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Neutrosophic quadruple ideals in neutrosophic quadruple BCI-algebras.

1 Introduction

To deal with incomplete, inconsistent and indeterminate information, Smarandache introduced the notion of neutrosophic sets (see ([1], [2] and [3]). In fact, neutrosophic set is a useful mathematical tool which extends the notions of classic set, (intuitionistic) fuzzy set and interval valued (intuitionistic) fuzzy set. Neutrosophic set theory has useful applications in several branches (see for e.g., [4], [5], [6] and [7]).

In [8], Smarandache considered an entry (i.e., a number, an idea, an object etc.) which is represented by a known part (a) and an unknown part (bT, cI, dF) where T, I, F have their usual neutrosophic logic meanings and a, b, c, d are real or complex numbers, and then he introduced the concept of neutrosophic quadruple numbers. Neutrosophic quadruple algebraic structures and hyperstructures are discussed in [9] and [10]. Recently, neutrosophic set theory has been applied to the BCK/BCI-algebras on various aspects (see for e.g., [11], [12] [13], [14], [15], [16], [17], [18], [19] and [20].) Using the notion of neutrosophic quadruple numbers based on a set, Jun et al. [21] constructed neutrosophic quadruple BCK/BCI-algebras. They investigated several properties, and considered ideal and positive implicative ideal in neutrosophic quadruple BCK-algebra, and closed ideal in neutrosophic quadruple BCI-algebra. Given subsets A and B of a neutrosophic quadruple BCK/BCI-algebra, they considered sets NQ(U, V) which consists of neutrosophic quadruple BCK/BCI-numbers with a condition. They provided conditions for the set NQ(U, V) to be a (positive implicative) ideal of a neutrosophic quadruple BCK-algebra, and the set NQ(U, V) to be a (closed) ideal of a neutrosophic quadruple BCI-algebra. They gave an example to show that the set {0} is not a positive implicative ideal in a neutrosophic quadruple BCK-algebra, and then they considered conditions for the set {0} to be a positive implicative ideal in a neutrosophic quadruple BCK-algebra. Muhiuddin et al. [22] discussed several properties and (implicative) neutrosophic quadruple ideals in (implicative) neutrosophic quadruple BCK-algebras.

In this paper, we introduce the notions of (regular) neutrosophic quadruple ideal and neutrosophic quadruple q-ideal in neutrosophic quadruple BCI-algebras, and investigate related properties. Given nonempty subsets A and B of a BCI-algebra S, we consider conditions for the set NQ(U, V) to be a (regular) neutrosophic quadruple ideal of NQ(S) and a neutrosophic quadruple q-ideal of NQ(S).

2 Preliminaries

We begin with the following definitions and properties that will be needed in the sequel.

A nonempty set S with a constant 0 and a binary operation * is called a BCI-algebra if for all x, y, z [member of] S the following conditions hold ([23] and [24]):

(I) (((x * y) * (x * z)) * (z * y) = 0),

(II) ((x * (x * y)) * y = 0),

(III) (x * x = 0),

(IV) (x * y = 0, y * x = 0 [??] x = y).

If a BCI-algebra S satisfies the following identity:

(V) ([for all]x [member of] S) (0 * x = 0),

then S is called a BCK-algebra. Define a binary relation [less than or equal to] on X by letting x * y = 0 if and only if x [less than or equal to] y. Then (S, [less than or equal to]) is a partially ordered set.

Theorem 2.1. Let S be a BCK/BCI-algebra. Then following conditions are hold:

([for all]x [member of] S) (x * 0 = x), (2.1)

([for all]x, y, z [member of] S) (x [less than or equal to] y [??] x * z [less than or equal to] y * z, z * y [less than or equal to] z * x), (2.2)

([for all]x, y, z [member of] S) ((x * y) * z = (x * z) * y), (2.3)

([for all]x, y, z [member of] S) ((x * z) * (y * z) [less than or equal to] x * y) (2.4)

where x [less than or equal to] y if and only if x * y = 0.

Any BCI-algebra S satisfies the following conditions (see [25]):

([for all]x, y [member of] S)(x * (x * (x * y)) = x * y); (2.5)

([for all]x, y [member of] S)(0 * (x * y) = (0 * x) * (0 * y)); (2.6)

([for all]x, y [member of] S)(0 * (0 * (x * y)) = (0 * y) * (0 * x)): (2.7)

A nonempty subset A of a BCK/BCI-algebra S is called a subalgebra of S if x * y [member of] A for all x, y [member of] A. A subset I of a BCK/BCI-algebra S is called an ideal of S if it satisfies:

0 [member of] I, (2.8)

([for all]x [member of] S) ([for all]y [member of] I) (x * y [member of] I [??] x [member of] I). (2.9)

An ideal I of a BCI-algebra S is said to be regular (see [26]) if it is also a subalgebra of S.

It is clear that every ideal of a BCK-algebra is regular (see [26]).

A subset I of a BCI-algebra S is called a q-ideal of S (see [27]) if it satisfies (2.8) and

([for all]x, y, z [member of] S)(x * (y * z) e I, y e I ^ x * z e I). (2.10)

We refer the reader to the books [25, 28] for further information regarding BCK/BCI-algebras, and to the site "http://fs.gallup.unm.edu/neutrosophy.htm" for further information regarding neutrosophic set theory.

We consider neutrosophic quadruple numbers based on a set instead of real or complex numbers.

Definition 2.2 ([21]). Let S be a set. A neutrosophic quadruple S-number is an ordered quadruple (a, xT, y^ zF) where a, x, y, z [member of] S and T, I, F have their usual neutrosophic logic meanings.

The set of all neutrosophic quadruple S-numbers is denoted by NQ(S), that is,

NQ(S) := {(a, xT, yI, zF) | a, x, y, z [member of] S},

and it is called the neutrosophic quadruple set based on S. If S is a BCK/BCI-algebra, a neutrosophic quadruple S-number is called a neutrosophic quadruple BCK/BCI-number and we say that NQ(S) is the neutrosophic quadruple BCK/BCI-set.

Let S be a BCK/BCI-algebra. We define a binary operation [??] on NQ(S) by

(a, xT, yI, zF) [??] (b, uT, vI, wF) = (a * b, (x * u)T, (y * v)I, (z * w)F)

for all (a, xT, yI, zF), (b, uT, vI, wF) [member of] NQ(S). Given [a.sub.1], [a.sub.2], [a.sub.3], [a.sub.4] [member of] S, the neutrosophic quadruple BCK/BCI-number ([a.sub.1], [a.sub.2]T, [a.sub.3]I, [a.sub.4]F) is denoted by a, that is,

[??] = ([a.sub.1], [a.sub.2]T, [a.sub.3]I, [a.sub.4]F),

and the zero neutrosophic quadruple BCK/BCI-number (0, 0T, 0I, OF) is denoted by [??], that is,

[??] = (0, 0T, 0I, 0F).

We define an order relation [much less than] and the equality "=" on NQ(S) as follows:

[mathematical expression not reproducible]

for all [??],[??] [member of] NQ(S). It is easy to verify that "[much less than]" is an equivalence relation on NQ(S).

Theorem 2.3 ([21]). If S is a BCK/BCI-algebra, then (NQ(S); [??], [??]) is a BCK/BCI-algebra.

We say that (NQ(S); [??], [??]) is a neutrosophic quadruple BCK/BCl-algebra, and it is simply denoted by NQ(S).

Let S be a BCK/BCI-algebra. Given nonempty subsets A and B of S, consider the set

NQ(U, V) := {(a, xT, yI, zF) [member of] NQ(S) | a, x G[member of] U & y, z [member of] V},

which is called the neutrosophic quadruple (U, V)-set.

The set NQ(U, U) is denoted by NQ(U), and it is called the neutrosophic quadruple U-set.

3 (Regular) neutrosophic quadruple ideals

Definition 3.1. Given nonempty subsets U and V of a BCI-algebra S, if the neutrosophic quadruple (U, V)-set NQ(U, V) is a (regular) ideal of a neutrosophic quadruple BCI-algebra NQ(S), we say NQ(U, V) is a (regular) neutrosophic quadruple ideal of NQ(S).

Question 1. If U and V are subalgebras of a BCI-algebra S, then is the neutrosophic quadruple (U, V)-set NQ(U, V) a neutrosophic quadruple ideal of NQ(S)

The answer to Question 1 is negative as seen in the following example.

Example 3.2. Consider a BCI-algebra S = {0,1, a, b, c} with the binary operation *, which is given in Table 1. Then the neutrosophic quadruple BCI-algebra NQ(S) has 625 elements. Note that U = {0, a} and V = {0, b}

are subalgebras of S. The neutrosophic quadruple (U, V)-set NQ(U, V) consists of the following elements:

[mathematical expression not reproducible]

If we take (1, aT, bI, 0F) [member of] NQ(S), then (1, aT, bI, OF) [member of] NQ(U,V) and

[mathematical expression not reproducible].

Hence the neutrosophic quadruple (U, V)-set NQ(U, V) is not a neutrosophic quadruple ideal of NQ(S).

We consider conditions for the neutrosophic quadruple (U, V)-set NQ(U, V) to be a regular neutrosophic quadruple ideal of NQ(S).

Lemma 3.3 ([21]). If U and V are subalgebras (resp., ideals) of a BCI-algebra S, then the neutrosophic quadruple (U, V)-set NQ(U, V) is a neutrosophic quadruple subalgebra (resp., ideal) of NQ(S).

Theorem 3.4. Let U and V be subalgebras of a BCI-algebra S such that

([for all]x, y [member of] S)(x [member of] U (resp., V), y [not member of] U (resp., V) [??] y * x [not member of] U (resp., V)). (3.1)

Then the neutrosophic quadruple (U, V)-set NQ(U, V) is a regular neutrosophic quadruple ideal of NQ(S).

Proof. By Lemma 3.3, NQ(U, V) is a neutrosophic quadruple subalgebra of NQ(S). Hence it is clear that [mathematical expression not reproducible]. Also,

[mathematical expression not reproducible],

and so [mathematical expression not reproducible] for some i = 1, 2 and T = 3, 4. It follows from (3.1) that [y.sub.i] * [x.sub.i] [not member of] or [y.sub.j] * [x.sub.j] [not member of] V for some i = 1, 2 and T = 3, 4. This is a contradiction, and so [??] [member of] NQ(U, V). Thus NQ(U, V) is a neutrosophic quadruple ideal of NQ(S), and therefore NQ(U, V) is a regular neutrosophic quadruple ideal of NQ(S). ?

Corollary 3.5. Let U be a subalgebra of a BCI-algebra S such that

([for all]x, y [member of] S)(x [member of] U, y [not member of] U [??] y * x [not member of] U). (3.2)

Then the neutrosophic quadruple U-set NQ(U) is a regular neutrosophic quadruple ideal of NQ(S).

Theorem 3.6. Let U and V be subsets of a BCI-algebra S. If any neutrosophic quadruple ideal NQ(U, V) of NQ(S) satisfies [mathematical expression not reproducible] is a regular neutrosophic quadruple ideal of NQ(S).

Proof. For any [??], [member of] NQ(U, V), we have

[mathematical expression not reproducible].

Since NQ(U, V) is an ideal of NQ(S), it follows that [mathematical expression not reproducible]. Hence NQ(U, V) is a neutrosophic quadruple subalgebra of NQ(S), and therefore NQ(U, V) is a regular neutrosophic quadruple ideal of NQ(S).

Corollary 3.7. Let U be a subset of a BCI-algebra S. If any neutrosophic quadruple ideal NQ(U) of NQ(S) satisfies [mathematical expression not reproducible] for all [??] [member of] NQ(U), then NQ(U) is a regular neutrosophic quadruple ideal of NQ(S).

Theorem 3.8. If U and V are ideals of a finite BCI-algebra S, then the neutrosophic quadruple (U, V)-set NQ(U, V) is a regular neutrosophic quadruple ideal of NQ(S).

Proof. By Lemma 3.3, NQ(U, V) is a neutrosophic quadruple ideal of NQ(S). Since S is finite, NQ(S) is also finite. Assume that [absolute value of NQ(S)] = n. For any element [??] [member of] NQ(U, V), consider the following n +1 elements:

[mathematical expression not reproducible].

Then there exist natural numbers p and q with p > q such that

[mathematical expression not reproducible].

Hence

[mathematical expression not reproducible]

Since NQ(U, V) is an ideal of NQ(S), it follows that [mathematical expression not reproducible]. Therefore NQ(U, V) is a regular neutrosophic quadruple ideal of NQ(S) by Theorem 3.6.

Corollary 3.9. If U is an ideal of a finite BCI-algebra S, then the neutrosophic quadruple U-set NQ(U) is a regular neutrosophic quadruple ideal of NQ(S).

4 Neutrosophic quadruple q-ideals

Definition 4.1. Given nonempty subsets U and V of S, if the neutrosophic quadruple (U, V)-set NQ(U, V) is a q-ideal of a neutrosophic quadruple BCI-algebra NQ(S), we say NQ(U, V) is a neutrosophic quadruple q-ideal of NQ(S).

Example 4.2. Consider a BCI-algebra S = {0,1, a} with the binary operation *, which is given in Table 2. Then the neutrosophic quadruple BCI-algebra NQ(S) has 81 elements. If we take U = {0,1} and V = {0,1}, then

[mathematical expression not reproducible]

is a neutrosophic quadruple q-ideal of NQ(S) where

[mathematical expression not reproducible]

Theorem 4.3. For any nonempty subsets U and V of a BCI-algebra S, if the neutrosophic quadruple (U, V)-set NQ(U, V) is a neutrosophic quadruple q-ideal of NQ(S), then it is both a neutrosophic quadruple subalgebra and a neutrosophic quadruple ideal of NQ(S).

Proof. Assume that NQ(U,V) is a neutrosophic quadruple q-ideal of NQ(S). Since [??] [member of] NQ(U,V), we have 0 [member of] U and 0 [member of] V. Let x, y, z [member of] S be such that x * (y * z) [member of] U [intersection] V and y G U n V. Then (y, yT, yI, yF) [member of] NQ(U,V) and

[mathematical expression not reproducible].

Since NQ(U, V) is a neutrosophic quadruple q-ideal of NQ(S), it follows that

(x * z, (x * z)T, (x * z)I, (x * z)F) = (x, xT, xI, xF) [??] (z, zT, zl, zF) G NQ(U, V).

Hence x * z G U [intersection] V, and therefore U and V are q-ideals of S. Since every q-ideal is both a subalgebra and an ideal, it follows from Lemma 3.3 that NQ(U, V) is both a neutrosophic quadruple subalgebra and a neutrosophic quadruple ideal of NQ(S).

The converse of Theorem 4.3 is not true as seen in the following example.

Example 4.4. Consider a BCI-algebra S = {0, a, b, c} with the binary operation *, which is given in Table 3.

Then the neutrosophic quadruple BCI-algebra NQ(S) has 256 elements. If we take A = {0} and B = {0}, then NQ(U, V) = {[??]} is both a neutrosophic quadruple subalgebra and a neutrosophic quadruple ideal of

NQ(S). If we take [??] := (c, bT, 0I, aF), [??] := (a, bT, 0I, cF) e NQ(S), then

[mathematical expression not reproducible].

But

[mathematical expression not reproducible].

Therefore NQ(U, V) is not a neutrosophic quadruple q-ideal of NQ(S).

We provide conditions for the neutrosophic quadruple (U, V)-set NQ(U, V) to be a neutrosophic quadruple q-ideal.

Theorem 4.5. If U and V are q-ideals of a BCI-algebra S, then the neutrosophic quadruple (U, V)-set NQ(U, V) is a neutrosophic quadruple q-ideal of NQ(S).

Proof. Suppose that U and V are q-ideals of a BCI-algebra S. Obviously, [mathematical expression not reproducible] be elements of NQ(S) be such that [mathematical expression not reproducible], and

[mathematical expression not reproducible],

that is, [x.sub.i] * ([y.sub.i] * [z.sub.i]) [member of] U and [x.sub.j] * ([y.sub.j] * [z.sub.j]) [member of] B for i = 1, 2 and T = 3, 4. It follows from (2.10) that [x.sub.i] * [z.sub.i] [member of] U and [x.sub.j] * [z.sub.j] e V for i = 1, 2 and T = 3, 4. Thus

[mathematical expression not reproducible], (4.1)

and therefore NQ(U, V) is a neutrosophic quadruple q-ideal of NQ(S).

Corollary 4.6. If A is a q-ideal of a BCI-algebra S, then the neutrosophic quadruple U-set NQ(U) is a neutrosophic quadruple q-ideal of NQ(S).

Corollary 4.7. If {0} is a q-ideal of a BCI-algebra S, then the neutrosophic quadruple (U, V)-set NQ(U, V) is a neutrosophic quadruple q-ideal of NQ(S) for any ideals U and V of S.

Corollary 4.8. If {0} is a q-ideal of a BCI-algebra S, then the neutrosophic quadruple U-set NQ(U) is a neutrosophic quadruple q-ideal of NQ(S) for any ideal U of S.

Theorem 4.9. Let U and V be ideals of a BCI-algebra S such that

([for all]x, y, z [member of] S)(x * (y * z) [member of] U [intersection] V [??] (x * y) * z [member of] U [intersection] V). (4.2)

Then the neutrosophic quadruple (U, V)-set NQ(U, V) is a neutrosophic quadruple q-ideal of NQ(S).

Proof. It is clear that [??] [member of] NQ(U, V). Let [??] = ([x.sub.1], [x.sub.2]T, [x.sub.3]I, [x.sub.4]F), y = ([y.sub.1], [y.sub.2]T, [y.sub.3]I, [y.sub.4]F) and [??] = ([z.sub.1], [z.sub.2]T, [z.sub.3]I, [z.sub.4]F) be elements of NQ(s) be such that [mathematical expression not reproducible]. Then [y.sub.1], [y.sub.2] [member of] U, [y.sub.3], [y.sub.4] [member of] V and

[mathematical expression not reproducible],

that is, [x.sub.i] * ([y.sub.i] * [z.sub.i]) [member of] U and [x.sub.j] * ([y.sub.j] * [z.sub.j]) [member of] V for i = 1, 2 and T = 3, 4. It follows from (2.3) and (4.2) that ([x.sub.i] * [z.sub.j]) * [y.sub.i] = ([x.sub.j] * [y.sub.i]) * [z.sub.i] [member of] U and ([x.sub.j] * [y.sub.j]) * [y.sub.j] = ([x.sub.j] * [y.sub.j]) * [z.sub.j] [member of] V for i = 1, 2 and T = 3, 4. Since U and V are ideals of S, we have [x.sub.i] * [z.sub.i] [member of] U and [x.sub.j] * [z.sub.j] [member of] V for i = 1, 2 and T = 3,4. Thus

[mathematical expression not reproducible], (4.3)

and therefore NQ(U, V) is a neutrosophic quadruple q-ideal of NQ(S).

Corollary 4.10. Let U be an ideal of a BCI-algebra S such that

([for all]x, y, z [member of] S)(x * (y * z) [member of] U [??] (x * y) * z [member of] U). (4.4)

Then the neutrosophic quadruple U-set NQ(U) is a neutrosophic quadruple q-ideal of NQ(S).

Theorem 4.11. Let U and V be ideals of a BCI-algebra S such that

([for all]x, y [member of] S)(x * (0 * y) [member of] U [intersection] V [??] x * y [member of] U [intersection] V). (4.5)

Then the neutrosophic quadruple (U, V)-set NQ(U, V) is a neutrosophic quadruple q-ideal of NQ(S).

Proof. Assume that x * (y * z) [member of] U [intersection] V for all x, y, z [member of] S. Note that

[mathematical expression not reproducible]

Thus (x * y) * (0 * z) [member of] U [intersection] V since U and V are ideals of S. It follows from (4.9) that (x * y) * z [member of] U [intersection] V. Using Theorem 4.9, NQ(U, V) is a neutrosophic quadruple q-ideal of NQ(S).

Corollary 4.12. Let U be an ideal of a BCI-algebra S such that

([for all]x, y [member of] S)(x * (0 * y) [member of] U [??] x * y [member of] U). (4.6)

Then the neutrosophic quadruple U-set NQ(U) is a neutrosophic quadruple q-ideal of NQ(S).

Theorem 4.13. Let U and V be ideals of a BCI-algebra S such that

([for all]x, y [member of] S)(x [member of] U [intersection] U [??] x * y [member of] U [intersection] V). (4.7)

Then the neutrosophic quadruple (U, V)-set NQ(U, V) is a neutrosophic quadruple q-ideal of NQ(S).

Proof. Assume that x * (y * z) [member of] U [intersection] V and y [member of] U [member of] V for all x, y, z [member of] S. Using (2.3) and (4.7), we get (x * z) * (y * z) = (x * (y * z)) * z [member of] U [intersection] V and y * z [member of] U [intersection] V. Since U and V are ideals of S, it follows that x * z [member of] U [intersection] V. Hence U and V are q-ideals of S, and therefore NQ(U, V) is a neutrosophic quadruple q-ideal of NQ(S) by Theorem 4.5.

Corollary 4.14. Let U be an ideal of a BCI-algebra S such that

([for all]x, y [member of] S)(x [member of] U [??] x * y [member of] U). (4.8)

Then the neutrosophic quadruple U-set NQ(U) is a neutrosophic quadruple q-ideal of NQ(S).

Theorem 4.15. Let U, V, I and T be ideals of a BCI-algebra S such that I [subset not equal to] U and J [subset not equal to] V. If I and T are q-ideals of S, then the neutrosophic quadruple (U, V)-set NQ(U, V) is a neutrosophic quadruple q-ideal of NQ(S).

Proof. Let x, y, z [member of] S be such that x * (0 * y) [member of] U [intersection] V. Then

(x * (x * (0 * y))) * (0 * y) = (x * (0 * y)) * (x * (0 * y)) = 0 [member of] I [intersection] J

by (2.3) and (III). Since I and T are q-ideals of S, it follows from (2.3) and (2.10) that

(x * y) * (x * (0 * y)) = (x * (x * (0 * y))) * y [member of] I [intersection] T [subset not equal to] U [intersection] V

Since U and V are ideals of S, we have x * y [member of] U [intersection] V. Therefore NQ(U, V) is a neutrosophic quadruple q-ideal of NQ (S) by Theorem 4.11.

Corollary 4.16. Let U and I be ideals of a BCI-algebra S such that I [subset not equal to] U. If I is a q-ideal of S, then the neutrosophic quadruple U-set NQ(U) is a neutrosophic quadruple q-ideal of NQ(S).

Theorem 4.17. Let U, V, I and T be ideals of a BCI-algebra S such that I [subset not equal to] U, J [subset not equal to] V and

([for all]x, y, z [member of] S)(x * (y * z) [member of] I [intersection] J [??] (x * y) * z [member of] I [intersection] J). (4.9)

Then the neutrosophic quadruple (U, V)-set NQ(U, V) is a neutrosophic quadruple q-ideal of NQ(S).

Proof. Let x, y, z [member of] S be such that x * (y * z) [member of] I [intersection] T and y [member of] I [intersection] J. Then

(x * z) * y = (x * y) * z [member of] I [intersection] J

by (2.3) and (4.9). Since I and T are ideals of S, it follows that x * z [member of] I [intersection] J. This shows that I and T are q-ideals of S. Therefore NQ(U, V) is a neutrosophic quadruple q-ideal of NQ(S) by Theorem 4.15.

Corollary 4.18. Let U and I be ideals of a BCI-algebra S such that I [subset not equal to] U and

([for all]x, y, z [member of] S)(x * (y * z) [member of] I [??] (x * y) * z [member of] I). (4.10)

Then the neutrosophic quadruple U-set NQ(U) is a neutrosophic quadruple q-ideal of NQ(S).

Theorem 4.19. Let U, V, I and T be ideals of a BCI-algebra S such that I [subset not equal to] U, T [subset not equal to] V and

([for all]x, y [member of] S)(x [member of] I [intersection] T [??] x * y [member of] I [intersection] J). (4.11)

Then the neutrosophic quadruple (U, V)-set NQ(U, V) is a neutrosophic quadruple q-ideal of NQ(S).

Proof. By the proof of Theorem 4.13, we know that I and T are q-ideals of S. Hence NQ(U, V) is a neutrosophic quadruple q-ideal of NQ(S) by Theorem 4.15.

Corollary 4.20. Let U and I be ideals of a BCI-algebra S such that I C U and

([for all]x, y [member of] S)(x [member of] I [??] x * y [member of] I). (4.12)

Then the neutrosophic quadruple A-set NQ(U) is a neutrosophic quadruple q-ideal of NQ(S).

Theorem 4.21. Let U, V, I and T be ideals of a BCI-algebra S such that I [subset not equal to] U, J [subset not equal to] V and

([for all]x, y [member of] S)(x * (0 * y) [member of] I [intersection] J [??] x * y [member of] I [intersection] J). (4.13)

Then the neutrosophic quadruple (U, V)-set NQ(U, V) is a neutrosophic quadruple q-ideal of NQ(S).

Proof. Assume that x * (y * z) [member of] I [intersection] J For all x, y, z [member of] S. Then (x * y) * z [member of] I [intersection] T by the proof of Theorem 4.11. It follows from Theorem 4.17 that neutrosophic quadruple (U, V)-set NQ(U, V) is a neutrosophic quadruple q-ideal of NQ(S).

Corollary 4.22. Let U and I be ideals of a BCI-algebra S such that I C U and

([for all]x, y [member of] S)(x * (0 * y) [member of] I [??] x * y [member of] I). (4.14)

Then the neutrosophic quadruple U-set NQ(U) is a neutrosophic quadruple q-ideal of NQ(S).

Future Work: Using the results of this paper, we will apply it to another algebraic structures, for example, MV-algebras, BL-algebras, MTL-algebras, [R.sub.0]-algebras, hoops, (ordered) semigroups and (semi, near) rings etc.

Acknowledgements: We are very thankful to the reviewer(s) for careful detailed reading and helpful comments/suggestions that improve the overall presentation of this paper.

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Received: December 16, 2018. Accepted: March 31, 2019.

G. Muhiuddin (1), Florentin Smarandache (2), Young Bae Jun (3)*

(1) Department of Mathematics, University of Tabuk, Tabuk 71491, Saudi Arabia.

E-mail: chishtygm@gmail.com

(2) Mathematics & Science Department, University of New Mexico, 705 Gurley Ave., Gallup, NM 87301, USA.

E-mail: fsmarandache@gmail.com

(3) Department of Mathematics Education, Gyeongsang National University, Jinju 52828, Korea.

E-mail: skywine@gmail.com

* Correspondence: Young Bae Jun (skywine@gmail.com)
Table 1: Cayley table for the binary operation

*    0    1    a    b    c

0    0    0    a    b    c
1    1    0    a    b    c
a    a    a    0    c    b
b    b    b    c    0    a
c    c    c    b    a    0

Table 2: Cayley table for the binary operation "*"

*      0     1     a

0      0     0     a
1      1     0     a
a      a     a     0

Table 3: Cayley table for the binary operation "*"

*      0     a     b     c

0      0     c     b     a
a      a     0     c     b
b      b     a     0     c
c      c     b     a     0
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