Annalsof Fuzzy Mathematicsand Informatics Volume14,No.1,(July2017),pp.43–54 ISSN:2093–9310(printversion) ISSN:2287–6235(electronicversion) http://www.afmi.or.kr
@FMI
c KyungMoonSaCo. http://www.kyungmoon.com
Thecategoryofneutrosophiccrispsets
K.Hur,P.K.Lim,J.G.Lee,J.Kim
Received18Februry2017; Revised1March2017; Accepted19March2017 Abstract. Weintroducethecategory NCSet consistingofneutrosophiccrispsetsandmorphismsbetweenthem.Andwestudy NCSet in thesenseofatopologicaluniverseandprovethatitisCartesianclosed over Set,where Set denotesthecategoryconsistingofordinarysetsand ordinarymappingsbetweenthem.
2010AMSClassification: 03E72,18B05
Keywords: Neutrosophiccrispset,Cartesianclosedcategory,Topologicaluniverse,Neutrosophiccrispset.
CorrespondingAuthor: J.G.Lee(jukolee@wku.ac.kr)
I
1. Introduction
n1965,Zadeh[20]hadintroducedaconceptofafuzzysetasthegeneralization ofacrispset.In1986,Atanassove[1]proposedthenotionofintuitionisticfuzzyset asthegeneralizationoffuzzysetsconsideringthedegreeofmembershipandnonmembership.In1998Smarandache[19]introducedtheconceptofaneutrosophicset consideringthedegreeofmembership,thedegreeofindeterminacyandthedegree ofnon-membership.Moreover,Salamaetal.[15, 16, 18]appliedtheconceptof neutrosophiccrispsetstotopologyandrelation.
Afterthattime,manyresearchers[2, 3, 4, 5, 7, 8, 10, 12, 13, 14]haveinvestigatedfuzzysetsinthesenseofcategorytheory,forinstance, Set(H), Setf (H), Setg(H), Fuz(H).Amongthem,thecategory Set(H)isthemostusefuloneasthe ”standard”category,because Set(H)isverysuitablefordescribingfuzzysetsand mappingsbetweenthem.Inparticular,Carrega[2],Dubuc[3],Eytan[4],Goguen [5],Pittes[12],Ponasse[13, 14]hadstudied Set(H)intoposview-point.However Huretal.investigated Set(H)intopologicalview-point.Moreover,Huretal.[8] introducedthecategory ISet(H)consistingofintuitionisticH-fuzzysetsandmorphismsbetweenthem,andstudied ISet(H)inthesenseoftopologicaluniverse. Recently,Limetal[10]introducedthenewcategory VSet(H)andinvestigatedit inthesenseoftopologicaluniverse.
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TheconceptofatopologicaluniversewasintroducedbyNel[11],whichimpliesa Cartesianclosedcategoryandaconcretequasitopos.Furthermoretheconcepthas alreadybeenuptoeffectiveuseforseveralareasofmathematics.
Inthispaper,first,weobtainsomepropertiesofneutrosophiccrispsetsproposed bySalamaandSmarandache[17]in2015.Second,weintroducethecategory NCSet consistingofneutrosophiccrispsetsandmorphismsbetweenthem.Andweprove thatthecategory NCSet istopologicalandcotopologicalover Set (SeeTheorem 4.6 andCorollary 4.8),where Set denotesthecategoryconsistingofordinarysets andordinarymappingsbetweenthem.Furthermore,weprovethatfinalepisinks in NCSet arepreservedbypullbacks(SeeTheorem 4.10)and NCSet isCartesian closedover Set (SeeTheorem 4.15).
2. Preliminaries
Inthissection,welistsomebasicdefinitionsandwell-knownresultsfrom[6, 9, 11] whichareneededinthenextsections.
Definition2.1 ([9]). Let A beaconcretecategoryand((Yj ,ξj ))J afamilyofobjects in A indexedbyaclassJ.Foranyset X,let(fj : X → Yj )J beasourceofmappings indexedby J .Thenan A-structure ξ on X issaidtobeinitialwithrespectto(in short,w.r.t.)(X, (fj ), (Yj ,ξj ))J ,ifitsatisfiesthefollowingconditions: (i)foreach j ∈ J , fj :(X,ξ) → (Yj ,ξj )isan A-morphism, (ii)if(Z,ρ)isan A-objectand g : Z → X isamappingsuchthatforeach j ∈ J , themapping fj ◦ g :(Z,ρ) → (Yj ,ξj )isan A-morphism,then g :(Z,ρ) → (X,ξ)is an A-morphism. Inthiscase,(fj :(X,ξ) → (Yj ,ξj ))J iscalledaninitialsourcein A Dualnotion:cotopologicalcategory.
Result2.2 ([9],Theorem1.5). Aconcretecategory A istopologicalifandonlyif itiscotopological.
Result2.3 ([9],Theorem1.6). Let A beatopologicalcategoryover Set,thenitis completeandcocomplete.
Definition2.4 ([9]). Let A beaconcretecategory.
(i)The A-fibreofaset X istheclassofall A-structureson X (ii) A issaidtobeproperlyfibredover Set,itsatisfiesthefollowings: (a)(Fibre-smallness)foreachset X,the A-fibreof X isaset, (b)(Terminalseparatorproperty)foreachsingletonset X,the A-fibreof X haspreciselyoneelement, (c)if ξ and η are A-structuresonaset X suchthat id :(X,ξ) → (X,η)and id :(X,η) → (X,ξ)are A-morphisms,then ξ = η.
Definition2.5 ([6]) Acategory A issaidtobeCartesianclosed,ifitsatisfiesthe followingconditions:
(i)foreach A-object A and B,thereexistsaproduct A × B in A, (ii)exponentialobjectsexistin A,i.e.,foreach A-object A,thefunctor A ×− : A → A hasarightadjoint,i.e.,forany A-object B,thereexistan A-object BA anda A-morphism eA,B : A × BA → B (calledtheevaluation)suchthatforany 44
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A-object C andany A-morphism f : A × C → B,thereexistsaunique A-morphism f : C → BA suchthatthediagramcommutes:
Definition2.6 ([6]). Acategory A iscalledatopologicaluniverseover Set,ifit satisfiesthefollowingconditions: (i) A iswell-structured,i.e.(a) A isconcretecategory;(b) A satisfiesthefibresmallnesscondition;(c) A hastheterminalseparatorproperty, (ii) A iscotopologicalover Set, (iii)finalepisinksin A arepreservedbypullbacks,i.e.,foranyepisink(gj : Xj → Y )J andany A-morphism f : W → Y ,thefamily(ej : Uj → W )J , obtainedby takingthepullback f and gj ,foreach j ∈ J ,isagainafinalepisink.
3. Neutrosophiccrispsets
In[17],SalamaandSmarandacheintroducedtheconceptofaneutrosophiccrisp setinaset X anddefinedtheinclusionbetweentwoneutrosophiccrispsets,the intersection[union]oftwoneutrosophiccrispsets,thecomplementofaneutrosophic crispset,neutrosophiccrispempty[resp.,whole]setasmorethantwotypes.And theystudiedsomepropertiesrelatedtoneutrosophiccrispsetoperations.However, byselectingonlyonetype,wedefinetheinclusion,theintersection[union],and neutrosophiccrispempty[resp.,whole]setagainandfindsomeproperties.
Definition3.1. Let X beanon-emptyset.Then A iscalledaneutrosophiccrisp set(inshort,NCS)in X if A hastheform A =(A1,A2,A3), where A1,A2, and A3 aresubsetsof X, Theneutrosophiccrispempty[resp.,whole]set,denotedby φN [resp., XN ]isan NCSin X definedby φN =(φ,φ,X)[resp., XN =(X,X,φ)].Wewilldenotethe setofallNCSsin X as NCS(X).
Inparticular,SalamaandSmarandache[17]classifiedaneutrosophiccrispsetas thefollowings.
Aneutrosophiccrispset A =(A1,A2,A3)in X iscalleda: (i)neutrosophiccrispsetofType1(inshort,NCS-Type1),ifitsatisfies
A1 ∩ A2 = A2 ∩ A3 = A3 ∩ A1 = φ, (ii)neutrosophiccrispsetofType2(inshort,NCS-Type2),ifitsatisfies
A1 ∩ A2 = A2 ∩ A3 = A3 ∩ A1 = φ and A1 ∪ A2 ∪ A3 = X, (iii)neutrosophiccrispsetofType 3 (inshort,NCS-Type 3),ifitsatisfies
A1 ∩ A2 ∩ A3 = φ and A1 ∪ A2 ∪ A3 = X WewilldenotethesetofallNCSs-Type1[resp.,Type2andType3]as NCS1(X) [resp., NCS2(X)and NCS3(X)].
Definition3.2. Let A =(A1,A2,A3),B =(B1,B2,B3) ∈ NCS(X).Then
(i) A issaidtobecontainedin B,denotedby A ⊂ B,if A1 ⊂ B1, A2 ⊂ B2 and A3 ⊃ B3, (ii) A issaidtoequalto B,denotedby A = B,if A ⊂ B and B ⊂ A, (iii)thecomplementof A,denotedby Ac,isanNCSin X definedas: Ac =(A3,Ac 2,A1), 45
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(iv)theintersectionof A and B,denotedby A ∩ B,isanNCSin X definedas:
A ∩ B =(A1 ∩ B1,A2 ∩ B2,A3 ∪ B3),
(v)theunionof A and B,denotedby A ∪ B,isanNCSin X definedas:
A ∪ B =(A1 ∪ B1,A2 ∪ B2,A3 ∩ B3).
Let(Aj )j∈J ⊂ NCS(X),where Aj =(Aj,1,Aj,2,Aj,3).Then (vi)theintersectionof(Aj )j∈J ,denotedby j∈J Aj (simply, Aj ),isanNCS in X definedas:
Aj =( Aj,1, Aj,2, Aj,3),
(vii)thetheunionof(Aj )j∈J ,denotedby j∈J Aj (simply, Aj ),isanNCSin X definedas:
Aj =( Aj,1, Aj,2, Aj,3)
ThefollowingsaretheimmediateresultsofDefinition 3.2
Proposition3.3. Let A,B,C ∈ NCS(X).Then (1) φN ⊂ A ⊂ XN , (2) if A ⊂ B and B ⊂ C,then A ⊂ C, (3) A ∩ B ⊂ A and A ∩ B ⊂ B, (4) A ⊂ A ∪ B and B ⊂ A ∪ B, (5) A ⊂ B ifandonlyif A ∩ B = A, (6) A ⊂ B ifandonlyif A ∪ B = B. AlsothefollowingsaretheimmediateresultsofDefinition 3.2.
Proposition3.4. Let A,B,C ∈ NCS(X).Then (1)(Idempotentlaws): A ∪ A = A, A ∩ A = A, (2)(Commutativelaws): A ∪ B = B ∪ A, A ∩ B = B ∩ A, (3)(Associativelaws): A ∪ (B ∪ C)=(A ∪ B) ∪ C, A ∩ (B ∩ C)=(A ∩ B) ∩ C, (4)(Distributivelaws): A ∪ (B ∩ C)=(A ∪ B) ∩ (A ∪ C), A ∩ (B ∪ C)=(A ∩ B) ∪ (A ∩ C), (5)(Absorptionlaws): A ∪ (A ∩ B)= A, A ∩ (A ∪ B)= A, (6)(DeMorgan’slaws):(A ∪ B)c = Ac ∩ Bc , (A ∩ B)c = Ac ∪ Bc , (7)(Ac)c = A, (8)(8a) A ∪ φN = A, A ∩ φN = φN , (8b) A ∪ XN = XN , A ∩ XN = A, (8c) X c N = φN , φc N = XN , (8d) ingeneral, A ∪ Ac = XN , A ∩ Ac = φN .
Proposition3.5. Let A ∈ NCS(X) andlet (Aj )j∈J ⊂ NCS(X).Then (1)( Aj )c = Ac j , ( Aj )c = Ac j , (2) A ∩ ( Aj )= (A ∩ Aj ), A ∪ ( Aj )= (A ∪ Aj ).
Proof. (1) Aj =(Aj,1,Aj,2,Aj,3).Then Aj =( Aj,1, Aj,2, Aj,3).Thus ( Aj )c =( Aj,3, ( Aj,2)c , Aj,1)=( Aj,3, Ac j,2, Aj,1)= Ac j . Similarly,thesecondpartisproved. (2)Let A =(A1,A2,A3).Then A ∪ ( Aj )=(A1 ∪ ( Aj,1),A2 ∪ ( Aj,2),A3 ∩ ( Aj,3)) 46
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=( (A1 ∪ Aj,1), (A2 ∪ Aj,2), (A3 ∩ Aj,3) = (A ∪ Aj ).
Similarly,thefirstpartisproved.
Definition3.6. Let f : X → Y beamapping,andlet A =(A1,A2,A3) ∈ NCS(X) and B =(B1,B2,B3) ∈ NCS(Y ).Then (i)theimageof A under f ,denotedby f (A),isanNCSin Y definedas: f (A)=(f (A1),f (A2),f (A3)), (ii)thepreimageof B,denotedby f 1(B),isanNCSin X definedas: f 1(B)=(f 1(B1),f 1(B2),f 1(B3))
Proposition3.7. Let f : X → Y beamappingandlet A,B,C ∈ NCS(X), (Aj )j∈J ⊂ NCS(X) and D,E,F ∈ NCS(Y ), (Dk)k∈K ⊂ NCS(Y ).Thenthe followingshold:
(1) if B ⊂ C,then f (B) ⊂ f (C) andif E ⊂ F ,then f 1(E) ⊂ f 1(F ) (2) A ⊂ f 1f (A)) andif f isinjective,then A = f 1f (A)), (3) f (f 1(D)) ⊂ D andif f issurjective,then f (f 1(D))= D, (4) f 1( Dk)= f 1(Dk), f 1( Dk)= f 1(Dk), (5) f ( Aj )= f (Aj ), f ( Aj ) ⊂ f (Aj ), (6) f (A)= φN ifandonlyif A = φN andhence f (φN )= φN ,inparticularif f issurjective,then f (XN )= YN , (7) f 1(YN )= YN , f 1(φN )= φ
Definition3.8 ([17]). Let A =(A1,A2,A3) ∈ NCS(X),where X isasethaving atleastdistinctthreepoints.Then A iscalledaneutrosophiccrisppoint(inshort, NCP)in X,if A1, A2 and A3 aredistinctsingletonsetsin X
Let A1 = {p1}, A2 = {p2} and A3 = {p3},where p1 = p2 = p3 ∈ X.Then A =(A1,A2,A3)isanNCPin X.Inthiscase,wewilldenote A as p =(p1,p2,p3). Furthermore,wewilldenotethesetofallNCPsin X as NCP (X).
Definition3.9. Let A =(A1,A2,A3) ∈ NCS(X)andlet p =(p1,p2,p3) ∈ NCP (X).Then p issaidtobelongto A,denotedby p ∈ A,if {p1}⊂ A1, {p2}⊂ A2 and {p3}c ⊃ A3,i.e., p1 ∈ A1, p2 ∈ A2 and p3 ∈ Ac 3
Proposition3.10. Let A =(A1,A2,A3) ∈ NCS(X).Then A = {p ∈ NCP (X): p ∈ A}.
Proof. Let p =(p1,p2,p3) ∈ NCP (X).Then {p ∈ NCP (X): p ∈ A} =( {p1 ∈ X : p1 ∈ A1}, {p2 ∈ X : p2 ∈ A2}, {p3 ∈ X : p3 ∈ Ac 3} = A.
Proposition3.11. Let A =(A1,A2,A3),B =(B1,B2,B3) ∈ NCS(X).Then A ⊂ B ifandonlyif p ∈ B,foreach p ∈ A.
Proof. Suppose A ⊂ B andlet p =(p1,p2,p3) ∈ A.Then
A1 ⊂ B1,A2 ⊂ B2,A3 ⊃ B3 47
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Proposition3.12. Let (Aj )j∈J ⊂ NCS(X) andlet p ∈ NCP (X)
(1) p ∈ Aj ifandonlyif p ∈ Aj foreach j ∈ J (2) p ∈ Aj ifandonlyifthereexists j ∈ J suchthat p ∈ Aj
Proof. Let Aj =(Aj,1,Aj,2,Aj,3)foreach j ∈ J andlet p =(p1,p2,p3). (1)Suppose p ∈ Aj .Then p1 ∈ Aj,1,p2 ∈ Aj,2,p3 ∈ Ac j,3 Thus p1 ∈ Aj,1,p2 ∈ Aj,2,p3 ∈ Ac j,3, foreach j ∈ J .So p ∈ Aj foreach j ∈ J Wecaneasilyseethatthesufficientconditionholds. (2)supposethenecessaryconditionholds.Thenthereexists j ∈ J suchthat p1 ∈ Aj,1,p2 ∈ Aj,2,p3 ∈ Ac j,3.
Thus p1 ∈ Aj,1,p2 ∈ Aj,2,p3 ∈ ( Aj,3)c So p ∈ Aj Wecaneasilyprovethatthenecessaryconditionholds.
Definition3.13. Let f : X → Y beaninjectivemapping,where X,Y aresets havingatleastdistinctthreepoints.Let p =(p1,p2,p3) ∈ NCP (X).Thenthe imageof p under f ,denotedby f (p),isanNCPin Y definedas: f (p)=(f (p1),f (p2),f (p3)).
Remark3.14. InDefinition 3.13,ifeither X or Y hastwopoints,or f isnot injective,then f (p)isnotanNCPin Y
Definition3.15 ([17]). Let A =(A1,A2,A3) ∈ NCS(X)and B =(B1,B2,B3) ∈ NCS(Y ).ThentheCartesianproductof A and B,denotedby A × B,isanNCS in X × Y definedas: A × B =(A1 × B1,A2 × B2,A3 × B3).
4. Propertiesof NCSet
Definition4.1. Apair(X,A)iscalledaneutrosophiccrispspace(inshort,NCSp), if A ∈ NCS(X).
Definition4.2. Apair(X,A)iscalledaneutrosophiccrispspace-Type j (inshort, NCSp-Type j),if A ∈ NCSj (X), j =1, 2, 3.
Definition4.3. Let(X,AX ),(Y,AY )betwoNCSpsorNCSps-Type j, j =1, 2, 3 andlet f : X → Y beamapping.Then f :(X,AX ) → (Y,AY )iscalledamorphism, if AX ⊂ f 1(AY ), equivalently, AX,1 ⊂ f 1(AY,1),AX,2 ⊂ f 1(AY,2)and AX,3 ⊃ f 1(AY,3), where AX =(AX,1,AX,2,AX,3)and AY =(AY,1,AY,2,AY,3).
Inparticular, f :(X,AX ) → (Y,AY )iscalledanepimorphism[resp.,amonomorphismandanisomorphism],ifitissurjective[resp.,injectiveandbijective].
FromDefinitions 3.9, 4.3 andProposition 3.11,itisobviousthat f :(X,AX ) → (Y,AY )isamorphism ifandonlyif p =(p1,p2,p3) ∈ f 1(AY ),foreach p =(p1,p2,p3) ∈ AX ,i.e., f (p1) ∈ AY,1, f (p2) ∈ AY,2, f (p3) / ∈ AY,3,i.e., 48
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f (p)=(f (p1),f (p2),f (p3)) ∈ AY .
ThefollowingisanimmediateresultofDefinitions 4.3.
Proposition4.4. ForeachNCSporeachNCSps-Type j (X,AX ), j =1, 2, 3,the identitymapping id :(X,AX ) → (X,AX ) isamorphism.
Proposition4.5. Let (X,AX ), (Y,AY ), (Z,AZ ) beNCSpsorNCSps-Type j, j = 1, 2, 3 andlet f : X → Y , g : Y → Z bemappings.If f :(X,AX ) → (Y,AY ) and f :(Y,AY ) → (Z,AZ ) aremorphisms,then g ◦ f :(X,AX ) → (Z,AZ ) isa morphism.
Proof. Let AX =(AX,1,AX,2,AX,3), AY =(AY,1,AY,2,AY,3)and AZ =(AZ,1,AZ,2, AZ,3).Thenbythehypotheses, AX ⊂ f 1(AY )and AY ⊂ g 1(AZ ).Thusby Definition 4.3, AX,1 ⊂ f 1(AY,1), AX,2 ⊂ f 1(AY,2), AX,3 ⊃ f 1(AY,3) and AY,1 ⊂ g 1(AZ,1), AY,2 ⊂ g 1(AZ,2), AY,3 ⊃ g 1(AZ,3). So AX,1 ⊂ f 1(g 1(AZ,1)), AX,2 ⊂ f 1(g 1(AZ,2)), AX,3 ⊃ f 1(g 1(AZ,3)). Hence AX,1 ⊂ (g ◦ f ) 1(AZ,1), AX,2 ⊂ (g ◦ f ) 1(AZ,2), AX,3 ⊃ (g ◦ f ) 1(AZ,2). Therefore g ◦ f isamorphism.
FromPropositions 4.4 and 4.5,wecanformtheconcretecategory NCSet [resp., NCSetj]consistingofNCSs[resp.,-Type j, j =1, 2, 3]andmorphismsbetween them.Every NCSet [resp., NCSetj, j =1, 2, 3]-morphismwillbecalleda NCSet [resp., NCSetj, j =1, 2, 3]-mapping.
Theorem4.6. Thecategory NCSet istopologicalover Set.
Proof. Let X beanysetandlet((Xj ,Aj ))j∈J beanyfamiliesofNCSpsindexed byaclass J .Suppose(fj : X → (Xj ,Aj ))J isasourceofordinarymappings.We definetheNCS AX in X by AX = f 1 j (Aj )and AX =(AX,1,AX,2,AX,3). Thenclearly, AX,1 = f 1 j (Aj,1), AX,2 = f 1 j (Aj,2), AX,3 = f 1 j (Aj,3). Thus(X,AX )isanNCSpand AX,1 ⊂ f 1 j (Aj,1), AX,2 ⊂ f 1 j (Aj,2)and AX,3 ⊃ f 1 j (Aj,3).Soeach fj :(X,AX ) → (Xj ,Aj )isan NCSet-mapping. Nowlet(Y,AY )beanyNCSpandsuppose g : Y → X isanordinarymapping forwhich fj ◦ g :(Y,AY ) → (Xj ,Aj )isa NCSet-mappingforeach j ∈ J .Thenfor each j ∈ J , AY ⊂ (fj ◦ g) 1(Aj )= g 1(f 1 j (Aj )).Thus AY ⊂ (fj ◦ g) 1(Aj )= g 1( f 1 j (Aj ))= g 1(AX ) So g :(Y,AY ) → (X,AX )isan NCSet-mapping.Hence(fj :(X,AX ) → (Xj ,Aj )J isaninitialsourcein NCSet.Thiscompletestheproof.
Example4.7. (1)Let X beaset,let(Y,AY )beanNCSpandlet f : X → Y be anordinarymapping.Thenclearly,thereexistsauniqueNCS AX in X forwhich f :(X,AX ) → (Y,AY )isan NCSet-mapping.Infact, AX = f 1(AY ). Inthiscase, AX iscalledtheinverseimageunder f oftheNCSstructure AY 49
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(2)Let((Xj ,Aj ))j∈J beanyfamilyofNCSpsandlet X =Πj∈J Xj .Foreach j ∈ J ,let prj : X → Xj betheordinaryprojection.ThenthereexistsauniqueNCS AX in X forwhich prj :(X,AX → (Xj ,Aj )isan NCSet-mappingforeach j ∈ J . Inthiscase, AX iscalledtheproductof(Aj )j∈J ,denotedby AX =ΠAj =(ΠAj,1, ΠAj,2, ΠAj,3) and(ΠXj , ΠAj )iscalledtheproductNCSpof((Xj ,Aj ))j∈J . Infact, AX = j∈J pr 1 j (Aj ).
Inparticular,if J = {1, 2},then A1 × A2 =(A1,1 × A2,1,A1,2 × A2,2,A1,3 × A2,3), where A1 =(A1,1,A1,2,A1,3) ∈ NCS(X1)and A2 =(A2,1,A2,2,A2,3) ∈ NCS(X2). ThefollowingisobviousfromResult 2.2.Butweshowdirectlyit.
Corollary4.8. Thecategory NCSet iscotopologicalover Set
Proof. Let X beanysetandlet((Xj ,Aj ))J beanyfamilyofNCSpsindexedbya class J .Suppose(fj : Xj → X)J isasinkofordinarymappings.Wedefine AX as AX = fj (Aj ),where AX =(AX,1,AX,2,AX,3)and Aj =(Aj,1,Aj,2,Aj,3).Then clearly, AX ∈ NCS(X)andeach fj :(Xj ,Aj ) → (X,AX )isan NCSet-mapping. NowforeachNCSp(Y,AY ),let g : X → Y beanordinarymappingforwhich each g ◦ fj :(Xj ,Aj ) → (Y,AY )isan NCSet-mapping.Thenclearlyforeach j ∈ J , Aj ⊂ (g ◦ fj ) 1(AY ),i.e., Aj ⊂ f 1 j (g 1(AY )).
Thus Aj ⊂ f 1 j (g 1(AY )).So fj ( Aj ) ⊂ fj ( f 1 j (g 1(AY ))).ByProposition 3.7 andthedefinitionof AX ,
fj ( Aj )= fj (Aj )= AX and fj ( f 1 j (g 1(AY )))= (fj ◦ f 1 j )(g 1(AY ))= g 1(AY ) Hence AX ⊂ g 1(AY ).Therefore g :(X,AX ) → (Y,AY )isan NCSet-mapping. Thiscompletestheproof.
ThefollowingisprovedsimilarlyastheproofofTheorem 4.6 Corollary4.9. Thecategory NCSetj istopologicalover Set for j =1, 2, 3. ThefollowingisprovedsimilarlyastheproofofCorollary 4.8 Corollary4.10. Thecategory NCSetj iscotopologicalover Set for j =1, 2, 3.
Theorem4.11. Finalepisinksin NCSet areprservedbypullbacks. Proof. Let(gj :(Xj ,Aj ) → (Y,AY ))J beanyfinalepisinkin NCSet andlet f : (W,AW ) → (Y,AY )beany NCSet-mapping.Foreach j ∈ J ,let Uj = {(w,xj ) ∈ W × Xj : f (w)= gj (xj )} Foreach j ∈ J ,wedefinetheNCS AUj =(AUj,1 ,AUj,2 ,AUj,3 )in Uj by: AUj,1 = AW,1 × Aj,1,AUj,2 = AW,2 × Aj,2,AUj,3 = AW,3 × Aj,3 Foreach j ∈ J ,let ej : Uj → W and pj : Uj → Xj beordinaryprojectionsof Uj Thenclearly, AUj,1 ⊂ e 1 j (AW,1),AUj,2 ⊂ e 1 j (AW,2),AUj,3 ⊃ e 1 j (AW,3) 50
J.G.Leeetal./Ann.FuzzyMath.Inform. 14 (2017),No.1,43–54 and AUj,1 ⊂ p 1 j (Aj,1),AUj,2 ⊂ p 1 j (Aj,2),AUj,3 ⊃ p 1 j (Aj,3). Thus AUj ⊂ e 1 j (AW )and AUj ⊂ p 1 j (Aj ).So ej :(Uj ,AUj ) → (W,AW )and pj :(Uj ,AUj ) → (Xj ,Aj )are NCSet-mappings.Moreover, gh ◦ ph = f ◦ ej foreach j ∈ J ,i.e.,thediagramisapullbacksquarein NCSet: pj (Uj ,AUj )(Xj ,Aj ) ✲ ej gj (W,AW ) ❄ ❄ f ✲ (Y,AY )
Nowinordertoprovethat(ej )J isanepisinkin NCSet,i.e.,each ej issurjective, let w ∈ W .Since(gj )J isanepisink,thereexists j ∈ J suchthat gj (xj )= f (w) forsome xj ∈ Xj .Thus(w,xj ) ∈ Uj and w = ej (w,xj ).So(ej )J isanepisinkin NCSet
Finally,letusshowthat(ej )J isfinalin NCSet.Let A∗ W bethefinalstructure in W w.r.t.(ej )J andlet w =(w1,w2,w3) ∈ AW .Since f :(W,AW ) → (Y,AY )is an NCSet-mapping,byDefinition 3.9, w1 ∈ AW,1 ∩ f 1(AY,1), w2 ∈ AW,2 ∩ f 1(AY,2)and w3 ∈ Ac W,3 ∩ (f 1(AY,3))c Thus w1 ∈ AW,1,f (w1) ∈ AY,1, w2 ∈ AW,2,f (w2) ∈ AY,2 and w3 ∈ Ac W,3,f (w3) ∈ Ac Y,3. Since(gj )J isfinal,
w1 ∈ AW,1,xj,1 ∈ J xj,1 ∈g 1 j (f (w))
Aj,1, w2 ∈ AW,2,xj,2 ∈ J xj,2 ∈g 1 j (f (w))
Aj,2 and w3 ∈ Ac W,3,xj,3 ∈ ( J xj,3 ∈g 1 j (f (w))
Aj,3)c . So(w1,xj,1) ∈ AUj,1 ,(w2,xj,2) ∈ AUj,2 and(w3,xj,3) ∈ Ac Uj,1 .Since A∗ W isthe finalstructurein W w.r.t.(ej )J , w ∈ A∗ W ,i.e., AW ⊂ A∗ W .Ontheotherhand, since(ej :(Uj ,AUj ) → (W,AW ))J isfinal,1W :(W,A∗ W ) → (W,AW )isan NCSetmappingandthus A∗ W ⊂ AW .Hence A∗ W = AW .Therefore(ej )J isfinal.This completestheproof.
ThefollowingisprovedsimilarlyastheproofofTheorem 4.9 Corollary4.12. Finalepisinksin NCSetj areprservedbypullbacks,for J =1, 2, 3. Foranysingletonset {a},NCS A{a} [resp.,NCS-Type jA{a},j ,for j =1, 2, 3] on {a} isnotunique,thecategory NCSet [resp., NCSetj,for j =1, 2, 3]isnot properlyfibredover Set.ThenbyDefinition 2.6,Corollary 4.8 andTheorem 4.11 [resp.,Corollaries 4.10 and 4.12],wehavethefollowingresult.
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Theorem4.13. Thecategory NCSet [resp., NCSetj,for j =1, 2, 3]satisfies alltheconditionsofatopologicaluniverseover Set excepttheterminalseparator property.
ThefollowingisanimmediateresultofDefinitions 3.9 and 3.15
Proposition4.14. Let p =(p1,p2,p3),q =(q1,q2,q3) ∈ NCP (X) andlet A = (A1,A2,A3),B =(B1,B2,B3) ∈ NCS(X).Then (p,q) ∈ A × B ifandonlyif (p1,q1) ∈ A1 × B1, (p2,q2) ∈ A2 × B2 and (p3,q3) ∈ (A2 × B2)c,i.e., p3 ∈ Ac 3 or q3 ∈ Bc 3.
Theorem4.15. Thecategory NCSet isCartesianclosedover Set Proof. Itisclearthat NCSet hasproductsbyTheorem 4.6.Thenitissufficientto seethat NCSet hasexponentialobjects.
ForanyNCSps X =(X,AX )and Y =(Y,AY ),let Y X bethesetofallordinary mappingsfrom X to Y .WedefinetheNCS AY X =(AY X ,1,AY X ,2,AY X ,3)in Y X by:foreach f =(f1,f2,f3) ∈ Y X , f ∈ AY X ifandonlyif f (x) ∈ AY ,foreach x =(x1,x2,x3) ∈ NCP (X),i.e., f1 ∈ AY X ,1,f2 ∈ AY X ,2,f3 / ∈ AY X ,3 ifandonlyif f1(x1) ∈ AY,1,f2(x2) ∈ AY,2,f3(x3) / ∈ AY,3 Infact,
AY X ,1 = {f1 ∈ Y X : f1(x1) ∈ AY,1 foreach x1 ∈ X}, AY X ,2 = {f2 ∈ Y X : f2(x2) ∈ AY,2 foreach x2 ∈ X}, AY X ,3 = {f3 ∈ Y X : f3(x3) / ∈ AY,3 forsome x3 ∈ X}
Thenclearly,(Y X ,AY X )isanNCSp. Let YX =(Y X ,AY X ).Thenbythedefinitionof AY X , AY X ,1 ⊂ f 1(AY,1), AY X ,2 ⊂ f 1(AY,2)and AY X ,3 ⊃ f 1(AY,3). Wedefine eX,Y : X × Y X → Y by eX,Y (x,f )= f (x),foreach(x,f ) ∈ X × Y X Let(x,f ) ∈ AX ×AY X ,where x =(x1,x2,x3), f =(f1,f2,f3).ThenbyProposition 4.14 andthedefinitionof eX,Y , (x1,f1) ∈ AX,1 × AY X ,1,(x2,f2) ∈ AX,2 × AY X ,2,(x3,f3) ∈ (AX,3 × AY X ,3)c and eX,Y (x1,f1)= f1(x1), eX,Y (x2,f2)= f2(x2), eX,Y (x3,f3)= f3(x3). Thusbythedefinitionof AY X , (x1,f1) ∈ f 1(AY,1) × f 1(AY,1), (x2,f2) ∈ f 1(AX,2) × f 1(AX,2), (x3,f3) ∈ (f 1(AX,3) × (f 1(AX,3))c So(x1,f1) ∈ e 1 X,Y (AY,1),(x2,f2) ∈ e 1 X,Y (AY,2)and(x3,f3) ∈ (e 1 X,Y (AY,3))c.Hence AX × AY X ⊂ e 1 X,Y (AY ).Therefore eX,Y : X × YX → Y isan NCSet-mapping. Forany Z =(Z,AZ ) ∈ NCSet,let h : X × Z → Y bean NCSet-mapping.We define h : Z → Y X by[h(z)](x)= h(x,z),foreach z ∈ Z andeach x ∈ X.Let (x,z) ∈ AX × AZ ,where x =(x1,x2,x3)and z =(z1,z2,z3).Since h : X × Z → Y isan NCSet-mapping, 52
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AX,1 × AZ,1 ⊂ h 1(AY,1), AX,2 × AZ,2 ⊂ h 1(AY,2), AX,3 × AZ,3 ⊃ h 1(AY,1).
ThenbyProposition 4.14, (x1,z1) ∈ h 1(AY,1),(x2,z2) ∈ h 1(AY,2),(x3,z3) ∈ (h 1(AY,3))c . Thus h((x1,z1)) ∈ AY,1, h((x2,z2)) ∈ AY,2, h((x3,z3)) ∈ (AY,3)c
Bythedefinitionof h, [h(z1)](x1) ∈ AY,1,[h(z2)](x2) ∈ AY,2,[h(z3)](x3) ∈ (AY,3)c
Bythedefinitionof AY X , [h(z1)](AZ,1) ⊂ AY X ,1,[h(z2)](AZ,2) ⊂ AY X ,2,[h(z3)](AZ,3) ⊃ AY X ,3. So AZ ⊂ h 1(AY X ).Hence h : Z → YX isan NCSet-mapping.Furthermore, h istheunique NCSet-mappingsuchthat eX,Y ◦ (1X × h)= h.Thiscompletesthe proof.
ThefollowingisprovedsimilarlyastheproofofTheorem 4.15 Corollary4.16. Thecategory NCSetj isCartesianclosedover Set for j =1, 2, 3
5. Conclusions
Foranon-emptyset X,bydefininganeutrosophiccrispset A =(A1,A2,A3)and anintuitionisticcrispset A =(A1,A2)in X,respectivelyasfollows:
(i) A1 ⊂ X,A2 ⊂ X,A3 ⊂ X, (ii) A1 ⊂ Ac 3,A3 ⊂ Ac 2, and (i) A1 ⊂ X,A2 ⊂ X, (ii) A1 ⊂ Ac 2, wecanformanothercategories NCSet∗ and ICSet.Furthermore,wewillstudy theminviewpointsofatopologicaluniverseandobtainsomerelationshipbetween them.
Acknowledgements. WeareverygratefulforthepositiveadvicesProf. SmarandacheandProf.Agboolahavegivenus.
References
[1] K.Atanassov,Intuitionisticfuzzysets,FuzzySetsandSystems20(1986)87–96.
[2] J.C.Carrega,Thecategory Set(H)and Fzz(H),Fuzzysetsandsystems9(1983)327–332.
[3] E.J.Dubuc,ConcretequasitopoiApplicationsofSheaves,Proc.Dunham1977,Lect.Notes inMath.753(1979)239–254.
[4] M.Eytan,Fuzzysets:atopologicalpointofview,Fuzzysetsandsystems5(1981)47–67.
[5] J.A.Goguen,CategoriesofV-sets,Bull.Amer.Math.Soc.75(1969)622–624.
[6] H.Herrlich,Catesianclosedtopologicalcategories,Math.Coll.Univ.CapeTown9(1974) 1–16.
[7] K.Hur,ANoteonthecategory Set(H),HonamMath.J.10(1988)89–94.
[8] K.Hur,H.W.KangandJ.H.Ryou,IntutionisticH-fuzzysets,J.KoreaSoc.Math.Edu. Ser.B:PureAppl.Math.12(1)(2005)33–45.
[9] C.Y.Kim,S.S.Hong,Y.H.HongandP.H.Park,AlgebrasinCartesianclosedtopological categories,LectureNoteSeriesVol.11985.
[10] P.K.Lim,S.R.KimandK.Hur,Thecategory VSet(H),InternationalJournalofFuzzy LogicandIntelligentSystems10(1)(2010)73–81.
[11] L.D.Nel,TopologicaluniversesandsmoothGelfandNaimarkduality,mathematicalapplicationsofcategorytheory,Prc.A.M.S.Spec.SessopnDenver,1983,ContemporaryMathematics 30(1984)224–276.
53
J.G.Leeetal./Ann.FuzzyMath.Inform. 14 (2017),No.1,43–54
[12] A.M.Pittes,Fuzzysetsdonotformatopos,FuzzysetsandSystems8(1982)338–358.
[13] D.Ponasse,Someremarksonthecategory Fuz(H)ofM.Eytan,FuzzysetsandSystems9 (1983)199–204.
[14] D.Ponasse,Categoricalstudiesoffuzzysets,FuzzysetsandSystems28(1988)235–244.
[15] A.A.Salama,SaidBroumiandFlorentinSmarandache,NeutrosophicCrispOpenSetand NeutrosophicCrispContinuityviaNeutrosophicCrispIdeals,inNeutrosophicTheoryandIts Applications.CollectedPapers,Vol.I,EuropaNovaasbl,pp.199-205,Brussels,EU2014.See http://fs.gallup.unm.edu/NeutrosophicTheoryApplications.pdf
[16] A.A.Salama,SaidBroumiandFlorentinSmarandache,SomeTypesofNeutrosophicCrisp SetsandNeutrosophicCrispRelations,inNeutrosophicTheoryandItsApplications.Collected Papers,Vol.I,EuropaNovaasbl,pp.378-385,Brussels,EU2014.
[17] A.A.SalamaandF.Smarandache,NeutrosophicCrispSetTheory,TheEducationalPublisher Columbus,Ohio2015.
[18] A.A.Salama,FlorentinSmarandacheandValeriKroumov,NeutrosophicCrispSetsandNeutrosophicCrispTopologicalSpaces,inNeutrosophicTheoryandItsApplications.Collected Papers,Vol.I,EuropaNovaasbl,pp.206-212,Brussels,EU2014.
[19] F.Smarandache,NeutrosophyNeutrisophicProperty,Sets,andLogic,AmerResPress,Rehoboth,USA1998.
[20] L.A.Zadeh,Fuzzysets,InformationandControl8(1965)338–353.
K.Hur (kulhur@wku.ac.kr)
DivisionofMathematicsandInformationalStatistics,InstituteofBasicNaturalScience,WonkwangUniversity,460,Iksan-daero,Iksan-Si,Jeonbuk54538,Korea
P.K.Lim (pklim@wku.ac.kr)
DivisionofMathematicsandInformationalStatistics,InstituteofBasicNaturalScience,WonkwangUniversity,460,Iksan-daero,Iksan-Si,Jeonbuk54538,Korea
J.G.Lee (jukolee@wku.ac.kr)
DivisionofMathematicsandInformationalStatistics,InstituteofBasicNaturalScience,WonkwangUniversity,460,Iksan-daero,Iksan-Si,Jeonbuk54538,Korea
J.Kim (junhikim@wku.ac.kr)
DepartmentofMathematicsEducation,WonkwangUniversity,460,Iksan-daero, Iksan-Si,Jeonbuk54538,Korea
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