Introduction to Plithogenic Hypersoft Subgroup

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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

, 2020

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.

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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...

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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

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(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 ).

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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

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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

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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

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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

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Γ(big,red,expensive)=          b3(0 87, 1, 0 66),b10(0 61
2)
b3(0.15,
.72
.47),b10
b3
.
.
.
        
U willbedenotedasPNHS(U ). Example3.8. InExample3.2letcorrespondingPNHSis
, 0 25, 0
,b12(0 32, 0 7, 0 83)
0
, 0
(0.77, 0.4, 0.48),b12(0.37, 0.18, 0.2)
(0
76, 0.17, 0
29),b10(0.5, 0.71, 0.98),b12(1, 0.35, 0
67)

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

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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

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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.

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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).

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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

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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

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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

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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 )}

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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)

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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 ).

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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

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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 )}

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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).

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4. Conclusions

Hypersoftsettheoryismoregeneralthansoftsettheoryandithasahugeareaofapplications.Thatiswhywehaveadoptedandimplementeditinplithogenicenvironmentsothatwe canintroducevariousalgebraicstructures.Becauseofthis,thenotionsofplithogenichypersoftsubgroupshavebecomegeneralthanfuzzy,intuitionisticfuzzy,neutrosophicsubgroups, andplithogenicsubgroups.Again,wehaveintroducedfunctionsindifferentplithogenichypersoftenvironments.Hence,homomorphismcanbeintroducedanditseffectsonthesenewly definedplithogenichypersoftsubgroupscanbestudied.Inthefuture,toextendthisstudy onemayintroducegeneralT-normandT-conormandfurthergeneralizeplithogenichypersoft subgroups.Also,onemayextendthesenotionsbyintroducingdifferentnormalversionsof plithogenichypersoftsubgroupsandbystudyingtheeffectsofhomomorphismonthem.

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Received:December13,2019.Accepted:May02,2020

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