 3 I , x 4 F ) , y ˜ = ( y 1 , y 2 T , y 3 I , y 4 F ) , and z ˜ = ( z 1 , z 2 T , z 3 I , z 4 F ) be elements of N Q ( X ) such that ( x ˜ y ˜ ) z ˜ N Q ( A , B ) and y ˜ z ˜ N Q ( A , B ) . Then
( x ˜ y ˜ ) z ˜ = ( ( x 1 y 1 ) z 1 , ( ( x 2 y 2 ) z 2 ) T , ( ( x 3 y 3 ) z 3 ) I , ( ( x 4 y 4 ) z 4 ) F ) N Q ( A , B ) ,
and
y ˜ z ˜ = ( y 1 z 1 , ( y 2 z 2 ) T , ( y 3 z 3 ) I , ( y 4 z 4 ) F ) N Q ( A , B ) ,
so ( x 1 y 1 ) z 1 A , ( x 2 y 2 ) z 2 A , ( x 3 y 3 ) z 3 B , ( x 4 y 4 ) z 4 B , y 1 z 1 A , y 2 z 2 A , y 3 z 3 B , and y 4 z 4 B . Note that
( ( ( x i z i ) z i ) ( y i z i ) ) ( ( x i y i ) z i ) = 0 A ( or B )
for i = 1 , 2 , 3 , 4 . Since ( x i y i ) z i A for i = 1 , 2 and ( x j y j ) z j B for j = 3 , 4 , it follows from Equation (23) that ( ( x i z i ) z i ) ( y i z i ) A for i = 1 , 2 , and ( ( x j z j ) z j ) ( y j z j ) B for j = 3 , 4 . Moreover, since y i z i A for i = 1 , 2 , and y j z j B for j = 3 , 4 , we have x 1 z 1 A , x 2 z 2 A , x 3 z 3 B , and x 4 z 4 B by Equation (22). Hence,
x ˜ z ˜ = ( x 1 z 1 , ( x 2 z 2 ) T , ( x 3 z 3 ) I , ( x 4 z 4 ) F ) N Q ( A , B ) .
Therefore, N Q ( A , B ) is a positive implicative ideal of N Q ( X ) . ☐
Theorem 13.
Let A and B be subsets of a B C K -algebra X such that N Q ( A , B ) is a positive implicative ideal of N Q ( X ) . Then the set
Ω a ˜ : = { x ˜ N Q ( X ) x ˜ a ˜ N Q ( A , B ) }
is an ideal of N Q ( X ) for any a ˜ N Q ( X ) .
Proof. 
Obviously, 0 ˜ Ω a ˜ . Let x ˜ , y ˜ N Q ( X ) be such that x ˜ y ˜ Ω a ˜ and y ˜ Ω a ˜ . Then ( x ˜ y ˜ ) a ˜ N Q ( A , B ) and y ˜ a ˜ N Q ( A , B ) . Since N Q ( A , B ) is a positive implicative ideal of N Q ( X ) , it follows from Equation (11) that x ˜ a ˜ N Q ( A , B ) and therefore that x ˜ Ω a ˜ . Hence, Ω a ˜ is an ideal of N Q ( X ) . ☐
Combining Theorems 12 and 13, we have the following corollary.
Corollary 5.
If A and B are subsets of a B C K -algebra X satisfying 0 A B and the condition expressed in Equation (22), then the set Ω a ˜ in Equation (24) is an ideal of N Q ( X ) for all a ˜ N Q ( X ) .
Theorem 14.
For any subsets A and B of a B C K -algebra X, if the set Ω a ˜ in Equation (24) is an ideal of N Q ( X ) for all a ˜ N Q ( X ) , then N Q ( A , B ) is a positive implicative ideal of N Q ( X ) .
Proof. 
Since 0 ˜ Ω a ˜ , we have 0 ˜ = 0 ˜ a ˜ N Q ( A , B ) . Let x ˜ , y ˜ , z ˜ N Q ( X ) be such that ( x ˜ y ˜ ) z ˜ N Q ( A , B ) and y ˜ z ˜ N Q ( A , B ) . Then x ˜ y ˜ Ω z ˜ and y ˜ Ω z ˜ . Since Ω z ˜ is an ideal of N Q ( X ) , it follows that x ˜ Ω z ˜ . Hence, x ˜ z ˜ N Q ( A , B ) . Therefore, N Q ( A , B ) is a positive implicative ideal of N Q ( X ) . ☐
Theorem 15.
For any ideals A and B of a B C K -algebra X and for any a ˜ N Q ( X ) , if the set Ω a ˜ in Equation (24) is an ideal of N Q ( X ) , then N Q ( X ) is a positive implicative B C K -algebra.
Proof. 
Let Ω be any ideal of N Q ( X ) . For any x ˜ , y ˜ , z ˜ N Q ( X ) , assume that ( x ˜ y ˜ ) z ˜ Ω and y ˜ z ˜ Ω . Then x ˜ y ˜ Ω z ˜ and y ˜ Ω z ˜ . Since Ω z ˜ is an ideal of N Q ( X ) , it follows that x ˜ Ω z ˜ . Hence, x ˜ z ˜ Ω , which shows that Ω is a positive implicative ideal of N Q ( X ) . Therefore, N Q ( X ) is a positive implicative B C K -algebra. ☐
In general, the set { 0 ˜ } is an ideal of any neutrosophic quadruple B C K -algebra N Q ( X ) , but it is not a positive implicative ideal of N Q ( X ) as seen in the following example.
Example 4.
Consider a B C K -algebra X = { 0 , 1 , 2 } with the binary operation ∗, which is given in Table 3.
Then the neutrosophic quadruple B C K -algebra N Q ( X ) has 81 elements. If we take a ˜ = ( 2 , 2 T , 2 I , 2 F ) and b ˜ = ( 1 , 1 T , 1 I , 1 F ) in N Q ( X ) , then
( a ˜ b ˜ ) b ˜ = ( ( 2 1 ) 1 , ( ( 2 1 ) 1 ) T , ( ( 2 1 ) 1 ) I , ( ( 2 1 ) 1 ) F ) = ( 1 1 , ( 1 1 ) T , ( 1 1 ) I , ( 1 1 ) F ) = ( 0 , 0 T , 0 I , 0 F ) = 0 ˜ ,
and b ˜ b ˜ = 0 ˜ . However,
a ˜ b ˜ = ( 2 1 , ( 2 1 ) T , ( 2 1 ) I , ( 2 1 ) F ) = ( 1 , 1 T , 1 I , 1 F ) 0 ˜ .
Hence, { 0 ˜ } is not a positive implicative ideal of N Q ( X ) .
We now provide conditions for the set { 0 ˜ } to be a positive implicative ideal in the neutrosophic quadruple B C K -algebra.
Theorem 16.
Let N Q ( X ) be a neutrosophic quadruple B C K -algebra. If the set
Ω ( a ˜ ) : = { x ˜ N Q ( X ) x ˜ a ˜ }
is an ideal of N Q ( X ) for all a ˜ N Q ( X ) , then { 0 ˜ } is a positive implicative ideal of N Q ( X ) .
Proof. 
We first show that
( x ˜ , y ˜ N Q ( X ) ) ( ( x ˜ y ˜ ) y ˜ = 0 ˜ x ˜ y ˜ = 0 ˜ ) .
Assume that ( x ˜ y ˜ ) y ˜ = 0 ˜ for all x ˜ , y ˜ N Q ( X ) . Then x ˜ y ˜ y ˜ , so x ˜ y ˜ Ω ( y ˜ ) . Since y ˜ Ω ( y ˜ ) and Ω ( y ˜ ) is an ideal of N Q ( X ) , we have x ˜ Ω ( y ˜ ) . Thus, x ˜ y ˜ , that is, x ˜ y ˜ = 0 ˜ . Let u ˜ : = ( x ˜ y ˜ ) y ˜ . Then
( ( x ˜ u ˜ ) y ˜ ) y ˜ = ( ( x ˜ y ˜ ) y ˜ ) u ˜ = 0 ˜ ,
which implies, based on Equations (3) and (26), that
( x ˜ y ˜ ) ( ( x ˜ y ˜ ) y ˜ ) = ( x ˜ y ˜ ) u ˜ = ( x ˜ u ˜ ) y ˜ = 0 ˜ ,
that is, x ˜ y ˜ ( x ˜ y ˜ ) y ˜ . Since ( x ˜ y ˜ ) y ˜ x ˜ y ˜ , it follows that
( x ˜ y ˜ ) y ˜ = x ˜ y ˜ .
If we put y ˜ = x ˜ ( y ˜ ( y ˜ x ˜ ) ) in Equation (27), then
x ˜ ( x ˜ ( y ˜ ( y ˜ x ˜ ) ) ) = ( x ˜ ( x ˜ ( y ˜ ( y ˜ x ˜ ) ) ) ) ( x ˜ ( y ˜ ( y ˜ x ˜ ) ) ) ( y ˜ ( y ˜ x ˜ ) ) ( x ˜ ( y ˜ ( y ˜ x ˜ ) ) ) ( y ˜ ( y ˜ x ˜ ) ) ( x ˜ y ˜ ) = ( y ˜ ( x ˜ y ˜ ) ) ( y ˜ x ˜ ) = ( ( y ˜ ( x ˜ y ˜ ) ) ( y ˜ x ˜ ) ) ( y ˜ x ˜ ) ( x ˜ ( x ˜ y ˜ ) ) ( y ˜ x ˜ ) .
On the other hand,
( ( x ˜ ( x ˜ y ˜ ) ) ( y ˜ x ˜ ) ) ( x ˜ ( x ˜ ( y ˜ ( y ˜ x ˜ ) ) ) ) = ( ( x ˜ ( x ˜ ( x ˜ ( y ˜ ( y ˜ x ˜ ) ) ) ) ) ( x ˜ y ˜ ) ) ( y ˜ x ˜ ) ) = ( ( x ˜ ( y ˜ ( y ˜ x ˜ ) ) ) ( x ˜ y ˜ ) ) ( y ˜ x ˜ ) ) ( y ˜ ( y ˜ ( y ˜ x ˜ ) ) ) ( y ˜ x ˜ ) ) = 0 ˜ ,
so ( ( x ˜ ( x ˜ y ˜ ) ) ( y ˜ x ˜ ) ) ( x ˜ ( x ˜ ( y ˜ ( y ˜ x ˜ ) ) ) ) = 0 ˜ , that is,
( ( x ˜ ( x ˜ y ˜ ) ) ( y ˜ x ˜ ) ) x ˜ ( x ˜ ( y ˜ ( y ˜ x ˜ ) ) ) .
Hence,
x ˜ ( x ˜ ( y ˜ ( y ˜ x ˜ ) ) ) = ( ( x ˜ ( x ˜ y ˜ ) ) ( y ˜ x ˜ ) ) .
If we use y ˜ x ˜ instead of x ˜ in Equation (28), then
y ˜ x ˜ = ( y ˜ x ˜ ) 0 ˜ = ( y ˜ x ˜ ) ( ( y ˜ x ˜ ) ( y ˜ ( y ˜ ( y ˜ x ˜ ) ) ) ) = ( ( y ˜ x ˜ ) ( ( y ˜ x ˜ ) y ˜ ) ) ( y ˜ ( y ˜ x ˜ ) ) = ( y ˜ x ˜ ) ( y ˜ ( y ˜ x ˜ ) ) ,
which, by taking x ˜ = y ˜ x ˜ , implies that
y ˜ ( y ˜ x ˜ ) = ( y ˜ ( y ˜ x ˜ ) ) ( y ˜ ( y ˜ ( y ˜ x ˜ ) ) ) = ( y ˜ ( y ˜ x ˜ ) ) ( y ˜ x ˜ ) .
It follows that
( y ˜ ( y ˜ x ˜ ) ) ( x ˜ y ˜ ) = ( ( y ˜ ( y ˜ x ˜ ) ) ( y ˜ x ˜ ) ) ( x ˜ y ˜ ) ( x ˜ ( y ˜ x ˜ ) ) ( x ˜ y ˜ ) = ( x ˜ ( x ˜ y ˜ ) ) ( y ˜ x ˜ ) ,
so,
y ˜ x ˜ = ( y ˜ ( y ˜ ( y ˜ x ˜ ) ) ) 0 ˜ = ( y ˜ ( y ˜ ( y ˜ x ˜ ) ) ) ( ( y ˜ x ˜ ) y ˜ ) ( ( y ˜ x ˜ ) ( ( y ˜ x ˜ ) y ˜ ) ) ( y ˜ ( y ˜ x ˜ ) ) = ( y ˜ x ˜ ) ( y ˜ ( y ˜ x ˜ ) ) ( y ˜ x ˜ ) x ˜ .
Since ( y ˜ x ˜ ) x ˜ y ˜ x ˜ , it follows that
( y ˜ x ˜ ) x ˜ = y ˜ x ˜ .
Based on Equation (29), it follows that
( ( x ˜ z ˜ ) ( y ˜ z ˜ ) ) ( ( x ˜ y ˜ ) z ˜ ) = ( ( ( x ˜ z ˜ ) z ˜ ) ( y ˜ z ˜ ) ) ( ( x ˜ y ˜ ) z ˜ ) ( ( x ˜ z ˜ ) y ˜ ) ( ( x ˜ y ˜ ) z ˜ ) = 0 ˜ ,
that is, ( x ˜ z ˜ ) ( y ˜ z ˜ ) ( x ˜ y ˜ ) z ˜ . Note that
( ( x ˜ y ˜ ) z ˜ ) ( ( x z ˜ ) ( y ˜ z ˜ ) ) = ( ( x ˜ y ˜ ) z ˜ ) ( ( x ( y ˜ z ˜ ) ) z ˜ ) ( x ˜ y ˜ ) ( x ˜ ( y ˜ z ˜ ) ) ( y ˜ z ˜ ) y ˜ = 0 ˜ ,
which shows that ( x ˜ y ˜ ) z ˜ ( x ˜ z ˜ ) ( y ˜ z ˜ ) . Hence, ( x ˜ y ˜ ) z ˜ = ( x ˜ z ˜ ) ( y ˜ z ˜ ) . Therefore, N Q ( X ) is a positive implicative, so { 0 ˜ } is a positive implicative ideal of N Q ( X ) . ☐

4. Conclusions

We have considered a neutrosophic quadruple B C K / B C I -number on a set and established neutrosophic quadruple B C K / B C I -algebras, which consist of neutrosophic quadruple B C K / B C I -numbers. We have investigated several properties and considered ideal theory in a neutrosophic quadruple B C K -algebra and a closed ideal in a neutrosophic quadruple B C I -algebra. Using subsets A and B of a neutrosophic quadruple B C K / B C I -algebra, we have considered sets N Q ( A , B ) , which consist of neutrosophic quadruple B C K / B C I -numbers with a condition. We have provided conditions for the set N Q ( A , B ) to be a (positive implicative) ideal of a neutrosophic quadruple B C K -algebra, and the set N Q ( A , B ) to be a (closed) ideal of a neutrosophic quadruple B C I -algebra. We have provided an example to show that the set { 0 ˜ } is not a positive implicative ideal in a neutrosophic quadruple B C K -algebra, and we have considered conditions for the set { 0 ˜ } to be a positive implicative ideal in a neutrosophic quadruple B C K -algebra.

Author Contributions

Y.B.J. and S.-Z.S. initiated the main idea of this work and wrote the paper. F.S. and H.B. provided examples and checked the content. All authors conceived and designed the new definitions and results, and have read and approved the final manuscript for submission.

Acknowledgments

The authors wish to thank the anonymous reviewers for their valuable suggestions. The second author, Seok-Zun Song, was supported under the framework of international cooperation program managed by the National Research Foundation of Korea (2017K2A9A1A01092970).

Conflicts of Interest

The authors declare no conflict of interest.

References

  1. Smarandache, F. Neutrosophy, Neutrosophic Probability, Set, and Logic, ProQuest Information & Learning, Ann Arbor, Michigan, USA, p. 105, 1998. Available online: http://fs.gallup.unm.edu/eBook-neutrosophics6.pdf (accessed on 1 September 2007).
  2. Smarandache, F. A Unifying Field in Logics: Neutrosophic Logic. Neutrosophy, Neutrosophic Set, Neutrosophic Probability; American Reserch Press: Rehoboth, NM, USA, 1999. [Google Scholar]
  3. Smarandache, F. Neutrosophic set—A generalization of the intuitionistic fuzzy set. Int. J. Pure Appl. Math. 2005, 24, 287–297. [Google Scholar]
  4. Garg, H. Linguistic single-valued neutrosophic prioritized aggregation operators and their applications to multiple-attribute group decision-making. J. Ambient Intell. Humaniz. Comput. 2018, in press. [Google Scholar] [CrossRef]
  5. Garg, H. Non-linear programming method for multi-criteria decision making problems under interval neutrosophic set environment. Appl. Intell. 2017, in press. [Google Scholar] [CrossRef]
  6. Garg, H. Some New Biparametric Distance Measures on Single-Valued Neutrosophic Sets with Applications to Pattern Recognition and Medical Diagnosis. Information 2017, 8, 162. [Google Scholar] [CrossRef]
  7. Garg, H. Novel single-valued neutrosophic aggregated operators under Frank norm operation and its application to decision-making process. Int. J. Uncertain. Quantif. 2016, 6, 361–375. [Google Scholar]
  8. Garg, H.; Garg, N. On single-valued neutrosophic entropy of order α. Neutrosophic Sets Syst. 2016, 14, 21–28. [Google Scholar]
  9. Saeid, A.B.; Jun, Y.B. Neutrosophic subalgebras of BCK/BCI-algebras based on neutrosophic points. Ann. Fuzzy Math. Inform. 2017, 14, 87–97. [Google Scholar]
  10. Jun, Y.B. Neutrosophic subalgebras of several types in BCK/BCI-algebras. Ann. Fuzzy Math. Inform. 2017, 14, 75–86. [Google Scholar]
  11. Jun, Y.B.; Kim, S.J.; Smarandache, F. Interval neutrosophic sets with applications in BCK/BCI-algebra. Axioms 2018, 7, 23. [Google Scholar] [CrossRef]
  12. Jun, Y.B.; Smarandache, F.; Bordbar, H. Neutrosophic N -structures applied to BCK/BCI-algebras. Information 2017, 8, 128. [Google Scholar] [CrossRef]
  13. Jun, Y.B.; Smarandache, F.; Song, S.Z.; Khan, M. Neutrosophic positive implicative N -ideals in BCK/BCI-algebras. Axioms 2018, 7, 3. [Google Scholar] [CrossRef]
  14. Khan, M.; Anis, S.; Smarandache, F.; Jun, Y.B. Neutrosophic N -structures and their applications in semigroups. Ann. Fuzzy Math. Inform. 2017, 14, 583–598. [Google Scholar]
  15. Öztürk, M.A.; Jun, Y.B. Neutrosophic ideals in BCK/BCI-algebras based on neutrosophic points. J. Inter. Math. Virtual Inst. 2018, 8, 1–17. [Google Scholar]
  16. Song, S.Z.; Smarandache, F.; Jun, Y.B. Neutrosophic commutative N -ideals in BCK-algebras. Information 2017, 8, 130. [Google Scholar] [CrossRef]
  17. Agboola, A.A.A.; Davvaz, B.; Smarandache, F. Neutrosophic quadruple algebraic hyperstructures. Ann. Fuzzy Math. Inform. 2017, 14, 29–42. [Google Scholar]
  18. Akinleye, S.A.; Smarandache, F.; Agboola, A.A.A. On neutrosophic quadruple algebraic structures. Neutrosophic Sets Syst. 2016, 12, 122–126. [Google Scholar]
  19. Iséki, K. On BCI-algebras. Math. Semin. Notes 1980, 8, 125–130. [Google Scholar]
  20. Iséki, K.; Tanaka, S. An introduction to the theory of BCK-algebras. Math. Jpn. 1978, 23, 1–26. [Google Scholar]
  21. Huang, Y. BCI-Algebra; Science Press: Beijing, China, 2006. [Google Scholar]
  22. Meng, J.; Jun, Y.B. BCK-Algebras; Kyungmoonsa Co.: Seoul, Korea, 1994. [Google Scholar]
Table 1. Cayley table for the binary operation “∗”.
Table 1. Cayley table for the binary operation “∗”.
0a
000
aa0
Table 2. Cayley table for the binary operation “⊙”.
Table 2. Cayley table for the binary operation “⊙”.
0 ˜ 1 ˜ 2 ˜ 3 ˜ 4 ˜ 5 ˜ 6 ˜ 7 ˜ 8 ˜ 9 ˜ 10 ˜ 11 ˜ 12 ˜ 13 ˜ 14 ˜ 15 ˜
0 ˜ 0 ˜ 0 ˜ 0 ˜ 0 ˜ 0 ˜ 0 ˜ 0 ˜ 0 ˜ 0 ˜ 0 ˜ 0 ˜ 0 ˜ 0 ˜ 0 ˜ 0 ˜ 0 ˜
1 ˜ 1 ˜ 0 ˜ 1 ˜ 0 ˜ 1 ˜ 0 ˜ 1 ˜ 0 ˜ 1 ˜ 0 ˜ 1 ˜ 0 ˜ 1 ˜ 0 ˜ 1 ˜ 0 ˜
2 ˜ 2 ˜ 2 ˜ 0 ˜ 0 ˜ 2 ˜ 2 ˜ 0 ˜ 0 ˜ 2 ˜ 2 ˜ 0 ˜ 0 ˜ 2 ˜ 2 ˜ 0 ˜ 0 ˜
3 ˜ 3 ˜ 2 ˜ 1 ˜ 0 ˜ 3 ˜ 2 ˜ 1 ˜ 0 ˜ 3 ˜ 2 ˜ 1 ˜ 0 ˜ 3 ˜ 2 ˜ 1 ˜ 0 ˜
4 ˜ 4 ˜ 4 ˜ 4 ˜ 4 ˜ 0 ˜ 0 ˜ 0 ˜ 0 ˜ 4 ˜ 4 ˜ 4 ˜ 4 ˜ 0 ˜ 0 ˜ 0 ˜ 0 ˜
5 ˜ 5 ˜ 4 ˜ 5 ˜ 4 ˜ 1 ˜ 0 ˜ 1 ˜ 0 ˜ 5 ˜ 4 ˜ 5 ˜ 4 ˜ 1 ˜ 0 ˜ 1 ˜ 0 ˜
6 ˜ 6 ˜ 6 ˜ 4 ˜ 4 ˜ 2 ˜ 2 ˜ 0 ˜ 0 ˜ 6 ˜ 6 ˜ 4 ˜ 4 ˜ 2 ˜ 2 ˜ 0 ˜ 0 ˜
7 ˜ 7 ˜ 6 ˜ 5 ˜ 4 ˜ 3 ˜ 2 ˜ 1 ˜ 0 ˜ 7 ˜ 6 ˜ 5 ˜ 4 ˜ 3 ˜ 2 ˜ 1 ˜ 0 ˜
8 ˜ 8 ˜ 8 ˜ 8 ˜ 8 ˜ 8 ˜ 8 ˜ 8 ˜ 8 ˜ 0 ˜ 0 ˜ 0 ˜ 0 ˜ 0 ˜ 0 ˜ 0 ˜ 0 ˜
9 ˜ 9 ˜ 8 ˜ 8 ˜ 8 ˜ 9 ˜ 8 ˜ 9 ˜ 8 ˜ 9 ˜ 0 ˜ 1 ˜ 0 ˜ 1 ˜ 0 ˜ 1 ˜ 0 ˜
10 ˜ 10 ˜ 10 ˜ 8 ˜ 8 ˜ 10 ˜ 10 ˜ 8 ˜ 8 ˜ 2 ˜ 2 ˜ 0 ˜ 2 ˜ 2 ˜ 2 ˜ 0 ˜ 0 ˜
11 ˜ 11 ˜ 10 ˜ 9 ˜ 8 ˜ 11 ˜ 10 ˜ 9 ˜ 8 ˜ 3 ˜ 2 ˜ 1 ˜ 0 ˜ 3 ˜ 2 ˜ 1 ˜ 0 ˜
12 ˜ 12 ˜ 12 ˜ 12 ˜ 12 ˜ 8 ˜ 8 ˜ 8 ˜ 8 ˜ 4 ˜ 4 ˜ 4 ˜ 4 ˜ 0 ˜ 0 ˜ 0 ˜ 0 ˜
13 ˜ 13 ˜ 12 ˜ 13 ˜ 12 ˜ 9 ˜ 8 ˜ 9 ˜ 8 ˜ 5 ˜ 4 ˜ 5 ˜ 4 ˜ 1 ˜ 0 ˜ 1 ˜ 0 ˜
14 ˜ 14 ˜ 14 ˜ 12 ˜ 12 ˜ 10 ˜ 10 ˜ 8 ˜ 8 ˜ 6 ˜ 6 ˜ 4 ˜ 4 ˜ 2 ˜ 2 ˜ 0 ˜ 0 ˜
15 ˜ 15 ˜ 14 ˜ 13 ˜ 12 ˜ 11 ˜ 10 ˜ 9 ˜ 8 ˜ 7 ˜ 6 ˜ 5 ˜ 4 ˜ 3 ˜ 2 ˜ 1 ˜ 0 ˜
Table 3. Cayley table for the binary operation “∗”.
Table 3. Cayley table for the binary operation “∗”.
012
0000
1100
2210

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