On Neutrosophic Quadruple Algebraic Structures

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On Neutrosophic Quadruple Algebraic Structures

S.A. Akinleye1, F. Smarandache2, A.A.A. Agboola3

1_{Department of Mathematics, Federal University of Agriculture, Abeokuta, Nigeria. E-mail: akinleye [email protected]}

2_{2Department of Mathematics & Science, University of New Mexico, 705 Gurley Ave., Gallup, NM 87301, USA. E-mail: [email protected]} 3_{Department of Mathematics, Federal University of Agriculture, Abeokuta, Nigeria. E-mail: [email protected]}

Abstract. In this paper we present the concept of

neutro-sophic quadruple algebraic structures. Specially, we study neutrosophic quadruple rings and we present their elementary properties.

Keywords: Neutrosophy, neutrosophic quadruple number, neutrosophic quadruple semigroup, neutrosophic quadruple group, neu-trosophic quadruple ring, neuneu-trosophic quadruple ideal, neuneu-trosophic quadruple homomorphism.

1 Introduction

The concept of neutrosophic quadruple numbers was introduced by Florentin Smarandache [3]. It was shown in [3] how arithmetic operations of addition, subtraction, mul-tiplication and scalar mulmul-tiplication could be performed on the set of neutrosophic quadruple numbers. In this paper, we studied neutrosophic sets of quadruple numbers togeth-er with binary optogeth-erations of addition and multiplication and the resulting algebraic structures with their elementary properties are presented. Specially, we studied neutrosoph-ic quadruple rings and we presented their basneutrosoph-ic properties.

Definition 1.1 [3]

A neutrosophic quadruple number is a number of the form , , , � , where , , � have their usual neutro-sophic logic meanings and , , , ∈ ℝ or ℂ. The set defined by

={ , , , � ∶ , , , ∈ ℝ or ℂ} (1) is called a neutrosophic set of quadruple numbers. For a neutrosophic quadruple number , , , � , represent-ing any entity which may be a number, an idea, an object, etc., is called the known part and , , � is called the unknown part.

Definition 1.2 Let

= , , , � ,

= , , , � ∈ .

We define the following:

+ = (2)

+ , + , + , + �

− = (3)

− , − , − , − � .

= , , , � ∈

and let � be any scalar which may be real or complex, the scalar product �. is defined by

�. = �. , , , � =

� , � , � , � � (4)

If � = , then we have . = , , , and for any non-zero scalars m and n and b =

, T, I, F , we have:

+ = + ,

= ,

− = − , − , − , − � .

Definition 1.4 [3][Absorbance Law]

Let � be a set endowed with a total order < , named “x prevailed by y” or “x less strong than y” or “x

less preferred than y”. ≤ is considered as “ prevailed by or equal to ” or “ less strong than or equal to ” or “ less preferred than or equal to ”.

For any elements , ∈ �, with ≤ , absorbance law is deﬁned as

∙ = ∙ = absorb ,

= max{ , } = (5)

which means that the bigger element absorbs the smaller element (the big ﬁsh eats the small ﬁsh). It is clear from (5) that

∙ = = , = { , } = (6)

and

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Analogously, if > , we say that “ prevails to ” or “ is stronger than ” or “ is preferred to ”. Also, if ≥ , we say that “ prevails or is equal to ” or “ is stronger than or equal to ” or “ is preferred or equal to ”.

Deﬁnition 1.5

Consider the set { , , �}. Suppose in an optimistic way we consider the prevalence order > > �. Then we have:

= = max{ , } = , (8)

� = � = max{ , �} = , (9)

� = � = max{ , �} = , (10)

= = , (11)

= = , (12)

�� = � = �. (13)

Analogously, suppose in a pessimistic way we consider the prevalence order < < �. Then we have:

= = { , } = , (14)

� = � = { , �} = �, (15)

� = � = { , �} = �, (16)

= = , (17)

= = , (18)

�� = � = �. (19)

= , , , � ,

= , , , � ∈ .

Then (20)

. = , , , � . , , , �

= , +

+ , + +

+ + , + ,

+ + + + � .

2 Main Results

All neutrosophic quadruple numbers to be considered in this section will be real neutrosophic quadruple numbers i.e , , , ∈ ℝ for any neutrosophic quadruple number

, , , � ∈ .

Theorem 2.1

, + is an abelian group.

Proof.

Suppose that

= , , , � ,

= , , ,

= , , , � ∈

are arbitrary.

It can easily be shown that

+ = + ∙ + + =

+ + ∙ + , , , = , , , =

and

+ − = − + = , , , .

Thus, = , , , is the additive identity element in

, + and for any ∈ , − is the additive inverse. Hence, , + is an abelian group.

Theorem 2.2

, . is a commutative monoid.

Proof.

Let

= , , , � ,

= , , ,

= , , , �

be arbitrary elements in . It can easily be shown that

= ∙ = ∙ ∙ , , , = .

Thus, = , , , is the multiplicative identity ele-ment in , . . Hence, , . is a commutative monoid.

Theorem 2.3

, . is not a group.

Proof.

Let

= , , , �

be any arbitrary element in .

Since we cannot ﬁnd any element = , , , � ∈

such that = = = , , , , it follows that − does not exist in for any given , , , ∈ ℝ

and consequently, , . cannot be a group.

Example 1.

Let � = { , , , � ∶ , , , ∈ ℤ�}. Then �, +

is an abelian group.

Example 2.

Let

× , . = { [

, , , � , , , ℎ�

, , , � , , , � ] :

a, b, c, d, e, f, g, h, i, j, k, l, m, n, p, q ∈ ℝ}

Then × , . is a non-commutative monoid.

Theorem 2.4

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Proof.

It is clear that , + is an abelian group and , .

is a semigroup. To complete the proof, suppose that

= , , , � ,

= , , ,

= , , , � ∈

are arbitrary. It can easily be shown that + = +

, + = + and = . Hence, , +, .

is a commutative ring.

From now on, the ring , +, . will be called neutro-sophic quadruple ring and it will be denoted by . The zero element of will be denoted by (0, 0, 0, 0) and the unity of will be denoted by (1, 0, 0, 0).

Example 3.

(i) Let � be as deﬁned in EXAMPLE 1. Then �, +, .

is a commutative neutrosophic quadruple ring called a neu-trosophic quadruple ring of integers modulo .

It should be noted that ℤ� has �_{elements and}

for ℤ we have

ℤ =

= { , , , , , , , , , , , , , , , , , , , � , , , , � , , , , � , , , , , , , , � , , , , , , , , , , , , � , , , , � , , , , � , , , , , , , , � }.

(ii) Let _× be as deﬁned in EXAMPLE 2. Then

× , . is a non-commutative neutrosophic quadruple

ring.

Definition 2.5

Let be a neutrosophic quadruple ring.

(i) An element ∈ is called idempotent if = . (ii) An element ∈ is called nilpotent if there

∈ +_{such that} �₌ _. Example 4.

(i) In ℤ , , , , � and , , , are idempo-tent elements.

(ii) In ℤ , , , , � is a nilpotent element.

Definition 2.6

is called a neutrosophic quadruple integral do-main if for , ∈ , = implies that =

or = .

Example 5.

ℤ the neutrosophic quadruple ring of integers is a neutrosophic quadruple integral domain.

Definition 2.7

An element ∈ is called a zero divisor if there

exists a nonzero element ∈ such that = . For example in ℤ , , , , � and , , , are zero divisors even though ℤ has no zero divisors.

This is one of the distinct features that characterize neutrosophic quadruple rings.

Definition 2.8

Let be a neutrosophic quadruple ring and let be a nonempty subset of . Then is called a neu-trosophic quadruple subring of if , +, . is itself a neutrosophic quadruple ring. For example, ℤ is a neutrosophic quadruple subring of ℤ for =

, , ,···.

Theorem 2.9

Let be a nonempty subset of a neutrosophic quad-ruple ring . Then is a neutrosophic quadruple subring if and only if for all , ∈ , the following conditions hold:

(i) − ∈

and

(ii) ∈ .

Proof.

Same as the classical case and so omitted.

Definition 2.10

Let be a neutrosophic quadruple ring. Then the set

= { ∈ : = ∀ ∈ }

is called the centre of .

Theorem 2.11

Then is a neutrosophic quadruple subring of .

Proof.

Same as the classical case and so omitted.

Theorem 2.12

Let be a neutrosophic quadruple ring and let

� be families of neutrosophic quadruple subrings of

. Then

⋂

�= �

is a neutrosophic quadruple subring of .

Definition 2.13

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each ∈ , then the smallest such positive integer is called the characteristic of . If no such positive integer exists, then is said to have characteristic zero. For example, ℤ has characteristic zero and ℤ�

has characteristic .

Definition 2.14

Let be a nonempty subset of a neutrosophic quad-ruple ring . is called a neutrosophic quadruple ideal of if for all x, y ∈ , ∈ , the following conditions hold:

(i) − ∈ .

(ii) ∈ and ∈ .

Example 6.

(i) ℤ is a neutrosophic quadruple ideal of

ℤ . (ii) Let

=

{ , , , , , , , , , , , � , , , , � }

be a subset of ℤ . Then is a neutrosophic quadruple ideal.

Theorem 2.15

Let and be neutrosophic quadruple ideals of and let

{ �}�=�

be a family of neutrosophic quadruple ideals of . Then:

(i) + = .

(ii) + = for all ∈ . (iii)

⋂

�= �

is a neutrosophic quadruple ideal of . (iv) + is a neutrosophic quadruple ideal of

.

Definition 2.16

Let be a neutrosophic quadruple ideal of . The set

/ = { + ∶ ∈ }

is called a neutrosophic quadruple quotient ring.

If + and + are two arbitrary elements of

/ and if ⊕ and ⊙ are two binary operations on

/ deﬁned by:

+ ⊕ + = + + ,

+ ⊙ + = + ,

it can be shown that ⊕ and ⊙ are well deﬁned and that (NQR/NQJ, ⊕, ⊙) is a neutrosophic quadruple ring.

Example 7.

Consider the neutrosophic quadruple ring ℤ and its neutrosophic quadruple ideal ℤ . Then

ℤ ℤ =

{ ℤ , , , , + ℤ , , , ,

+ ℤ , , , , + ℤ , , , , �

+ ℤ , , , , � + ℤ , , , , �

+ ℤ , , , , + ℤ , , , , �

+ ℤ , , , , + ℤ , , , ,

+ ℤ , , , , � + ℤ , , , , �

+ ℤ , , , , � + ℤ , , , , +

ℤ , , , , � + ℤ }.

which is clearly a neutrosophic quadruple ring.

Definition 2.17

Let and be two neutrosophic quadruple rings and let � ∶ → be a mapping deﬁned for all , ∈ as follows:

(i) � + = � + � .

(ii) � = � � .

(iii) � = , � = and � � = �.

(iv) � , , , = , , , .

Then � is called a neutrosophic quadruple homomor-phism. Neutrosophic quadruple monomorphism, endomor-phism, isomorendomor-phism, and other morphisms can be deﬁned in the usual way.

Definition 2.18

Let � ∶ → be a neutrosophic quadruple ring homomorphism.

(i) The image of � denoted by � is deﬁned by the set � = { ∈ ∶ = � , for some ∈

}.

(ii) The kernel of � denoted by � is deﬁned by the set � = { ∈ ∶ � = , , , }.

Theorem 2.19

Let � ∶ → be a neutrosophic quadruple ring homomorphism. Then:

(i) � is a neutrosophic quadruple subring of . (ii) � is not a neutrosophic quadruple ideal of .

Proof.

(i) Clear.

(ii) Since , , � cannot have image (0,0,0,0) under �, it follows that the elements , , , , , , , , , , , �

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Example 8.

Consider the projection map

� ∶ ℤ × ℤ → ℤ

deﬁned by � , = for all , ∈ ℤ .

It is clear that � is a neutrosophic quadruple homo-morphism and its kernel is given as

� =

{{ , , , , , , , , , , , , , , , , ( , , , , , , , ), ( , , , , , , , ), ( , , , , , , , � ), ( , , , , , , , � ), ( , , , , , , , � ), ( , , , , , , , ), ( , , , , , , , � ), ( , , , , , , , ), ( , , , , , , , ), ( , , , , , , , � ), ( , , , , , , , � ), ( , , , , , , , � ), , , , , , , , , , , , , , , , � }.

Theorem 2.20

Let φ: NQR(Z) → NQR(Z)/NQR(nZ) be a mapping de-fined by φ(x) = x + NQR(nZ) for all x ∈ NQR(Z) and n =

, , , … . Then φ is not a neutrosophic quadruple ring homomorphism.

References

[1] A.A.A. Agboola, On Refined Neutrosophic Algebraic Struc-turesI, Neutrosophic Sets and Systems 10 (2015), 99-101. [2] F. Smarandache, Neutrosophy/Neutrosophic Probability, Set,

and Logic, American Research Press, Rehoboth, USA, 1998, http://fs.gallup.unm.edu/eBook-otherformats.htm

[3] F. Smarandache, Neutrosophic Quadruple Numbers, Refined Neutrosophic Quadruple Numbers, Absorbance Law, and the Multiplication of Neutrosophic Quadruple Numbers, Neutrosophic Sets and Systems 10 (2015), 96-98.

[4] F. Smarandache, (t,i,f) - Neutrosophic Structures and

I-Neutrosophic Structures, I-Neutrosophic Sets and Systems 8 (2015), 3-10.

[5] F. Smarandache, n-Valued Refined Neutrosophic Logic and

Its Applications in Physics, Progress in Physics 4 (2013), 143-146.