NeutrosophicSetsandSystems,Vol.33
UniversityofNewMexico
IntroductiontoPlithogenicHypersoftSubgroup
SudiptaGayen1 ,FlorentinSmarandache2 ,SripatiJha1 ,ManoranjanKumarSingh3 ,Said Broumi4 andRanjanKumar5,∗
1 NationalInstituteofTechnologyJamshedpur,India;sudi23dipta@gmail.com 2 UniversityofNewMexico,USA
3 MagadhUniversity,BodhGaya,India
4 FacultyofScienceBenMSik,UniversityHassanII,Morocco
5 JainDeemedtobeUniversity,Jayanagar,Bengaluru,India;ranjank.nit52@gmail.com ∗ Correspondence:ranjank.nit52@gmail.com
Abstract Inthisarticle,someessentialaspectsofplithogenichypersoftalgebraicstructureshavebeenanalyzed.Herethenotionsofplithogenichypersoftsubgroupsi.e.plithogenicfuzzyhypersoftsubgroup,plithogenic intuitionisticfuzzyhypersoftsubgroup,plithogenicneutrosophichypersoftsubgrouphavebeenintroducedand studied.Fordoingthatwehaveredefinedthenotionsofplithogeniccrisphypersoftset,plithogenicfuzzyhypersoftset,plithogenicintuitionisticfuzzyhypersoftset,andplithogenicneutrosophichypersoftsetandalso giventheirgraphicalillustrations.Furthermore,byintroducingfunctionindifferentplithogenichypersoftenvironments,somehomomorphicpropertiesofplithogenichypersoftsubgroupshavebeenanalyzed.
Keywords: Hypersoftset;Plithogenicset;Plithogenichypersoftset;Plithogenichypersoftsubgroup —————————————————————————————————————————-
ALISTOFABBREVIATIONS
US signifiesuniversalset.
CS signifiescrispset.
FS signifiesfuzzyset.
IFS signifiesintuitionisticfuzzyset.
NS signifiesneutrosophicset.
PS signifiesplithogenicset.
SS signifiessoftset.
HS signifieshypersoftset.
CHS signifiescrisphypersoftset.
FHS signifiesfuzzyhypersoftset.
IFHS signifiesintuitionisticfuzzyhypersoftset.
NHS signifiesneutrosophichypersoftset.
Gayenetal.,IntroductiontoPlithogenicHypersoftSubgroup
PHS signifiesplithogenichypersoftset.
PCHS signifiesplithogeniccrisphypersoftset.
PFHS signifiesplithogenicfuzzyhypersoftset.
PIFHS signifiesplithogenicintuitionisticfuzzyhypersoftset.
PNHS signifiesplithogenicneutrosophichypersoftset.
CG signifiescrispgroup.
FSG signifiesfuzzysubgroup.
IFSG signifiesintuitionisticfuzzysubgroup.
NSG signifiesneutrosophicsubgroup.
DAF signifiesdegreeofappurtenancefunction.
DCF signifiesdegreeofcontradictionfunction.
PSG signifiesplithogenicsubgroup.
PCHSG signifiesplithogeniccrsiphypersoftsubgroup.
PFHSG signifiesplithogenicfuzzyhypersoftsubgroup.
PIFHSG signifiesplithogenicintuitionisticfuzzyhypersoftsubgroup.
PNHSG signifiesplithogenicneutrosophichypersoftsubgroup.
DMP signifiesdecisionmakingproblem.
ρ(U ) signifiespowersetof U
1. Introduction
FS[1]theorywasfirstinitiatedbyZadehtohandleuncertainreal-lifesituationsmore preciselythanCSs.Gradually,someothersettheorieslikeIFS[2],NS[3],Pythagorean FS[4],PS[5],etc.,haveemerged.Thesesetsareabletohandleambiguoussituationsmore appropriatelythanFSs.NStheorywasintroducedbySmarandachewhichwasgeneralizations ofIFSandFS.Hehasalsointroducedneutrosophicprobability,measure[6,7],psychology[8], pre-calculusandcalculus[9],etc.Presently,NStheoryisvastlyusedinvariouspureaswell asappliedfields.Forinstance,inmedicaldiagnosis[10,11],shortestpathproblem[12–20], DMP[21–26],transportationproblem[27,28],forecasting[29],mobileedgecomputing[30], abstractalgebra[31],patternrecognitionproblem[32],imagesegmentation[33],internetof things[34],etc.AnothersettheoryofprofoundimportanceisPStheorywhichisextensively usedinhandlingvariousuncertainsituations.ThissettheoryismoregeneralthanCS,FS,IFS, andNStheory.Gradually,plithogenicprobabilityandstatistics[35],plithogeniclogic[35],etc., haveevolvedwhicharegeneralizationsofcrispprobability,statistics,andlogic.Smarandache hasalsointroducedthenotionsofplithogenicnumber,plithogenicmeasurefunction,bipolar PS,tripolarPS,multipolarPS,complexPS,refinedPS,etc.Presently,PStheoryisextensively usedinnumerousresearchdomains.
ThenotionofSS[36]theoryisanotherfundamentalsettheory.Presently,SStheoryhas becomeoneofthemostpopularbranchesinmathematicsforitshugeareasofapplicationsin variousresearchfields.Forinstance,nowadaysinDMP[37],abstractalgebra[38–40],etc.,it iswidelyused.Again,thereexistconceptslikevaguesets[41,42],roughset[43],hardset[44], etc.,whicharewellknownfortheirvastapplicationsinvariousdomains.Gradually,basedon SStheorythenotionsoffuzzySS[45],intuitionisticSS[46],neutrosophicSS[47]theory,etc., havebeenintroducedbyvariousresearchers.Infuzzyabstractalgebra,thenotionsofFSG[48], IFSG[49],NSG[31],etc.,havebeendevelopedandstudiedbydifferentmathematicians.SS theoryhasopenedsomenewwindowsofopportunitiesforresearchersworkingnotonlyin appliedfieldsbutalsoinpurefields.Asaresult,thenotionsofsoftFSG[39],softIFSG[50], softNSG[51],etc.,wereintroduced.Lateron,Smarandachehasproposedtheconceptof HS[52]theorywhichisageneralizationofSStheory.Also,hehasextendedandintroduced theconceptofHSintheplithogenicenvironmentandgeneralizedthatfurther.Asaresult,a newbranchhasemergedwhichcanbeafruitfulresearchfieldforitspromisingpotentials.The followingTable1containssomesignificantcontributionsinSSandPStheorybynumerous researchers.
Table1. SignificanceandinfluencesofPS&SStheoryinvariousfields.
Author&referencesYearContributionsinvariousfields
Majhietal.[53]2002AppliedSStheoryinaDMP.
Fengetal.[54]2010DescribedanadjustableapproachtofuzzySSbased DMPwithsomeexamples.
aman[55]2011DefinedfuzzysoftaggregationoperatorwhichallowstheconstructionofmoreefficientDMP.
Broumietal.[56]2014DefinedneutrosophicparameterizedSSandneutrosophicparameterizedaggregationoperatorandapplieditinDMP.
Broumietal.[57]2014Definedinterval-valuedneutrosophicparameterized SSareductionmethodforit.
Delietal.[58]2014Introducedneutrosophicsoftmulti-settheoryand studiedsomeofitsproperties.
Deli&Naim[59]2015IntroducedintuitionisticfuzzyparameterizedSS andstudiedsomeofitsproperties.
Smarandache[60]2018IntroducedphysicalPS.
Smarandache[61]2018Studiedaggregationplithogenicoperatorsinphysicalfields.
continued...
Gayenetal.,IntroductiontoPlithogenicHypersoftSubgroup
Author&referencesYearContributionsinvariousfields
Gayenetal.[62]2019Introducedthenotionsofplithogenicsubgroupsand studiedsomeoftheirhomomorphicproperties.
Abdel-Bassetetal.[63]2019Describedanovelmodelforevaluationofhospital medicalcaresystemsbasedonPSs.
Abdel-Bassetetal.[64]2019DescribedanovelplithogenicTOPSIS-CRITIC modelforsustainablesupplychainriskmanagement.
Abdel-Bassetetal.[65]2019Proposedahybridplithogenicdecision-makingapproachwithqualityfunctiondeployment.
ThisChapterhasbeensystematizedasthefollowing:InSection2,literaturereviewsof FS,IFS,NS,FSG,IFSG,NSG,PS,PHS,etc.,arementioned.InSection3,theconceptsof PCHS,PFHS,PIFHS,andPNHShavebeenredefinedinadifferentwayandtheirgraphical illustrationshavebeengiven.Also,thenotionsofPFHSG,PIFHSG,andPNHSGhavebeen introducedandfurthertheeffectsofhomomorphismonthosenotionsarestudied.Finally,in Section4,theconclusionisgivenmentioningsomescopesoffutureresearches.
2. LiteratureSurvey
Inthissegment,someimportantnotionslike,FS,IFS,NS,FSG,IFSG,NSG,etc.,have beendiscussed.WehavealsomentionedPS,SS,HSandsomeaspectsofPHS.Thesenotions willplayvitalrolesindevelopingtheconceptsofPHSGs.
Definition2.1. [1]Let U beaCS.Afunction σ : U → [0, 1]iscalledaFS.
Definition2.2. [2]Let U beaCS.AnIFS γ of U iswrittenas γ = {(m,tγ (m),fγ (m)): m ∈ U },where tγ (m)and fγ (m)aretwoFSsof U, whicharecalledthedegreeofmembership andnon-membershipofany m ∈ U. Here ∀m ∈ U,tγ (m)and fγ (m)satisfytheinequality 0 ≤ tγ (m)+ fγ (m) ≤ 1
Definition2.3. [3]Let U beaCS.ANS η of U isdenotedas η = {(m,tη (m),iη (m),fη (m)): m ∈ U },where tη (m),iη (m),fη (m): U →] 0, 1+[arethecorrespondingdegreeoftruth, indeterminacy,andfalsityofany m ∈ U. Here ∀m ∈ Utη (m), iη (m)and fη (m)satisfythe inequality 0 ≤ tη (m)+ iη (m)+ fη (m) ≤ 3+
2.1. Fuzzy,Intuitionisticfuzzy&Neutrosophicsubgroup
Definition2.4. [48]AFSofaCG U iscalledasaFSGiff ∀m,u ∈ U, theconditions mentionedbelowaresatisfied: (i) α(mu) ≥ min{α(m),α(u)} Gayenetal.,IntroductiontoPlithogenicHypersoftSubgroup
(ii) α(m 1) ≥ α(m).
Definition2.5. [49]AnIFS γ = {(m,tγ (m),fγ (m)): m ∈ U } ofaCG U iscalledanIFSG iff ∀m,u ∈ U,
(i) tγ (mu 1) ≥ min{tγ (m),tγ (u)}
(ii) fγ (mu 1) ≤ max{fγ (m),fγ (u)}
ThesetofalltheIFSGof U willbedenotedasIFSG(U ).
Definition2.6. [31]Let U beaCGand δ beaNSof U.δ iscalledaNSGof U iffthe conditionsmentionedbelowaresatisfied:
(i) δ(mu) ≥ min{δ(m),δ(u)},i.e. tδ (mu) ≥ min{tδ (m),tδ (u)}, iδ (mu) ≥ min{iδ (m),iδ (u)} and fδ (mu) ≤ max{fδ (m),fδ (u)}
(ii) δ(m 1) ≥ δ(m)i.e. tδ (m 1) ≥ tδ (u), iδ (m 1) ≥ iδ (u)and fδ (m 1) ≤ fδ (u).
Theorem2.1. [66]Let g beahomomorphismofaCG U1 intoanotherCG U2.Then preimageofanIFSG γ of U2 i.e. g 1(γ) isanIFSGof U1.
Theorem2.2. [66]Let g beasurjectivehomomorphismofaCG U1 toanotherCG U2.Then theimageofanIFSG γ of U1 i.e. g(γ) isanIFSGof U2
Theorem2.3. [31]ThehomomorphicimageofanyNSGisaNSG.
Theorem2.4. [31]ThehomomorphicpreimageofanyNSGisaNSG.
SomemorereferencesinthedomainsofFSG,IFSG,NSG,etc.,whichcanbehelpfulto variousotherresearchersare[67–71].
2.2. Plithogenicset&Plithogenichypersoftset
Definition2.7. [5]Let U beaUSand P ⊆ U. APSisdenotedas Ps =(P,ψ,Vψ ,a,c), where ψ beanattribute, Vψ istherespectiverangeofattributesvalues, a : P × Vψ → [0, 1]s isthe DAFand c : Vψ × Vψ → [0, 1]t isthecorrespondingDCF.Here s,t ∈{1, 2, 3}.
InDefinition2.7,for s =1and t =1 a willbecomeaFDAFand c willbecomeaFDCF.In general,weconsideronlyFDAFandFDCF.Also, ∀(ui,uj ) ∈ Vψ × Vψ ,c satisfies c(ui,ui)=0 and c(ui,uj )= c(uj ,ui).
Definition2.8. [36]Let U beaUS, VA beasetofattributevalues.Thentheorderedpair (Γ,U )iscalledaSSover U, whereΓ: VA → ρ(U ).
Gayenetal.,IntroductiontoPlithogenicHypersoftSubgroup
Definition2.9. [52]Let U beaUS.Let r1,r2,...,rn be n attributesandcorresponding attributevaluesetsarerespectively D1,D2,...,Dn (where Di ∩ Dj = φ,for i = j and i,j ∈ {1, 2,...,n}).Let Vψ = D1 × D2 ×···× Dn.Thentheorderedpair(Γ,Vψ )iscalledaHSof U, whereΓ: Vψ → ρ(U ).
Definition2.10. [72]AUS UC istermedasacrispUSif ∀u ∈ UC , u fullybelongsto UC i.e.membershipof u is1.
Definition2.11. [72]AUS UF istermedasafuzzyUSif ∀u ∈ UF , u partiallybelongsto UF i.e.membershipof u belongingto[0, 1].
Definition2.12. [72]AUS UIF istermedasanintuitionisticfuzzyUSif ∀u ∈ UIF , u partiallybelongsto UIF andalsopartiallydoesnotbelongto UIF i.e.membershipof u belongingto[0, 1] × [0, 1].
Definition2.13. [72]AUS UN istermedasanneutrosophicUSif ∀u ∈ UN , u hastruth belongingness,indeterminacybelongingness,andfalsitybelongingnessto UN i.e.membership of u belongingto[0, 1] × [0, 1] × [0, 1].
Definition2.14. [72]AUS UP overanattributevalueset ψ istermedasaplithogenicUSif ∀u ∈ UP , u belongsto UP withsomedegreeonthebasisofeachattributevalue.Thisdegree canbecrisp,fuzzy,intuitionisticfuzzy,orneutrosophic.
Definition2.15. [52]Let UC beacrispUSand ψ = {r1,r2,...,rn} beasetof n attributes withattributevaluesetsrespectivelyas D1,D2,...,Dn (where Di ∩ Dj = φ for i = j and i,j ∈{1, 2,...,n}).Also,let Vψ = D1 × D2 ×···× Dn .Then(Γ,Vψ ),whereΓ: Vψ → ρ(UC ) istermedasaCHSover UC . Definition2.16. [52]Let UF beafuzzyUSand ψ = {r1,r2,...,rn} beasetof n attributes withattributevaluesetsrespectivelyas D1,D2,...,Dn (where Di ∩ Dj = φ for i = j and i,j ∈{1, 2,...,n}).Also,let Vψ = D1 × D2 ×···× Dn .Then(Γ,Vψ ),whereΓ: Vψ → ρ(UF ) iscalledaFHSover UF Definition2.17. [52]Let UIF beanintuitionisticfuzzyUSand ψ = {r1,r2,...,rn} beaset of n attributeswithattributevaluesetsrespectivelyas D1,D2,...,Dn (where Di ∩ Dj = φ for i = j and i,j ∈{1, 2,...,n}).Also,let Vψ = D1 × D2 ×···× Dn .Then(Γ,Vψ ),where Γ: Vψ → ρ(UIF )iscalledanIFHSover UIF Definition2.18. [52]Let UN beaneutrosophicUSand ψ = {r1,r2,...,rn} beasetof n attributeswithattributevaluesetsrespectivelyas D1,D2,...,Dn (where Di ∩ Dj = φ for i = j and i,j ∈{1, 2,...,n}).Also,let Vψ = D1 ×D2 ×···×Dn .Then(Γ,Vψ ),whereΓ: Vψ → ρ(UN ) iscalledaNHSover UN Gayenetal.,IntroductiontoPlithogenicHypersoftSubgroup
Definition2.19. [52]Let UP beaplithogenicUSand ψ = {r1,r2,...,rn} beasetof n attributeswithattributevaluesetsrespectivelyas D1,D2,...,Dn (where Di ∩ Dj = φ for i = j and i,j ∈{1, 2,...,n}).Also,let Vψ = D1 ×D2 ×···×Dn .Then(Γ,Vψ ),whereΓ: Vψ → ρ(UP ) iscalledaPHSover UP .
Further,dependingonsomeonespreferencesPHScanbecategorizedasPCHS,PFHS, PIFHS,andPNHS.In[52],Smarandachehaswonderfullyintroducedandillustratedthese categorieswithproperexamples.
Inthenextsection,wehavementionedanequivalentstatementofDefinition2.19and describeditscategoriesinadifferentway.Also,wehavegivensomegraphicalrepresentationsof PCHS,PFHS,PIFHS,andPNHS.Again,wehaveintroducedfunctionsintheenvironmentsof PFHS,PIFHS,andPNHS.Furthermore,wehaveintroducedthenotionsofPFHSG,PIFHSG, andPNHSGandstudiedtheirhomomorphiccharacteristics.
3. ProposedNotions
AsanequivalentstatementtoDefinition2.19,wecanconcludethat ∀M ∈ range(Γ)and ∀i ∈{1, 2,...,n}, ∃ai : M × Di → [0, 1]s (s =1, 2or3)suchthat ∀(m,d) ∈ M × Di,ai(m,d) representtheDAFsof m totheset M onthebasisoftheattributevalue d.Thenthepair (Γ,Vψ )iscalledaPHS. So,basedonsomeonesrequirementonemaychoose s =1, 2or3andfurther,dependingon thesechoicesPHScanbecategorizedasPFHS,PIFHS,andPNHS.Also,bydefiningDAF as ai : M × Di →{0, 1}, thenotionofPCHScanbeintroduced.Thefollowingsarethose aforementionednotions: Let ψ = {r1,r2,...,rn} beasetof n attributesandcorrespondingattributevaluesetsare respectively D1,D2,...,Dn (where Di ∩ Dj = φ,for i = j and i,j ∈{1, 2,...,n}).Let Vψ = D1 × D2 ×···× Dn and(Γ,Vψ )beaHSover U, whereΓ: Vψ → ρ(U ).
Definition3.1. Thepair(Γ,Vψ )iscalledaPCHSif ∀M ∈ range(Γ)and ∀i ∈{1, 2,...,n} ∃aCi : M × Di →{0, 1} suchthat ∀(m,d) ∈ M × Di,aCi (m,d)=1.
AsetofallthePCHSsoveraset U willbedenotedasPCHS(U ).
Example3.2. Letaballoonsellerhasaset U = {b1,b2,...,b20} ofatotalof20balloons somewhichareofdifferentsize,color,andcost.Also,letfortheaforementionedattributes correspondingattributevaluesetsare D1 = {small,medium,large}, D2 = {red,orange,blue} and D3 = {small,medium,large}.Letapersoniswillingtobuysomeballoonshavingthe attributesasbig,redandexpensive.Letsassume(Γ,Vψ )beaHSover U, whereΓ: Vψ → ρ(U ) and Vψ = D1 × D2 × D3.Also,letΓ(big,red,expensive)= {b3,b10,b12} Gayenetal.,IntroductiontoPlithogenicHypersoftSubgroup
ThencorrespondingPCHSwillbeΓ(big,red,expensive)= {b3(1, 1, 1),b10(1, 1, 1),b12(1, 1, 1)}. ItsgraphicalrepresentationisshowninFigure1.
Figure1. PCHSaccordingtoExample3.2
Definition3.3. Thepair(Γ,Vψ )iscalledaPFHSif ∀M ∈ range(Γ)and ∀i ∈ {1, 2,...,n}, ∃aFi : M × Di → [0, 1]suchthat ∀(m,d) ∈ M × Di,aFi (m,d) ∈ [0, 1]. AsetofallthePFHSsoveraset U willbedenotedasPFHS(U ).
Example3.4. InExample3.2letcorrespondingPFHSisΓ(big,red,expensive)= {b3(0 75, 0 3, 0 8),b10(0 45, 0 57, 0 2),b12(0 15, 0 57, 0 95)}.Itsgraphicalrepresentationis showninFigure2.
Figure2. PFHSaccordingtoExample3.4
Gayenetal.,IntroductiontoPlithogenicHypersoftSubgroup
Definition3.5. Thepair(Γ,Vψ )iscalledaPIFHSif ∀M ∈ range(Γ)and ∀i ∈ {1, 2,...,n}, ∃aIF i : M ×Di → [0, 1]×[0, 1]suchthat ∀(m,d) ∈ M ×Di,aFi (m,d) ∈ [0, 1]×[0, 1].
AsetofallthePIFHSsoveraset U willbedenotedasPIFHS(U ).
Example3.6. InExample3.2letcorrespondingPIFHSis
Γ(big,red,expensive)= b3(0.87, 0.52, 0.66),b10(0.6, 0.52, 0.2),b12(0.33, 0.2, 0.83) b3(0.3, 0.4, 0.72),b10(0.5, 0.19, 0.98),b12(1, 0.72, 0.3)
ItsgraphicalrepresentationisshowninFigure3. Figure3. PIFHSaccordingtoExample3.6
Definition3.7. Thepair(Γ,Vψ )iscalledaPNHSif ∀M ∈ range(Γ)and ∀i ∈ {1, 2,...,n}, ∃aNi : M × Di → [0, 1] × [0, 1] × [0, 1]suchthat ∀(m,d) ∈ M × Di,aNi (m,d) ∈ [0, 1] × [0, 1] × [0, 1].
AsetofallthePNHSsoveraset
ItsgraphicalrepresentationisshowninFigure4. Gayenetal.,IntroductiontoPlithogenicHypersoftSubgroup
Figure4. PNHSaccordingtoExample3.8
3.1. Images&PreimagesofPFHS,PIFHS&PNHSunderafunction
Let U1 and U2 betwoCSsand ∀i,j ∈{1, 2,...,n}, Di and Pj areattributevaluesets consistingofsomeattributevalues.Again,let gij : U1 × Di → U2 × Pj aresomefunctions. Thenthefollowingscanbedefined: Definition3.9. Let(Γ1,V 1 ψ ) ∈ PFHS(U1)and(Γ2,V 2 ψ ) ∈ PFHS(U2),where V 1 ψ = D1 × D2 × ···× Dn and V 2 ψ = P1 × P2 ×···× Pn.Also,let ∀M ∈ range(Γ1), aFi : M × Di → [0, 1]arethe correspondingFDAFs.Again,let ∀N ∈ range(Γ2), bFj : N × Pj → [0, 1]arethecorresponding FDAFs.Thentheimagesof(Γ1,V 1 ψ )underthefunctions gij : U1 × Di → U2 × Pj arePFHS over U2 andtheyaredenotedas gij (Γ1,V 1 ψ ),wherethecorrespondingFDAFsaredefinedas: gij (aFi )(n,p)= max aFi (m,d)if(m,d) ∈ g 1 ij (n,p) 0otherwise Thepreimagesof(Γ2,V 2 ψ )underthefunctions gij : U1 ×Di → U2 ×Pj arePFHSsover U1,which aredenotedas g 1 ij (Γ2,V 2 ψ )andthecorrespondingFDAFsaredefinedas g 1 ij (bFj )(m,d)= bFj (gij (m,d)).
Definition3.10. Let(Γ1,V 1 ψ ) ∈ PIFHS(U1)and(Γ2,V 2 ψ ) ∈ PIFHS(U2),where V 1 ψ = D1 × D2 ×···× Dn and V 2 ψ = P1 × P2 ×···× Pn.Also,let ∀M ∈ range(Γ1), aIFi : Gayenetal.,IntroductiontoPlithogenicHypersoftSubgroup
M × Di → [0, 1] × [0, 1]with aIFi (m,d)= {((m,d),aT IFi (m,d),aF IFi (m,d)):(m,d) ∈ M × Di} arethecorrespondingIFDAFs.Again,let ∀N ∈ range(Γ2), bIFj : N × Pj → [0, 1] × [0, 1] with bIFj (n,p)= {((n,p),bT IFj (n,p),bF IFj (n,p)):(n,p) ∈ N × Pj } arethecorresponding IFDAFs.Thentheimagesof(Γ1,V 1 ψ )underthefunctions gij : U1 × Di → U2 × Pj are PIFHSover U2,whicharedenotedas gij (Γ1,V 1 ψ )andthecorrespondingIFDAFsaredefined as: gij (aIFi )(n,p)=(gij (aT IFi )(n,p),gij (aF IFi )(n,p)),where gij (a T IFi )(n,p)= max a T IFi (m,d)if(m,d) ∈ g 1 ij (n,p) 0otherwise and gij (a F IFi )(n,p)= min a F IFi (m,d)if(m,d) ∈ g 1 ij (n,p) 1otherwise
Thepreimagesof(Γ2,V 2 ψ )underthefunctions gij : U1 × Di → U2 × Pj arePIFHSsover U1, whicharedenotedas g 1 ij (Γ2,V 2 ψ )andthecorrespondingIFDAFsaredefinedas g 1 ij (bIFj )(m,d)=(g 1 ij (bT IFj )(m,d),g 1 ij (bF IFj )(m,d)),where g 1 ij (bT IFj )(m,d)= bT IFj (gij (m,d)) and g 1 ij (bF IFj )(m,d)= bF IFj (gij (m,d)) Definition3.11. Let(Γ1,V 1 ψ ) ∈ PNHS(U1)and(Γ2,V 2 ψ ) ∈ PNHS(U2),where V 1 ψ = D1 ×D2 × ···×Dn and V 2 ψ = P1×P2×···×Pn.Also,let ∀M ∈ range(Γ1), aNi : M ×Di → [0, 1]×[0, 1]×[0, 1] with aNi (m,d)= {((m,d),aT Ni (m,d),aI Ni (m,d),aF Ni (m,d)):(m,d) ∈ M × Di} arethecorrespondingNDAFs.Again,let ∀N ∈ range(Γ2), bNj : N × Pj → [0, 1] × [0, 1] × [0, 1]with bNj (n,p)= {((n,p),bT Ni (n,p),bI Ni (n,p),bF Ni (n,p)):(n,p) ∈ N × Pi} arethecorresponding NDAFs.Thentheimagesof(Γ1,V 1 ψ )underthefunctions gij : U1 × Di → U2 × Pj arePNHS over U2,whicharedenotedas gij (Γ1,V 1 ψ )andthecorrespondingNDAFsaredefinedas: gij (aNi )(n,p)=(gij (aT Ni )(n,p),gij (aI Ni )(n,p),gij (aF Ni )(n,p)),where gij (a T Ni )(n,p)= max a T Ni (m,d)if(m,d) ∈ g 1 ij (n,p) 0otherwise , gij (a I Ni )(n,p)= max a I Ni (m,d)if(m,d) ∈ g 1 ij (n,p) 0otherwise , and gij (a F Ni )(n,p)= min a F Ni (m,d)if(m,d) ∈ g 1 ij (n,p) 1otherwise
Gayenetal.,IntroductiontoPlithogenicHypersoftSubgroup
Thepreimagesof(Γ2,V 2 ψ )underthefunctions gij : U1 × Di → U2 × Pj arePNHSover U1, whicharedenotedas g 1 ij (Γ2,V 2 ψ )andthecorrespondingNDAFsaredefinedas g 1 ij (bNj )(m,d)=(g 1 ij (bT Nj )(m,d),g 1 ij (bI Nj )(m,d),g 1 ij (bF Nj )(m,d)),where g 1 ij (bT Nj )(m,d)= bT Nj (gij (m,d)), g 1 ij (bI Nj )(m,d)= bI Nj (gij (m,d))and g 1 ij (bF Nj )(m,d)= bF Nj (gij (m,d)).
Inthenextsegment,wehavedefinedplithogenichypersoftsubgroupsinfuzzy,intuitionistic fuzzy,andneutrosophicenvironments.Wehavealso,analyzedtheirhomomorphicproperties.
3.2. PlithogenicHypersoftSubgroup
3.2.1. PlithogenicFuzzyHypersoftSubgroup
Definition3.12. Letthepair(Γ,Vψ )beaPFHSofaCG U ,where Vψ = D1 × D2 ×···× Dn and ∀i ∈{1, 2,...,n},Di areCGs.Then(Γ,Vψ )iscalledaPFHSGof U ifandonlyif ∀M ∈ range(Γ), ∀(m1,d), (m2,d ) ∈ M × Di and ∀aFi : M × Di → [0, 1],theconditionsmentioned belowaresatisfied:
(i) aFi ((m1,d) · (m2,d )) ≥ min{aFi (m1,d),aFi (m2,d )} and (ii) aFi (m1,d) 1 ≥ aFi (m1,d).
AsetofallPFHSGofaCG U isdenotedasPFHSG(U ).
Example3.13. Let U = {e,m,u,mu} betheKleins4-groupand ψ = {r1,r2} isasetof twoattributesandcorrespondingattributevaluesetsarerespectively, D1 = {1,i, 1, i} and D2 = {1,w,w2},whicharetwocyclicgroups.Let Vψ = D1 × D2 and(Γ,Vψ )beaHSover U, whereΓ: Vψ → ρ(U )suchthattherangeofΓi.e.R(Γ)= {{e,m}, {e,u}, {e,mu}}.Letfor M = {e,m}, aF1 : M × D1 → [0, 1]isdefinedinTable2and aF2 : M × D2 → [0, 1]isdefined inTable3respectively.
Table2. Membershipvaluesof aF1 aF1 1 i 1 i e 0 4 0 2 0 4 0 2 m 0.2 0.2 0.2 0.2
Table3. Membershipvaluesof aF2 aF2 1 w w2 e 0 8 0 5 0 5 m 0.6 0.5 0.5
Letfor M = {e,u}, aF1 : M × D1 → [0, 1]isdefinedinTable4and aF2 : M × D2 → [0, 1]is definedinTable5respectively.
Gayenetal.,IntroductiontoPlithogenicHypersoftSubgroup
Table4. Membershipvaluesof aF1 aF1 1 i 1 i e 0.8 0.2 0.7 0.2 u 0 5 0 2 0 5 0 2
Table5. Membershipvaluesof aF2 aF2 1 w w2 e 0.7 0.4 0.4 u 0 3 0 3 0 3
Letfor M = {e,mu}, aF1 : M × D1 → [0, 1]isdefinedinTable6and aF2 : M × D2 → [0, 1]is definedinTable7respectively.
Table6. Membershipvaluesof aF2 aF1 1 i 1 i e 0 9 0 2 0 4 0 2 mu 0 7 0 2 0 7 0 2
Table7. Membershipvaluesof aF1 aF2 1 w w2 e 1 0 3 0 3 mu 0 2 0 2 0 2
Here,forany M ∈ range(Γ)and ∀i ∈{1, 2},aFi satisfyDefinition3.12.Hence,(Γ,Vψ ) ∈ PFHSG(U ).
Proposition3.1. Let U beaCGand (Γ,Vψ ) ∈ PFHSG(U ),where Vψ = D1 × D2 ×···× Dn and ∀i ∈{1, 2,...,n},Di areCGs.Thenforany M ∈ range(Γ), ∀(m,d) ∈ M × Di and ∀aFi : M × Di → [0, 1],thefollowingsaresatisfied:
(i) aFi (e,di e) ≥ aFi (m,d),where e and di e aretheneutralelementsof U and Di (ii) aFi (m,d) 1 = aFi (m,d)
Proof. (i)Let e and di e betheneutralelementsof U and Di.Then ∀(m,d) ∈ M × Di, aFi (e,di e)= aFi ((m,d) (m,d) 1),
≥ min{aFi (m,d),aFi (m,d) 1} (byDefinition3.12)
≥ min{aFi (m,d),aFi (m,d)} (byDefinition3.12) ≥ aFi (m,d)
(ii)Let U beagroupand(Γ,Vψ ) ∈ PFHSG(U ).ThenbyDefinition3.12, aFi (m,d) 1 ≥ aFi (m,d)(3.1)
Again, aFi (m,d)= aFi ((m,d) 1) 1 ≥ aFi (m,d) 1 (3.2)
Hence,fromEquation3.1andEquation3.2, aFi (m,d) 1 = aFi (m,d).
Gayenetal.,IntroductiontoPlithogenicHypersoftSubgroup
Proposition3.2. Letthepair (Γ,Vψ ) beaPFHSofaCG U ,where Vψ = D1 × D2 ×···× Dn and ∀i ∈{1, 2,...,n},Di areCGs.Then (Γ,Vψ ) iscalledaPFHSGof U ifandonlyif ∀M ∈ range(Γ), ∀(m1,d), (m2,d ) ∈ M × Di and ∀aFi : M × Di → [0, 1], aFi ((m1,d) (m2,d ) 1) ≥ min{aFi (m1,d),aFi (m2,d )}.
Proof. Let U beaCGand(Γ,Vψ ) ∈ PFHSG(U ).ThenbyDefinition3.12andProposition3.1
aFi ((m1,d) · (m2,d ) 1) ≥ min{aFi (m1,d),aFi (m2,d ) 1} =min{aFi (m1,d),aFi (m2,d )}
Conversely,let aFi ((m1,d) (m2,d ) 1) ≥ min{aFi (m1,d),aFi (m2,d )}.Also,let e and di e be theneutralelementsof U and Di.Then,
aFi (m,d) 1 = aFi ((e,di e) · (m,d) 1)
≥ min{aFi (e,di e),aFi (m,d)} =min{aFi ((m,d) (m,d) 1),aFi (m,d)} ≥ min{aFi (m,d),aFi (m,d),aFi (m,d)} = aFi (m,d)(3.3)
Now, aFi ((m1,d) (m2,d ))= aFi ((m1,d) ((m2,d ) 1) 1 ) ≥ min{aFi (m1,d),aFi (m2,d ) 1} =min{aFi (m1,d),aFi (m2,d )} (byEquation3.3)(3.4)
Hence,byEquation3.3andEquation3.4,(Γ,Vψ ) ∈ PFHSG(U ).
Proposition3.3. IntersectionoftwoPFHSGsisalsoaPFHSG.
Theorem3.4. ThehomomorphicimageofaPFHSGisaPFHSG.
Proof. Let U1 and U2 betwoCGsand ∀i,j ∈{1, 2,...,n},Di and Pj areattributevaluesets consistingofsomeattributevaluesandlet gij : U1 × Di → U2 × Pj arehomomorphisms.Also, let(Γ1,V 1 ψ ) ∈ PFHSG(U1),where V 1 ψ = D1 × D2 ×···× Dn.Again,let ∀M ∈ range(Γ1), aFi : M × Di → [0, 1]arethecorrespondingFDAFs. Assuming(n1,p1), (n2,p2) ∈ U2 ×Pj ,if g 1 ij (n1,p1)= φ and g 1 ij (n2,p2)= φ,then gij (Γ1,V 1 ψ ) ∈ PFHSG(U2).
Letsassumethat ∃(m1,d1), (m2,d2) ∈ U1 × Di suchthat gij (m1,d1)=(n1,p1)and Gayenetal.,IntroductiontoPlithogenicHypersoftSubgroup
gij (m2,d2)=(n2,p2).Then gij (aFi )(n1,p1) (n2,p2) 1 =max (n1 ,p1 ) (n2 ,p2 ) 1 =gij (m,d) aFi (m,d)
≥ aFi (m1,d1) · (m2,d2) 1
≥ min{aFi (m1,d1),aFi (m2,d2)} (as(Γ1,V 1 ψ ) ∈ PFHSG(U1))
≥ min{ max (n1 ,p1 )=gij (m1 ,d1 ) aFi (m1,d1), max (n2 ,p2 )=gij (m2 ,d2 ) aFi (m2,d2)}
≥ min{gij (aFi )(n1,p1),gij (aFi )(n2,p2)}
Hence, gij (Γ1,V 1 ψ ) ∈ PFHSG(U2).
Theorem3.5. ThehomomorphicpreimageofaPFHSGisaPFHSG.
Proof. Let U1 and U2 betwoCGsand ∀i,j ∈{1, 2,...,n},Di and Pj areattributevaluesets consistingofsomeattributevaluesandlet gij : U1×Di → U2×Pj arehomomorphisms.Also,let (Γ2,V 2 ψ ) ∈ PFHSG(U2), V 2 ψ = P1×P2×···×Pn.Again, ∀N ∈ range(Γ2), bFj : N ×Pj → [0, 1]are thecorrespondingFDAFs.Letsassume(m1,d1), (m2,d2) ∈ U1 ×Di.As gij isahomomorphism thefollowingscanbeconcluded:
g 1 ij (bFi )(m1,d1) · (m2,d2) 1
= bFi (gij ((m1,d1) · (m2,d2) 1))
= bFi (gij (m1,d1) · gij (m2,d2) 1)(As gij isahomomorphism)
≥ min{bFi (gij (m1,d1)),bFi (gij (m2,d2))} (As(Γ2,V 2 ψ ) ∈ PFHSG(U2))
≥ min{g 1 ij (bFi )(m1,d1),g 1 ij (bFi )(m2,d2)} Then g 1 ij (Γ2,V 2 ψ ) ∈ PFHSG(U1).
3.2.2. PlithogenicIntuitionisticFuzzyHypersoftSubgroup
Definition3.14. Letthepair(Γ,Vψ )beaPIFHSofaCG U ,where Vψ = D1 × D2 ×···× Dn and ∀i ∈{1, 2,...,n},Di areCGs.Then(Γ,Vψ )iscalledaPIFHSGof U ifandonlyif ∀M ∈ range(Γ), ∀(m1,d), (m2,d ) ∈ M × Di and ∀aIFi : M × Di → [0, 1] × [0, 1]with aIFi (m,d)= {((m,d),aT IFi (m,d),aF IFi (m,d)):(m,d) ∈ M × Di},thesubsequentconditionsarefulfilled:
(i) aT IFi ((m1,d) · (m2,d )) ≥ min{aT IFi (m1,d),aT IFi (m2,d )} (ii) aT IFi (m1,d) 1 ≥ aT IFi (m1,d) (iii) aF IFi ((m1,d) (m2,d )) ≤ max{aF IFi (m1,d),aF IFi (m2,d )} (iv) aF IFi (m1,d) 1 ≤ aF IFi (m1,d)
AsetofallPIFHSGofaCG U isdenotedasPIFHSG(U ). Gayenetal.,IntroductiontoPlithogenicHypersoftSubgroup
Example3.15. Let U = S3 beaCGand ψ = {r1,r2} isasetoftwoattributesandcorrespondingattributevaluesetsarerespectively, D1 = A3 and D2 = S2,whicharerespectivelyanalternatinggroupoforder3andasymmetricgroupoforder2.Let Vψ = D1 × D2 and(Γ,Vψ )bea HSover U, whereΓ: Vψ → ρ(U )suchthattherangeofΓi.e.R(Γ)= {{(1), (13)}, {(1), (23)}}. Letfor M = {(1), (13)}, aIF1 : M × D1 → [0, 1] × [0, 1]isdefinedinTable8–9and aIF2 : M × D2 → [0, 1] × [0, 1]isdefinedinTable10–11respectively.
Table8. Membershipvaluesof aIF1 aT IF1 (1) (123) (132) (1) 0.4 0.5 0.5 (13) 0.2 0.2 0.2
Table10. Membershipvaluesof aIF2 aT IF2 (1) (12) (1) 0.8 0.4 (13) 0 3 0 3
Table9. Non-membership valuesof aIF1 aF IF1 (1) (123) (132) (1) 0.4 0.7 0.7 (13) 0 8 0 8 0 8
Table11. Non-membership valuesof aIF2 aF IF2 (1) (12) (1) 0.4 0.8 (13) 0 9 0 9
Letfor M = {(1), (23)} aIF1 : M × D1 → [0, 1] × [0, 1]isdefinedinTable12–13and aIF2 : M × D2 → [0, 1] × [0, 1]isdefinedinTable14–15respectively.
Table12. Membershipvaluesof aIF1 aT IF1 (1) (123) (132) (1) 0 6 0 4 0 4 (23) 0 5 0 4 0 4
Table13. Non-membership valuesof aIF1 aF IF1 (1) (123) (132) (1) 0 4 0 7 0 7 (23) 0 6 0 7 0 7
Table15. Non-membership valuesof aIF2 aF IF2 (1) (12) (1) 0 5 0 9 (23) 0 8 0 9 Here,forany M ∈ range(Γ)and ∀i ∈{1, 2},aIF i satisfyDefinition3.14.Hence,(Γ,Vψ ) ∈ PIFHSG(U ).
Table14. Membershipvaluesof aIF2 aT IF2 (1) (12) (1) 0 7 0 6 (23) 0 7 0 6
Gayenetal.,IntroductiontoPlithogenicHypersoftSubgroup
Proposition3.6. Let U beaCGand (Γ,Vψ ) ∈ PIFHSG(U ),where Vψ = D1 × D2 ×···× Dn and ∀i ∈{1, 2,...,n},Di areCGs.Thenforany M ∈ range(Γ) and ∀(m,di) ∈ M × Di and ∀aIFi : M × Di → [0, 1] × [0, 1] with aIFi (m,d)= {((m,d),aT IFi (m,d),aF IFi (m,d)):(m,d) ∈ M × Di},thesubsequentconditionsaresatisfied:
(i) aT IFi (e,di e) ≥ aT IFi (m,d),where e and di e aretheneutralelementsof U and Di (ii) aT IFi (m,d) 1 = aT IFi (m,d) (iii) aF IFi (e,di e) ≤ aF IFi (m,d),where e and di e aretheneutralelementsof U and Di (iv) aF IFi (m,d) 1 = aF IFi (m,d)
Proof. Here,(i)and(ii)canbeeasilyprovedusingProposition3.1.
(iii)Let e and di e betheneutralelementsof U and Di.Then ∀(m,d) ∈ M × Di,
a F IFi (e,di e)= a F IFi ((m,d) · (m,d) 1)
≤ max{a F IFi (m,d),a F IFi (m,d) 1} (byDefinition3.14)
≤ max{a F IFi (m,d),a F IFi (m,d)} (byDefinition3.14)
≤ a F IFi (m,d) (iv)Let U beaCGand(Γ,Vψ ) ∈ PFHSG(U).ThenbyDefinition3.14, a F IFi (m,d) 1 ≤ a F IFi (m,d)(3.5) Again, a F IFi (m,d)= a F IFi ((m,d) 1) 1 ≤ a F IFi (m,d) 1 (3.6) Hence,byEquation3.5andEquation3.6, aF IFi (m,d) 1 = aF IFi (m,d)
Proposition3.7. Letthepair (Γ,Vψ ) beaPIFHSofaCG U ,where Vψ = D1 × D2 ×···× Dn and ∀i ∈{1, 2,...,n},Di areCGs.Then (Γ,Vψ ) iscalledaPIFHSGof U ifandonlyif ∀M ∈ range(Γ), ∀(m1,d), (m2,d ) ∈ M × Di and ∀aIFi : M × Di → [0, 1] × [0, 1] with aIFi (m,d)= {((m,d),aT IFi (m,d),aF IFi (m,d)):(m,d) ∈ M × Di},thesubsequentconditionsarefulfilled: (i) aT IFi ((m1,d) · (m2,d ) 1) ≥ min{aT IFi (m1,d),aT IFi (m2,d )} and (ii) aF IFi ((m1,d) · (m2,d ) 1) ≤ max{aF IFi (m1,d),aF IFi (m2,d )}
Proof. Here,(i)canbeprovedusingProposition3.2. (ii)Let U beaCGand(Γ,Vψ ) ∈ PFHSG(U ).ThenbyDefinition3.14andProposition3.6 a F IFi ((m1,d) (m2,d ) 1) ≤ max{a F IFi (m1,d),a F IFi (m2,d ) 1} ≤ max{a F IFi (m1,d),a F IFi (m2,d )}
Conversely,let aF IFi ((m1,d) (m2,d ) 1) ≤ max{aF IFi (m1,d),aF IFi (m2,d )}.Also,let e and di e betheneutralelementsof U and Di.Then a F IFi (m,d) 1 = a F IFi ((e,di e) (m,d) 1)
≤ max{a F IFi (e,di e),a F IFi (m,d)}
≤ max{a F IFi ((m,d) (m,d) 1),a F IFi (m,d)}
≤ max{a F IFi (m,d),a F IFi (m,d),a F IFi (m,d)}
= a F IFi (m,d)(3.7)
Now, a F IFi ((m1,d) · (m2,d ))= a F IFi ((m1,d) · ((m2,d ) 1) 1 )
≤ max{a F IFi (m1,d),a F IFi (m2,d ) 1} =max{a F IFi (m1,d),a F IFi (m2,d )} (byEquation3.7)(3.8)
Hence,byEquation3.7andEquation3.8,(Γ,Vψ ) ∈ PFHSG(U ).
Proposition3.8. IntersectionoftwoPIFHSGsisalsoaPIFHSG.
Theorem3.9. ThehomomorphicimageofaPIFHSGisaPIFHSG.
Proof. Let U1 and U2 betwoCGsand ∀i,j ∈{1, 2,...,n},Di and Pj areattributevaluesets consistingofsomeattributevaluesandlet gij : U1 × Di → U2 × Pj arehomomorphisms. Also,let(Γ1,V 1 ψ ) ∈ PIFHSG(U1),where V 1 ψ = D1 × D2 ×···× Dn.Again,let ∀M ∈ range(Γ1)and aIFi : M ×Di → [0, 1]×[0, 1]with aIFi (m,d)= {((m,d),aT IFi (m,d),aF IFi (m,d)): (m,d) ∈ M × Di} arethecorrespondingIFDAFs.Assuming(n1,p1), (n2,p2) ∈ U2 × Pj ,if g 1 ij (n1,p1)= φ and g 1 ij (n2,p2)= φ,then gij (Γ1,V 1 ψ ) ∈ PIFHSG(U2).Letsassumethat ∃(m1,d1), (m2,d2) ∈ U1 × Di suchthat gij (m1,d1)=(n1,p1)and gij (m2,d2)=(n2,p2).Then byTheorem3.4
gij (a T IFi )(n1,p1) (n2,p2) 1 ≥ min{gij (a T IFi )(n1,p1),gij (a T IFi )(n2,p2)} Again, gij (a F IFi )(n1,p1) · (n2,p2) 1 =min (n1 ,p1 ) (n2 ,p2 ) 1 =gij (m,d) a F IFi (m,d)
≤ a F IFi (m1,d1) (m2,d2) 1
≤ max{a F IFi (m1,d1),a F IFi (m2,d2)} (as(Γ1,V 1 ψ ) ∈ PIFHSG(U1))
≤ max{ min (n1 ,p1 )=gij (m1 ,d1 ) a F IFi (m1,d1), min (n2 ,p2 )=gij (m2 ,d2 ) a F IFi (m2,d2)}
≤ max{gij (a F IFi )(n1,p1),gij (a F IFi )(n2,p2)} (byDefinition3.10)
Gayenetal.,IntroductiontoPlithogenicHypersoftSubgroup
Hence, gij (Γ1,V 1 ψ ) ∈ PIFHSG(U2).
Theorem3.10. ThehomomorphicpreimageofaPIFHSGisaPIFHSG.
Proof. Let U1 and U2 betwoCGsand ∀i,j ∈{1, 2,...,n}, Di and Pj areattributevaluesets consistingofsomeattributevaluesandlet gij : U1 × Di → U2 × Pj arehomomorphisms.Also, let(Γ2,V 2 ψ ) ∈ PIFHS(U2),where V 2 ψ = P1 × P2 ×···× Pn.Again,let ∀N ∈ range(Γ2),bIFj : N × Pj → [0, 1] × [0, 1]with bIFj (n,p)= {((n,p),bT IFj (n,p),bF IFj (n,p)):(n,p) ∈ N × Pj } arethecorrespondingIFDAFs.Letsassume(m1,d1), (m2,d2) ∈ U1 × Di.Since, gij isa homomorphism,byTheorem3.5 g 1 ij (bT IFj )(m1,d1) (m2,d2) 1 ≥ min{g 1 ij (bT IFj )(m1,d1),g 1 ij (bT IFj )(m2,d2)}.
Again, g 1 ij (bF IFj )(m1,d1) · (m2,d2) 1 = bF IFj (gij ((m1,d1) · (m2,d2) 1))
= bF IFj (gij (m1,d1) gij (m2,d2) 1)(As gij isahomomorphism)
≤ max{bF IFj (gij (m1,d1)),bF IFj (gij (m2,d2))} (As(Γ2,V 2 ψ ) ∈ PIFHSG(U2))
≤ max{g 1 ij (bF IFj )(m1,d1),g 1 ij (bF IFj )(m2,d2)}
Hence, g 1 ij (Γ2,V 2 ψ ) ∈ PIFHSG(U1).
3.2.3. PlithogenicNeutrosophicHypersoftSubgroup
Definition3.16. content.Letthepair(Γ,Vψ )beaPNHSofaCG U ,where Vψ = D1 × D2 × ···× Dn and ∀i ∈{1, 2,...,n},Di areCGs.Then(Γ,Vψ )iscalledaPNHSGof U ifandonly if ∀M ∈ range(Γ), ∀(m1,d), (m2,d ) ∈ M × Di and ∀aNi : M × Di → [0, 1] × [0, 1] × [0, 1], with aNi (m,d)= {((m,d),aT Ni (m,d),aI Ni (m,d),aF Ni (m,d)):(m,d) ∈ M × Di},thesubsequent conditionsarefulfilled:
(i) aT Ni ((m1,d) · (m2,d ) 1) ≥ min{aT Ni (m1,d),aT Ni (m2,d )} (ii) aT Ni (m1,d) 1 ≥ aT Ni (m1,d) (iii) aI Ni ((m1,d) (m2,d ) 1) ≥ min{aI Ni (m1,d),aI Ni (m2,d )} (iv) aI Ni (m1,d) 1 ≥ aI Ni (m1,d) (v) aF Ni ((m1,d) (m2,d ) 1) ≤ max{aF Ni (m1,d),aF Ni (m2,d )} (vi) aF Ni (m1,d) 1 ≤ aF Ni (m1,d)
AsetofallPNHSGofaCG U isdenotedasPNHSG(U ).
Gayenetal.,IntroductiontoPlithogenicHypersoftSubgroup
Example3.17. Let D6 = {e,m,u,mu,um,mum} beadihedralgroupoforder6and ψ = {r1,r2} isasetoftwoattributesandcorrespondingattributevaluesetsarerespectively, D1 = {1,w,w2} and D2 = A3,whicharerespectivelyacyclicgroupoforder3andanalternating groupoforder3.Let Vψ = D1 × D2 and(Γ,Vψ )beaHSover U, whereΓ: Vψ → ρ(U )such thattherangeofΓi.e.R(Γ)= {{e,mu,um}, {e,mum}} Letfor M = {e,mu},aN1 : M × D1 → [0, 1] × [0, 1] × [0, 1]isdefinedinTable16–18and aN2 : M × D2 → [0, 1] × [0, 1] × [0, 1]isdefinedinTable19–21respectively.
Table16. Truthvaluesof aN1 aT N1 1 w w2 e 0.7 0.5 0.5 mu 0 3 0 3 0 3 um 0 3 0 3 0 3
Table18. Falsityvaluesof aN1 aF N1 1 w w2 e 0 3 0 5 0 5 mu 0.7 0.7 0.7 um 0.7 0.7 0.7
Table17. Indeterminacyvaluesof aN1 aI N1 1 w w2 e 0 8 0 4 0 4 mu 0.5 0.4 0.4 um 0.5 0.4 0.4
Table19. Truthvaluesof aN2 aT N2 (1) (123) (132) e 0.7 0.2 0.2 mu 0 1 0 1 0 1 um 0 1 0 1 0 1 Table20. Indeterminacyvaluesof aN2 aI N2 (1) (123) (132) e 0 8 0 5 0 5 mu 0.8 0.5 0.5 um 0.8 0.5 0.5
Table21. Falsityvaluesof aN2 aF N2 (1) (123) (132) e 0.3 0.8 0.8 mu 0 9 0 9 0 9 um 0 9 0 9 0 9
Letfor M = {e,mum},aN1 : M × D1 → [0, 1] × [0, 1] × [0, 1]isdefinedinTable22–24and aN2 : M × D2 → [0, 1] × [0, 1] × [0, 1]isdefinedinTable25–27respectively.
Table22. Truthvaluesof aN1 aT N1 1 w w2 e 0 8 0 4 0 4 mum 0.2 0.2 0.2
Table23. Indeterminacyvaluesof aN1 aI N1 1 w w2 e 0.8 0.6 0.6 mum 0 7 0 6 0 6
Table24. Falsityvaluesof aN1 aF N1 1 w w2 e 0 2 0 6 0 6 mum 0.8 0.8 0.8
Table26. Indeterminacyvaluesof aN2 aI N2 (1) (123) (132) e 0 5 0 2 0 2 mum 0 1 0 1 0 1
Table25. Truthvaluesof aN2 aT N2 (1) (123) (132) e 0.9 0.8 0.8 mum 0 9 0 8 0 8
Table27. Falsityvaluesof aN2 aF N2 (1) (123) (132) e 0.1 0.2 0.2 mum 0 1 0 2 0 2
Here,forany M ∈ range(Γ)and ∀i ∈{1, 2},aNi satisfyDefinition3.16.Hence,(Γ,Vψ ) ∈ PNHSG(U ).
Proposition3.11. Letthepair (Γ,Vψ ) beaPNHSofaCG U ,where Vψ = D1 × D2 ×···× Dn and ∀i ∈{1, 2,...,n},Di areCGs.Then (Γ,Vψ ) iscalledaPNHSGof U ifandonlyif ∀M ∈ range(Γ), ∀(m1,d), (m2,d ) ∈ M × Di and ∀aNi : M × Di → [0, 1] × [0, 1] × [0, 1],with aNi (m,d)= {((m,d),aT Ni (m,d),aI Ni (m,d),aF Ni (m,d)):(m,d) ∈ M ×Di}.Thenthesubsequent conditionsaresatisfied:
(i) aT Ni (e,d) ≥ aT Ni (m,d),where e istheneutralelementof U (ii) aT Ni (m,d) 1 = aT Ni (m,d) (iii) aI Ni (e,d) ≥ aI Ni (m,d),where e istheneutralelementof U . (iv) aI Ni (m,d) 1 = aI Ni (m,d) (v) aF Ni (e,d) ≤ aF Ni (m,d),where e istheneutralelementof U . (vi) aF Ni (m,d) 1 = aF Ni (m,d)
Proof. ThiscanbeprovedusingProposition3.1andProposition3.6.
Proposition3.12. Letthepair (Γ,Vψ ) beaPNHSofaCG U ,where Vψ = D1 × D2 ×···× Dn and ∀i ∈{1, 2,...,n},Di areCGs.Then (Γ,Vψ ) iscalledaPNHSGof U ifandonlyif ∀M ∈ range(Γ), ∀(m1,d), (m2,d ) ∈ M × Di and ∀aNi : M × Di → [0, 1] × [0, 1] × [0, 1],with aNi (m,d)= {((m,d),aT Ni (m,d),aI Ni (m,d),aF Ni (m,d)):(m,d) ∈ M ×Di}.Thenthesubsequent conditionsarefulfilled:
(i) aT Ni ((m1,d) · (m2,d ) 1) ≥ min{aT Ni (m1,d),aT Ni (m2,d )} (ii) aI Ni ((m1,d) (m2,d ) 1) ≥ min{aI Ni (m1,d),aI Ni (m2,d )} (iii) aF Ni ((m1,d) (m2,d ) 1) ≤ max{aF Ni (m1,d),aF Ni (m2,d )}
Gayenetal.,IntroductiontoPlithogenicHypersoftSubgroup
Proof. ThiscanbeprovedusingProposition3.2andProposition3.7.
Proposition3.13. IntersectionoftwoPNHSGsisalsoaPNHSG. Theorem3.14. ThehomomorphicimageofaPNHSGisaPNHSG.
Proof. Let U1 and U2 betwoCGsand ∀i,j ∈{1, 2,...,n},Di and Pj areattributevaluesets consistingofsomeattributevaluesandlet gij : U1 × Di → U2 × Pj arehomomorphisms.Also, let(Γ1,V 1 ψ ) ∈ PNHSG(U1),where V 1 ψ = D1 × D2 ×···× Dn.Again,let ∀M ∈ range(Γ1),aNi : M × Di → [0, 1] × [0, 1] × [0, 1]with aNi (m,d)= {((m,d),aT Ni (m,d),aI Ni (m,d),aF Ni (m,d)): (m,d) ∈ M × Di} arethecorrespondingNDAFs. Assuming(n1,p1), (n2,p2) ∈ U2 × Pj ,if g 1 ij (n1,p1)= φ and g 1 ij (n2,p2)= φ then gij (Γ1,V 1 ψ ) ∈ NHSG(U2).Letsassumethat ∃(m1,d1), (m2,d2) ∈ U1 ×Di suchthat gij (m1,d1)=(n1,p1)and gij (m2,d2)=(n2,p2).ThenbyTheorem3.4andTheorem3.9,wecanprovethefollowings: gij (a T Ni )(n1,p1) · (n2,p2) 1 ≥ min{gij (a T Ni )(n1,p1),gij (a T Ni )(n2,p2)}, gij (a I Ni )(n1,p1) · (n2,p2) 1 ≥ min{gij (a I Ni )(n1,p1),gij (a I Ni )(n2,p2)}, and gij (a F Ni )(n1,p1) (n2,p2) 1 ≤ max{gij (a F Ni )(n1,p1),gij (a F Ni )(n2,p2)}. Hence, gij (Γ1,V 1 ψ ) ∈ PNHSG(U2).
Theorem3.15. ThehomomorphicpreimageofaPNHSGisaPNHSG.
Proof. Let U1 and U2 betwoCGsand ∀i,j ∈{1, 2,...,n}Di and Pj areattributevaluesets consistingofsomeattributevaluesandlet gij : U1 × Di → U2 × Pj arehomomorphisms.Also, let(Γ2,V 2 ψ ) ∈ PNHSG(U2),where V 2 ψ = P1 × P2 ×···× Pn.Again,let ∀N ∈ range(Γ2),bNj : N × Pj → [0, 1] × [0, 1] × [0, 1]with bNj (n,p)= {((n,p),bT Nj (n,p),bI Nj (n,p),bF Nj (n,p)):(n,p) ∈ N × Pj } arethecorrespondingIFDAFs.Letsassume(m1,d1), (m2,d2) ∈ U1 × Di.Since gij isahomomorphismbyTheorem3.5andTheorem3.10,thefollowingscanbeproved: g 1 ij (bT Nj )(m1,d1) · (m2,d2) 1 ≥ min{g 1 ij (bT Nj )(m1,d1),g 1 ij (bT Nj )(m2,d2)}, g 1 ij (bI Nj )(m1,d1) · (m2,d2) 1 ≥ min{g 1 ij (bI Nj )(m1,d1),g 1 ij (bI Nj )(m2,d2)}, and g 1 ij (bF Nj )(m1,d1) (m2,d2) 1 ≤ max{g 1 ij (bF Nj )(m1,d1),g 1 ij (bF Nj )(m2,d2)} Hence, g 1 ij (Γ2,V 2 ψ ) ∈ PNHSG(U1).
Gayenetal.,IntroductiontoPlithogenicHypersoftSubgroup
4. Conclusions
Hypersoftsettheoryismoregeneralthansoftsettheoryandithasahugeareaofapplications.Thatiswhywehaveadoptedandimplementeditinplithogenicenvironmentsothatwe canintroducevariousalgebraicstructures.Becauseofthis,thenotionsofplithogenichypersoftsubgroupshavebecomegeneralthanfuzzy,intuitionisticfuzzy,neutrosophicsubgroups, andplithogenicsubgroups.Again,wehaveintroducedfunctionsindifferentplithogenichypersoftenvironments.Hence,homomorphismcanbeintroducedanditseffectsonthesenewly definedplithogenichypersoftsubgroupscanbestudied.Inthefuture,toextendthisstudy onemayintroducegeneralT-normandT-conormandfurthergeneralizeplithogenichypersoft subgroups.Also,onemayextendthesenotionsbyintroducingdifferentnormalversionsof plithogenichypersoftsubgroupsandbystudyingtheeffectsofhomomorphismonthem.
References
1. L.A.Zadeh.Fuzzysets. InformationandControl,8(3):338–358,1965.
2. K.T.Atanassov.Intuitionisticfuzzysets. FuzzySetsandSystems,20(1):87–96,1986.
3. F.Smarandache.Aunifyingfieldinlogics:neutrosophiclogic.In Philosophy,pages1–141.American ResearchPress,1999.
4. R.R.Yager.Pythagoreanfuzzysubsets.In 2013JointIFSAWorldCongressandNAFIPSAnnualMeeting, pages57–61.IEEE,2013.
5. F.Smarandache. Plithogenicset,anextensionofcrisp,fuzzy,intuitionisticfuzzy,andneutrosophicsetsrevisited.InfiniteStudy,2018.
6. F.Smarandache. Anintroductiontotheneutrosophicprobabilityappliedinquantumphysics.InfiniteStudy, 2000.
7. F.Smarandache. Introductiontoneutrosophicmeasure,neutrosophicintegral,andneutrosophicprobability. InfiniteStudy,2013.
8. F.Smarandache. Neutropsychicpersonality:Amathematicalapproachtopsychology.InfiniteStudy,2018.
9. F.SmarandacheandH.E.Khalid. Neutrosophicprecalculusandneutrosophiccalculus.InfiniteStudy,2015.
10. S.Broumi,I.Deli,andF.Smarandache.N-valuedintervalneutrosophicsetsandtheirapplicationinmedical diagnosis. CriticalReview,CenterforMathematicsofUncertainty,CreightonUniversity,Omaha,NE,USA, 10:45–69,2015.
11. Gaurav,M.Kumar,K.Bhutani,andS.Aggarwal.Hybridmodelformedicaldiagnosisusingneutrosophic cognitivemapswithgeneticalgorithms.In 2015IEEEInternationalConferenceonFuzzySystems,pages 1–7.IEEE,2015.
12. R.Kumar,A.Dey,F.Smarandache,andS.Broumi. Chapter6:Astudyofneutrosophicshortestpath problem,NeutrosophicGraphTheoryandAlgorithms,pages144–175.IGI-Global,2019.
13. R.Kumar,S.A.Edalatpanah,S.Jha,S.Broumi,R.Singh,andA.Dey.Amultiobjectiveprogramming approachtosolveintegervaluedneutrosophicshortestpathproblems. NeutrosophicSetsandSystems, 24:134–149,2019.
14. R.Kumar,S.A.Edalatpanah,S.Jha,andR.Singh.Anovelapproachtosolvegaussianvaluedneutrosophic shortestpathproblems. InternationalJournalofEngineeringandAdvancedTechnology,8(3):347–353,2019.
15. R.Kumar,S.A.Edalatpanah,S.Jha,S.Gayen,andR.Singh.Shortestpathproblemsusingfuzzyweighted arclength. InternationalJournalofInnovativeTechnologyandExploringEngineering,8(6):724–731,2019.
Gayenetal.,IntroductiontoPlithogenicHypersoftSubgroup
16. R.Kumar,S.A.Edaltpanah,S.Jha,S.Broumi,andA.Dey.Neutrosophicshortestpathproblem. NeutrosophicSetsandSystems,23:5–15,2018.
17. R.Kumar,S.Jha,andR.Singh.Shortestpathprobleminnetworkwithtype-2triangularfuzzyarclength. JournalofAppliedResearchonIndustrialEngineering,4(1):1–7,2017.
18. R.Kumar,S.Jha,andR.Singh.Adifferentappraochforsolvingtheshortestpathproblemundermixed fuzzyenvironment. InternationalJournalofFuzzySystemApplications,9(2):132–161,2020.
19. S.Broumi,A.Bakal,M.Talea,F.Smarandache,andL.Vladareanu.Applyingdijkstraalgorithmforsolving neutrosophicshortestpathproblem.In 2016InternationalConferenceonAdvancedMechatronicSystems, pages412–416.IEEE,2016.
20. S.Broumi,A.Dey,M.Talea,A.Bakali,F.Smarandache,D.Nagarajan,M.Lathamaheswari,andR.Kumar. Shortestpathproblemusingbellmanalgorithmunderneutrosophicenvironment. Complex&Intelligent Systems,pages1–8,2019.
21. M.Abdel-Basset,A.Atef,andF.Smarandache.Ahybridneutrosophicmultiplecriteriagroupdecision makingapproachforprojectselection. CognitiveSystemsResearch,57:216–227,2019.
22. M.Abdel-Basset,M.Mohamed,Y.Zhou,andI.M.Hezam.Multi-criteriagroupdecisionmakingbasedon neutrosophicanalytichierarchyprocess. JournalofIntelligentandFuzzySystems,33(6):4055–4066,2017.
23. P.Majumdar.Neutrosophicsetsanditsapplicationstodecisionmaking.In ComputationalIntelligencefor BigDataAnalysis,pages97–115.Springer,2015.
24. J.Ye.Similaritymeasuresbetweenintervalneutrosophicsetsandtheirapplicationsinmulticriteriadecisionmaking. JournalofIntelligent&FuzzySystems,26(1):165–172,2014.
25. H.Zhang,J.Wang,andX.Chen.Intervalneutrosophicsetsandtheirapplicationinmulticriteriadecision makingproblems. TheScientificWorldJournal,2014:1–16,2014.
26. S.Broumi,M.Talea,F.Smarandache,andA.Bakali.Decision-makingmethodbasedontheintervalvalued neutrosophicgraph.In 2016FutureTechnologiesConference,pages44–50.IEEE,2016.
27. R.Kumar,S.A.Edalatpanah,S.Jha,andR.Singh.Apythagoreanfuzzyapproachtothetransportation problem. Complex&IntelligentSystems,5(2):255–263,2019.
28. J.Pratihar,R.Kumar,A.Dey,andS.Broumi. Chapter7:Transportationprobleminneutrosophicenvironment,NeutrosophicGraphTheoryandAlgorithms,pages180–212.IGI-Global,2019.
29. M.Abdel-Basset,M.Mohamed,andF.Smarandache.Aaefinedapproachforforecastingbasedonneutrosophictimeseries. Symmetry,11(4):457,2019.
30. M.Abdel-Basset,G.Manogaran,andM.Mohamed.Aneutrosophictheorybasedsecurityapproachforfog andmobile-edgecomputing. ComputerNetworks,157:122–132,2019.
31. V.C¸etkinandH.Ayg¨un.Anapproachtoneutrosophicsubgroupanditsfundamentalproperties. Journal ofIntelligent&FuzzySystems,29(5):1941–1947,2015.
32. I.K.VlachosandG.D.Sergiadis.Intuitionisticfuzzyinformation-applicationstopatternrecognition. PatternRecognitionLetters,28(2):197–206,2007.
33. Y.GuoandH.D.Cheng.Newneutrosophicapproachtoimagesegmentation. PatternRecognition, 42(5):587–595,2009.
34. M.Abdel-Basset,N.A.Nabeeh,H.A.El-Ghareeb,andA.Aboelfetouh.Utilisingneutrosophictheoryto solvetransitiondifficultiesofiot-basedenterprises. EnterpriseInformationSystems,pages1–21,2019.
35. F.Smarandache. Plithogeny,plithogenicset,logic,probability,andstatistics.InfiniteStudy,2017.
36. D.Molodtsov.Softsettheory–firstresults. Computers&MathematicswithApplications,37(4-5):19–31, 1999.
37. P.K.Maji,R.Biswas,andA.R.Roy.Softsettheory. Computers&MathematicswithApplications,45(45):555–562,2003.
38. H.Akta¸sandN.C¸a˘gman.Softsetsandsoftgroups. Informationsciences,177(13):2726–2735,2007.
Gayenetal.,IntroductiontoPlithogenicHypersoftSubgroup
39. A.Ayg¨unoluandH.Ayg¨un.Introductiontofuzzysoftgroups. Computers&MathematicswithApplications, 58(6):1279–1286,2009.
40. M.Shabir,M.Ali,M.Naz,andF.Smarandache.Softneutrosophicgroup. NeutrosophicSetsandSystems, 1:13–25,2013.
41. H.BustinceandP.Burillo.Vaguesetsareintuitionisticfuzzysets. Fuzzysetsandsystems,79(3):403–405, 1996.
42. D.H.HongandC.H.Choi.Multicriteriafuzzydecision-makingproblemsbasedonvaguesettheory. Fuzzy setsandsystems,114(1):103–113,2000.
43. M.Kryszkiewicz.Roughsetapproachtoincompleteinformationsystems. Informationsciences,112(14):39–49,1998.
44. Z.Pawlak.Hardandsoftsets.In Roughsets,fuzzysetsandknowledgediscovery,pages130–135.Springer, 1994.
45. A.R.RoyandP.K.Maji.Afuzzysoftsettheoreticapproachtodecisionmakingproblems. Journalof ComputationalandAppliedMathematics,203(2):412–418,2007.
46. P.K.Maji.Moreonintuitionisticfuzzysoftsets.In InternationalWorkshoponRoughSets,FuzzySets, DataMining,andGranular-SoftComputing,pages231–240.Springer,2009.
47. P.K.Maji. Neutrosophicsoftset.InfiniteStudy,2013.
48. A.Rosenfeld.Fuzzygroups. JournalofMathematicalAnalysisandApplications,35(3):512–517,1971.
49. R.Biswas.Intuitionisticfuzzysubgroups.In MathematicalForum,volume10,pages37–46,1989.
50. F.Karaaslan,K.Kaygisiz,andN.C¸a˘gman.Onintuitionisticfuzzysoftgroups. JournalofNewResultsin Science,2(3):72–86,2013.
51. T.BeraandN.K.Mahapatra.Introductiontoneutrosophicsoftgroups. NeutrosophicSetsandSystems, 13:118–127,2016.
52. F.Smarandache. Extensionofsoftsettohypersoftset,andthentoplithogenichypersoftset.InfiniteStudy, 2018.
53. P.K.Maji,A.R.Roy,andR.Biswas.Anapplicationofsoftsetsinadecisionmakingproblem. Computers &MathematicswithApplications,44(8-9):1077–1083,2002.
54. F.Feng,Y.B.Jun,X.Liu,andL.Li.Anadjustableapproachtofuzzysoftsetbaseddecisionmaking. JournalofComputationalandAppliedMathematics,234(1):10–20,2010.
55. N.C¸a˘gman,S.Enginoglu,andF.Citak.Fuzzysoftsettheoryanditsapplications. IranianJournalofFuzzy Systems,8(3):137–147,2011.
56. S.Broumi,I.Deli,andF.Smarandache.Neutrosophicparametrizedsoftsettheoryanditsdecisionmaking. InternationalFrontierScienceLetters,1(1):1–11,2014.
57. S.Broumi,I.Deli,andF.Smarandache.Intervalvaluedneutrosophicparameterizedsoftsettheoryandits decisionmaking. JournalofNewResultsinScience,3(7):58–71,2014.
58. I.Deli,S.Broumi,andM.Ali.Neutrosophicsoftmulti-settheoryanditsdecisionmaking. Neutrosophic SetsandSystems,5:65–76,2014.
59. I.DeliandN.C¸a˘gman.Intuitionisticfuzzyparameterizedsoftsettheoryanditsdecisionmaking. Applied SoftComputing,28:109–113,2015.
60. F.Smarandache.Physicalplithogenicset.In APSMeetingAbstracts,2018.
61. F.Smarandache.Aggregationplithogenicoperatorsinphysicalfields. BulletinoftheAmericanPhysical Society,63(13),2018.
62. S.Gayen,F.Smarandache,S.Jha,M.K.Singh,S.Broumi,andR.Kumar. Chapter8:Introductionto plithogenicsubgroup,NeutrosophicGraphTheoryandAlgoritm,pages209–233.IGI-Global,October2019.
63. M.Abdel-Basset,M.El-hoseny,A.Gamal,andF.Smarandache.Anovelmodelforevaluationhospital medicalcaresystemsbasedonplithogenicsets. Artificialintelligenceinmedicine,100:1–8,2019.
Gayenetal.,IntroductiontoPlithogenicHypersoftSubgroup
64. M.Abdel-BassetandR.Mohamed.Anovelplithogenictopsis-criticmodelforsustainablesupplychainrisk management. JournalofCleanerProduction,247,2019.
65. M.Abdel-Basset,R.Mohamed,A.N.H.Zaied,andF.Smarandache.Ahybridplithogenicdecision-making approachwithqualityfunctiondeploymentforselectingsupplychainsustainabilitymetrics. Symmetry, 11(7):903,2019.
66. P.K.Sharma.Homomorphismofintuitionisticfuzzygroups.In InternationalMathematicalForum,volume6,pages3169–3178,2011.
67. S.Gayen,S.Jha,andM.Singh.Ondirectproductofafuzzysubgroupwithananti-fuzzysubgroup. InternationalJournalofRecentTechnologyandEngineering,8:1105–1111,2019.
68. P.S.Das.Fuzzygroupsandlevelsubgroups. JournalofMathematicalAnalysisandApplications,84(1):264–269,1981.
69. S.Gayen,F.Smarandache,S.Jha,andR.Kumar. Chapter10:Interval-valuedneutrosophicsubgroupbased oninterval-valuedtriplet-norm,NeutrosophicSetsinDecisionAnalysisandOperationsResearch,pages 215–243.IGI-Global,December2019.
70. R.Biswas.Rosenfeld’sfuzzysubgroupswithinterval-valuedmembershipfunctions. FuzzySetsandSystems, 63(1):87–90,1994.
71. C.Y.Xu.Homomorphismofintuitionisticfuzzygroups.In 2007InternationalConferenceonMachine LearningandCybernetics,volume2,pages1178–1183.IEEE,2007.
72. S.Rana,M.Qayyum,M.Saeed,andF.Smarandache.Plithogenicfuzzywholehypersoftset:constructionof operatorsandtheirapplicationinfrequencymatrixmultiattributedecisionmakingtechnique. Neutrosophic SetsandSystems,28(1):34–50,2019.
Received:December13,2019.Accepted:May02,2020