1. bookVolume 6 (2021): Issue 1 (January 2021)
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Some Characterizations of Neutrosophic Submodules of an R-module

Published Online: 31 Dec 2020
Page range: 359 - 372
Received: 23 Aug 2019
Accepted: 27 Feb 2020
Journal Details
License
Format
Journal
First Published
01 Jan 2016
Publication timeframe
2 times per year
Languages
English
Abstract

In this paper, we investigate some new characterizations in the algebraic nature of neutrosophic submodule defined over a classical module using single valued neutrosophic set. We additionally characterized the neutrosophic point and determined the algebraic properties of neutrosophic point using the operations defined on neutrosophic submodule. Finley we examined a neutrosophic set as a generator of a neutrosophic submodule and derived some related concepts.

Keywords

Introduction

A fuzzy set represents vague concepts and contexts expressed in natural language by means of graded membership of elements in [0,1] which is introduced by Lotfi A. Zadeh in 1965 [14, 25]. In 1986, Attanassov put forward intuitionistic fuzzy set hypothesis as a delineation of a set in which every segment is corresponding with a participation grades and non enrollment [3]. In 1995, Smarandache outlined neutrosophic set in which each element of a set is represented by three differing types of membership values and objective is to narrow the gap between the vague, ambiguous and inexact real world situations [5,6,7, 23]. Neutrosophic set theory gives a thorough scientific stage in which wispy and uncertain hypothetical phenomena can be managed by hierarchal membership of components.

The algebraic structure in pure mathematics cloning with uncertainty has been studied by some authors. In 1971, Azriel Rosenfield bestowed a seminal paper on fuzzy subgroup and W.J. Liu developed the idea of fuzzy normal subgroup and fuzzy subring. Consolidating neutrosophic set hypothesis with algebraic structures is a rising pattern in the region of mathematical research. In 2011, Isaac.P, P.P.John [12] recognized some algebraic nature of intuitionistic fuzzy submodule of a classical module. Neutrosophic algebraical structures and its properties provide us a solid mathematical foundation to clarify connected scientific ideas in designing, information mining and economics. In this paper we discuss about the generators of a neutrosophic submodule and some related results.

Preliminaries
Definition 2.1

[2] Let R be a commutative ring with unity. A module M over R, denoted as MR is an abelian group with a law of composition written ‘+’and a scalar multiplication R × MM, written (r,v) ⇝ rv, that satisfy these axioms

1v = v

(rs)v = r(sv)

(r + s)v = rv + sv

r(v + v) = rv + rvr,sR and v,v‘ ∈ M.

Definition 2.2

[2] A submodule N of MR is a nonempty subset of MR that is closed under addition and scalar multiplication.

Definition 2.3

[21, 24] A neutrosophic set P of the universal set X (NS(X)) is defined as P={(η,tP(η),iP(η),fP(η)):ηX} P = \{ (\eta ,{t_P}(\eta ),{i_P}(\eta ),{f_P}(\eta )):\eta \in X\} where tP,iP, fP : X → (0,1+). The three components tP, iP and fP represent membership value (Percentage of truth), indeterminacy (Percentage of indeterminacy) and non membership value (Percentage of falsity) respectively. These components are functions of non standard unit interval (0,1+) [18].

Remark 2.1

[10,21] If the components of a neutrosophic set P, tP,iP, fP : X → [0,1], then P is known as single valued neutrosophic set(SVNS).

Remark 2.2

In this paper, we discuss about the algebraic structure MR-module with underlying set as SVNS. For simplicity SVNS will be called neutrosophic set.

Remark 2.3

UX denotes the set of all neutrosophic subset of X or neutrosophic power set of X.

Definition 2.4

[17, 21, 22] Let P, QUX. Then P is contained in Q, denoted as PQ if and only if P(η) ⩽ Q(η) ∀ ηX, this means that tP(η)tQ(η),iP(η)iQ(η),fP(η)fQ(η),ηX {t_P}(\eta ) \le {t_Q}(\eta ), {i_P}(\eta ) \le {i_Q}(\eta ), {f_P}(\eta ) \ge {f_Q}(\eta ), \forall \eta \in X

Definition 2.5

[13, 19, 21] The complement of P = {(x,tP(x),iP(x), fP(x) : xX} ∈ UX is denoted by PC and defined as PC = {x, fP(x), 1 − iP(x), tP(x) : xX}.

Remark 2.4

(PC)C = P where PUX

Definition 2.6

[8, 13, 21] Let P, QUX

The union C = {η,tC(η), iC(η), fC(η) : ηX} of P and Q [17] is denoted by C = PQ where tC(η)=tP(η)tQ(η)iC(η)=iP(η)iQ(η)fC(η)=fP(η)fQ(η) \matrix{{{t_C}(\eta ) = {t_P}(\eta ) \vee {t_Q}(\eta )} \cr {{i_C}(\eta ) = {i_P}(\eta ) \vee {i_Q}(\eta )} \cr {{f_C}(\eta ) = {f_P}(\eta ) \wedge {f_Q}(\eta )} \cr }

The intersection C = {η, tC(η), iC(η), fC(η) : ηX} of P and Q [17] is denoted by C = P ∩ Q where tC(η)=tP(η)tQ(η)iC(η)=iP(η)iQ(η)fC(η)=fP(η)fQ(η) \matrix{{{t_C}(\eta ) = {t_P}(\eta ) \wedge {t_Q}(\eta )} \cr {{i_C}(\eta ) = {i_P}(\eta ) \wedge {i_Q}(\eta )} \cr {{f_C}(\eta ) = {f_P}(\eta ) \vee {f_Q}(\eta )} \cr }

Definition 2.7

[17,22] For any P = {(η,tP(η), iP(η), fP(η)) : ηX} ∈ UX, the support P* of P can be defined as P*={ηX,tP(η)>0,iP(η)>0,fP(η)<1} {P^*} = \{ \eta \in X,{t_P}(\eta ) > 0,{i_P}(\eta ) > 0,{f_P}(\eta ) < 1\}

Definition 2.8

[1,16] Let P = {(η, tP(η), iP(η), fP(η)) : ηR} be an NS(R). Then P is called a neutrosophic ideal of R if it satisfies the following conditions ∀ η,θR

tP(ηθ) ≥ tP(η) ∧ tP(θ)

iP(ηθ) ≥ iP(η) ∧ iP(θ)

fP(ηθ) ≥ fP(η) ∨ fP(θ)

tP(ηθ) ≥ tP(η) ∨ tP(θ)

iP(ηθ) ≥ iP(η) ∨ iP(θ)

fP(ηθ) ≤ fP(η) ∧ fP(θ)

Remark 2.4

We denote the set of all neutrosophic ideals of R by U(R)

Neutrosophic submodule
Definition 3.1

[8, 9] A neutrosophic subset PUMR is called a neutrosophic submodule of MR if

tP(0) = 1, iP(0) = 1, fP(0) = 0

tP(η + θ) ≥ tP(η) ∧ tP(θ)

iP(η + θ) ≥ iP(η) ∧ iP(θ)

fP(η + θ) ≤ fP(η) ∨ fP(θ), for all η, θ in MR

tP(γη) ≥ tP(η)

iP(γη) ≥ iP(η)

fP(γη) ≤ fP(η), for all η in MR, for all γ in R

Remark 3.1

We denote neutrosophic submodules over MR using single valued neutrosophic set by U(M).

Remark 3.2

If PU(M), then the neutrosophic components of P can be denoted as (tP(η), iP(η), fP(η)) ∀ ηMR.

Definition 3.2

[8] A neutrosophic subset γP = {η,tγP(η), iγP(η), fγP(η) : ηMRR} of MR where PUM defined as follows tγP(η)={tP(θ):θMR,η=γθ}iγP(η)={iP(θ):θMR,η=γθ}fγP(η)={fP(θ):θMR,η=γθ} \matrix{{{t_{\gamma P}}(\eta ) = \vee \{ {t_P}(\theta ):\theta \in {M_R},\eta = \gamma \theta \} } \cr {{i_{\gamma P}}(\eta ) = \vee \{ {i_P}(\theta ):\theta \in {M_R},\eta = \gamma \theta \} } \cr {{f_{\gamma P}}(\eta ) = \wedge \{ {f_P}(\theta ):\theta \in {M_R},\eta = \gamma \theta \} } \cr }

Proposition 3.1

Let P = {η,tP(η), iP(η), fP(η); ηMR} ∈ UMR, then tγP(γη) ≥ tP(η), iγP(γη) ≥ iP(η) and fγP(γη) ≤ fP(η).

Proof

We have tγP(γη)={tP(θ):θMR,γη=γθ}tP(η),ηMR {t_{\gamma P}}(\gamma \eta ) = \vee \{ {t_P}(\theta ):\theta \in {M_R},\gamma \eta = \gamma \theta \} \ge {t_P}(\eta ),\forall \eta \in {M_R} Similarly iγP(γη) ≥ iP(η). Also fγP(γη)={fP(θ):θMR,γη=γθ}fP(η),ηMR {f_{\gamma P}}(\gamma \eta ) = \wedge \{ {f_P}(\theta ):\theta \in {M_R},\gamma \eta = \gamma \theta \} \le {f_P}(\eta ),\forall \eta \in {M_R}

Definition 3.3

[8] Let P = {η, tP(η), iP(η), fP(η); ηMR} ∈ UMR, then P={η,tP(η),iP(η),fP(η);ηMR}UMR - P = \{ \eta ,{t_{ - P}}(\eta ),{i_{ - P}}(\eta ),{f_{ - P}}(\eta );\eta \in {M_R}\} \in {U^{{M_R}}} where tP(η)=tP(η),iP(η)=iP(η),fP(η)=fP(η),ηMR {t_{ - P}}(\eta ) = {t_P}( - \eta ), {i_{ - P}}(\eta ) = {i_P}( - \eta ), {f_{ - P}}(\eta ) = {f_P}( - \eta ), \forall \eta \in {M_R}

Proposition 3.2

[8] If P = {η, tP(η), iP(η), fP(η); ηMR} ∈ UMR, then 1.P = P and (−1)P = −P

Theorem 3.1

[8] Let PUMR, then PU(M) if and only if the following properties are satisfiedη,θMR, γ,βR i)tP(0)=1,iP(0)=1,fP(0)=0ii)tP(γη+βθ)tP(η)tP(θ),iP(γη+βθ)iP(η)iP(θ),fP(γη+βθ)fP(η)fP(θ) \matrix{{i)\;\;{t_P}(0) = 1,{i_P}(0) = 1,{f_P}(0) = 0} \cr {ii)\;\;{t_P}(\gamma \eta + \beta \theta ) \ge {t_P}(\eta ) \wedge {t_P}(\theta ), {i_P}(\gamma \eta + \beta \theta ) \ge {i_P}(\eta ) \wedge {i_P}(\theta ), {f_P}(\gamma \eta + \beta \theta ) \le {f_P}(\eta ) \vee {f_P}(\theta )} \cr }

Theorem 3.2

Let PU(M). Then P* is a neutrosophic submodule of MR.

Proof

Given PU(M) and P* = {ηMR, tP(η) > 0, iP(η) > 0, fP(η) < 1}. Let η, θP*. Then tP(η)>0,iP(η)>0,fP(η)<1tP(θ)>0,iP(θ)>0,fP(θ)<1 \matrix{{{t_P}(\eta ) > 0,{i_P}(\eta ) > 0,{f_P}(\eta ) < 1} \cr {{t_P}(\theta ) > 0,{i_P}(\theta ) > 0,{f_P}(\theta ) < 1} \cr }

To prove that γη + β θP* where γ,βR

⇒ to prove that tP(γη + β θ) > 0, iP(γη + β θ) > 0, fP(γη + β θ) < 1

Now tP(γη+βθ)tP(γη)tP(βθ)tP(η)tP(θ)>0 \matrix{{{t_P}(\gamma \eta + \beta \theta )} \hfill & \ge \hfill & {{t_P}(\gamma \eta ) \wedge {t_P}(\beta \theta )} \hfill \cr {} \hfill & \ge \hfill & {{t_P}(\eta ) \wedge {t_P}(\theta )} \hfill \cr {} \hfill & > \hfill & 0 \hfill \cr }

In the same way, we can prove the other two inequalities. Hence the proof.

Definition 3.4

Let Pi, iJ be an arbitrary non empty family of UMR, then

i∈J Pi = {η,ti∈JPi (η), ii∈JPi (η), fi∈JPi (η) : ηMR} where tiJPi(η)=iJtPi(η)iiJPi(η)=iJiPi(η)fiJPi(η)=iJfPi(η) \matrix{{{t_{\bigcap\nolimits_{i \in J} {P_i}}}(\eta ) = \mathop \wedge \limits_{i \in J} {t_{{P_i}}}(\eta )} \cr {{i_{\bigcap\nolimits_{i \in J} {P_i}}}(\eta ) = \mathop \wedge \limits_{i \in J} {i_{{P_i}}}(\eta )} \cr {{f_{\bigcap\nolimits_{i \in J} {P_i}}}(\eta ) = \mathop \vee \limits_{i \in J} {f_{{P_i}}}(\eta )} \cr }

i∈J Pi = {η, ti∈JPi(η), ii∈JPi(η), fi∈JPi(η) : ηMR} where tiJPi(η)=iJtPi(η)iiJPi(η)=iJiPi(η)fiJPi(η)=iJfPi(η) \matrix{{{t_{\bigcup\nolimits_{i \in J} {P_i}}}(\eta ) = \mathop \vee \limits_{i \in J} {t_{{P_i}}}(\eta )} \cr {{i_{\bigcup\nolimits_{i \in J} {P_i}}}(\eta ) = \mathop \vee \limits_{i \in J} {i_{{P_i}}}(\eta )} \cr {{f_{\bigcup\nolimits_{i \in J} {P_i}}}(\eta ) = \mathop \wedge \limits_{i \in J} {f_{{P_i}}}(\eta )} \cr }

Proposition 3.3

Let Pi, iJ be an arbitrary non empty family of UMR, then γ(∪iJ Pi) = ∪i J(γPi) for γR

Proof

Consider γi∈J Pi = {η,tγi∈JPi(η), iγi∈JPi(η), fγi∈JPi (η) : ηMR, γR}

Now tγiJPi(η)={tiJPi(θ):θMR,η=γθ}={iJtPi(θ):θMR,η=γθ}=iJtγPi(η)=tiJγPi(η) \matrix{{{t_{\gamma \bigcup\nolimits_{i \in J} {P_i}}}(\eta )} \hfill & { = \vee \{ {t_{\bigcup\nolimits_{i \in J} {P_i}}}(\theta ):\theta \in {M_R},\eta = \gamma \theta \} } \hfill \cr {} \hfill & { = \vee \{ \mathop \vee \limits_{i \in J} {t_{{P_i}}}(\theta ):\theta \in {M_R},\eta = \gamma \theta \} } \hfill \cr {} \hfill & { = \mathop \vee \limits_{i \in J} {t_{\gamma {P_i}}}(\eta )} \hfill \cr {} \hfill & { = {t_{\bigcup\nolimits_{i \in J} \gamma {P_i}}}(\eta )} \hfill \cr }

Similarly iγi∈JPi (η) = ii∈JγPi (η)

Now fγiJPi(η)={fiJPi(θ):θMR,η=γθ}={iJfPi(θ):θMR,η=γθ}=iJfγPi(η)=fiJγPi(η) \matrix{{{f_{\gamma \bigcup\nolimits_{i \in J} {P_i}}}(\eta )} \hfill & { = \wedge \{ {f_{\bigcup\nolimits_{i \in J} {P_i}}}(\theta ):\theta \in {M_R},\eta = \gamma \theta \} } \hfill \cr {} \hfill & { = \wedge \{ \mathop \wedge \limits_{i \in J} {f_{{P_i}}}(\theta ):\theta \in {M_R},\eta = \gamma \theta \} } \hfill \cr {} \hfill & { = \mathop \wedge \limits_{i \in J} {f_{\gamma {P_i}}}(\eta )} \hfill \cr {} \hfill & { = {f_{\bigcup\nolimits_{i \in J} \gamma {P_i}}}(\eta )} \hfill \cr }

Hence γ(∪iJ Pi) = ∪i J(γPi) for γR

Theorem 3.3

Let Pi, iJ be an arbitrary non empty family of U(M), theniJ PiU(M)

Proof

We have ∩iJ Pi = {η,ti∈JPi (η), ii∈JPi (η), fi∈JPi (η) : ηMR} and tiJPi(0)=iJtPi(0)=1iiJPi(0)=iJiPi(0)=1fiJPi(0)=iJfPi(0)=0 \matrix{{{t_{\bigcap\nolimits_{i \in J} {P_i}}}(0) = \mathop \wedge \limits_{i \in J} {t_{{P_i}}}(0) = 1} \cr {{i_{\bigcap\nolimits_{i \in J} {P_i}}}(0) = \mathop \wedge \limits_{i \in J} {i_{{P_i}}}(0) = 1} \cr {{f_{\bigcap\nolimits_{i \in J} {P_i}}}(0) = \mathop \vee \limits_{i \in J} {f_{{P_i}}}(0) = 0} \cr }

Now tiJPi(γη+βθ)=iJtPi(γη+βθ)iJ(tPi(η)tPi(θ))=[iJtPi(η)][iJtPi(θ)]=tiJPi(η)tiJPi(θ) \matrix{{{t_{\bigcap\nolimits_{i \in J} {P_i}}}(\gamma \eta + \beta \theta )} \hfill & { = \mathop \wedge \limits_{i \in J} {t_{{P_i}}}(\gamma \eta + \beta \theta )} \hfill \cr {} \hfill & { \ge \mathop \wedge \limits_{i \in J} ({t_{{P_i}}}(\eta ) \wedge {t_{{P_i}}}(\theta ))} \hfill \cr {} \hfill & { = [\mathop \wedge \limits_{i \in J} {t_{{P_i}}}(\eta )] \wedge [\mathop \wedge \limits_{i \in J} {t_{{P_i}}}(\theta )]} \hfill \cr {} \hfill & { = {t_{\bigcap\nolimits_{i \in J} {P_i}}}(\eta ) \wedge {t_{\bigcap\nolimits_{i \in J} {P_i}}}(\theta )} \hfill \cr } in the same way we can derive iiJPi(η+θ)iiJPi(η)iiJPi(θ)fiJPi(η+θ)fiJPi(η)fiJPi(θ) \matrix{{{i_{\bigcap\nolimits_{i \in J} {P_i}}}(\eta + \theta ) \ge {i_{\bigcap\nolimits_{i \in J} {P_i}}}(\eta ) \wedge {i_{\bigcap\nolimits_{i \in J} {P_i}}}(\theta )} \cr {{f_{\bigcap\nolimits_{i \in J} {P_i}}}(\eta + \theta ) \le {f_{\bigcap\nolimits_{i \in J} {P_i}}}(\eta ) \vee {f_{\bigcap\nolimits_{i \in J} {P_i}}}(\theta )} \cr } Hence ∩iJ PiU(M)

Definition 3.5

[20] Let P,QUMR, then the sum P+Q={η,tP+Q(η),tP+Q(η),tP+Q(η):ηMR}UMR P + Q = \{ \eta ,{t_{P + Q}}(\eta ),{t_{P + Q}}(\eta ),{t_{P + Q}}(\eta ):\eta \in {M_R}\} \in {U^{{M_R}}} defined as follows tP+Q(η)={tP(θ)tQ(ϑ)|η=θ+ϑ,θ,ϑMR}iP+Q(η)={iP(θ)iQ(ϑ)|η=θ+ϑ,θ,ϑMR}fP+Q(η)={fP(θ)fB(ϑ)|η=θ+ϑ,θ,ϑMR} \matrix{{{t_{P + Q}}(\eta ) = \vee \{ {t_P}(\theta ) \wedge {t_Q}(\vartheta )|\eta = \theta + \vartheta ,\theta ,\vartheta \in {M_R}\} } \cr {{i_{P + Q}}(\eta ) = \vee \{ {i_P}(\theta ) \wedge {i_Q}(\vartheta )|\eta = \theta + \vartheta ,\theta ,\vartheta \in {M_R}\} } \cr {{f_{P + Q}}(\eta ) = \wedge \{ {f_P}(\theta ) \vee {f_B}(\vartheta )|\eta = \theta + \vartheta ,\theta ,\vartheta \in {M_R}\} } \cr }

Definition 3.6

Let Pi, iJ be an arbitrary family of U(M) where Pi = {η, tPi(η), iPi(η), fPi(η) : ηM} for each iJ. Then iJPi={η,tiJPi(η),iiJPi(η),fiJPi(η):ηMR} \sum\limits_{i \in J} {P_i} = \{ \eta ,{t_{\mathop {\sum }\limits_{i \in J} {P_i}}}(\eta ),{i_{\mathop {\sum }\limits_{i \in J} {P_i}}}(\eta ),{f_{\mathop {\sum }\limits_{i \in J} {P_i}}}(\eta ):\eta \in {M_R}\} where tiJPi(η)={iJtPi(ηi):ηiMR,iJηi=η}ηMRiiJPi(x)={iJiPi(Mi):ηiMR,iJηi=η}ηMRfiJPi(x)={iJfPi(ηi):ηiMR,iJηi=η}ηMR \matrix{{{t_{\mathop {\sum }\limits_{i \in J} {P_i}}}(\eta ) = \vee \{ \mathop \wedge \limits_{i \in J} {t_{{P_i}}}({\eta _i}):{\eta _i} \in {M_R},\sum\limits_{i \in J} {\eta _i} = \eta \} \forall \eta \in {M_R}} \cr {{i_{\mathop {\sum }\limits_{i \in J} {P_i}}}(x) = \vee \{ \mathop \wedge \limits_{i \in J} {i_{{P_i}}}({M_i}):{\eta _i} \in {M_R},\sum\limits_{i \in J} {\eta _i} = \eta \} \forall \eta \in {M_R}} \cr {{f_{\mathop {\sum }\limits_{i \in J} {P_i}}}(x) = \wedge \{ \mathop \vee \limits_{i \in J} {f_{{P_i}}}({\eta _i}):{\eta _i} \in {M_R},\sum\limits_{i \in J} {\eta _i} = \eta \} \forall \eta \in {M_R}} \cr } where, in iJηi \mathop {\sum }\limits_{i \in J} {\eta _i} , at most finitely ηi ≠ 0.

Theorem 3.4

If P,QU(M), then P + QU(M)

Proof

It is enough to prove P + Q satisfies the properties listed below ∀η,θMR,γ,βR

tP+Q(0) = 1, iP+Q(0) = 1, fP+Q(0) = 0.

tA+B(γη + β θ) ≥ tP+Q(η) ∧ tP+Q(θ), iP+Q(γη + β θ) ≥ iP+Q(η) ∧ iP+Q(θ), fA+B(γη + β θ) ≤ fP+Q(η) ∨ fP+Q(θ)

From the definition 3.5, property 1 is obvious because P,QU(M).

Consider tP+Q(η)tP+Q(θ)={tP(η1)tQ(η2):η=η1+η2}{tP(θ1)tQ(θ2):θ=θ1+θ2}{tP(γη1)tQ(γη2):γη=γη1+γη2}{tP(βθ1)tQ(βθ2):βθ=βθ1+βθ2}={[tP(γη1)tP(βθ1)][tQ(γη2)tQ(βθ2)]:γη=γη1+γη2,βθ=βθ1+βθ2}{tP(γη1+βθ1)tQ(γη2+βθ2):γη+βθ=γη1+βθ1+γη2+βθ2}tP+Q(γη+βθ)whereγη+βθ=γ(η1+η2)+β(θ1+θ2)η1,η2,θ1,θ2MR \matrix{{{t_{P + Q}}(\eta ) \wedge {t_{P + Q}}(\theta )} \hfill & { = \mathop \vee \{ {t_P}({\eta _1}) \wedge {t_Q}({\eta _2}):\eta = {\eta _1} + {\eta _2}\} \wedge \mathop \vee \{ {t_P}({\theta _1}) \wedge {t_Q}({\theta _2}):\theta = {\theta _1} + {\theta _2}\} } \hfill \cr {} \hfill & { \le \mathop \vee \{ {t_P}(\gamma {\eta _1}) \wedge {t_Q}(\gamma {\eta _2}):\gamma \eta = \gamma {\eta _1} + \gamma {\eta _2}\} \wedge \mathop \vee \{ {t_P}(\beta {\theta _1}) \wedge {t_Q}(\beta {\theta _2}):\beta \theta = \beta {\theta _1} + \beta {\theta _2}\} } \hfill \cr {} \hfill & { = \mathop \vee \{ [{t_P}(\gamma {\eta _1}) \wedge {t_P}(\beta {\theta _1})] \wedge [{t_Q}(\gamma {\eta _2}) \wedge {t_Q}(\beta {\theta _2})]:\gamma \eta = \gamma {\eta _1} + \gamma {\eta _2},\beta \theta = \beta {\theta _1} + \beta {\theta _2}\} } \hfill \cr {} \hfill & { \le \mathop \vee \{ {t_P}(\gamma {\eta _1} + \beta {\theta _1}) \wedge {t_Q}(\gamma {\eta _2} + \beta {\theta _2}):\gamma \eta + \beta \theta = \gamma {\eta _1} + \beta {\theta _1} + \gamma {\eta _2} + \beta {\theta _2}\} } \hfill \cr {} \hfill & { \le {t_{P + Q}}(\gamma \eta + \beta \theta )\; where\; \gamma \eta + \beta \theta = \gamma ({\eta _1} + {\eta _2}) + \beta ({\theta _1} + {\theta _2}) \forall {\eta _1},{\eta _2},{\theta _1},{\theta _2} \in {M_R}} \hfill \cr }

Similarly, iP+Q(γη + β θ) ≥ iP+Q(η) ∧ iP+Q(θ), fP+Q(γη + β θ) ≤ fP+Q(η) ∧ fP+Q(θ) ⇒ P+ ∈ U(M).

Corollary 3.4.1

Let Pi, iJ be a family of neutrosophic submodules of an MR. Then iJPiU(M) \mathop {\sum }\limits_{i \in J} {P_i} \in U(M) .

Definition 3.7

For any ηX, the neutrosophic point N^{η} {\hat N_{\{ \eta \} }} is defined as N^{η}(s)={(s,tN^{η},iN^{η},fN^{η}):sX} {\hat N_{\{ \eta \} }}(s) = \{ (s,{t_{{{\hat N}_{\{ \eta \} }}}},{i_{{{\hat N}_{\{ \eta \} }}}},{f_{{{\hat N}_{\{ \eta \} }}}}):s \in X\} where N^{η}(s)={(1,1,0)η=s(0,0,1)ηs {\hat N_{\{ \eta \} }}(s) = \left\{ {\matrix{{(1,1,0)} & {\eta = s} \cr {(0,0,1)} & {\eta \ne s} \cr } } \right.

Remark 3.3

Let X be a non empty set. The neutrosophic point N^{0} {\hat N_{\{ 0\} }} in X is defined as N^{0}(x)={(x,tN^{0},iN^{0},fN^{0}):xX} {\hat N_{\{ 0\} }}(x) = \{ (x,{t_{{{\hat N}_{\{ 0\} }}}},{i_{{{\hat N}_{\{ 0\} }}}},{f_{{{\hat N}_{\{ 0\} }}}}):x \in X\} where N^{0}(x)={(1,1,0)x=0(0,0,1)x0 {\hat N_{\{ 0\} }}(x) = \left\{ {\matrix{{(1,1,0)} & {x = 0} \cr {(0,0,1)} & {x \ne 0} \cr } } \right.

Theorem 3.5

Let PU(M). P=N^{0}P*={0} P = {\hat N_{\{ 0\} }} \Leftrightarrow {P^*} = \{ 0\}

Proof

If P=N^{0} P = {\hat N_{\{ 0\} }} , and P* = {ηMR,tP(η) > 0, iP(η) > 0, fP(η) < 1} = {0}.

Conversely, if P* = {0} ⇒ tP(0) > 0, iP(0) > 0, fP(η) < 1 and tP(η) = 0, iP(η) = 0 and fP(η) = 1 ∀ η ≠ 0. Therefore P(η)={(1,1,0)η=0(0,0,1)η0=N^{0} P(\eta ) = \left\{ {\matrix{{(1,1,0)} & {\eta = 0} \cr {(0,0,1)} & {\eta \ne 0} \cr } } \right. = {\hat N_{\{ 0\} }}

Neutrosophic Submodule Generated by Neutrosophic Set

In this section we study about the U(M) of MR generated by single valued neutrosophic set defined over a classical module.

Definition 4.1

Let p = {η,tP(η), iP(η), fP(η) : ηMR} ∈ UM. Then the U(M) of MR generated by neutrosophic set P can be denoted and defined as P={Q|PQ:QU(M)} \langle P\rangle = \cap \{ Q|P \subseteq Q:Q \in U(M)\}

Remark 4.1

If Q = 〈P〉, then P is called generator of Q.

Theorem 4.1

Let Pi = {(η,tPi(η),iPi(η), fPi(η) : iJ, ηMR} be an arbitrary non empty family of NS(MR). Then 〈∪i JPi〉 = ∑iJ Pi

Proof

By a corollary 3.4.1, we can write iJPi={η,tiJPi(η),iiJPi(η),fiJPi(η):ηMR}U(M) \sum\limits_{i \in J} {P_i} = \{ \eta ,{t_{\mathop {\sum }\limits_{i \in J} {P_i}}}(\eta ),{i_{\mathop {\sum }\limits_{i \in J} {P_i}}}(\eta ),{f_{\mathop {\sum }\limits_{i \in J} {P_i}}}(\eta ):\eta \in {M_R}\} \in U(M) where, for all η in MR tiJPi(η)={iJtPi(ηi):ηiMR,iJηi=η}iiJPi(η)={iJiPi(ηi):ηiMR,iJηi=η}fiJPi(η)={iJfPi(ηi):ηiMR,iJηi=η} \matrix{{{t_{\mathop {\sum }\limits_{i \in J} {P_i}}}(\eta ) = \vee \{ \mathop \wedge \limits_{i \in J} {t_{{P_i}}}({\eta _i}):{\eta _i} \in {M_R},\sum\limits_{i \in J} {\eta _i} = \eta \} } \cr {{i_{\mathop {\sum }\limits_{i \in J} {P_i}}}(\eta ) = \vee \{ \mathop \wedge \limits_{i \in J} {i_{{P_i}}}({\eta _i}):{\eta _i} \in {M_R},\sum\limits_{i \in J} {\eta _i} = \eta \} } \cr {{f_{\mathop {\sum }\limits_{i \in J} {P_i}}}(\eta ) = \wedge \{ \mathop \vee \limits_{i \in J} {f_{{P_i}}}({\eta _i}):{\eta _i} \in {M_R},\sum\limits_{i \in J} {\eta _i} = \eta \} } \cr } where, in ∑iJ ηi finitely ηis ≠ 0

So we can conclude for all η in MR

tPi(η)tiJPi(η),ηMR {t_{{P_i}}}(\eta ) \le {t_{\mathop {\sum }\limits_{i \in J} {P_i}}}(\eta ),\forall \eta \in {M_R}

iPi(η)iiJPi(η),ηMR {i_{{P_i}}}(\eta ) \le {i_{\mathop {\sum }\limits_{i \in J} {P_i}}}(\eta ),\forall \eta \in {M_R}

fPi(η)fiJPi(η),ηMR {f_{{P_i}}}(\eta ) \ge {f_{\mathop {\sum }\limits_{i \in J} {P_i}}}(\eta ),\forall \eta \in {M_R}

Hence Pi ⊆ ∑iJ Pi, ∀iJ.

Now to prove that ∑iJ Pi is the least neutrosophic submodule and ∑iJ Pi contains all Pis P_i^\prime s .

Let Q = {η,tQ(η), iQ(η), fQ(η) : ηMR} ∈ U(M) and PiQ,∀iJ, which means that tPi(η)tQ(η),iPi(η)iQ(η),fPi(η)fQ(η)iJ {t_{{P_i}}}(\eta ) \le {t_Q}(\eta ), {i_{{P_i}}}(\eta ) \le {i_Q}(\eta ), {f_{{P_i}}}(\eta ) \ge {f_Q}(\eta ) \forall i \in J

Let ηMR where ∑iJ ηi = η and only finitely ηis0 {\eta _i^\prime}s \ne 0 , then tiJPi(η)={iJtPi(ηi):ηiMR,iJηi=η}{iJtQ(ηi):ηiM,iJηi=η}{tQ(iJηi):ηiMR,iJηi=η}=tQ(η) \matrix{{{t_{\sum\nolimits_{i \in J} {P_i}}}(\eta )} \hfill & { = \vee \{ { \wedge _{i \in J}}{t_{{P_i}}}({\eta _i}):{\eta _i} \in {M_R},\sum\limits_{i \in J} {\eta _i} = \eta \} } \hfill \cr {} \hfill & { \le \vee \{ { \wedge _{i \in J}}{t_Q}({\eta _i}):{\eta _i} \in M,\sum\limits_{i \in J} {\eta _i} = \eta \} } \hfill \cr {} \hfill & { \le \vee \{ {t_Q}(\sum\limits_{i \in J} {\eta _i}):{\eta _i} \in {M_R},\sum\limits_{i \in J} {\eta _i} = \eta \} } \hfill \cr {} \hfill & { = {t_Q}(\eta )} \hfill \cr }

In the same way, iiJ Pi (η) iQ(η), fiJ Pi (η) ≥ fQ(η).

⇒ ∑iJ PiQ. Hence ∑iJ PiU(M) is the smallest one and contains all Pis P_i^\prime s . Therefore ∑iJ Pi is the smallest U(M) which contains ∪i JPi ⊆ ∑iJ Pi. Hence 〈 ∪i JPi = ∑iJ Pi

Definition 4.2

Let CU(R) and PNS(MR). Define the operations CP and CP as NS(MR) as follows

CP (η) = (η,tCP(η),iCP(η), fCP(η)) ∀ ηM where tCP(η)={tC(γ)tP(θ):γR,θM,γθ=η}iCP(η)={iC(γ)iP(θ):γR,θM,γθ=η}fCP(η)={fC(γ)fP(θ):γR,θM,γθ=η} \matrix{{{t_{C\circledcirc P}}(\eta ) = \vee \{ {t_C}(\gamma ) \wedge {t_P}(\theta ):\gamma \in R,\theta \in M,\gamma \theta = \eta \} } \cr {{i_{C\circledcirc P}}(\eta ) = \vee \{ {i_C}(\gamma ) \wedge {i_P}(\theta ):\gamma \in R,\theta \in M,\gamma \theta = \eta \} } \cr {{f_{C\circledcirc P}}(\eta ) = \wedge \{ {f_C}(\gamma ) \vee {f_P}(\theta ):\gamma \in R,\theta \in M,\gamma \theta = \eta \} } \cr }

CP (η) = (η,tCP(η), iCP(η), fCP(η)) ∀ ηM where tCP(η)={i=1n(tC(γi)tP(ηi):γiR,ηiM,i=1nγiηi=η,1in,nN}iCA(η)={i=1n(iC(γi)iA(ηi):γiγ,ηiM,i=1nγiηi=η,1in,nN}fCA(η)={i=1n(fC(γi)fA(ηi):γiγ,ηiM,i=1nγiηi=η,1in,nN} \matrix{{{t_{C\circledast P}}(\eta ) = \vee \{ \wedge _{i = 1}^n({t_C}({\gamma _i}) \wedge {t_P}({\eta _i}):{\gamma _i} \in R,{\eta _i} \in M,\sum\limits_{i = 1}^n {\gamma _i}{\eta _i} = \eta ,1 \le i \le n,n \in N\} } \cr {{i_{C\circledast A}}(\eta ) = \vee \{ \wedge _{i = 1}^n({i_C}({\gamma _i}) \wedge {i_A}({\eta _i}):{\gamma _i} \in \gamma ,{\eta _i} \in M,\sum\limits_{i = 1}^n {\gamma _i}{\eta _i} = \eta ,1 \le i \le n,n \in N\} } \cr {{f_{C\circledast A}}(\eta ) = \wedge \{ \vee _{i = 1}^n({f_C}({\gamma _i}) \vee {f_A}({\eta _i}):{\gamma _i} \in \gamma ,{\eta _i} \in M,\sum\limits_{i = 1}^n {\gamma _i}{\eta _i} = \eta ,1 \le i \le n,n \in N\} } \cr }

Theorem 4.2

Let PUMR, then

γR, N^{γ}P=γP {\hat N_{\{ \gamma \} }}\circledcirc P = \gamma P

γR, ηM, N^{γ}P(η)={η,tN^{γ}P(η),iN^{γ}P(η),fN^{γ}P(η)} \eta \in M,{\hat N_{\{ \gamma \} }}\circledast P(\eta ) = \{ \eta ,{t_{{{\hat N}_{\{ \gamma \} }}\circledast P}}(\eta ),{i_{{{\hat N}_{\{ \gamma \} }}\circledast P}}(\eta ),{f_{{{\hat N}_{\{ \gamma \} }}\circledast P}}(\eta )\} where 1 ≤ in, nN tN^{γ}P(η)={i=1ntP(ηi):ηiM,γi=1nηi=η}iN^{γ}P(η)={i=1niP(ηi):ηiM,γi=1nηi=η}fN^{γ}P(η)={i=1nfP(ηi):ηiMR,γi=1nηi=η} \matrix{{{t_{{{\hat N}_{\{ \gamma \} }}\circledast P}}(\eta ) = \vee \{ \wedge _{i = 1}^n{t_P}({\eta _i}):{\eta _i} \in M,\gamma \sum\limits_{i = 1}^n {\eta _i} = \eta \} } \cr {{i_{{{\hat N}_{\{ \gamma \} }}\circledast P}}(\eta ) = \vee \{ \wedge _{i = 1}^n{i_P}({\eta _i}):{\eta _i} \in M,\gamma \sum\limits_{i = 1}^n {\eta _i} = \eta \} } \cr {{f_{{{\hat N}_{\{ \gamma \} }}\circledast P}}(\eta ) = \wedge \{ \vee _{i = 1}^n{f_P}({\eta _i}):{\eta _i} \in {M_R},\gamma \sum\limits_{i = 1}^n {\eta _i} = \eta \} } \cr }

Proof

(1) The neutrosophic point N^{γ} {\hat N_{\{ \gamma \} }} , for any γR, is defined as N^{γ}(ς)={(ς,tN^{γ},iN^{γ},fN^{γ}):ςR} {\hat N_{\{ \gamma \} }}(\varsigma ) = \{ (\varsigma ,{t_{{{\hat N}_{\{ \gamma \} }}}},{i_{{{\hat N}_{\{ \gamma \} }}}},{f_{{{\hat N}_{\{ \gamma \} }}}}):\varsigma \in R\} where N^{γ}(ς)={(1,1,0)γ=ς(0,0,1)γς {\hat N_{\{ \gamma \} }}(\varsigma ) = \left\{ {\matrix{{(1,1,0)} & {\gamma = \varsigma } \cr {(0,0,1)} & {\gamma \ne \varsigma } \cr } } \right.

Consider N^{γ}P(η)={(η,tN^{γ}P(η),iN^{γ}P(η),fN^{γ}P(η))} {\hat N_{\{ \gamma \} }}\circledcirc P(\eta ) = \{ (\eta ,{t_{{{\hat N}_{\{ \gamma \} }}\circledcirc P}}(\eta ),{i_{{{\hat N}_{\{ \gamma \} }}\circledcirc P}}(\eta ),{f_{{{\hat N}_{\{ \gamma \} }}\circledcirc P}}(\eta ))\} ηMR, γR, we have tN^{γ}P(η)={tN^{γ}(ς)tP(θ):ςR,θMR,ςθ=η}={tP(θ):θM,γ=η}=tγP(η) \matrix{{{t_{{{\hat N}_{\{ \gamma \} }}\circledcirc P}}(\eta )} \hfill & { = \vee \{ {t_{{{\hat N}_{\{ \gamma \} }}}}(\varsigma ) \wedge {t_P}(\theta ):\varsigma \in R,\theta \in {M_R},\varsigma \theta = \eta \} } \hfill \cr {} \hfill & { = \vee \{ {t_P}(\theta ):\theta \in M,\gamma = \eta \} } \hfill \cr {} \hfill & { = {t_{\gamma P}}(\eta )} \hfill \cr }

Similarly we get, iN^{γ}P(η)=iγP(η) {i_{{{\hat N}_{\{ \gamma \} }}\circledcirc P}}(\eta ) = {i_{\gamma P}}(\eta ) , fN^{γ}P(η)=fγP(η) {f_{{{\hat N}_{\{ \gamma \} }}\circledcirc P}}(\eta ) = {f_{\gamma P}}(\eta )

(2) Now consider, for any γR,ηMR, 1 ≤ in, nN tN^{γ}P(η)={i=1n(tN^{γ}(γi)tP(ηi):riR,ηiM,i=1nγiηi=η}={i=1ntP(ηi):ηiMR,γi=1nηi=η} \matrix{{{t_{{{\hat N}_{\{ \gamma \} }}\circledast P}}(\eta )} \hfill & { = \vee \{ \wedge _{i = 1}^n({t_{{{\hat N}_{\{ \gamma \} }}}}({\gamma _i}) \wedge {t_P}({\eta _i}):{r_i} \in R,{\eta _i} \in M,\sum\limits_{i = 1}^n {\gamma _i}{\eta _i} = \eta \} } \hfill \cr {} \hfill & { = \vee \{ \wedge _{i = 1}^n{t_P}({\eta _i}):{\eta _i} \in {M_R},\gamma \sum\limits_{i = 1}^n {\eta _i} = \eta \} } \hfill \cr }

Similarly we get iN^{γ}P(η)={i=1niP(ηi):ηiMR,γi=1nηi=η}fN^{γ}P(η)={i=1nfP(ηi):ηiMR,γi=1nηi=η} \matrix{{{i_{{{\hat N}_{\{ \gamma \} }}\circledast P}}(\eta ) = \vee \{ \wedge _{i = 1}^n{i_P}({\eta _i}):{\eta _i} \in {M_R},\gamma \sum\limits_{i = 1}^n {\eta _i} = \eta \} } \cr {{f_{{{\hat N}_{\{ \gamma \} }}P}}(\eta ) = \wedge \{ \vee _{i = 1}^n{f_P}({\eta _i}):{\eta _i} \in {M_R},\gamma \sum\limits_{i = 1}^n {\eta _i} = \eta \} } \cr }

Theorem 4.3

If PU(R) and QU(M), then PQU(M)

Proof

From the definition of PQ, we can write, for all 1 ≤ in, nN tPQ(0)={i=1n(tP(γi)tQ(ηi):γiR,ηiMR,i=1nγiηi=0,}=1whenγi=ηi=0i,sincetP(0)tP(γ)γR \matrix{{{t_{P\circledast Q}}(0)} \hfill & { = \vee \{ \wedge _{i = 1}^n({t_P}({\gamma _i}) \wedge {t_Q}({\eta _i}):{\gamma _i} \in R,{\eta _i} \in {M_R},\sum\limits_{i = 1}^n {\gamma _i}{\eta _i} = 0,\} } \hfill \cr {} \hfill & { = 1\; when\; {\gamma _i} = {\eta _i} = 0 \forall i, since\; {t_P}(0) \ge {t_P}(\gamma ) \forall \gamma \in R} \hfill \cr }

Similarly iPQ(0) = 1 and fPQ(0) = 0.

Then prove that tPQ(η + θ) ≥ tPQ(η) ∧ tPQ(θ) for η,θMR,1 ≤ in,nN tPQ(η+θ)={i=1n(tP(γi)tQ(zi):γiR,ziM,i=1nγizi=η+θ}{i=1n(tP(ςi)tQ(ηi+θi):ςiR,ηi,θiM,i=1nςi(ηi+θi)=η+θ,i}{i=1n(tP(ςi)(tQ(ηi)tQ(θi))):ςiR,ηi,θiM,i=1nςi(ηi+θi)=η+θ,i}={i=1n(tP(ςi)(tQ(ηi))(tP(ςi))tQ(θi)):ςiR,ηi,θiM,i=1nςi(ηi+θi)=η+θ}{i=1n(tP(ςi)tQ(ηi)):ςiR,ηiM,i=1nςiηi=η}{i=1n(tP(ςi)tQ(θi)):ςiR,θiM,i=1nςiθi=θ}=tPQ(η)tPQ(θ) \matrix{{{t_{P\circledast Q}}(\eta + \theta )} \hfill & { = \vee \{ \wedge _{i = 1}^n({t_P}({\gamma _i}) \wedge {t_Q}({z_i}):{\gamma _i} \in R,{z_i} \in M,\sum\limits_{i = 1}^n {\gamma _i}{z_i} = \eta + \theta \} } \hfill \cr {} \hfill & { \ge \vee \{ \wedge _{i = 1}^n({t_P}({\varsigma _i}) \wedge {t_Q}({\eta _i} + {\theta _i}):{\varsigma _i} \in R, {\eta _i},{\theta _i} \in M,\sum\limits_{i = 1}^n {\varsigma _i}({\eta _i} + {\theta _i}) = \eta + \theta ,\forall i\} } \hfill \cr {} \hfill & { \ge \vee \{ \wedge _{i = 1}^n({t_P}({\varsigma _i}) \wedge ({t_Q}({\eta _i}) \wedge {t_Q}({\theta _i}))):{\varsigma _i} \in R, {\eta _i},{\theta _i} \in M,\sum\limits_{i = 1}^n {\varsigma _i}({\eta _i} + {\theta _i}) = \eta + \theta ,\forall i\} } \hfill \cr {} \hfill & { = \vee \{ \wedge _{i = 1}^n({t_P}({\varsigma _i}) \wedge ({t_Q}({\eta _i})) \wedge ({t_P}({\varsigma _i})) \wedge {t_Q}({\theta _i})):{\varsigma _i} \in R, {\eta _i}, {\theta _i} \in M,\sum\limits_{i = 1}^n {\varsigma _i}({\eta _i} + {\theta _i}) = \eta + \theta \} } \hfill \cr {} \hfill & { \ge \vee \{ \wedge _{i = 1}^n({t_P}({\varsigma _i}) \wedge {t_Q}({\eta _i})):{\varsigma _i} \in R,{\eta _i} \in M,\sum\limits_{i = 1}^n {\varsigma _i}{\eta _i} = \eta \} \wedge \vee \{ \wedge _{i = 1}^n({t_P}({\varsigma _i}) \wedge {t_Q}({\theta _i})):{\varsigma _i} \in R,{\theta _i} \in M,\sum\limits_{i = 1}^n {\varsigma _i}{\theta _i} = \theta \} } \hfill \cr {} \hfill & { = {t_{PQ}}(\eta ) \wedge {t_{P\circledast Q}}(\theta )} \hfill \cr }

Similarly we can prove that iPQ(η + θ) ≥ iPQ(η) ∧ iPQ(θ) η, θM and fPQ(η + θ) ≤ fPQ(η) ∨ fP⊙Q(θ) η, θMR.

Now for all 1 ≤ in, nN tPQ(γη)={i=1n(tP(γi)tQ(ηi)):γiR,ηiMR,i=1nγiηi=γη}{i=1n(tP(γςi)tQ(θi)):ςiR,θiMR,γi=1nςiθi=γη}[whenγ=1]{i=1n(tP(ςi)tQ(θi)):ςiR,θiMR,i=1nςiθi=η,1in,nN}[sincePU(R)tP(γςi)tP(γ)tP(ςi)tP(ςi)]=tPQ(η) \matrix{{{t_{P\circledast Q}}(\gamma \eta )} \hfill & = \hfill & { \vee \{ \wedge _{i = 1}^n({t_P}({\gamma _i}) \wedge {t_Q}({\eta _i})):{\gamma _i} \in R,{\eta _i} \in {M_R},\sum\limits_{i = 1}^n {\gamma _i}{\eta _i} = \gamma \eta \} } \hfill \cr {} \hfill & \ge \hfill & { \vee \{ \wedge _{i = 1}^n({t_P}(\gamma {\varsigma _i}) \wedge {t_Q}({\theta _i})):{\varsigma _i} \in R,{\theta _i} \in {M_R},\gamma \sum\limits_{i = 1}^n {\varsigma _i}{\theta _i} = \gamma \eta \} [ when\; \gamma = 1]} \hfill \cr {} \hfill & \ge \hfill & { \vee \{ \wedge _{i = 1}^n({t_P}({\varsigma _i}) \wedge {t_Q}({\theta _i})):{\varsigma _i} \in R,{\theta _i} \in {M_R},\sum\limits_{i = 1}^n {\varsigma _i}{\theta _i} = \eta ,1 \le i \le n,n \in N\} } \hfill \cr {} \hfill & {} \hfill & {[ since\; P \in U(R) \Rightarrow {t_P}(\gamma {\varsigma _i}) \ge {t_P}(\gamma ) \vee {t_P}({\varsigma _i}) \ge {t_P}({\varsigma _i}) ]} \hfill \cr {} \hfill & = \hfill & {{t_{P\circledast Q}}(\eta )} \hfill \cr }

Similarly, iPQ() ≥ iPQ(η) and fPQ() ≤ fPQ(η).

Hence PQU(M).

Theorem 4.4

Let PUMR and corresponding to P, define QUMR such that Q = {η, tQ(η), iQ(η), fQ(η) : ηMR} where tQ(η)={1η=0{i=1ntP(ηi):i=1nγiηi=η,ηiMR,γiR}otherwiseiQ(η)={1η=0{i=1niP(ηi):i=1nγiηi=η,ηiM,γiR}otherwisefQ(η)={0η=0{i=1nfP(ηi):i=1nγiηi=η,ηiMR,γiR}otherwise \matrix{{{t_Q}(\eta ) = \left\{ {\matrix{1 \hfill & {\eta = 0} \hfill \cr { \vee \{ \wedge _{i = 1}^n{t_P}({\eta _i}):\sum\nolimits_{i = 1}^n {\gamma _i}{\eta _i} = \eta ,{\eta _i} \in {M_R},{\gamma _i} \in R\} } \hfill & {otherwise} \hfill \cr } } \right.} \hfill \cr {{i_Q}(\eta ) = \left\{ {\matrix{1 \hfill & {\eta = 0} \hfill\cr { \vee \{ \wedge _{i = 1}^n{i_P}({\eta _i}):\sum\nolimits_{i = 1}^n {\gamma _i}{\eta _i} = \eta ,{\eta _i} \in M,{\gamma _i} \in R\} } \hfill & {otherwise} \hfill \cr } } \right.} \hfill \cr {{f_Q}(\eta ) = \left\{ {\matrix{0 \hfill & {\eta = 0} \hfill \cr { \wedge \{ \vee _{i = 1}^n{f_P}({\eta _i}):\sum\nolimits_{i = 1}^n {\gamma _i}{\eta _i} = \eta ,{\eta _i} \in {M_R},{\gamma _i} \in R\} } \hfill & {otherwise} \hfill \cr } } \right.} \cr } where 1 ≤ in, nN. Then QU(M) and 〈P〉 = Q.

Proof

From the definition of Q, tP(η) ≤ tQ(η), iP(η) ≤ iQ(η) and fP(η) ≥ fQ(η) ∀ηMR, then PQ. We know tQ(0) = 1, iQ(0) = 1 and fQ(0) = 0. Let γR, ηMR.

If γη = 0 then tQ(γη)=1tQ(η),iQ(γη)=1iQ(η)andfQ(γη)=0fQ(η) {t_Q}(\gamma \eta ) = 1 \ge {t_Q}(\eta ), {i_Q}(\gamma \eta ) = 1 \ge {i_Q}(\eta )\; and\; {f_Q}(\gamma \eta ) = 0 \le {f_Q}(\eta )

Suppose γη ≠ 0 then η ≠ 0 and ∀ 1 ≤ in,nN tQ(γη)={i=1ntP(ηi):i=1nγiηi=γη,ηiMR,γiR,}{i=1ntP(ηi):i=1nγςiηi=γη,ηiMR,ςiR}{i=1ntP(ηi):γi=1nςiηi=γη,ηiMR,ςiR}{i=1ntP(ηi):i=1nςiηi=η,ηiMR,ςiR}(whenγ=1)=tQ(η) \matrix{{{t_Q}(\gamma \eta )} \hfill & = \hfill & { \vee \{ \wedge _{i = 1}^n{t_P}({\eta _i}):\sum\limits_{i = 1}^n {\gamma _i}{\eta _i} = \gamma \eta ,{\eta _i} \in {M_R},{\gamma _i} \in R, \} } \hfill \cr {} \hfill & \ge \hfill & { \vee \{ \wedge _{i = 1}^n{t_P}({\eta _i}):\sum\limits_{i = 1}^n \gamma {\varsigma _i}{\eta _i} = \gamma \eta ,{\eta _i} \in {M_R},{\varsigma _i} \in R\} } \hfill \cr {} \hfill & \ge \hfill & { \vee \{ \wedge _{i = 1}^n{t_P}({\eta _i}):\gamma \sum\limits_{i = 1}^n {\varsigma _i}{\eta _i} = \gamma \eta ,{\eta _i} \in {M_R},{\varsigma _i} \in R\} } \hfill \cr {} \hfill & \ge \hfill & { \vee \{ \wedge _{i = 1}^n{t_P}({\eta _i}):\sum\limits_{i = 1}^n {\varsigma _i}{\eta _i} = \eta ,{\eta _i} \in {M_R},{\varsigma _i} \in R\} } \hfill \cr {} \hfill & {} \hfill & {(when\gamma = 1)} \hfill \cr {} \hfill & = \hfill & {{t_Q}(\eta )} \hfill \cr }

In the same way we can show that iQ(γη) ≥ iQ(η) and fQ(γη) ≤ fQ(η)

Suppose η, θ and η + θ ≠ 0, ∀ 1 ≤ in, nN, then tQ(η+θ)={i=1ntA(zi):i=1nγizi=η+θ,ziMR,γiR}{i=1ntP(zi):zi=ηi+θi,i=1nγi(ηi+θi)=η+θ,ηi,θiM,γiR}{(i=1ntP(ηi))(i=1ntP(θi)):zi=ηi+θi,i=1nγiηi+i=1nγiθi=η+θ,ηi,θiMR,γiR}{(i=1ntP(ηi):i=1nγiηi=η,ηiMR,γiR}{(i=1ntP(θi):i=1nγiθi=θ,θiM,γiR}=tQ(η)tQ(θ) \matrix{{{t_Q}(\eta + \theta )} \hfill & = \hfill & { \vee \{ \wedge _{i = 1}^n{t_A}({z_i}):\sum\limits_{i = 1}^n {\gamma _i}{z_i} = \eta + \theta ,{z_i} \in {M_R},{\gamma _i} \in R\} } \hfill \cr {} \hfill & \ge \hfill & { \vee \{ \wedge _{i = 1}^n{t_P}({z_i}):{z_i} = {\eta _i} + {\theta _i},\sum\limits_{i = 1}^n {\gamma _i}({\eta _i} + {\theta _i}) = \eta + \theta ,{\eta _i},{\theta _i} \in M,{\gamma _i} \in R\} } \hfill \cr {} \hfill & \ge \hfill & { \vee \{ ( \wedge _{i = 1}^n{t_P}({\eta _i})) \wedge ( \wedge _{i = 1}^n{t_P}({\theta _i})):{z_i} = {\eta _i} + {\theta _i},\sum\limits_{i = 1}^n {\gamma _i}{\eta _i} + \sum\limits_{i = 1}^n {\gamma _i}{\theta _i} = \eta + \theta ,} \hfill \cr {} \hfill & {} \hfill & {{\eta _i},{\theta _i} \in {M_R},{\gamma _i} \in R\} } \hfill \cr {} \hfill & \ge \hfill & { \vee \{ ( \wedge _{i = 1}^n{t_P}({\eta _i}):\sum\limits_{i = 1}^n {\gamma _i}{\eta _i} = \eta ,{\eta _i} \in {M_R},{\gamma _i} \in R\} \wedge } \hfill \cr {} \hfill & {} \hfill & { \vee \{ ( \wedge _{i = 1}^n{t_P}({\theta _i}):\sum\limits_{i = 1}^n {\gamma _i}{\theta _i} = \theta ,{\theta _i} \in M,{\gamma _i} \in R\} } \hfill \cr {} \hfill & = \hfill & {{t_Q}(\eta ) \wedge {t_Q}(\theta )} \hfill \cr }

Similarly, iQ(η + θ) ≥ iQ(η) ∧ iQ(θ) and fQ(η + θ) ≤ fQ(η) ∨ fQ(θ).

QU(M) and PQ.

Now consider S = {η,tS(η),iS(η), fS(η) : ηMR} ∈ U(M) and which contains P. Now to prove that QS. From the assumption, PS,

tP(η)tS(η),iP(η)iS(η)PndfP(η)fS(η). {t_P}(\eta ) \le {t_S}(\eta ), {i_P}(\eta ) \le {i_S}(\eta )\; Pnd\; {f_P}(\eta ) \ge {f_S}(\eta ).

Now for all 1 ≤ in, nN tQ(η)={i=1ntP(ηi):i=1nγiηi=η,ηiMR,γiR}{i=1ntS(ηi):i=1nγiηi=η,ηiMR,γiR}[weknowtS(η)=tS(i=1nγiηi)i=1ntS(γiηi)i=1ntS(ηi)tS(η)(i=1ntS(ηi))]tS(η) \matrix{{{t_Q}(\eta )} \hfill & = \hfill & { \vee \{ \wedge _{i = 1}^n{t_P}({\eta _i}):\sum\limits_{i = 1}^n {\gamma _i}{\eta _i} = \eta ,{\eta _i} \in {M_R},{\gamma _i} \in R\} } \hfill \cr {} \hfill & \le \hfill & { \vee \{ \wedge _{i = 1}^n{t_S}({\eta _i}):\sum\limits_{i = 1}^n {\gamma _i}{\eta _i} = \eta ,{\eta _i} \in {M_R},{\gamma _i} \in R\} } \hfill \cr {} \hfill & {} \hfill & {[we\; know \;{t_S}(\eta ) = {t_S}(\sum\limits_{i = 1}^n {\gamma _i}{\eta _i}) \ge \wedge _{i = 1}^n{t_S}({\gamma _i}{\eta _i}) \ge \wedge _{i = 1}^n{t_S}({\eta _i}) \Rightarrow {t_S}(\eta ) \le \vee ( \wedge _{i = 1}^n{t_S}({\eta _i}))]} \hfill \cr {} \hfill & \le \hfill & {{t_S}(\eta )} \hfill \cr }

We can derive in the same pattern, iQ(η) ≤ iS(η) and fQ(η) ≥ fS(η). ⇒ QC. Thus we can conclude 〈P〉 = Q.

Conclusion

Neutrosophic submodule is one of the generalizations of a classical algebraic structure, module. The study of neutrosophic submodule give extra promptitude to the classic algebraic structures rather than fuzzy or intuitionistic fuzzy sets because of the investigation of three different level graded functions of each element in [0,1]. This paper has developed a method to identify generator of U(M) and derived algebraic results with the help of some algebraic operators as neutrosophic sets. This work are often extended to the generators of arbitrary nonempty family of neutrosophic submodules and structure preserving properties like isomorphism of neutrosophic submodules. Neutrosophic submodules provide us a solid mathematical foundation to clarify connected scientific ideas in image processing, control theory and economic science.

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