InternationalJournalofPureandAppliedMathematics
Volume118No.62018,527-539
ISSN:1311-8080(printedversion);ISSN:1314-3395(on-lineversion)
url: http://www.ijpam.eu SpecialIssue ijpam.eu
A.Solairaju1 andS.Thiruveni2
1DepartmentofMathematics, JamalMohamedCollege, Trichy,Tamilnadu-620020. solairama@yahoo.co.in 2DepartmentofMathematics, MotherTeresaWomen’sUniversity, Kodaikanal,Tamilnadu-India thiruveni.adhi@gmail.com
Abstract
Inthispaper,thenotionofNeutrosophicfuzzyidealsofnear-ringsareintroduced anddiscussedsomealgebraicpropertieslikeunion,intersection,homomorphicimage andpreimageofneutrosophicfuzzyidealsofnear-rings.Furtherwediscussaboutthe directproductofNeutrosophicfuzzyidealsofnear-rings.
AMSSubjectClassification: 03E72,13A15,16Y30
KeyWordsandPhrases: Neutrosophicfuzzyset,Neutrosophicfuzzyidealof near-rings
1Introduction
TheconceptoffuzzysetwasintroducedbyZadeh[12]in1965.Thenotionfuzzysubnearringsandfuzzyidealsofnear-ringswasintroducedbyAbou zaid[1].Thisconceptisfurther discussedbyKim[6]andDutta[5].V.ChinnaduraiandS.Kadalarasi[4]discussedthedirect productofn(n=1,2,....,k)fuzzysubnearring,fuzzyidealandfuzzyR-subgroups.The conceptofneutrosophywasintroducedbyFlorentinSmarandache[10]asanewbranchof philosophy.NeutrosophyisabaseofNeutrosophiclogicwhichisanextensionoffuzzylogic inwhichindeterminancyisincluded.InNeutrosophiclogic,eachpropositionisestimated tohavethepercentageoftruthinasubsetT,percentageofindeterminancyinasubsetI, andthepercentageoffalsityinasubsetF.Thetheoryofneutrosophicsethaveachieved greatsuccessinvariousfieldslikemedicaldiagnosis,imageprocessing,decisionmaking problem,roboticsandsoon.I.Arockiarani[3]considertheneutrosophicsetwithvalue fromthesubsetof[0,1]andetxtendedtheresearchinfuzzyneutrosophicset.J.Martina JencyandI.Arockiarani[8]initiatetheconceptofsubgroupoidsinfuzzyneutrosophicset. Inthispaperweintroducethenotionofneutrosophicfuzzysubnear-ring,neutrosophic fuzzyidealsofnear-ringsanddiscussedsomealgebraicproperties.Alsowediscussedabout thedirectproductofneutrosophicneutrosophicfuzzyidealsofnear-rings.
2Preliminaries
Definition1. 1.(N, +)isagroup 2.(N,.)isasemigroup
3.(x + y).z = x.z + y.z for x,y,z ∈ N
Definition2. Anidealofanear-ring N isasubgroup I of N suchthat 1.(I, +)isanormalsubgroupof(N, +) 2. IN ⊆ I 3. y(x + i) yx ∈ I,for x,y ∈ N and i ∈ I I isaleftidealof N if I satisfies(i)and(ii)andrightidealof N if I satisfies(i)and(iii).
Definition3. Afuzzyset µ inanear-ringNissaidtobefuzzysubnear-ringofNif 1. µ(x y) ≥ min(µ(x),µ(y)),forallx,y ∈ N 2. µ(xy) ≥ min(µ(x),µ(y)),forall x,y ∈ N
Definition4. Afuzzyset µ inanear-ringNissaidtobefuzzyidealofNif
(i) µ(x y) ≥ min(µ(x),µ(y)),forall x,y ∈ N (ii) µ(y + x y) ≥ µ(x),forall x,y ∈ N (iii) µ(xy) ≥ µ(y),forall x,y ∈ N (iv) µ((x + z)y xy ≥ µ(z),forall x,y ∈ N Afuzzysetwith(i)-(iii)iscalledfuzzyleftidealofNwhereasfuzzysetwith(i),(ii)and (iv)iscalledafuzzyrightidealof N
Definition5. ANeutrosophicfuzzysetAontheuniverseofdiscourseXcharacterizedbyatruthmembershipfunction TA(x),aindeterminacyfunctionIA(x)andafalsity membershipfunction FA(x)isdefinedas A = {<x,TA(x),IA(x),FA(x) >: x ∈ X} where TA , IA,FA : X → [0, 1]and0 ≤ TA(x)+ IA(x)+ FA(x) ≤ 3
Definition6. Let A and B beneutrosophicfuzzysetsof X. Then, 1. A ∪ B = {<x,TA∪B (x),IA∪B (x),FA∪B (x) >},where TA∪B (x)= max(TA(x),TB (x),IA∪B (x)= min(IA(x),IB (x),FA∪B (x))= min(FA(x),FB (x)) ,forall x∈X.
A
B =(TA
B (x),IA∩B (x)),FA∩B (x) >},where TA∩B (x)= min(TA(x),TB (x)),IA∩B (x)= max(IA(x),IB (x)),FA∩B (x)= max
3Neutrosophicfuzzyidealsofnear-rings
Definition8. Let N beanear-ring.ANeutrosophicfuzzyset A inanear-ring N is calledneutrosophicfuzzyidealof N if
(i) TA(x y)≥min(TA(x),TA(y)),IA(x y)≤max(IA(x),IA(y)),FA(x y)≤max(FA(x),FA(y))
(ii) TA(y + x y)≥TA(x),IA(y + x y)≤IA(x),FA(y + x y)≤FA(x)
(iii) TA(xy)≥TA(y),IA(xy)≤IA(y),FA(xy)≤FA(y)
(iv) TA((x + z)y xy)≥TA(z),IA((x + z)y xy)≤IA(z),FA((x + z)y xy)≤FA(z). ANeutrosophicfuzzysubsetwiththeabovecondition(i)-(iii)iscalledaneutrosophic fuzzyleftidealof N whereaswith(i),(ii)and(iv)iscalledneutrosophicfuzzyrightideal of N
Theorem9. Let A and B beneutrosophicfuzzyidealof N. If A ⊂ B,then A ∪ B isaneutrosophicfuzzyidealof N.
Proof. LetAandBareneutrosophicfuzzyidealsofN.Letx,y,z∈ N. (i)
TA∪B (x y)= max(TA(x y),TB (x y))
≥ max(min(TA(x),TA(y)),min(TB (x),TB (y))
= min{max(TA(x),TB (x)),max(TA(y),TB (y))}
= min(TA∪B (x),TA∪B (y))
IA∪B (x y)= min(IA(x y),IB (x y))
≤ min(max(IA(x),IA(y)),max(IB (x),IB (y))
= max{min(IA(x),IB (x)),max(IA(y),IB (y))}
= max(IA∪B (x),IA∪B (y))
FA∪B (x y)= min(FA(x y),FB (x y))
≤ min(max(FA(x),FA(y)),max(FB (x),FB (y))
(ii)
= max{min(FA(x),FB (x)),max(FA(y),FB (y))}
= max(FA∪B (x),FA∪B (y))
TA∪B (y + x y)= max(TA(y + x y),TB (y + x y))
≥ max(TA(x),TB (x))
= TA∪B (x)
IA∪B (y + x y)= min(IA(y + x y),IB (y + x y))
≤ min(IA(x),IB (x))
= IA∪B (x)
FA∪B (y + x y)= min(FA(y + x y),FB (y + x y))
≤ min(FA(x),FB (x))
= FA∪B (x)
(iii)
TA∪B (xy)= max(TA(xy),TB (xy)) ≥ max(TA(y),TB (y))= TA∪B (y)
IA∪B (xy)= min(IA(xy),IB (xy)) ≤ min(IA(y),IB (y))= IA∪B (y)
FA∪B (xy)= min(FA(xy),FB (xy)) ≤ min(FA(y),FB (y))= FA∪B (y) (iv)
TA∪B ((x + z)y xy)= max{TA((x + z)y xy),TB ((x + z)y xy)} ≥ max(TA(z),TB (z))= TA∪B (z)
IA∪B ((x + z)y xy)= min{IA((x + z)y xy),IB ((x + z)y xy)}
≤ min(IA(z),IB (z))= IA∪B
TA∪B ((x + z)y xy)= min{FA((x + z)y xy),FB ((x + z)y xy))} ≤ min(FA(z),FB (z))= FA∪B (z)
Therefore, A ∪ Bisaneutrosophicfuzzyidealof N.
Theorem10. Let A and B beneutrosophicfuzzyidealofN.Then A ∩ B isa neutrosophicfuzzyidealofN.
Proof. Let A and B areneutrosophicfuzzyidealsof N .Let x,y,z ∈ N (i)
TA∩B (x y)= min(TA(x y),TB (x y))
≥ min(min(TA(x),TA(y)),min(TB (x),TB (y))
= min{min(TA(x),TB (x)),min(TA(y),TB (y))}
= min(TA∩B (x),TA∩B (y))
IA∩B (x y)= max(IA(x y),IB (x y))
≤ max(max(IA(x),IA(y)),max(IB (x),IB (y))
= max{max(IA(x),IB (x)),max(IA(y),IB (y))}
= max(IA∩B (x),IA∩B (y))
FA∩B (x y)= max(FA(x y),FB (x y))
≤ max(max(FA(x),FA(y)),max(FB (x),FB (y))
= max{max(FA(x),FB (x)),max(FA(y),FB (y))}
= max(FA∩B (x),FA∩B (y))
(ii)
TA∩B (y + x y)= min(TA(y + x y),TB (y + x y))
≥ min(TA(x),TB (x))
= TA∩B (x)
IA∩B (y + x y)= max(IA(y + x y),IB (y + x y))
≤ max(IA(x),IB (x))
= IA∩B (x)
FA∩B (y + x y)= max(FA(y + x y),FB (y + x y))
≤ max(FA(x),FB (x))
= FA∩B (x)
(iii)
TA∩B (xy)= min(TA(xy),TB (xy)) ≥ min(TA(y),TB (y))= FA∩B (y)
IA∩B (xy)= max(IA(xy),IB (xy)) ≤ max(IA(y),IB (y))= IA∩B (y)
FA∩B (xy)= max(FA(xy),FB (xy)) ≤ max(FA(y),FB (y))= FA∩B (y)
(iv)
TA∩B ((x + z)y xy)= min{TA((x + z)y xy),TB ((x + z)y xy))}
≥ min(TA(z),TB (z))= TA∩B (z)
IA∩B ((x + z)y xy)= max{IA((x + z)y xy),IB ((x + z)y xy))}
≤ max(IA(z),IB (z))= IA∩B (z)
FA∩B ((x + z)y xy)= max{FA((x + z)y xy),FB ((x + z)y xy))}
≤ max(FA(z),FB (z))= FA∩B (z)
Therefore, A ∩ B isaneutrosophicfuzzyidealof N.
Corollary11. If A1,A2,....,An areneutrosophicfuzzyidealsof N, then A = n i=1 Ai isaneutrosophicfuzzyidealofN.
Lemma12. Forall a,b ∈ I and i isanypositiveinteger,if a = b, then (i) ai ≤ bi (ii) [min(a,b)]i = min(ai,bi) (iii) [max(a,b)]i = max(ai,bi)
Theorem13. Let A beaNeutrosophicfuzzyidealof N Then Am = {<x,TAm (x),IAm (x),FAm (x) >: x∈N } isaNeutrosophicfuzzyidealof N m,where m isapositiveintegerand
TAm (x)=(TA(x))m,IAm (x)=(IA(x))mFAm (x)=(FA(x))m
Proof. Let A beaNeutrosophicfuzzyidealof N .Let x,y,z∈N. (i)
TAm (x y)=(TA(x y))m
≥ [min(TA(x),TA(y))]m = min((TA(x))m , (TA(y))m)
= min(TAm (x),TAm (y))
IAm (x y)=(IA(x y))m
≤ [max(IA(x),IA(y))]m = max((IA(x))m , (IA(y))m)
= max(IAm (x),IAm (y))
FAm (x y)=(FA(x y))m
≤ [max(FA(x),FA(y))]m = max((FA(x))m , (FA(y))m)
= max(FAm (x),FAm (y))
(ii)
TAm (y + x y)=(TA(y + x y))m ≥ (TA(x))m = TAm (x)
IAm (y + x y)=(IA(y + x y))m ≤ (IA(x))m = IAm (x)
FAm (y + x y)=(FA(y + x y))m ≤ (FA(x))m = FAm (x)
(iii)
T m A (xy)=(TA(xy))m ≥ (TA(y))m = T m A (y), I m A (xy)=(IA(xy))m ≤ (IA(y))m = I m A (y),
F m A (xy)=(FA(xy))m ≤ (FA(y))m = F m A (y).
(iv)
TAm ((x + z)y xy)=(TA((x + z)y xy))m ≥ (TA(z))m = TAm (z)
IAm ((x + z)y xy)=(IA((x + z)y xy))m ≤ (IA(z))m = IAm (z)
FAm ((x + z)y xy)=(FA((x + z)y xy))m ≤ (FA(z))m = FAm (z)
Therefore, Am isaNeutrosophicfuzzyidealof N m
4DirectproductofNeutorsophicfuzzyidealsofnear-rings
Definition14. Let A and B beneutrosophicfuzzysubsetsofnear-rings N1and N2 respectively.Then,thedirectproductofneutrosophicfuzzysubsetsofnear-ringsis definedby A×B : N1×N2 → [0, 1]suchthat A×B = {< (x,y),TA×B (x,y),IA×B (x,y),FA×B (x,y) >: x ∈ N1,y ∈ N2} where,TA×B (x,y)= min(TA(x),TB (y)),IA×B (x,y)= max(IA(x),IB (y)) and FA×B (x,y)= max(FA(x),FB (y))
Definition15. Let A and B beneutrosophicfuzzysubsetsofnear-rings N1and N2 respectively.Then A × B isaneutrosophicfuzzyidealof N1 × N2 ifitsatisfiesthe followingconditions:
(i)
TA×B ((x1,x2) (y1,y2)) ≥ min(TA×B (x1,x2),TA×B (y1,y2)), IA×B ((x1,x2) (y1,y2)) ≤ max(IA×B (x1,x2),IA×B (y1,y2)), FA×B ((x1,x2) (y1,y2)) ≤ max(FA×B (x1,x2),FA×B (y1,y2)) (ii)
TA×B ((y1,y2)+(x1,x2) (y1,y2)) ≥ min(TA×B (x1,x2)), IA×B ((y1,y2)+(x1,x2) (y1,y2)) ≤ max(IA×B (x1,x2)), FA×B ((y1,y2)+(x1,x2) (y1,y2)) ≤ max(FA×B (x1,x2))
(iii)
TA×B ((x1,x2)(y1,y2)) ≥ TA×B (y1,y2),
IA×B ((x1,x2)(y1,y2)) ≤ IA×B (y1,y2), FA×B ((x1,x2)(y1,y2)) ≤ FA×B (y1,y2)
(iv)
TA×B ([(x1,x2)+(z1,z2)](y1,y2) (x1,x2)(y1,y2)) ≥ TA×B (z1,z2),
IA×B ([(x1,x2)+(z1,z2)](y1,y2) (x1,x2)(y1,y2)) ≤ IA×B (z1,z2)
FA×B ([(x1,x2)+(z1,z2)](y1,y2) (x1,x2)(y1,y2)) ≤ TA×B (z1,z2)
Theorem16. Let A and B beneutrosophicfuzzyidealsof N1,N2 respectively.Then A × B isaneutrosophicfuzzyidealof N1 × N2
Proof. Let A and B beneutrosophicfuzzyidealsof N1,N2 respectively. Let(x1,x2), (y1,y2), (z1,z2) ∈ N1 × N2 (i)
TA×B ((x1,x2) (y1,y2))= TA×B (x1 y1,x2 y2)
= min(TA(x1 y1),TB (x2 y2)
≥ min{min[TA(x1),TA(y1))],min[TB (x2),TB (y2))]
= min{min[TA(x1),TB (x2)],min[TA(y1),TB (y2)]
= min(TA×B (x1,x2),TA×B (y1,y2))
IA×B ((x1,x2) (y1,y2))= IA×B (x1 y1,x2 y2)
= max(IA(x1 y1),IB (x2 y2)
≤ max{max[IA(x1),IA(y1))],max[IB (x2),IB (y2))]
= max{max[IA(x1),IB (x2)],max[IA(y1),IB (y2)]
= max(IA×B (x1,x2),IA×B (y1,y2))
FA×B ((x1,x2) (y1,y2))= FA×B (x1 y1,x2 y2)
= max(FA(x1 y1),FB (x2 y2)
≤ max{max[FA(x1),FA(y1))],max[FB (x2),FB (y2))]
= max{max[FA(x1),FB (x2)],max[FA(y1),FB (y2)]
= max(FA×B (x1,x2),FA×B (y1,y2))
(ii)
TA×B ((y1,y2)+((x1,x2)) (y1,y2))= TA×B B((y1 + x1 y1), (y2 + x2 y2))
= min[TA(y1 + x1 y1),TB (y2 + x2 y2)]
≥ min(TA(x1),TB (x2))
= TA×B (x1,x2)IA×B ((y1,y2)+((x1,x2)) (y1,y2))
= IA×B ((y1 + x1 y1), (y2 + x2 y2))
= max[IA(y1 + x1 y1),IB (y2 + x2 y2)]
≤ max(IA(x1),IB (x2))
= IA×B (x1,x2)FA×B ((y1,y2)+((x1,x2)) (y1,y2))
= FA×B ((y1 + x1 y1), (y2 + x2 y2))
= max[FA(y1 + x1 y1),FB (y2 + x2 y2)]
≤ max(FA(x1),FB (x2))
= FA×B (x1,x2)
(iii)
TA×B (x1,x2)(y1,y2)= TA×B (x1y1,x2y2) = min(TA(x1y1),TB (x2y2)) ≥ min(TA(y1),TB (y2))
= TA×B ((y1,y2)
IA×B (x1,x2)(y1,y2)
= IA×B (x1y1,x2y2)
= max(IA(x1y1),IB (x2y2))
≤ max(IA(y1),IB (y2))
= IA×B (y1,y2)
FA×B (x1,x2)(y1,y2)
= FA×B (x1y1,x2y2)
= max(FA(x1y1),FB (x2y2)) ≤ max(FIA(y1),FB (y2))
= FA×B (y1,y2)
(iv)
TA×B ([(x1,x2)+(z1,z2) (y1,y2) (x1,x2)(y1,y2)= TA×B ([x1 + z1]y1 x1y1, [x2 + z2]y2 x2y2)
= min(TA[x1 + z1]y1 x1y1), TB [x2 + z2]y2 x2y2) ≥ min(TA(z1),TB (z2)
= TA×B (z1,z2)
IA×B ([(x1,x2)+(z1,z2) (y1,y2) (x1,x2)(y1,y2)= IA×B ([x1 + z1]y1 x1y1, [x2 + z2]y2 x2y2)
= max(IA(x1 + z1]y1 x1y1), IB [x2 + z2]y2 x2y2) ≤ max(IA(z1),IB (z2)
= IA×B (z1,z2)
FA×B ([(x1,x2)+(z1,z2) (y1,y2) (x1,x2)(y1,y2)= FA×B ([x1 + z1]y1 x1y1, [x2 + z2]y2 x2y2)
= max(FA(x1 + z1]y1 x1y1), FB [x2 + z2]y2 x2y2)
≤ max(FA(z1),FB (z2)
= FA×B (z1,z2) Therefore A
Definition17. Let R and S betwonear-rings.Thenthemapping f : R → S is callednear-ringhomomorphismifforall x,y∈R, thefollowingholds
(i) f (x + y)= f (x)+ f (y) (ii) f (xy)= f (x)f (y)
Definition18. Let X and Y betwonon-emptysetsand f : X → Y beafunction.
(i)IfBisaNeutrosophicfuzzysetinY,thenthepreimageofBunder f ,denotedby f 1(B),istheNeutrosophicfuzzysetinXdefinedby
f 1 (B)= {< (x),f 1 (TB (x)) ,f 1 (IB (x)) ,f 1 (FB (x)) >: x ∈ X} where f 1 (TB (x))= TB (f (x))andsoon.
(ii)IfAisaNeutrosophicfuzzysetinX,thentheimageofAunder f ,denotedby f (A), istheNeutrosophicfuzzysetinYdefinedby f (A)= {(y),f (T )A (y) ,f (IA (y)),f (F )A (y): y ∈ Y },where f (TA (y))= sup x∈f 1 (y) TA (x) iff 1(y)=0N 0 otherwise f (IA (y))= sup x∈f 1 (y) IA (x) iff 1(y)=0N 0 otherwise f (FA (y))= inf x∈f 1 (y) FA (x) iff 1(y)=0N 1 otherwise where f (FA (y))=(1 f (1 FA))(y)
Theorem19. Let N and N beanytwonear-ringsand f beahomomorphismof N onto N .If A isaNeutrosophicfuzzyidealof N ,then f 1(A) isaNeutrosophicfuzzy idealof N.
Proof. Letx,y,z∈ N. (i)
f 1(TA)(x y)= TA(f (x y))
= TA(f (x) f (y))
≥ min(TA(f (x),TA(f (y))))
= min(f 1(TA)(x),f 1(TA)(y))
f 1(IA)(x y)= IA(f (x y))
= IA(f (x) f (y))
≤ max(IA(f (x),IA(f (y))))
= max(f 1(IA)(x),f 1(IA)(y),f 1(FA)(x y))
= FA(f (x y))
= FA(f (x) f (y))
≤ max(FA(f (x),FA(f (y))))
= max(f 1(FA)(x),f 1(FA)(y))
(ii)
f 1(TA)(y + x y))= TA(f (y + x y))
= TA(f (y)+ f (x) f (y)) ≥ min(TA(f (x)))
= min(f 1(TA)(x)f 1(IA)(y + x y))
= IA(f (y + x y))
= IA(f (y)+ f (x) f (y)) ≤ max(IA(f (x)))
= max(f 1(IA)(x),f 1(FA)(y + x y))
= FA(f (y + x y))
= FA(f (y)+ f (x) f (y)) ≤max(FA(f (x)))
= max(f 1(FA)(x))
(iii)
(iv)
f 1(TA)(xy)= TA(f (xy))
= TA(f (x)f (y)) ≥ min(TA(f (y))) = min(f 1(TA)(y)) f 1(IA)(xy)= IA(f (xy))
= IA(f (x)f (y)) ≤ max(IA(f (y))) = max(f 1(IA)(y))
f 1(FA)(xy)= FA(f (xy))
= FA(f (x)f (y)) ≤max(FA(f (y))) = max(f 1(FA)(y))
f 1(TA)((x + z)y xy)= TA(f [(x + z)y xy])
= TA([f (x)+ f (z)]f (y) f (x)f (y)) ≥ min(TA(f (z)))
= min(f 1(TA)(z)f 1(IA)((x + z)y xy))
= IA(f [(x + z)y xy])
= IA([f (x)+ f (z)]f (y) f (x)f (y))
≤ max(IA(f (z))
= maxf 1(IA)(z),f 1(FA)((x + z)y xy))
= FA(f [(x + z)y xy])
= FA([f (x)+ f (z)]f (y) f (x)f (y))
≤ max(FA(f (z)))
= maxf 1(FA)(z)
Therefore, f 1(A)isaNeutrosophicfuzzyidealof N.
Theorem20. Let N1 and N2 beanytwonear-ringsand fbe ahomomorphismof N1 onto N2.IfAisaNeutrosophicfuzzyidealof N1,thenf(A)isaNeutrosophicfuzzy idealof N2
Proof. Let y1,y2,y3∈N2 and x1,x2,x3∈N1
(i)
f (TA(y1 y2)=sup x1 ,x2 ∈f 1 (N2 ) TA(x1 x2) ≥ sup x1 ,x2 ∈f 1 (N2 ) min[(TA(x1),TA(x2)] = min(sup x1 ∈f 1 (N2 ) TA(x1), sup x2 ∈f 1 (N2 ) TA(x2)) = min(f (TA(y1),f (TA(y2))
(ii)
f (TA(y1 + y2 y1)=sup x1 ,x2 ∈f 1 (N2 ) TA(x1 + x2 x1) ≥ sup x1 ∈f 1 (N2 ) TA(x1) = f (TA(y1))
(iii)
f (TA(y1y2)=sup x1 ,x2 ∈f 1 (N2 ) TA(x1x2) ≥ sup x1 ∈f 1 (N2 ) TA(x2) = f (TA(y2))
(iv)
f (TA)[(y1 + y3)y2 y1y2]=sup x1 ,x2 ,x3 ∈f 1 (N2 ) TA[(x1+x3)x2 x1x2] ≥ sup x3 ∈f 1 (N2 ) TA(x3)= f (TA(y3)
Similarlywecanprovethat f (IA(y1 y2) ≥ min(f (IA(y1),f (IA(y2)), f (IA(y1 + y2 y1)≥f (IA(y1)), f (IA(y1y2)≥f (IA(y2)),f (IA[(y1 + y3)y2 y1y2)] ≥f (IA(y3)
.Also,wecanprovethat
f (FA(y1 y2) ≤ max(f (FA(y1),f (FA(y2)), f (FA(y1 + y2 y1) ≤ f (FA(y1)), f (FA(y1y2) ≤ f (FA(y2)), f (FA[(y1 + y3)(y2 y1y2)] ≤f (FA(y3)
Hence, f (A)isaNeutrosophicfuzzyidealof N2
6Conclusion:
Inthispaperwehavestudiedthenotionofneutrosophicfuzzysubnear-ring,neutrosophic fuzzyidealsofnear-ringsanddiscussedsomealgebraicproperties.Wehaveprovedthat unionoftwoneutrosophicfuzzyidealsofnear-ringisaneutrosophicfuzzyidealofthat near-ring.Alsowehaveprovedthatthepositiveintegralpowersofneutrosophicfuzzy idealofnear-ringisaneutrosophicfuzzyideal.Wehavedefinedthedirectproducton neutrosophicfuzzyidealsofnear-ringsandprovedthatthedirectproductofanytwo neutrosophicfuzzyidealsofnear-ringsisaneutrosophicfuzzyideal.Also,wecanextend theresultforfinitenumberofneutrosophicfuzzyidealsofnear-rings.
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