SomeResultsonMultigranulationNeutrosophic RoughSetsonaSingleDomain
HuZhao 1,* andHong-YingZhang 2
1 SchoolofScience,Xi’anPolytechnicUniversity,Xi’an710048,China
2 SchoolofMathematicsandStatistics,Xi’anJiaotongUniversity,Xi’an710049,China; Zhyemily@mail.xjtu.edu.cn
* Correspondence:zhaohu@xpu.edu.cnorzhaohu2007@yeah.net;Tel.:+86-187-0294-2326
Received:11August2018;Accepted:13September2018;Published:19September2018
Abstract: Asageneralizationofsinglevalueneutrosophicroughsets,theconceptofmulti-granulation neutrosophicroughsetswasproposedbyBoetal.,andsomebasicpropertiesofthepessimistic (optimistic)multigranulationneutrosophicroughapproximationoperatorswerestudied.However, theydidnotdoacomprehensivestudyonthealgebraicstructureofthepessimistic(optimistic) multigranulationneutrosophicroughapproximationoperators.Inthepresentpaper,wewillprovide thelatticestructureofthepessimisticmultigranulationneutrosophicroughapproximationoperators. Inparticular,intheone-dimensionalcase,forspecialneutrosophicrelations,thecompletelylattice isomorphicrelationshipbetweenupperneutrosophicroughapproximationoperatorsandlower neutrosophicroughapproximationoperatorsisproved.
Keywords: neutrosophicset;neutrosophicroughset;pessimistic(optimistic)multigranulation neutrosophicapproximationoperators;completelattice
1.Introduction
Inordertodealwithimpreciseinformationandinconsistentknowledge,Smarandache[1,2] firstintroducedthenotionofneutrosophicsetbyfusingatri-componentsetandthenon-standard analysis.Aneutrosophicsetconsistsofthreemembershipfunctions,whereeveryfunctionvalueis arealstandardornon-standardsubsetofthenonstandardunitinterval ]0 ,1+ [.Sincethen,many authorshavestudiedvariousaspectsofneutrosophicsetsfromdifferentpointsofview,forexample, inordertoapplytheneutrosophicideatologics,Rivieccio[3]proposedneutrosophiclogicswhichisa generalizationoffuzzylogicsandstudiedsomebasicproperties.GuoandCheng[4]andGuoand Sengur[5]obtainedgoodapplicationsinclusteranalysisandimageprocessingbyusingneutrosophic sets.SalamaandBroumi[6]andBroumiandSmarandache[7]firstintroducedtheconceptofrough neutrosophicsets,handledincompleteandindeterminateinformation,andstudiedsomeoperations andtheirproperties.
Inordertoapplyneutrosophicsetsconveniently,Wangetal.[8]proposedsinglevalued neutrosophicsetsbysimplifyingneutrosophicsets.Singlevaluedneutrosophicsetscanalsobe viewedasageneralizationofintuitionisticfuzzysets(Atanassov[9]).Singlevaluedneutrosophicsets havebecomeanewmajorlyresearchissue.Ye[10–12]proposeddecisionmakingbasedoncorrelation coefficientsandweightedcorrelationcoefficientofsinglevaluedneutrosophicsets,andgavean applicationofproposedmethods.MajumdarandSamant[13]studiedsimilarity,distanceandentropy ofsinglevaluedneutrosophicsetsfromatheoreticalaspect.
¸SahinandKüçük[14]gaveasubsethoodmeasureofsinglevaluedneutrosophicsetsbasedon distanceandshoweditseffectivenessthroughanexample.Weknowthatthere’sacertainconnection amongfuzzyroughapproximationoperatorsandfuzzyrelations(resp.,fuzzytopologies,information
Symmetry 2018, 10,417;doi:10.3390/sym10090417
www.mdpi.com/journal/symmetry
systems[15–17]).Hence,Yangetal.[18]firstlyproposedneutrosophicrelationsandstudiedsome kindsofkernelsandclosuresofneutrosophicrelations.Subsequentlytheyproposedsinglevalued neutrosophicroughsets[19]byfusingsinglevaluedneutrosophicsetsandroughsets(Pawlak,[20]), andtheystudiedsomepropertiesofsinglevalueneutrosophicupperandlowerapproximation operators.Asageneralizationofsinglevalueneutrosophicroughsets,BaoandYang[21]introduced p-dimensionsinglevaluedneutrosophicrefinedroughsets,andtheyalsogavesomepropertiesof p-dimensionsinglevaluedneutrosophicupperandlowerapproximationoperators.
Asanothergeneralizationofsinglevalueneutrosophicroughsets,Boetal.[22]proposedtheconcept ofmulti-granulationneutrosophicroughsetsandobtainedsomebasicpropertiesofthepessimistic (optimistic)multigranulationneutrosophicroughapproximationoperators.However,thelatticestructures ofthoseroughapproximationoperatorsinreferences[19,21,22],werenotwellstudied.Followingthis idea,ZhaoandZhang[23] gavethesupremumandinfimumofthe p-dimensionneutrosophicupper andlowerapproximationoperators,buttheydidnotstudytherelationshipbetweenthe p-dimension neutrosophicupperapproximationoperatorsandthe p-dimensionneutrosophiclowerapproximation operators,especiallyintheone-dimensionalcase.Inspiredbypaper[23],anaturalproblemis:Canthe latticestructureofpessimistic(optimistic)multigranulationneutrosophicapproximationoperators begiven?
Inthepresentpaper,westudythealgebraicstructureofoptimistic(pessimistic)multigranulation singlevaluedneutrosophicapproximationoperators.
Thestructureofthepaperisorganizedasfollows.Thenextsectionreviewssomebasicdefinitionsof neutrosophicsetsandone-dimensionalmulti-granulationroughsets.InSection 3,thelatticestructureof thepessimisticmultigranulationneutrosophicroughapproximationoperatorsarestudied.InSection 4, forspecialneutrosophicrelations,aone-to-onecorrespondencerelationshipbetweenneutrosophic upperapproximationoperatorsandlowerapproximationoperatorsisgiven.Finally,Section 5 concludesthisarticleandpointsoutthedeficienciesofthecurrentresearch.
2.Preliminaries
Inthissection,webrieflyrecallseveraldefinitionsofneutrosophicset(here“neutrosophicset” refersexclusivelyto“singlevalueneutrosophicset”)andone-dimensionalmulti-granulationroughset.
(
([11]). LetCandDbetwoneutrosophicsetsinX, if TC (a) ≤ TD (a), IC (a) ≥ ID (a) andFC (a) ≥ FD (a) foreach a ∈ X,thenwecalled C iscontainedin D, i.e., C D.If C D and D C,thenwecalled C isequal toD,denotedbyC = D.
Definition3 ([18]). LetAandBbetwoneutrosophicsetsinX,
(1) TheunionofAandBisasneutrosophicsetC,denotedbyA B,where ∀x ∈ X, TC (a)= max{TA (a), TB (a)}, IC (a)= min{IA (a), IB (a)}, and FC (a)= min{FA (a), FB (a)}.
(2) TheintersectionofAandBisaneutrosophicsetD,denotedbyA B,where ∀x ∈ X,
TD (a)= min{TA (a), TB (a)}, ID (a)= max{IA (a), IB (a)}, and FD (a)= max{FA (a), FB (a)}
Definition4 ([18]) AneutrosophicrelationRinXisdefinedasfollows: R = {< (a, b), TR (a, b), IR (a, b), FR (a, b) >| (a, b) ∈ X × X}, whereTR : X × X → [0,1], IR : X × X → [0,1], FR : X × X → [0,1] , and 0 ≤ supTR (a, b)+ supIR (a, b)+ supFR (a, b) ≤ 3.
ThefamilyofallneutrosophicrelationsinXwillbedenotedby SVNR(X),andthepair (X, R) iscalleda neutrosophicapproximationspace
Definition5 ([19]) Let (X, R) beaneutrosophicapproximationspace, ∀A ∈ SVNS(X),thelowerandupper approximationsof A withrespectto (X, R),denotedby R(A) and R(A),aretwoneutrosophicsetswhose membershipfunctionsaredefinedas: ∀a ∈ X,
TR(A) (a)= ∧ b∈X[FR (a, b) ∨ TA (b)], IR(A) (a)= ∨ b∈X[(1 IR (a, b)) ∧ IA (b)],
FR(A) (a)= ∨ b∈X[TR (a, b) ∧ FA (b)], TR(A) (a)= ∨ b∈X[TR (a, b) ∧ TA (b)],
IR(A) (a)= ∧ b∈X[IR (a, b) ∨ IA (b)], FR(A) (a)= ∧ b∈X[FR (a, b) ∨ FA (b)]
Thepair (R(A), R(A)) iscalledtheone-dimensionalmulti-granulationroughset(alsocalledsinglevalue neutrosophicroughsetorone-dimensionsinglevaluedneutrosophicrefinedroughset)ofAwithrespectto (X, R).R and Rarereferredtoastheneutrosophiclowerandupperapproximationoperators,respectively.
Lemma1 ([19]) LetR1 andR2 betwoneutrosophicrelationsinX, ∀A ∈ SVNS(X), wehave
(1) R1 R2(A)= R1(A) R2(A);
(2) R1 R2(A)= R1(A) R2(A); (3) R1 R2(A) R1(A) R2(A) R1(A) R2(A);
(4) R1 R2(A) R1(A) R2(A)
Inthissection,set M = {R1, R2, , Rn } = {Ri }i=1,n iscalledamultigranulationneutrosophic relationsseton X ifeach Ri isaneutrosophicrelationon X.Inthiscase,thepair (X, M) willbecalled an n-dimensionalmultigranulationneutrosophicapptoximationspace.
Definition6 ([22]) Let (X, M) bean n-dimensionalmultigranulationneutrosophicapptoximationspace.We definetwopairsofapproximationoperatorsasfollows,forall ∀A ∈ SVNS(X) anda ∈ X, MO (A)=(MO (A), MO (A)), MP (A)=(MP (A), MP (A)), where TMO (A) (a)= ∨n i=1TRi (A) (a), IMO (A) (a)= ∧n i=1 IRi (A) (a), FMO (A) (a)= ∧n i=1 FRi (A) (a) T MO (A) (a)= ∧n i=1TRi (A) (a), I MO (A) (a)= ∨n i=1 IRi (A) (a), F MO (A) (a)= ∨n i=1 FRi (A) (a).
TMP (A) (a)= ∧n i=1TRi (A) (a), IMP (A) (a)= ∨n i=1 IRi (A) (a), FMP (A) (a)= ∨n i=1 FRi (A) (a).
T MP (A) (a)= ∨n i=1TRi (A) (a), I MP (A) (a)= ∧n i=1 IRi (A) (a), F MP (A) (a)= ∧n i=1 FRi (A) (a).
Thenthepair MO (A)=(MO (A), MO (A)) iscalledanoptismisticmultigranulationneutrosophicrough set,andthepair MP (A)=(MP (A), MP (A)) iscalledanpessimisticmultigranulationneutrosophicroughset MO and MP arereferredtoastheoptimisticandpessimisticmultigranulationneutrosophicupperapproximation operators,respectively.Similarly, MO and MP arereferredtoastheoptimisticandpessimisticmultigranulation neutrosophiclowerapproximationoperators,respectively.
Remark1. If n = 1,thenthemultigranulationneutrosophicroughsetwilldegeneratedtoaone-dimensional multi-granulationroughset(seeDefinition5).Inthefollowing,thefamilyofallmultigranulationneutrosophic relationsseton X willbedenotedby n SVNR(X).Definedarelation on n SVNR(X) asfollows: M N ifandonlyifMi Ni,then (n SVNR(X), ) isaposet,whereM = {Mi }i=1,n andN = {Ni }i=1,n ∀ Mj j∈Λ ⊆ n SVNR(X),where Mj = Mj i i=1,n and Λ beaindexset,wecandefineunionand intersectionofMj asfollows: ∨ j∈Λ Mj = j∈Λ Mj i i=1,n , ∧ j∈Λ Mj = j∈Λ Mj i i=1,n , where T j∈Λ Mj i (a, b)= ∨ j∈Λ T Mj i (a, b), I j∈Λ Mj i (a, b)= ∧ j∈Λ I Mj i (a, b), F j∈Λ Mj i (a, b)= ∧ j∈Λ F Mj i (a, b), T j∈Λ Mj i (a, b)= ∧ j∈Λ T Mj i (a, b), I j∈Λ Mj i (a, b)= ∨ j∈Λ I Mj i (a, b), F j∈Λ Mj i (a, b)= ∨ j∈Λ F Mj i (a, b)
Then ∨ j∈Λ Mj and ∧ j∈Λ Mj aretwomultigranulationneutrosophicrelationson X,andweeasilyshowthat ∨ j∈Λ Mj and ∧ j∈Λ Mj areinfimumandsupremumof Mj j∈Λ ,respectively.Hencewecaneasilyobtainthe followingtheorem:
Theorem1. (n SVNR(X), , ∧, ∨) isacompletelattice, Xn = {Xn, Xn, ··· , Xn n } and ∅N = {∅N , ∅N , ··· , ∅N n } areitstopelementandbottomelement,respectively,where Xn and ∅N aretwo neutrosophicrelationsin X anddefinedasfollows: ∀(a, b) ∈ X × X, TXN (a, b)= 1, IXN (a, b)= 0, FXN (a, b)= 0 and T∅N (a, b)= 0, I∅N (a, b)= 1, F∅N (a, b)= 1.Inparticular, (SVNR(X), , , ) isacompletelattice.
Theorem2. Let M
Proof. Weonlyshowthatthecaseoftheoptimisticmultigranulationneutrosophicapproximationoperators. (1) ∀a ∈ X,byLemma1andDefinition6,wehavethefollowing:
TM∨NO (A) (a)
= ∨n i=1TRi Qi (A) (a)= ∨n i=1TRi (A) Qi (A) (a)
= ∨n i=1 TRi (A) (a) ∧ TQi (A) (a)
≤ ∨n i=1TRi (A) (a) ∧ ∨n i=1TQi (A) (a)
= TMO (A) (a) ∧ TNO (A) (a)
= TMO (A) NO (A) (a),
IM∨NO (A) (a)
= ∧n i=1 IRi Qi (A) (a)= ∧n i=1 IRi (A) Qi (A) (a)
= ∧n i=1 IRi (A) (a) ∨ IQi (A) (a)
≥ ∧n i=1 IRi (A) (a) ∨ ∧n i=1 IQi (A) (a)
= IMO (A) (a) ∨ INO (A) (a)
= IMO (A) NO (A) (a),
FM∨NO (A) (a)
= ∧n i=1 FRi Qi (A) (a)= ∧n i=1 FRi (A) Qi (A) (a)
= ∧n i=1 FRi (A) (a) ∨ FQi (A) (a)
≥ ∧n i=1 FRi (A) (a) ∨ ∧n i=1 FQi (A) (a) = FMO (A) (a) ∨ FNO (A) (a)
= FMO (A) NO (A) (a)
Hence, M ∨ NO (A) MO (A) NO (A) (2) ∀a ∈ X,byLemma1andDefinition6,wehavethefollowing:
T M∨NO (A) (a)
= ∧n i=1TRi Qi (A) (a)= ∧n i=1TRi (A) Qi (A) (a) = ∧n i=1 TRi (A) (a) ∨ TQi (A) (a)
≥ ∧n i=1TRi (A) (a) ∨ ∧n i=1TQi (A) (a)
= T MO (A) (a) ∨ T NO (A) (a)= T MO (A) NO (A) (a),
I M∨NO (A) (a)
= ∨n i=1 IRi Qi (A) (a)= ∨n i=1 IRi (A) Qi (A) (a) = ∨n i=1 IRi (A) (a) ∧ IQi (A) (a)
≤ ∨n i=1 IRi (A) (a) ∧ ∨n i=1 IQi (A) (a) = I MO (A) (a) ∧ I NO (A) (a)= I MO (A) NO (A) (a),
F M∨NO (A) (a)
= ∨n i=1 FRi Qi (A) (a)= ∨n i=1 FRi (A) Qi (A) (a)
= ∨n i=1 FRi (A) (a) ∧ FQi (A) (a)
≤ ∨n i=1 FRi (A) (a) ∧ ∨n i=1 FQi (A) (a) = F MO (A) (a) ∧ F NO (A) (a)= F MO (A) NO (A) (a).
Hence, M ∨ NO (A) MO (A) NO (A)
(3) ∀a ∈ X,byLemma1andDefinition6,wehavethefollowing:
TM∧NO (A) (a)
= ∨n i=1TRi Qi (A) (a) ≥∨n i=1TRi (A) Qi (A) (a)
= ∨n i=1 TRi (A) (a) ∨ TQi (A) (a)
= ∨n i=1TRi (A) (a) ∨ ∨n i=1TQi (A) (a)
= TMO (A) (a) ∨ TNO (A) (a) ≥ TMO (A) (a) ∧ TNO (A) (a),
IM∧NO (A) (a)
= ∧n i=1 IRi Qi (A) (a) ≤∧n i=1 IRi (A) Qi (A) (a)
= ∧n i=1 IRi (A) (a) ∧ IQi (A) (a)
= ∧n i=1 IRi (A) (a) ∧ ∧n i=1 IQi (A) (a)
= IMO (A) (a) ∧ INO (A) (a) ≤ IMO (A) (a) ∨ INO (A) (a),
FM∧NO (A) (a)
= ∧n i=1 FRi Qi (A) (a) ≤∧n i=1 FRi (A) Qi (A) (a)
= ∧n i=1 FRi (A) (a) ∧ FQi (A) (a)
= ∧n i=1 FRi (A) (a) ∧ ∧n i=1 FQi (A) (a)
= FMO (A) (a) ∧ FNO (A) (a) ≤ FMO (A) (a) ∨ FNO (A) (a).
Hence, M ∧ No (A) Mo (A) No (A) Mo (A) No (A) (4) ∀a ∈ X,byLemma1andDefinition6,wehavethefollowing:
T M∧NO (A) (a)
= ∧n i=1TRi Qi (A) (a) ≤∧n i=1TRi (A) Qi (A) (a)
= ∧n i=1 TRi (A) (a) ∧ TQi (A) (a)
= ∧n i=1TRi (A) (a) ∧ ∧n i=1TQi (A) (a)
= T MO (A) (a) ∧ T NO (A) (a)= T MO (A) NO (A) (a),
I M∧NO (A) (a)
= ∨n i=1 IRi Qi (A) (a) ≥∨n i=1 IRi (A) Qi (A) (a) = ∨n i=1 IRi (A) (a) ∨ IQi (A) (a) = ∨n i=1 IRi (A) (a) ∨ ∨n i=1 IQi (A) (a) = I MO (A) (a) ∨ T NO (A) (a)= I MO (A) NO (A) (a),
F M∧NO (A) (a)
= ∨n i=1 FRi Qi (A) (a) ≥∨n i=1 FRi (A) Qi (A) (a)
= ∨n i=1 FRi (A) (a) ∨ FQi (A) (a)
= ∨n i=1 FRi (A) (a) ∨ ∨n i=1 FQi (A) (a)
= F MO (A) (a) ∨ F NO (A) (a)= F MO (A) NO (A) (a)
Hence, M ∧ NO (A) MO (A) NO (A).
FromTheorem2,wecaneasilyobtainthefollowingcorollary:
Corollary1. Let M = {Ri }i=1,n and N = {Qi }i=1,n betwomultigranulationneutrosophicrelationsseton X.IfM N,then ∀A ∈ SVNS(X), NO (A) MO (A), NP (A) MP (A)), MO (A) NO (A), MP (A) NP (A)
Let HP n = MP | M ∈ n SVNR(X) and LP n = MP | M ∈ n SVNR(X) bethesetof pessimisticmultigranulationneutrosophicupperandlowerapproximationoperatorsin X,respectively.
• Definedarelation ˆ ≤ on HP n asfollows: MP ˆ ≤ NP ifandonlyif MP (A) NP (A) foreach A ∈ SVNS(X).Then (HP n , ˆ ≤) isaposet.
• Definedarelation ˆ ≤ on LP n asfollows: MP ˆ ≤ NP ifandonlyif NP (A) MP (A) foreach A ∈ SVNS(X).Then (LP n , ˆ ≤) isaposet.
Let HO n = MO | M ∈ n SVNR(X) and LO n = MO | M ∈ n SVNR(X) bethesetofoptimistic multigranulationneutrosophicupperandlowerapproximationoperatorsin X,respectively.
• Definedarelation ˆ ≤ on HO n asfollows: MO ˆ ≤ NO ifandonlyif MO (A) NO (A) foreach A ∈ SVNS(X).Then (HO n , ≤) isaposet.
• Definedarelation ˆ ≤ on LO n asfollows: MO ˆ ≤ NO ifandonlyif NO (A)) MO (A) foreach A ∈ SVNS(X).Then (LO n , ˆ ≤) isaposet.
Theorem3. (1) ∀ MP i i∈I ⊆(HP n , ˆ ≤) and I beaindexset,wecandefineunionandintersectionof MP i asfollows: ˆ ∨ i∈I MP i = ∨ i∈I Mi P , ˆ ∧ i∈I MP i = [ ∧ i∈I Mi ]P ,
where [ ∧ i∈I Mi ]= ∨ M ∈ n SVNR(X) |∀A ∈ SVNS(X), MP (A) i∈I MP i (A) Then ˆ ∨ i∈I MP i and ˆ ∧ i∈I MP i aresupremumandinfimumof MP i i∈I ,respectively. (2) ∀ MP i i∈I ⊆(LP n , ˆ ≤) andIbeaindexset,wecandefineunionandintersectionofMP i asfollows: ˆ ∨ i∈I MP i = ∨ i∈I Mi P , ˆ ∧ i∈I MP i =[ ∨ i∈I Mi ]P , where [ ∨ i∈I Mi ]= ∨ M ∈ n SVNR(X) |∀A ∈ SVNS(X), i∈I MP i (A) MP (A) . Then ˆ ∨ i∈I MP i and ˆ ∧ i∈I MP i aresupremumandinfimumof MP i i∈I ,respectively.
Proof. Weonlyshow(1). Let M = ∨ i∈I Mi,then Mi M foreach i ∈ I ByCorollary1,wehave Mi P (A) MP (A) for any A ∈ SVNS(X).Thus Mi P ˆ ≤ MP.If M isamultigranulationneutrosophicrelationssetsuchthat Mi P ˆ ≤ M P foreach i ∈ I,then A ∈ SVNS(X), Mi P (A) M P (A).Hence, MP (A)= ∨ i∈I Mi P (A)= i∈I Mi P (A) M P (A) Thus MP ˆ ≤ M P.So ˆ ∨ i∈I MP i = ∨ i∈I Mi P isthesupremumof MP i i∈I Let Q =[ ∧ i∈I Mi ],then ∀B ∈ SVNS(X),wehave QP (B)= [ ∧ i∈I Mi ]P (B) i∈I MP i (B) MP i (B).
Thus QP ˆ ≤ MP i foreach i ∈ I.If M∗ isamultigranulationneutrosophicrelationssetsuchthat M∗ P ˆ ≤ MP i foreach i ∈ I,then M∗ P (A) i∈I MP i (A).
Bytheconstructionof [ ∧ i∈I Mi ],wecaneasilyobtain M∗ [ ∧ i∈I Mi ]= Q.Hence, M∗ P ˆ ≤ [ ∧ i∈I Mi ]P = QP , So ˆ ∧ i∈I MP i = [ ∧ i∈I Mi ]P istheinfimumof MP i i∈I Remark2. (1) ∀A ∈ SVNS(X), ∀a ∈ X, wecancalculatethatthefollowingformulaholds
T ∅N P (A) (a)= 0, I ∅N P (A) (a)= 1, F ∅N P (A) (a)= 1, T ∅N P (A) (a)= 1, I ∅N P (A) (a)= 0, F ∅N P (A) (a)= 0. Hence, ∀M ∈ n SVNR(X), ∅N P (A) MP (A) and MP (A) ∅N P (A).Itshowsthat ∅N P ˆ ≤ MP and ∅N P ˆ ≤ MP,i.e., ∅N P isthebottomelementof (HP n , ˆ ≤) and ∅N P isthebottomelementof (LP n , ˆ ≤).By Theorem3,wehavethefollowingresult:Both (HP n , ˆ ≤, ˆ ∧, ˆ ∨) and (LP n , ˆ ≤, ˆ ∧, ˆ ∨) arecompletelattices
(2)Similarly,wecanprovethatboth (HO n , ˆ ≤, ˆ ∧, ˆ ∨) and (LO n , ˆ ≤, ˆ ∧, ˆ ∨) arecompletelatticesifwecanuse thegeneralizationformulaof
M ∨ NO (A) MO (A) NO (A) andM ∨ NO (A) MO (A) NO (A),
However,byTheorem2,weknownthat
M ∨ NO (A) MO (A) NO (A) andM ∨ NO (A) MO (A) NO (A)
So,naturally,thereisthefollowingproblem: Howtogivethesupremumandinfimumoftheoptimisticmultigranulationneutrosophicrough approximationoperators?
Intheone-dimensionalcase,forconvenience,wewilluse H = R | R ∈ SVNR(X) and L = {R | R ∈ SVNR(X)} todenotethesetofneutrosophicupperandlowerapproximationoperators in X,respectively.AccordingtoLemma1,Remark2andTheorem3,wehavethefollowingresult:both (H, ≤, ∧, ∨) and (L, ≤, ∧, ∨) arecompletelattices(itisalsotheone-dimensionalcaseofReference[23]).
Thus, ∀a ∈ X, TR1(A) (a) ≤ TR2(A) (a), IR1(A) (a) ≥ IR2(A) (a), andFR1(A) (a) ≥ FR2(A) (a)
Moreover, TR1 (a, a) ∧ TA (a) ≤ TR2 (a, a) ∧ TA (a), IR1 (a, a) ∨ IA (a) ≥ IR2 (a, a) ∨ IA (a), and FR1 (a, a) ∨ FA (a) ≥ FR2 (a, a) ∨ FA (a). Consideringthearbitrarinessof A,inparticular,take A = {< a, (1,0,0) >},wehave TR1 (a, a) ≤ TR2 (a, a), IR1 (a, a) ≥ IR2 (a, a) and FR1 (a, a) ≥ FR2 (a, a).
Hence,R1 R2 Similarly,wealsocanshowthatthefollowingresult: (2) If R1 ≤ R2,then R1 R2 So,by(1),(2)andCorollary1,wehave SVNR({a}) ∈A,i.e., A isnotan emptyfamily.
Now,wewillgivetherelationshipbetweencompletelattices (H, ≤, ∧, ∨) and (L, ≤, ∧, ∨)
Proposition1. If SVNR(X) ∈A,then [ i∈I Ri ]= i∈I Ri =[ i∈I Ri ],where I isaindexset,and Ri ∈ SVNR(X) foreachi ∈ I
Proof. Wefirstshowthat [ i∈I Ri ]= i∈I Ri.Let R beaneutrosophicrelationin X suchthat i∈I Ri (A) R(A) foreach A ∈ SVNS(X),then Ri ≥ R,thisisequivalentto Ri R since SVNR(X) ∈ A.Thus i∈I Ri R. Moreover,bytheconstructionof [ i∈I Ri ], wehave i∈I Ri [ i∈I Ri ]. Onthe otherhand,wecanshowthat i∈I Ri (A) i∈I Ri (A) foreach A ∈ SVNS(X).So
[ i∈I Ri ]= R ∈ SVNR(X) |∀A ∈ SVNS(X), i∈I Ri (A) R(A) i∈I Ri
Hence [ i∈I Ri ]= i∈I Ri. Now,weshowthat i∈I Ri =[ i∈I Ri ]. Let R beasinglevaluedneutrosophicrelationinsuch that i∈I Ri (A) R(A) foreach A ∈ SVNS(X),then Ri ≥ R,thisisequivalentto Ri R since SVNR(X) ∈A.Thus i∈I Ri R. Moreover,bytheconstructionof [ i∈I Ri ].Wehave i∈I Ri [ i∈I Ri ]
Ontheotherhand,wecanshowthat i∈I Ri (A) i∈I Ri (A) foreach A ∈ SVNS(X).So [ i∈I Ri ]= {R ∈ SVNR(X) |∀A ∈ SVNS(X), i∈I Ri (A) R(A)} i∈I Ri
Hence, [ i∈I Ri ]= i∈I Ri. Fromaboveproved,weknowthat [ i∈I Ri ]= i∈I Ri =[ j∈J Rj ]. Theorem4. If SVNR(X) ∈A,then (SVNR(X), , , ) and (H, ≤, ∧, ∨) arecompletelatticeisomorphism. Proof. Defineamapping φ12 : SVNR(X) → H asfollows: ∀R ∈ SVNR(X), φ12(R)= R Obviously, φ12 issurjective.If R1 = R2,noticethat SVNR(X) ∈A,weknowthat R1 = R2 So φ12 isone-one. ∀{Ri }i∈I ⊆ SVNR(X) and I beaindexset.ByTheorem3andProposition1,wehave φ12( i∈I Ri )= i∈I Ri = ∨ i∈I Ri = ∨ i∈Iφ12(Ri ), and φ12( i∈I Ri )= i∈I Ri = [ i∈I Ri ]= ∧ i∈I Ri = ∧ i∈Iφ12(Ri )
Hence, φ12 preservesarbitraryunionandarbitraryintersection.
Fromaboveproved,weknowthat (SVNR(X), , , ) and (H, ≤, ∧, ∨) arecomplete latticeisomorphism.
Theorem5. If SVNR(X) ∈A,then (SVNR(X), , , ) and (L, ≤, ∧, ∨) arecompletelatticeisomorphism.
Proof. Defineamapping φ13 : SVNR(X) → L asfollows:∀R ∈ SVNR(X), φ12(R)= R Obviously, φ13 issurjective.If R1 = R2,noticethat SVNR(X) ∈A,weknowthat R1 = R2 So φ13 isone-one. ∀{Ri }i∈I ⊆ SVNR(X) and I beanindexset.ByTheorem3andProposition1,wehave φ13( i∈I Ri )= i∈I Ri = ∨ i∈I Ri = ∨ i∈Iφ13(Ri ), and φ13( i∈I Ri )= i∈I Ri =[ i∈I Ri ] = ∧ i∈I Ri = ∧ i∈Iφ13(Ri ).
Hence, φ13 preservesarbitraryunionandarbitraryintersection.
Fromtheaboveproof,weknowthat (SVNR(X), , , ) and (L, ≤, ∧, ∨) arecomplete latticeisomorphism.
Theorem6. If SVNR(X) ∈A,then (H, ≤, ∧, ∨) and (L, ≤, ∧, ∨) arecompletelatticeisomorphism.
Proof. ThroughTheorems4and5,weimmediatelyknowthattheconclusionholds.Wecanalsoprove itbythefollowingway: Defineamapping φ23 : H → L asfollows: ∀R ∈ H, φ23(R)= R ThroughTheorems4and5, theremustbeoneandonlyone R ∈ SVNR(X) suchthat φ23(R)= R foreach R ∈ L Thisshows φ23 issurjective.If R1 = R2,noticethat SVNR(X) ∈A,weknowthat R1 = R2 So φ23 isone-one. ∀ Ri i∈I ⊆ H and I beaindexset.ThroughTheorem3andProposition1,wehave φ23( ∨ i∈I Ri )= φ23( i∈I Ri )= i∈I Ri = ∨ i∈I Ri = ∨ i∈Iφ13(Ri ), and φ13( ∧ i∈I Ri )= φ13([ i∈I Ri ])=[ i∈I Ri ] =[ i∈I Ri ] = ∧ i∈I Ri = ∧ i∈Iφ23(Ri ). Hence, φ23 preservesarbitraryunionandarbitraryintersection.So, (H, ≤, ∧, ∨) and (L, ≤, ∧, ∨) arecompletelatticeisomorphism.
Remark3. ThroughTheorems4–6,wecanascertainthat φ12,φ13 and φ23 areisomorphicmappingsamong completelattices.Moreover,thefollowingdiagramcancommute,i.e., φ23 ◦ φ12 = φ13 (seeFigure 1). Symmetry 2018, 10, x FOR PEER REVIEW 13 of 14 ((),,,) X SVNR 12φ 13φ (,,,) H ≤∨∧ 23φ (,,,) L ≤∨∧
Figure1. Correspondencerelationshipamongthreecompletelattices.
Figure 1. Correspondence relationship among three complete lattices
5. Conclusions
5.Conclusions
Following the idea of multigranulation neutrosophic rough sets on a single domain as introduced by Bo et al. (2018), we gave the lattice structure of the pessimistic multigranulation neutrosophic rough approximation operators. In the one-dimensional case, for each special () X SVNR , we gave a one-to-one correspondence relationship between complete lattices (,) H ≤ and (,) L ≤ Unfortunately, at the moment, we haven’t solved the following problems:
Followingtheideaofmultigranulationneutrosophicroughsetsonasingledomainasintroduced byBoetal.(2018),wegavethelatticestructureofthepessimisticmultigranulationneutrosophic roughapproximationoperators.Intheone-dimensionalcase,foreachspecial SVNR(X),wegavea
(1) Can the supremum and infimum of the optimistic multigranulation neutrosophic rough approximation operators be given?
(2) For any set , are (,) H ≤ and (,) L ≤ isomorphic between complete lattices?
Symmetry 2018, 10,417
one-to-onecorrespondencerelationshipbetweencompletelattices (H, ≤) and (L, ≤).Unfortunately, atthemoment,wehaven’tsolvedthefollowingproblems:
(1) Canthesupremumandinfimumoftheoptimisticmultigranulationneutrosophicrough approximationoperatorsbegiven?
(2) Foranyset,are (H, ≤) and (L, ≤) isomorphicbetweencompletelattices?
AuthorContributions: H.Z.providedtheideaofthepaperandprovedthetheorems.H.-Y.Z.provided helpfulsuggestions.
Funding: Thisresearchreceivednoexternalfunding.
Acknowledgments: TheworkispartlysupportedbytheNationalNaturalScienceFoundationofChina (GrantNo.61473181,11771263and11671007),theDoctoralScientificResearchFoundationofXi’anPolytechnic University(GrantNo.BS1426),theConstructionFundingofXi’anPolytechnicUniversityforMathematics (GrantNo.107090701),andtheScientificResearchProgramFundedbyShaanxiProvincialEducation Department(2018).
ConflictsofInterest: Theauthorsdeclarenoconflictofinterest.
References
1. Smarandache,F. Neutrosophy:NeutrosophicProbability,Set,andLogic;AmericanResearchPress:Rehoboth, MA,USA,1998.
2. Smarandache,F.Neutrosophicset—-Agenerializationoftheintuitionisticsfuzzysets. Int.J.PureAppl.Math. 2005, 24,287–297.
3. Rivieccio,U.Neutrosophiclogics:Prospectsandproblems. FuzzySetsSyst. 2008, 159,1860–1868.[CrossRef]
4. Guo,Y.;Cheng,H.D.Anewneutrosophicapproachtoimagesegmentation. PatternRecogn. 2009, 42,587–595. [CrossRef]
5. Guo,Y.;Sengur,A.NCM:Neutrosophicc-meansclusteringalgorithm. PatternRecogn. 2015, 48,2710–2724. [CrossRef]
6. Salama,A.A.;Broumi,S.Roughnessofneutrosophicsets. ElixirAppl.Math. 2014, 74,26833–26837.
7. Broumi,S.;Smarandache,F.Roughneutrosophicsets. Ital.J.PureAppl.Math. 2014, 32,493–502.
8. Wang,H.;Smarandache,F.;Zhang,Y.Q.;Sunderraman,R.Singlevaluedneutrosophicsets,Multispace Multistruct. GoogleSch. 2010, 4,410–413.
9. Atanassov,K.Intuitionisticfuzzysets. FuzzySetsSyst. 1986, 20,87–96.[CrossRef]
10. Ye,J.Multicriteriadecision-makingmethodusingthecorrelationcoefficientundersingle-valued neutrosophicenvironment. Int.J.Gen.Syst. 2013, 42,386–394.[CrossRef]
11. Ye,J.Improvedcorrelationcoefficientsofsinglevaluedneutrosophicsetsandintervalneutrosophicsetsfor multipleattributedecisionmaking. J.Intell.FuzzySyst. 2014, 27,2453–2462.
12. Ye,S.;Ye,J.Dicesimilaritymeasureamongsinglevaluedneutrosophicmultisetsanditsapplcationin medicaldiagnosis. NeutrosophicSetsSyst. 2014, 6,48–53.
13. Majumdar,P.;Samant,S.K.Onsimilarityandentropyofneutrosophicsets. J.Intell.FuzzySyst. 2014, 26, 1245–1252.
14. ¸Sahin,R.;Küçük,A.Subsethoodmeasureforsinglevaluedneutrosophicsets. J.Intell.FuzzySyst. 2015, 29, 525–530.[CrossRef]
15. Li,Z.W.;Cui,R.C. T-similarityoffuzzyrelationsandrelatedalgebraicstructures. FuzzySetsSyst. 2015, 275, 130–143.[CrossRef]
16. Li,Z.W.;Cui,R.C.Similarityoffuzzyrelationsbasedonfuzzytopologiesinducedbyfuzzyrough approximationoperators. Inf.Sci. 2015, 305,219–233.[CrossRef]
17.
Li,Z.W.;Liu,X.F.;Zhang,G.Q.;Xie,N.X.;Wang,S.C.Amulti-granulationdecision-theoreticroughsetmethod fordistributedfc-decisioninformationsystems:Anapplicationinmedicaldiagnosis. Appl.SoftComput. 2017, 56,233–244.[CrossRef]
18. Yang,H.L.;Guo,Z.L.;She,Y.H.;Liao,X.W.Onsinglevaluedneutrosophicrelations. J.Intell.FuzzySyst. 2016, 30,1045–1056.[CrossRef]
Symmetry 2018, 10,417
19. Yang,H.L.;Zhang,C.L.;Guo,Z.L.;Liu,Y.L.;Liao,X.W.Ahybridmodelofsinglevaluedneutrosophicsets androughsets:Singlevaluedneutrosophicroughsetmodel. SoftComput. 2017, 21,6253–6267.[CrossRef]
20. Pawlak,Z.Roughsets. Int.J.Comput.Inf.Sci. 1982, 11,341–356.[CrossRef]
21. Bao,Y.L.;Yang,H.L.Onsinglevaluedneutrosophicrefinedroughsetmodelanditsapplition. J.Intell. FuzzySyst. 2017, 33,1235–1248.[CrossRef]
22. Bo,C.X.;Zhang,X.H.;Shao,S.T.;Smarandache,F.Multi-GranulationNeutrosophicRoughSetsonaSingle DomainandDualDomainswithApplications. Symmetry 2018, 10,296.[CrossRef]
23. Zhao,H.;Zhang,H.Y.Aresultonsinglevaluedneutrosophicrefinedroughapproximationoperators. J.Intell. FuzzySyst. 2018,1–8.[CrossRef]
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