NEUTROSOPHICSOFTFILTERS
NAIMEDEMIRTAS¸
Abstract. Inthispaper,theconceptofneutrosophicsoftfilterandits basicpropertiesareintroduced.Later,wesetupaneutrosophicsofttopologywiththehelpofaneutrosophicsoftfilter.Wealsogivethenotions ofthegreatestlowerboundandtheleastupperboundofthefamilyof neutrosophicsoftfilters,neutrosophicsoftfiltersubbase andneutrosophic softfilterbaseandexploresomebasicpropertiesofthem.
1. Introduction
Wecannotsolvetheproblemsbyusingmathematicaltoolsgenerallyinthe sociallifesinceinmathematics,theconceptsarepreciseandnotsubjective.To dealwiththisproblem,researchersproposedseveralmethodssuchasfuzzyset theory[12],roughsettheory[7]andsoftsettheory[6].Theoriesoffuzzysets androughsetscanbeconsideredastoolsfordealingwithvaguenessbutboth ofthesetheorieshavetheirowndifficulties.Thereasonforthesedifficultiesis, possibly,theinadequacyoftheparametrizationtoolofthe theoryasmentioned byMolodtsov[6]in1999.Molodtsovinitiatedanovelconceptofsoftset theorywhichisacompletelynewapproachformodelinguncertainitiesand succesfullyapplieditintoseveraldirectionssuchassmoothnessoffunctions, gametheory,RiemannIntegration,theoryofmeasurement,andsoon.The fundamentalconceptsofneutrosophicsetwereintroducedbySmarandache [10].Thistheoryisageneralizationofclassicalsets,fuzzysettheory[12], intuitionisticfuzzysettheory[1],etc.Latersomeresearchers[8,9]studied basicconceptsandpropertiesofneutrosophicsets.Thenotionofneutrosophic softsetswasfirstdefinedbyMaji[5]andlater,DeliandBroumi[3]modified it.Bera[2]introducedtheconceptofneutrosophicsofttopologicalspaces. Also,neutrosophicsoftpointconceptandneutrosophicsoft Ti-spaceswere presentedbyG¨un¨uuzArasetal.[4].
Themainpurposeofthispaperistointroduceneutrosophicsoftfilters. Laterwestudysomebasicpropertiesofneutrosophicsoftfiltersandsetupa neutrosophicsofttopologywiththehelpofaneutrosophicsoftfilter.Some newnotionsinneutrosophicsoftfilterssuchasthegreatest lowerboundand
Keywordsandphrases. Neutrosophicsoftset,neutrosophicsofttopologicalspace,neutrosophicsoftfilter.
theleastupperboundofthefamilyofneutrosophicsoftfilters,neutrosophic softfiltersubbaseandneutrosophicsoftfilterbasewereintroduced.Also,we givesomebasicporpertiesoftheseconcepts.
2. Introduction
Wecannotsolvetheproblemsbyusingmathematicaltoolsgenerallyinthe sociallifesinceinmathematics,theconceptsarepreciseandnotsubjective.To dealwiththisproblem,researchersproposedseveralmethodssuchasfuzzyset theory[12],roughsettheory[7]andsoftsettheory[6].Theoriesoffuzzysets androughsetscanbeconsideredastoolsfordealingwithvaguenessbutboth ofthesetheorieshavetheirowndifficulties.Thereasonforthesedifficultiesis, possibly,theinadequacyoftheparametrizationtoolofthe theoryasmentioned byMolodtsov[6]in1999.Molodtsovinitiatedanovelconceptofsoftset theorywhichisacompletelynewapproachformodelinguncertainitiesand succesfullyapplieditintoseveraldirectionssuchassmoothnessoffunctions, gametheory,RiemannIntegration,theoryofmeasurement,andsoon.The fundamentalconceptsofneutrosophicsetwereintroducedbySmarandache [10].Thistheoryisageneralizationofclassicalsets,fuzzysettheory[12], intuitionisticfuzzysettheory[1],etc.Latersomeresearchers[8,9]studied basicconceptsandpropertiesofneutrosophicsets.Thenotionofneutrosophic softsetswasfirstdefinedbyMaji[5]andlater,DeliandBroumi[3]modified it.Bera[2]introducedtheconceptofneutrosophicsofttopologicalspaces. Also,neutrosophicsoftpointconceptandneutrosophicsoft Ti-spaceswere presentedbyG¨un¨uuzArasetal.[4].
Themainpurposeofthispaperistointroduceneutrosophicsoftfilters. Laterwestudysomebasicpropertiesofneutrosophicsoftfiltersandsetupa neutrosophicsofttopologywiththehelpofaneutrosophicsoftfilter.Some newnotionsinneutrosophicsoftfilterssuchasthegreatest lowerboundand theleastupperboundofthefamilyofneutrosophicsoftfilters,neutrosophic softfiltersubbaseandneutrosophicsoftfilterbasewereintroduced.Also,we givesomebasicporpertiesoftheseconcepts.
3. Preliminaries
Inthissection,wepresentthebasicdefinitionsandresults ofneutrosophic softsetsandneutrosophicsofttopologicalspacesthatwerequireinthenext sections.
Definition1. [3]Let X beaninitialuniversesetand E beasetofparameters. Let P (X)denotethesetofallneutrosophicsetsof X.Thenaneutrosophic softset( ∼ F,E)over X isasetdefinedbyasetvaluefunction ∼ F representing
amapping ∼ F : E → P (X),where ∼ F iscalledtheapproximatefunctionofthe neutrosophicsoftset( ∼ F,E).Inotherwords,theneutrosophicsoftsetisa parameterizedfamilyofsomeelementsoftheset P (X)andthereforeitcan bewrittenasasetoforderedpairs. ( ∼ F,E)= e, x,T∼ F (e)(x),I∼ F (e)(x),F∼ F (e)(x) : x ∈ X : e ∈ E , where T∼ F (e)(x),I∼ F (e)(x),F∼ F (e)(x) ∈ [0, 1]arerespectivelycalledthetruthmembership,indeterminacy-membershipandfalsity-membershipfunctionof ∼ F (e).Sincethesupremumofeach T,I,F is1,theinequality0 ≤ T∼ F (e)(x)+ I∼ F (e)(x)+ F∼ F (e)(x) ≤ 3isobvious.
Definition2. [2]Let( ∼ F,E)beaneutrosophicsoftsettheuniverseset X Thecomplementof( ∼ F,E)isdenotedby( ∼ F,E)c andisdefinedby: ( ∼ F,E)c = e, x,F∼ F (e)(x), 1 I∼ F (e)(x),T∼ F (e)(x) : x ∈ X : e ∈ E . Itisobviousthat ( ∼ F,E)c c =( ∼ F,E).
Definition3. [5]Let( ∼ F,E)and( ∼ G,E)betwoneutrosophicsoftsetsover theuniverseset X.( ∼ F,E)issaidtobeaneutrosophicsoftsubsetof( ∼ G,E) if T∼ F (e)(x) ≤ T∼ G(e)(x), I∼ F (e)(x) ≤ I∼ G(e)(x), F∼ F (e)(x) ≥ F∼ G(e)(x), ∀e ∈ E, ∀x ∈ X.Itisdenotedby( ∼ F,E) ⊆ ( ∼ G,E).( ∼ F,E)issaidtobeneutrosophic softequalto( ∼ G,E)if( ∼ F,E)isaneutrosophicsoftsubsetof( ∼ G,E)and( ∼ G,E) isaneutrosophicsoftsubsetof( ∼ F,E).Itisdenotedby( ∼ F,E)=( ∼ G,E).
Definition4. [4]Let( ∼ F1,E)and( ∼ F2,E)betwoneutrosophicsoftsetsover theuniverseset X.Thentheirunionisdenotedby( ∼ F1,E) ∪ ( ∼ F2 ,E)=( ∼ F3,E) andisdefinedby: ( ∼ F3,E)= e, x,T ∼ F3 (e)(x),I ∼ F3 (e)(x),F ∼ F3 (e)(x) : x ∈ X : e ∈ E , where
T ∼ F3 (e)(x)=max T ∼ F1 (e)(x),T ∼ F2 (e)(x) ,
I ∼ F3 (e)(x)=max I ∼ F1 (e)(x),I ∼ F2 (e)(x) ,
F ∼ F3 (e)(x)=min F ∼ F1 (e)(x),F ∼ F2 (e)(x)
Definition5. [4]Let( ∼ F1,E)and( ∼ F2,E)betwoneutrosophicsoftsetsover theuniverseset X.Thentheirintersectionisdenotedby( ∼ F1,E) ∩ ( ∼ F2,E)= ( ∼ F3,E)andisdefinedby: ( ∼ F3,E)= e, x,T ∼ F3 (e)(x),I ∼ F3 (e)(x),F ∼ F3 (e)(x) : x ∈ X : e ∈ E , where T ∼ F3 (e)(x)=min T ∼ F1 (e)(x),T ∼ F2 (e)(x) , I ∼ F3 (e)(x)=min I ∼ F1 (e)(x),I ∼ F2 (e)(x) , F ∼ F3 (e)(x)=max F ∼ F1 (e)(x),F ∼ F2 (e)(x)
Definition6. [4]Aneutrosophicsoftset( ∼ F,E)overtheuniverseset X issaid tobeanullneutrosophicsoftsetif T∼ F (e)(x)=0, I∼ F (e)(x)=0, F∼ F (e)(x)=1; ∀e ∈ E, ∀x ∈ X.Itisdenotedby0(X,E).
Definition7. [4]Aneutrosophicsoftset( ∼ F,E)overtheuniverseset X is saidtobeanabsoluteneutrosophicsoftsetif T∼ F (e)(x)=1, I∼ F (e)(x)=1, F∼ F (e)(x)=0; ∀e ∈ E, ∀x ∈ X.Itisdenotedby1(X,E). Clearly,0c (X,E) =1(X,E) and1c (X,E) =0(X,E)
Definition8. [4]Let NSS(X,E)bethefamilyofallneutrosophicsoftsets overtheuniverseset X and τ ⊆ NSS(X,E).Then τ issaidtobeaneutrosophicsofttopologyon X if: 1 0(X,E) and1(X,E) belongto τ , 2. theunionofanynumberofneutrosophicsoftsetsin τ belongsto τ , 3 theintersectionofafinitenumberofneutrosophicsoftsets in τ belongs to τ .
Then(X,τ,E)issaidtobeaneutrosophicsofttopologicalspaceover X Eachmemberof τ issaidtobeaneutrosophicsoftopenset.Aneutrosophic softset( ∼ F,E)iscalledaneutrosophicsoftclosedsetiffitscomplement( ∼ F,E)c isaneutrosophicsoftopenset.
Definition9. [4]Let NSS(X,E)bethefamilyofallneutrosophicsoftsets overtheuniverseset X.Thenneutrosophicsoftset xe (α,β,γ) iscalledaneutrosophicsoftpoint,forevery x ∈ X,0 <α,β,γ ≤ 1, e ∈ E andisdefinedas follows: xe (α,β,γ)(e ′)(y)= (α,β,γ) ife′ = eandy = x, (0, 0, 1) ife′ = eory = x.
Definition10. [4]Let( ∼ F,E)beaneutrosophicsoftsetovertheuniverseset X.Wesaythat xe (α,β,γ) ∈ ( ∼ F,E)readasbelongingtotheneutrosophicsoft set( ∼ F,E)whenever α ≤ T∼ F (e)(x), β ≤ I∼ F (e)(x)and F∼ F (e)(x) ≤ γ.
Definition11. [4]Let(X,τ,E)beaneutrosophicsofttopologicalspaceover X.Aneutrosophicsoftset( ∼ F,E)in(X,τ,E)iscalledaneutrosophicsoft neighborhoodoftheneutrosophicsoftpoint xe (α,β,γ) ∈ ( ∼ F,E),ifthereexistsa neutrosophicsoftopenset( ∼ G,E)suchthat xe (α,β,γ) ∈ ( ∼ G,E) ⊆ ( ∼ F,E).
Theorem1. [4]Let (X,τ,E) beaneutrosophicsofttopologicalspaceand ( ∼ F,E) beaneutrosophicsoftsetover X.Then ( ∼ F,E) isaneutrosophic softopensetifandonlyif ( ∼ F,E) isaneutrosophicsoftneighborhoodofits neutrosophicsoftpoints.
Theneighborhoodsystemofaneutrosophicsoftpoint xe (α,β,γ), denotedby U (xe (α,β,γ),E),isthefamilyofallitsneighborhoods.
Theorem2. [4]Theneighborhoodsystem U (xe (α,β,γ),E) at xe (α,β,γ) inaneutrosophicsofttopologicalspace (X,τ,E) hasthefollowingproperties:
1)If( ∼ F,E) ∈ U (xe (α,β,γ),E),then xe (α,β,γ) ∈ ( ∼ F,E),
2)If( ∼ F,E) ∈ U (xe (α,β,γ),E)and( ∼ F,E) ⊆ ( ∼ H,E)then( ∼ H,E) ∈ U (xe (α,β,γ),E),
3)If( ∼ F,E),( ∼ G,E) ∈ U (xe (α,β,γ),E)then( ∼ F,E) ∩ ( ∼ G,E) ∈ U (xe (α,β,γ),E),
4)If( ∼ F,E) ∈ U (xe (α,β,γ),E)thenthereexistsa( ∼ G,E) ∈ U (xe (α,β,γ),E)such that( ∼ G,E) ∈ U (ye′ (α′ ,β′,γ′),E)foreach ye′ (α′,β′,γ′ ) ∈ ( ∼ G,E).
Definition12. Let(X,τ,E)beaneutrosophicsofttopologicalspaceand G¸(xe (α,β,γ),E)beafamilyofsomeneutrosophicsoftneighborhoodsofneutrosophicsoftpoint xe (α,β,γ).If,foreachneutrosophicsoftneighborhood( ∼ G,E) of xe (α,β,γ),thereexistsa( ∼ H,E) ∈G¸(xe (α,β,γ),E)suchthat xe (α,β,γ) ∈ ( ∼ H,E) ⊆ ( ∼ G,E),thenwesaythatG¸(xe (α,β,γ),E)isaneutrosophicsoftneighborhood baseat xe (α,β,γ) Theorem3. Ifforeachneutrosophicsoftpoint xe (α,β,γ) therecorresponds afamily U (xe (α,β,γ),E) suchthattheproperties 1) - 4) inTheorem13are satisfied,thenthereisaunique τ neutrosophicsofttopologicalstructureover X suchthatforeach xe (α,β,γ), U (xe (α,β,γ),E) isthefamilyof τ -neutrosophic softneighborhoodsof xe (α,β,γ).
Proof. Let τ = ( ∼ G,E) ∈ NSS(X,E): xe (α,β,γ) ∈ ( ∼ G,E)=⇒ ( ∼ G,E) ∈ U(xe (α,β,γ),E) .
Itisclearthat, τ isaneutrosophicsofttopologyover X.Thefamily τ certainlysatisfiesaxioms2. and3. inDefinition8:for3.,thisfollowsimmediatelyfrom2)inTheorem13andfor2 ,from3)inTheorem13.The axiom1. inDefinition8isaresultof2)and3)inTheorem13.Itremains toshowthat,intheneutrosophicsofttopologydefinedby τ , U (xe (α,β,γ),E) isthesetof τ -neutrosophicsoftneighborhoodsof xe (α,β,γ) foreach xe (α,β,γ).It followsfrom2)inTheorem13thatevery τ -neutrosophicsoftneighborhood of xe (α,β,γ) belongsto U (xe (α,β,γ),E).Conversely,let( ∼ G1,E)beaneutrosophic softsetbelongingto U (xe (α,β,γ),E)andlet( ∼ G2,E)betheneutrosophicsoftset ofneutrosophicsoftpoints ye′ (α′ ,β′,γ′) suchthat( ∼ G1,E) ∈ U (ye′ (α′ ,β′,γ′),E).If wecanshowthat xe (α,β,γ) ∈ ( ∼ G2,E),( ∼ G2,E) ⊆ ( ∼ G1,E)and( ∼ G2,E) ∈ τ , thentheproofwillbecomplete.Sinceforeveryneutrosophicsoftpoint ye′ (α′ ,β′,γ′) ∈ ( ∼ G2,E)belongsto( ∼ G1,E)byreasonof1)inTheorem13and thehypothesis( ∼ G1,E) ∈ U (ye′ (α′ ,β′,γ′),E),weobtain( ∼ G2,E) ⊆ ( ∼ G1,E).Since ( ∼ G1,E) ∈ U (xe (α,β,γ),E)and( ∼ G2,E) ⊆ ( ∼ G1,E),wehave xe (α,β,γ) ∈ ( ∼ G2,E).It remainstoshowthat( ∼ G2,E) ∈ τ ,i.e.that( ∼ G2,E) ∈ U (ye′ (α′ ,β′,γ′),E)foreach ye′ (α′ ,β′,γ′) ∈ ( ∼ G2,E).If ye′ (α′ ,β′,γ′) ∈ ( ∼ G2,E)thenby4)inTheorem13there isaneutrosophicsoftset( ∼ G3,E)suchthatforeach ze′′ (α′′ ,β′′,γ′′) ∈ ( ∼ G3,E)we have( ∼ G1,E) ∈ U (ze′′ (α′′ ,β′′,γ′′),E).Since( ∼ G1,E) ∈ U (ze′′ (α′′ ,β′′,γ′′ ),E)meansthat
ze′′ (α′′ ,β′′,γ′′ ) ∈ ( ∼ G2,E),itfollowsthat( ∼ G3,E) ⊆ ( ∼ G2,E)andtherefore,by2)in Theorem13,that( ∼ G2,E) ∈ U (ye′ (α′ ,β′,γ′),E).
4. Neutrosophicsoftfilters
Definition13. Let ℵ⊆ NSS(X,E),then ℵ iscalledaneutrosophicsoftfilter on X if ℵ satisfiesthefollowingproperties: (ℵ1)0(X,E) / ∈ℵ, (ℵ2) ∀( ∼ F,E), ( ∼ G,E) ∈ℵ =⇒ ( ∼ F,E) ∩ ( ∼ G,E) ∈ℵ, (ℵ3) ∀( ∼ F,E) ∈ℵ and( ∼ F,E) ⊆ ( ∼ G,E)=⇒ ( ∼ G,E) ∈ℵ
Remark 1. Itfollowsfrom(ℵ1)and(ℵ2)thateveryfiniteintersectionsof neutrosophicsoftsetsof ℵ arenot0(X,E)
Proposition1. Thecondition (ℵ2) isequivalenttothefollowingtwoconditions:
(ℵ2a)Theintersectionoftwomembersof ℵ belongsto ℵ
(ℵ2b)1(X,E) belongsto ℵ.
Example 1 Thefamily ℵ = {1(X,E)} isaneutrosophicsoftfilterover X
Theorem4. Let 0(X,E) =( ∼ F,E) ∈ NSS(X,E).Thenthefamily ℵ ( ∼ F,E) = ( ∼ G,E):( ∼ F,E) ⊆ ( ∼ G,E) ∈ NSS(X,E) isaneutrosophicsoftfilterover X
Proof. Since1(X,E) ∈ℵ and0(X,E) / ∈ℵ, ∅ = ℵ = NSS(X,E).Suppose ( ∼ H1,E), ( ∼ H2,E) ∈ℵ,then( ∼ F,E) ⊆ ( ∼ H1,E),( ∼ F,E) ⊆ ( ∼ H2,E).Thus T∼ F (e)(x) ≤ min T ∼ H1 (e)(x),T ∼ H2 (e)(x) , I∼ F (e)(x) ≤ min I ∼ H1 (e)(x),I ∼ H2 (e)(x) and F∼ F (e)(x) ≤ max F ∼ H1 (e)(x),F ∼ H2 (e)(x) forall x ∈ X.So( ∼ F,E) ⊆ ( ∼ H1,E) ∩ ( ∼ H2,E)andhence( ∼ H1,E) ∩ ( ∼ H2,E) ∈ℵ. Theorem5. Let (X,τ,E) beaneutrosophicsofttopologicalspaceover X. Theneighborhoodsystem U (xe (α,β,γ),E) isaneutrosophicsoftfilterforevery neutrosophicsoftpoint xe (α,β,γ).Also,itiscalledneutrosophicsoftneighborhoodsfilteroftheneutrosophicsoftpoint xe (α,β,γ)
Proof. (ℵ1)By1)inTheorem13,since xe (α,β,γ) ∈ ( ∼ G,E),weobtain0(X,E) / ∈ U (xe (α,β,γ),E).
(ℵ2)Thisisclearlyseenby3)inTheorem13.
(ℵ3)Thisisclearlyseenby2)inTheorem13.
Now,wesetupaneutrosophicsofttopologywiththehelpofaneutrosophic softfilter.
Theorem6. If,forevery xe (α,β,γ),thereexistsaneutrosophicsoftfilter ℵ(xe (α,β,γ))= U (xe (α,β,γ),E) whichsatisfiesthefollowingtwoproperties,thenthereexistsa uniqueneutrosophicsofttopology τ suchthat ℵ(xe (α,β,γ)) consistsofthe τneutrosophicsoftneighborhoodsoftheneutrosophicsoftpoint xe (α,β,γ)
(1)Everyneutrosophicsoftsetintheneutrosophicsoftfilter ℵ(xe (α,β,γ)) containstheneutrosophicsoftpoint xe (α,β,γ),
(2)Forevery( ∼ G,E) ∈ℵ(xe (α,β,γ))thereexistsa( ∼ H,E) ∈ℵ(xe (α,β,γ))such thatforevery ye′ (α′,β′,γ′ ) ∈ ( ∼ H,E),( ∼ G,E) ∈ℵ(ye′ (α′ ,β′,γ′)).
Proof. Sincetheaxioms(ℵ1),(ℵ2),(ℵ3),(1)and(2)areequivalenttothe neighborhoodaxioms1) 4),byTheorem15,thereexistsaneutrosophic softtopology τ suchthat ℵ(xe (α,β,γ))consistsofthe τ -neutrosophicsoftneighborhoodsoftheneutrosophicsoftpoint xe (α,β,γ)
Example 2 Let(X,τ,E)beaneutrosophicsofttopologicalspaceand xe (α,β,γ) beaneutrosophicsoftpointover X.Since( ∼ G,E)cannotbeanelementof G¸(xe (α,β,γ),E)forevery( ∼ H,E) ∈G¸(xe (α,β,γ),E)and( ∼ H,E) ⊆ ( ∼ G,E),thenthe neutrosophicsoftneighborhoodbaseG¸(xe (α,β,γ),E)isnotaneutrosophicsoft filterover X
5. Comparisonofneutrosophicsoftfilters
Definition14. Let ℵ1 and ℵ2 beneutrosophicsoftfiltersover X.If ℵ1 ⊆ℵ2, then ℵ2 issaidtobefinerthat ℵ1 or ℵ1 coarserthan ℵ2
Ifalso ℵ1 = ℵ2,then ℵ2 isstrictlyfinerthan ℵ1 or ℵ1 isstrictlycoarser than ℵ2.Ifeither ℵ1 ⊆ℵ2 or ℵ2 ⊆ℵ1,then ℵ1 iscomparablewith ℵ2
Theorem7. Let (ℵi)i∈I beafamilyofneutrosophicsoftfiltersover X.Then ℵ = ∩ i∈I ℵi isaneutrosophicsoftfilterover X
Infact ℵ isthegreatestlowerboundofthefamily(ℵi)i∈I .
Proof. (ℵ1)Since0(X,E) / ∈ℵi foreach i ∈ I,then0(X,E) doesnotbelongto ℵ = ∩ i∈I ℵi
(ℵ2)Let( ∼ F,E), ( ∼ G,E) ∈ℵ = ∩ i∈I ℵi.Then( ∼ F,E), ( ∼ G,E) ∈ℵi foreach i ∈ I.Since( ∼ F,E) ∩ ( ∼ G,E) ∈ℵi foreach i ∈ I,soweobtain( ∼ F,E) ∩ ( ∼ G,E) ∈ ℵ = ∩ i∈I ℵi (ℵ3)Let( ∼ F,E) ∈ℵ = ∩ i∈I ℵi and( ∼ F,E) ⊆ ( ∼ G,E).Since( ∼ F,E) ∈ℵi for each i ∈ I and( ∼ F,E) ⊆ ( ∼ G,E),weget( ∼ G,E) ∈ℵi foreach i ∈ I.Hence ( ∼ G,E) ∈ℵ = ∩ i∈I ℵi.
Now,weinvestigatetheleastupperboundofthefamilyofneutrosophic softfiltersover X.
Theorem8. Let S ⊆ NSS(X,E).Thenthereexistsaneutrosophicsoftfilter ℵ whichcontainsthefamily S,if S hasthefollowingproperty:”Theallfinite intersectionsofneutrosophicsoftsetsof S arenot 0(X,E)”.
Proof. Let S = ( ∼ Fi,E): ∀i ∈ J (Jisfinite), ∩ i∈J ( ∼ Fi,E) =0(X,E) .Then wegivethefamilywhichconsistsoffiniteintersectionsofelementsof S; β = ( ∼ G,E): ∀i ∈ J (Jisfinite), ( ∼ Fi,E) ∈ Sand ( ∼ G,E)= ∩ i∈J( ∼ Fi,E)
Thenthefamily ℵ(S)= ( ∼ H,E):( ∼ G,E) ∈ βand ( ∼ G,E) ⊆ ( ∼ H,E) isa neutrosophicsoftfilterover X.
(ℵ1)0(X,E) ∈ β,forevery( ∼ H,E) ∈ℵ(S),( ∼ H,E) =0(X,E) andso0(X,E) / ∈ ℵ(S).
(ℵ2)Let( ∼ H1,E), ( ∼ H2,E) ∈ℵ(S).Thereexistneutrosophicsoftsets ( ∼ G1,E), ( ∼ G2,E) ∈ β suchthat( ∼ G1,E) ⊆ ( ∼ H1,E)and( ∼ G2,E) ⊆ ( ∼ H2,E). Fromthedefinitionof β,0(X,E) =( ∼ G1,E) ∩ ( ∼ G2,E) ∈ β.Since( ∼ G1,E) ∩ ( ∼ G2,E) ⊆ ( ∼ H1,E) ∩ ( ∼ H2,E),weobtain( ∼ H1,E) ∩ ( ∼ H2,E) ∈ℵ(S).
(ℵ3)Let( ∼ H1,E) ∈ℵ(S)and( ∼ H1,E) ⊆ ( ∼ H2,E).Thenthereexistsaneutrosophicsoftset( ∼ G,E) ∈ β suchthat( ∼ G,E) ⊆ ( ∼ H1,E).Since( ∼ H1,E) ⊆ ( ∼ H2,E),weobtain( ∼ H2,E) ∈ℵ(S).
Remark 2. Theneutrosophicsoftfilter ℵ(S)inTheorem26issaidtobe generatedby S and S issaidtobeneutrosophicsoftfiltersubbaseof ℵ(S).It isclearthat S ⊆ℵ(S).
Theorem9. Theneutrosophicsoftfilter ℵ(S) whichisgeneratedby S isthe coarsestneutrosophicsoftfilterwhichcontains S
Proof. Supposethat S ⊆ℵ1.ByTheorem26, S ⊆ β ⊆ℵ1.ByRemark27, forevery( ∼ H,E) ∈ℵ(S)thereexistsa( ∼ G,E) ∈ β suchthat( ∼ G,E) ⊆ ( ∼ H,E). Since β ⊆ℵ1,then( ∼ G,E) ∈ℵ1.Since ℵ1 isaneutrosophicsoftfilter,( ∼ H,E) ∈ ℵ1 by(ℵ3)inDefinition16.Henceweobtain ℵ(S) ⊆ℵ1.
Theorem10. Thefamily (ℵi)i∈I ofneutrosophicsoftfiltersover X hasa leastupperboundifandonlyifforallfinitesubfamilies (ℵi)1≤i≤n of (ℵi)i∈I andall ( ∼ Gi,E) ∈ℵi (1 ≤ i ≤ n), ( ∼ G1,E) ∩ ∩ ( ∼ Gn,E) =0(X,E) Proof. =⇒:Ifthereexistsaleastupperboundofthefamily(ℵi)i∈I ,by(ℵ1) and(ℵ2)inDefinition16,forallfinitesubfamilies(ℵi)1≤i≤n of(ℵi)i∈I andall ( ∼ Gi,E) ∈ℵi (1 ≤ i ≤ n),theintersection( ∼ G1,E) ∩ ... ∩ ( ∼ Gn,E) =0(X,E). ⇐=:Let( ∼ G1,E) ∩ ... ∩ ( ∼ Gn,E) =0(X,E) forallfinitesubfamilies(ℵi)1≤i≤n of(ℵi)i∈I andall( ∼ Gi,E) ∈ℵi (1 ≤ i ≤ n).Thentheneutrosophicsoftfilter ℵ(S)generatedby S = ∪ i∈I ℵi = ( ∼ F,E):(∃i ∈ I)( ∼ F,E) ∈ℵi
istheleastupperboundofthefamily(ℵi)i∈I byTheorem28.
Definition15. Let β ⊆ NSS(X,E),then β issaidtobeaneutrosophicsoft filterbaseon X if (β1) β = ∅ and0(X,E) / ∈ β (β2)Theintersectionoftwomembersof β containamemberof β.
Remark 3 β whichisinTheorem26isaneutrosophicsoftfilterbase.
Remark 4 Itisclearthat,everyneutrosophicsoftfilterisaneutrosophicsoft filterbase.
Example 3 Let(X,τ,E)beasofttopologicalspaceand xe (α,β,γ) beaneutrosophicsoftpointover X.TheneutrosophicsoftneighborhoodbaseG¸(xe (α,β,γ),E) isaneutrosophicsoftfilterbaseover X
(β1)Clearly,G¸(xe (α,β,γ),E) = ∅.Forevery( ∼ H,E) ∈G¸(xe (α,β,γ),E), xe (α,β,γ) ∈ ( ∼ H,E).Then( ∼ H,E) =0(X,E).Henceweobtain0(X,E) / ∈G¸(xe (α,β,γ),E).
(β2)Let( ∼ G,E), ( ∼ H,E) ∈ G¸(xe (α,β,γ),E).Since( ∼ G,E), ( ∼ H,E) ∈ U (xe (α,β,γ),E), weget( ∼ G,E) ∩ ( ∼ H,E) ∈ U (xe (α,β,γ),E).ByDefinition14,thereexistsa
( ∼ K,E) ∈G¸(xe (α,β,γ),E)suchthat( ∼ K,E) ⊆ ( ∼ G,E) ∩ ( ∼ H,E).Henceweget G¸(xe (α,β,γ),E)isaneutrosophicsoftfilterbaseofneutrosophicsoftneighborhoodsfilter U (xe (α,β,γ),E)byDefinition30.
Theorem11. Let ℵ beaneutrosophicsoftfilterover X and β ⊆ℵ.Then β isabaseof ℵ ifandonlyifeverymemberof ℵ containsamemberof β Proof. ItisobviousfromTheorem26.
Definition16. Twoneutrosophicsoftfilterbases β1 and β2 over X areequivalentifandonlyifeverymemberof β1 containsamemberof β2 andevery memberof β2 containsamemberof β1.
Remark 5. Twoequivalentneutrosophicsoftfilterbasesgeneratethesame neutrosophicsoftfilter.
Theorem12. Let (X,τ,E) beasofttopologicalspaceand xe (α,β,γ) beaneutrosophicsoftpointover X.IfG¸ 1(xe (α,β,γ),E) andG¸ 2(xe (α,β,γ),E) aredifferentneutrosophicsoftneighborhoodbasesof xe (α,β,γ),thenG¸ 1(xe (α,β,γ),E) and G¸ 2(xe (α,β,γ),E) aretwoequivalentneutrosophicsoftfilterbases.
Proof. Foreach( ∼ F1,E) ∈ G¸ 1(xe (α,β,γ),E),byExample33,( ∼ F1,E) ∈ U (xe (α,β,γ),E). Also,sinceG¸ 2(xe (α,β,γ),E) ⊆ U (xe (α,β,γ),E)thereexistsa( ∼ F2,E) ∈ G¸ 2(xe (α,β,γ),E) suchthat( ∼ F2,E) ⊆ ( ∼ F1,E).Similarly,foreach( ∼ F2,E) ∈G¸ 2(xe (α,β,γ),E),by Example33,( ∼ F2,E) ∈ U (xe (α,β,γ),E).SinceG¸ 1(xe (α,β,γ),E) ⊆ U (xe (α,β,γ),E), thereexistsa( ∼ F1,E) ∈G¸ 1(xe (α,β,γ),E)suchthat( ∼ F1,E) ⊆ ( ∼ F2,E).Hencewe obtainG¸ 1(xe (α,β,γ),E)andG¸ 2(xe (α,β,γ),E)areequivalentbyDefinition35.
Theorem13. Let β1, β2 beneutrosophicsoftfilterbasesand ℵ1, ℵ2 beneutrosophicsoftfiltersover X suchthat β1 ⊆ℵ1 and β2 ⊆ℵ2.Then ℵ2 ⊆ℵ1 if andonlyifeverymemberof β2 containsamemberof β1.
Proof. =⇒:Let ℵ2 ⊆ℵ1 and( ∼ G2,E) ∈ β2.Since β2 ⊆ℵ2 ⊆ℵ1, then( ∼ G2,E) ∈ ℵ1.Since β1 ⊆ℵ1,thereexistsa( ∼ G1,E) ∈ β1 suchthat( ∼ G1,E) ⊆ ( ∼ G2,E)by Theorem34.
⇐=:Let( ∼ F2,E) ∈ℵ2.FromTheorem34,thereexistsa( ∼ G2,E)such that( ∼ G2,E) ⊆ ( ∼ F2,E).Byhypothesis,thereexistsa( ∼ G1,E) ∈ β1 such that( ∼ G1,E) ⊆ ( ∼ G2,E).Thenweobtain( ∼ G1,E) ⊆ ( ∼ F2,E).Since β1 ⊆ℵ1, ( ∼ F2,E) ∈ℵ1 byDefinition30.Henceweobtain ℵ2 ⊆ℵ1.
6. Conclusion
Inthepresentstudy,wehaveintroducedneutrosophicsoftfilterswhichare definedoveraninitialuniversewithafixedsetofparameters.Wesetupa neutrosophicsofttopologywiththehelpofaneutrosophicsoftfilter.We furtherinvestigatesomeessentialfeaturesandbasicconceptsofneutrosophic softfilters.Weexpectthatresultsinthispaperwillbehelpfullforfuture studiesinneutrosophicsoftsets.
References
[1]AtanassovK.Intuitionisticfuzzysets.FuzzySetsSyst 1986;20:87-96.
[2]BeraT,MahapatraNK.Introductiontoneutrosophicsoft topologicalspace.Opsearch 2017;54:841-867.
[3]DeliI,BroumiS.Neutrosophicsoftrelationsandsomeproperties.AnnFuzzyMath Inform2015;9:169-182.
[4]G¨un¨uuzArasC¸, ¨ Ozt¨urkTY,BayramovS.Separationaxiomsonneutrosophicsoft topolologicalspaces.TurkJMath2019;43:498-510.
[5]MajiPK.Neutrosophicsoftset.AnnFuzzyMathInform2013;5:157-168.
[6]MolodtsovD.Softsettheory-firstresults.ComputMathAppl1999;37:19-31.
[7]PawlakZ.Roughsets.IntJComputInfSci1982;11:341-356.
[8]SalamaAA,AlagamyH.NeutrosophicFilters.IJCSEITR2013;3:307-312.
[9]SalmaAA,AlblowiSA.Neutrosophicsetandneutrosophic topologicalspaces.IOSRJ Math2012;3:31-35.
[10]SmarandacheF.Neutrosophicset,ageneralizationoftheintuitionisticfuzzysets.IntJ PureApplMath2005;24:287-297.
[11]Y¨ukselS¸,TozluN,G¨uzelErg¨ulZ.Softfilter.MathSci2014;8.
[12]ZadehLA.Fuzzysets.InfControl1965;8:338-353.
DepartmentofMathematics,MersinUniversity,33343,Mersin,TURKEY E-mailaddress: naimedemirtas@mersin.edu.tr