NeutroAlgebra of Neutrosophic Triplets using {Zn, x}

Neutrosophic Sets and Systems.Vol. 38,2020

UniversityofNewMexico

NeutroAlgebra of Neutrosophic Triplets using {Zn, x}

Vasantha Kandasamy1 , Ilanthenral Kandasamy1'* and Florentin Smarandache2

1 School of Computer Science and Engineering, VIT, Vellore 632014, India; vasantha.wb@vit.ac.in

2Department of Mathematics, University of New Mexico, 705 Gurley Avenue, Gallup, NM 87301, USA; smarand@unm.edu

*Correspondence: ilanthenral.k@vit.ac.in

Abstract. Smarandache in 2019 has generalized the algebraic structures to NeutroAlgebraic structures and AntiAlgebraic structures. In this paper, authors, for the first time, define the NeutroAlgebra of neutrosophic triplets group under usual+ and x, built using {Zn, x}, n a composite number, 5 < n < oo, which are not partial algebras. As idempotents in Zn alone are neutrals that contribute to neutrosophic triplets groups, we analyze them and build NeutroAlgebra of idempotents under usual+ and x, which are not partial algebras. We prove in this paper the existence theorem for NeutroAlgebra of neutrosophic triplet groups. This proves the neutrals assocaited withneutrosophic triplet groups in {Zn, X}underproduct is a NeutroAlgebra oftriplets. We also prove the non-existence theorem of NeutroAlgebra for neutrosophic triplets in case of Zn when n = 2p,3p and 4p (for some primes p). Several open problems are proposed. Further, the NeutroAlgebras of extended neutrosophic triplet groups have been obtained.

Keywords: neutrosophic triplets; neutrosophic extended triplets; neutrosophic triplet group; neutrosophic extended triplet group; NeutroAlgebra; partial algebra; NeutroAlgebra of neutrosophic triplets; NeutroAlgebra of neutrosophic extended triplets; AntiAlgebra

1. Introduction

The neutrosophic theory proposed by Smarandache in [1] has become a powerful tool in the study/analysis of real-world data as they are dominated by uncertainty, inconsistency, and indeterminacy. Neutrosophy deals with the neutralities and indeterminacies of real-world problems. The innovative concept of neutrosophic triplet groups was introduced by [2], which gives for any element a in (G, *), the anti(a) and neut(a) satisfying conditions

a*neut(a) = neut(a)*a= a

Kandasamy, V. and et. al., NeutroAlgebra of Neutrosophic Triplets

and a ∗ anti(a)= anti(a) ∗ a = neut(a)

where neut(a)isnottheidentityelementortheclassicalidentityofthegroup.Theycall (a,neut(a),anti(a))astheneutrosophictripletgroup.Theseneutrosophictripletsbuiltusing Zn arealwayssymmetricabouttheneutralelements.Forif(a, neut(a),anti(a))isneutrosophictripletthen(anti(a),neut(a),a)therebygivingaperfectsymmetryof a and anti(a) aboutthe neut(a).Thestudyofneutralitieshavebeencarriedoutbyseveralresearchersin neutrosophicalgebraicstructureslikeneutrosophictripletrings,groups,neutrosophicquadruplevectorspaces,neutrosophicsemiidempotents,dupletsandtripletsinneutrosophicrings, neutrosophictripletinbiaglebras,neutrosophictripletclassicalgroupandtheirapplications, tripletloops,subgroups,cancellablesemigroupsandAbel-Grassmangroupoids[2–24].

[13]hasdefinedaclassicalgroupstructureontheseneutrosophictripletgroupsandhas obtainedseveralinterestingpropertiesandgivenopenconjectures.Smarandache[2]defined theNeutrosophicExtendedTriplet,whentheneutralelementisallowedtobetheclassical unitelement.Zhangetalhasdefinedneutrosophicextendedtripletgroupandhaveobtained severalresultsin[25].Later[26]haveobtainedsomeresultsonneutrosophicextendedtriplet groupswithpartialorderdefinedonit.Moreresultsaboutneutrosophictripletgroupsand neutrosophicextendedtripletgroupscanbefoundin[25–32].

WeinthispaperstudytheverynewnotionofNeutroAlgebraintroducedby[33].Several interestingresultsareobtainedin[12,34–36],andtheyintroducedNeutroBCAlgebraand subNeutroBIAlgebraandsoon.NeutroAlgebrasandAntiAlgebrasintheclassicalnumber systemswerestudiedin[37].

HereweintroduceNeutroAlgebraundertheusualproductandsumincaseofidempotents inthesemigroups {Zn, ×},nacompositenumber,5 <n< ∞.Thisstudyisveryimportant foralltheneutrosophictripletsin {Zn, ×},happentobecontributedonlybytheidempotents, whicharetheonlyneutralsin {Zn, ×}.WeobtainNeutroAlgebrasunderusual+and × inthe caseofneutrosophictripletgroupsandneutrosophicextendedtripletgroups.Itispertinent tokeeponrecordwedefineclassicalproductonneutrosophictriplets,andtheyareclassical groupsunderproductofthesetriplets.Thispaperhassixsections.Sectiononeisintroductory innature,andbasicconceptsarerecalledinsectiontwo.Sectionthreeobtainstheexistence andnon-existencetheoremonNeutroAlgebrasunderusual+or × usingneutrosophictriplet groups.Insectionfour,asimilarstudyiscarriedoutinthecaseofneutrosophicextended tripletgroups.Thefifthsectionprovidesadiscussiononthistopic,andthefinalsectiongives theconclusionsbasedonourstudyandsomeopenconjectureswhichwillbetakenforfuture researchbytheauthors.

Kandasamy,V.andet.al.,NeutroAlgebraofNeutrosophicTriplets

BasicConcepts

Herewerecallsomebasicdefinitionswhichisimportanttomakethispaperaselfcontained one.

Definition2.1. Letusassumethat N isanemptysetandwithbinaryoperation*defined onit.Niscalledaneutrosophictripletset(NTS)ifforany a ∈ N ,thereexistsaneutralof “a”(denotedbyneut(a)),andanoppositeof“a”(denotedbyanti(a))satisfyingthefollowing conditions: a ∗ neut(a)= neut(a) ∗ a = a a ∗ anti(a)= anti(a) ∗ a = neut(a)

And,theneutrsophictripleisgivenby(a,neut(a),anti(a)).

Inaneutrosophictripletset(N,*), a ∈ N ,neut(a)andanti(a)maynotbeunique.

Inthedefinitiongivenin[2],theneutralelementcannotbeanunitelementintheusual sense,andthenthisrestrictionisremoved,usingtheconceptofaneutrosophicextendedtriplet in[26].

Theclassicalunitelementcanberegardedasaspecialneutralelement.Thenotionof neutrosophictripletgroupsandthatofneutrosophicextendedtripletgroupsaredistinctly dealtwithinthispaper.

Definition2.2. Letusassumethat(N,*)isaneutrosophictripletset.Then,Niscalleda neutrosophictripletgroup,ifitsatisfies:

(1) ClosureLaw,i.e., a ∗ b ∈ N, ∀a,b ∈ N ;

(2) Associativity,i.e.,(a ∗ b) ∗ c = a ∗ (b ∗ c), ∀a,b,c ∈ N Aneutrosophictripletgroup(N, ∗)issaidtobecommutative,if a ∗ b = b ∗ a, ∀a,b ∈ N

Let A beaconcept(asintermsofattribute,idea,proposition,ortheory).Bythe neutrosphicationprocess,wesplitthenon-emptyspaceintothreeregionstwooppositeones correspondingto A and antiA ,andoneneutral(indeterminate) neutA (alsodenoted neutroA )betweentheopposites,whichmayormaynotbedisjoint;dependingonthe application,buttheirunionequalsthewholespace.

ANeutroAlgebraisanalgebrathathasatleastoneneutrooperationoroneneutroaxiom (axiomthatistrueforsomeelements,indeterminateorfalsefortheotherelements)[33].A partialalgebrahasattheminimumonepartialoperation,andallitsaxiomsareclassical. Throughatheoremin[34],provedthatNeutroAlgebraisageneralizationofpartialalgebra, andalsogiveillustrationsofNeutroAlgebrasthatarenotpartialalgebras.Boolehasdefined thePartialAlgebra(basedonPartialFunction)asanalgebrawholeoperationispartially well-defined,andpartiallyundefined(thisundefinedgoesunderIndeterminacywithrespect Kandasamy,V.andet.al.,NeutroAlgebraofNeutrosophicTriplets

toNeutroAlgebra).Therefore,aPartialAlgebra(PartialFunction)hassomeelementsfor whichtheoperationisundefined(notouter-defined).SimilarlyanAntiAlgebraisanonempty setthatisendowedwithatleastoneanti-operation(oranti-function)oratleastoneantiaxiom.

3. NeutroAlgebrasofneutrosophictripletsusing {Zn, ×}

HereforthefirsttimeauthorsbuildNeutroAlgebrasusingneutrosophictripletsgroupbuilt usingthemodulointegers Zn; n acompositenumber.Neutrosophictripletgroupsandextendedneutrosophictripletgroupswerestudiedby[25,26].FirstwedefineNeutroAlgebra usingthenon-trivialidempotentsof Zn, n acompositenumber.Thisstudyismandatoryas alltheneutralelementsofneutrosophictripletsbuildusing Zn areonlythenon-trivialidempotentsof Zn.NextwegivetheexistenceandnonexistencetheoremsincaseofNeutroAlgebras fortheseneutrosophictripletsets.Wegivesomeinterestingpropertiesaboutthem.Further itisimportanttonoteunlessseveralopenconjecturesaboutidempotentsin Zn givenin[13], aresolvedorsomeprogressismadeinthatdirectionitwillnotbepossibletocompletelycharacterizeNeutroAlgebrasoftheneutrosophictripletgroupsorextendedneutrosophictriplet groups.Wewillbeusing[13]togetNeutroAlgebrasofidempotentsandNeutroAlgebraof neutrosophictripletsets.FirstweprovideexamplesofNeutroAlgebrausingsubsetsofthe semigroup {Zn, ×} andthenNeutroAlgebraofidempotentsin {Zn, ×}.

Example3.1. Let S = {Z15, ×} beasemigroupunderproductmodulo15.Nowconsiderthe subset A = {5, 10, 14}∈ S.TheCayleytablefor A isgiveninTable1,whereouter-defined elementsaredenotedby od Table1. CayleyTablefor A × 5 10 14 5 10 5 10 10 5 10 5 14 10 5 od

Weseethetablehasouter-definedelementsdenotedby od.So A isaNeutroAlgebrawhich isnotapartialalgebra,sincetheoperation14 × 14isouter-defined.14 × 14 ≡ 1(mod15), but1 / ∈{5, 10, 14}.ThereforeTable1isonlyaNeutroAlgebra.Everysubsetof S neednotbe aNeutroAlgebra.Fortake B = {3, 6, 9, 12} asubsetin S.ConsidertheCayleytableforBis giveninTable2. B isnotaNeutroAlgebraaseveryterminthecellisdefinedandassociativityaxiomis totallytrue.. Kandasamy,V.andet.al.,NeutroAlgebraofNeutrosophicTriplets

Table2. CayleyTablefor B

× 3 6 9 12 3 9 3 12 6 6 3 6 9 12 9 12 9 6 3 12 6 12 3 9

Clearly B isasubsemigroupof S,infactagroupunder × modulo15with6asitsmultiplicativeidentity,so S isaSmarandachesemigroup[10].

Consider C = {2, 7, 8} asubsetof S.TheCayleytableforCisgiveninTable3,thishas everycelltobeouter-defined.

Table3. CayleyTablefor C × 2 7 8 2 od od od 7 od od od 8 od od od

So C isnotaNeutroAlgebraorasubsemigroupbutanAntiAlgebrasincetheoperation × istotallyouter-definedunder × modulo15.

Thuswecancategoricallyputforththefollowingfacts. Everyclassicalalgebraicstructure A withbinaryoperationsdefinedonitissuchthatany propersubset B of A withinheritedoperationof A fallsunderthethreecategories;

(1) B canbeapropersubstructureofastrongerstructureofAwiththeinheritedoperationsof A

(2) B canonlybeaNeutroAlgebra,whichmaybeaPartialAlgebra,whensomeoperation isundefined,andallotheroperationsarewell-definedandallaxiomsaretrue.

(3) B canbeanAntiAlgebrawhenatleastoneoperationistotallyouter-defined.orat leastoneaxiomistotallyfalse.

UnderthesecircumstancesifonewantstogetaNeutroAlgebrawhichisnotapartial algebraforapropersubsetofaclassicalalgebraicstructureoneshouldexploitthespecial axiomssatisfiedbythem,tothisendwestudythepropertyofidempotentsinthesemigroup {Zn, ×}

WealsoincaseofneutrosophictripletgroupobtainaNeutroAlgebrawhichisnotapartial algebra.

Kandasamy,V.andet.al.,NeutroAlgebraofNeutrosophicTriplets

38,2020

FirstwegiveexamplesofNeutroAlgebrawhicharenotpartialalgebrasusingidempotents ofthesemigroupS= {Zn, ×}

Example3.2. LetS= {Z6, ×} bethesemigroupunderproductmodulo6.Thenontrivial idempotentsofSare V = {3, 4}.TheCayleytablefor V isgiveninTable4,

Table4. CayleyTablefor V × 3 4 3 3 od 4 od 4

So V isaNeutroAlgebraunder × butnotapartialalgebra.Forthesame V defineoperation +modulo6,theCayleytableforVisgiveninTable5and V isAntiAlgebraandnotapartial algebraeither.

Table5. CayleyTablefor V + 3 4 3 od od 4 od od

Supposewetake W = {0, 1, 3, 4} thecollectionoftrivialandnontrivialidempotentsofS, andifwetakeSasawholesetbutstudytheidempotentaxiomin W weseefromTable6.

Table6. CayleyTablefor W × 0 1 3 4 0 0 0 0 0 1 0 1 3 4 3 0 3 3 0 4 0 4 3 4

SupposewefindtheCayleytablefor W under+wegettheCayleytablegiveninthe followingTable7.

W itselfisaNeutroAlgebraunderusual+withseveralundefinedterms. W underusual productisasubsemigroupofidempotentsofS;whereasSundersumofidempotentsisa NeutroAlgebrawhichisnotapartialalgebraundertheaxiomofthepropertyofidempotency.

Nowifwetakeforanysubsetof S theaxiomofidempotentpropertywegetNeutroAlgebras whicharenotpartialalgebras.

Tothiseffectweprovideanexample. Kandasamy,V.andet.al.,NeutroAlgebraofNeutrosophicTriplets

Table7. CayleyTablefor W

+ 0 1 3 4 0 0 1 3 4 1 1 od 4 od 3 3 4 0 1 4 4 od 1 od

Example3.3. LetS= {Z42, ×} bethesemigroupunderproductmodulo42.Thetrivialand nontrivialidempotentsofSare B = {0, 1, 7, 15, 21, 22, 28, 36}.Wedefine+modulo42on thissetofidempotentskeepingtheresultantwhatweneedistheaxiomofidempotency.The CayleytableforBisgiveninTable8.

Table8. CayleyTablefor B + 0 1 7 15 21 22 28 36 0 0 1 7 15 21 22 28 36 1 1 od od od 22 od od od 7 7 od od 22 28 od od 1 15 15 od 22 od 36 od 1 od 21 21 22 28 36 0 1 7 15 22 22 od od od 1 od od od 28 28 od od 1 7 od od 22 36 36 od 1 od 15 od 22 od

Thus B isaNeutroAlgebrawhichisnotapartialalgebraundertheaxiomofidempotency. ThuswehavealargeclassofNeutroAlgebraswhicharenotpartialalgebras.

Asthemainthemeofthispaperisstudyofneutrosophictripletsusingmodulointegers {Zn, ×} andprovetheexistencetheoremandnon-existencetheoremofNeutroAlgebraof neutrosophictripletgroups.

InviewofallthesewehavethefollowingexistencetheoremofNeutroAlgebraofneutrosophic triplets.

Theorem3.4. Let S = {Zn, ×},nnotaprime, 5 <n< ∞.LetVbethecollectionofall nontrivialidempotentsthatisallneutralsofS,where0and1arenotinS.ThenVunder productisaNeutroAlgebraoftriplets.

Proof. Let W = {w1,w2,...,wt} bethenontrivialidempotentsof S.Itisprovedin[13] thatif Wi isthesetofallneutrosophictripletsofanontrivialidempotent wi in S which Kandasamy,V.andet.al.,NeutroAlgebraofNeutrosophicTriplets

servesastheneutralforthecollection Wi then {Wi, ×} isaneutrosophictripletclassical groupunderusualproductand i variesoverallneutrals;1 ≤ i ≤ t.If V isthecollectionofall neutrosophictriplets(this V willincludeall Wi fordifferentneutralsornontrivialidempotents in S),associatedwith S = {Zn, ×};then V isnotclosedunderusualproduct[13]andthere aremanyundefinedelementsunderusualproductso V isaNeutroAlgebraofneutrosophic triplets.Hencetheclaim.

InviewofthiswehavethefollowingpartialnonexistencetheoremofNeutroAlgebraof neutrosophictripletsunder+for Znp where n =2, 3and4forsomevaluesofPprovided intheTables9,10and11.Wehavefor Zn, n aproductofmorethantwoprimescanhave NeutroAlgebraofneutrosophictripletsunder+.

Theorem3.5. Let S = {Znp, ×};where n =2,3and4,(p aspecificprimeand np isnot asquareofaprime,primevaluesreferTables9,10,and11)beasemigroupunderproduct modulonp.IfVdenotesthecollectionofallidempotentsassociatedwiththenontrivial idempotentsof Znp then {V, +} isneveraNeutroAlgebraoftripletsfor n =2,3and4. Proof. Recallfrom[13]thattherearetwoidempotentsinallthethreecaseswhen n =2p or 3p or4p giveninTables9,10and11.

Table9. Idempotenttablefor Z2p S.no Z2p p p+1 1 Z6 3 4 2 Z10 5 6 3 Z14 7 8 4 Z22 11 12 5 Z26 13 14 6 Z34 17 18 7 Z38 19 20 8 Z46 23 24 9 Z58 29 30

Weseeanysumoftheidempotentsis1andproductis0. Herein Z3p and Z4p alsosumofidempotentsis1andthatproductis0.Tablesareprovided forthem[13].Incaseof2p thenontrivialidempotentsare p and p +1,clearlyundersumthis isaset.Thuswehaveprovedthenon-existenceofNeutroAlgebraofidempotentsunder’+’. Tothiseffectfirstprovideanexample. Kandasamy,V.andet.al.,NeutroAlgebraofNeutrosophicTriplets

Table10. Idempotenttablefor Z3p

S.No. Z3p p p+1 2p 2p+1 1 Z15 - 6 102 Z21 7 - - 15 3 Z33 - 12 224 Z39 13 - - 27 5 Z51 - 18 347 Z57 19 - - 39 8 Z69 - 24 469 Z159 - 54 106 -

Table11. Idempotenttablefor Z4p

S.No. Z4p p p +1 3p 3p +1 1 Z12 - 4 92 Z20 5 - - 16 3 Z28 - 8 214 Z44 - 12 335 Z52 13 - - 40 6 Z76 - 20 577 Z212 53 - - 160 8 Z388 97 - - 292 9 Z332 - 84 249 -

Example3.6. ConsiderthesemigroupS= {Z10, ×}.ThenontrivialidempotentsofSwhich contributetotheneutrosophictripletsetare; {6, 5} in Z10.Considertheneutrosophictriplet set V = {(5, 5, 5), (6, 6, 6), (8, 6, 2), (2, 6, 8), (4, 6, 4)}.Itisproved V \{(5, 5, 5)} isaneutrosophic tripletclassicalgroupunder × [13].NowtheCayleytableofVunderusualproduct × isgiven inTable12.

Table12. CayleyTablefor V

× (5,5,5) (6,6,6) (8,6,2) (2,6,8) (4,6,4) (5,5,5) (5,5,5) od od od od (6,6,6) od (6,6,6) (8,6,2) (2,6,8) (4,6,4) (8,6,2) od (8,6,2) (4,6,4) (6,6,6) (2,6,8) (2,6,8) od (2,6,8) (6,6,6) (4,6,4) (8,6,2) (4,6,4) od (4,6,4) (2,6,8) (8,6,2) (6,6,6)

Kandasamy,V.andet.al.,NeutroAlgebraofNeutrosophicTriplets

ClearlyVisaNeutroAlgebraunderusualproductandnotapartialalgebra.Sincewehave notincludedtheneutralsthatisnontrivialidempotentslike0and1wehavethistobeonly aNeutroAlgebraoftriplets.

Table13. CayleyTablefor V + (5,5,5) (6,6,6) (8,6,2) (2,6,8) (4,6,4) (5,5,5) od od od od od (6,6,6) od od od od od (8,6,2) od od od od od (2,6,8) od od od od od (4,6,4) od od od od od

Thustheneutrosophictripletscollectionyieldsonlyasetunderadditionwherenopairof neutrosophictripletsgivesundersumaneutrosophictriplet.HenceourclaimnoNeutroAlgebraneutrosophictripletsunderaddition.So V inTable13isanAntiAlgebra.Likewisethe cases3p and4p fromtables. Soifweincludethenontrivialidempotents0and1thenwecangetNeutroAlgebraof idempotentsunder+whichiscarriedoutinthefollowingsection.

Example3.7. Considerthesemigroup S = {Z105, ×} under × modulo105.Thenontrivial idempotentsare V = {15, 21, 36, 70, 85, 91}.LetMbethecollectionofallneutrosophictriplets usingtheidempotentsinV.Mcontainselementssay { (15,15,15),(21,21,21),(36,36,36), (30,15,60),(51,36,81)},fromtheCayleytableofMunder+weseetherearesomeundefined termsalsogiveninTable14.

Table14. CayleyTablefor M + (15,15,15) (21,21,21) (36,36,36) (30,15,60) (51,36,81) (15,15,15) od (36,36,36) od od od (21,21,21) (36,36,36) od od (51,36,81) od (36,36,36) od od od od od (30,15,60) od (51,36,81) od od od (51,36,81) od od od od od

HencewehaveaNeutroAlgebraofneutrosophictripletsunder+.

Weproposesomeopenproblemsinthisregardinthefinalsectionofthispaper. NowwefindwaystogetNeutroAlgebraofneutrosophictripletsunder+.Thepossibilityis byusingextendedneutrosophictripletsgroupwecanhaveforall Zn, n anycompositenumber Kandasamy,V.andet.al.,NeutroAlgebraofNeutrosophicTriplets

NeutroAlgebraofneutrosophictripletsunder+.Unlesstheconjecturesproposedin[13]is solvedcompletecharacterizationisnotpossible,onlypartialresultsandexamplestothat effectarepossible.

InthefollowingsectionwediscussNeutroAlgebraofextendedneutrosophictripletsets.

4. NeutroAlgebraofextendedneutrosophictripletsusing {Zn, ×}

InthissectionweprovetheexistenceofNeutroAlgebraofextendedneutrosophictriplets using {Zn, ×},formoreaboutextendedneutrosophictripletsrefer[2,26]underboth+and ×.Throughoutthissectionweassumethecollectionofidempotentscontainsboththetrivial idempotents1and0.Itisthusmandatorytheneutrosophictripletsetcollectioncontains(0, 0,0)and(1,1,1)apartfromtheneutrosophictripletsoftheform(a,1,anti a =inverseof a),where a isin Zn whichhasinversein Zn.

Wefirstprovethecollectionofalltrivialandnontrivialidempotentsin Zn isaNeutroAlgebraunder+andalsounder ×

Theorem4.1. Let S = {Zn, ×} bethesemigroupunderproductmodulo n, 5 <n< ∞.Let V = {Collectionofallidempotentsin Zn including0and1 }

(1) V \{0, 1} isaNeutroAlgebraofidempotentsunder × modulon.

(2) VisaNeutroAlgebraofidempotentsunder+modn.

Proof. Consider V \{0, 1} foreveryxin V \{1, 0} issuchthat x × x = x,so V \{1, 0} isa NeutroAlgebraunder ×.Hence(1)istrue.

Proofof(2):ToshowVisaNeutroAlgebraofidempotentsunder+.Since0isin V we haveforevery x ∈ V ;0+ x = x isin V ,howeverwedonotingeneralhavethesumoftwo idempotentstobeanidempotent.Forinstance1+1=2isnotanidempotentso(V, +)has undefinedelements,henceundefined.Thus(2)isproved.

Weprovideanexampletothiseffect.

Example4.2. Let S = {Z10,n, ×} bethesemigroupunder × modulo10.Thetrivialand nontrivialidempotentsare V = {0, 1, 5, 6}.ItiseasilyverifiedVisaNeutroAlgebraunder +,for6+6=2modulo10.HoweverVisnotaNeutroAlgebraunder ×,but V \{0, 1}isa NeutroAlgebraunder × modulo10.For6+5=1modulo10,so V \{1, 0} isaNeutroAlgebra. Nowtheneutrosophictripletsof S associatedwiththeidempotents V are N = { (0,0,0), (1,1,1),(5,1,5),(3,1,7),(7,1,3),(5,5,5),(6,6,6),(4,6,4),(2,6,8)and(8,6,2) }.We seeNunder+isaNeutroAlgebra,for(1,1,1)+(7,1,3)=(8,2,4)isnotinN.Nisnot aNeutroAlgebraunder+.But N \{ (0,0,0),(1,1,1),(5,1,5),(3,1,7),(7,1,3) } = W neutrosophictripletsformedbythenontrivialidempotents5and6isaNeutroAlgebraas(5, Kandasamy,V.andet.al.,NeutroAlgebraofNeutrosophicTriplets

5,5) × (2,6,8)=(0,0,0)whichisnotinW.Hencetheclaim.If {(0, 0, 0)} isadded,then theset V becomesaNeutroAlgebraunder+.

Table15. CayleyTablefor V + (0,0,0) (5,5,5) (6,6,6) (8,6,2) (2,6,8) (4,6,4) (0,0,0) (0,0,0) (5,5,5) (6,6,6) (8,6,2) (2,6,8) (4,6,4) (5,5,5) (5,5,5) od od od od od (6,6,6) (6,6,6) od od od od od (8,6,2) (8,6,2) od od od od od (2,6,8) (2,6,8) od od od od od (4,6,4) (4,6,4) od od od od od

Theorem4.3. Let S = {Zn, ×} beasemigroupunder × modulon,wherenisnotaprime and 5 <n< ∞.Let N = {collectionofallextendedneutrosophictripletsetincluding(0,0, 0),andallneutrosophictripletsassociatedwiththetrivialidempotent1}

(1) NisaNeutroAlgebraunder+ofextendedneutrosophictripletsset.

(2) N \{(0, 0, 0)} isaNeutroAlgebraofextendedneutrosophictripletsetunderproduct modulon.

Proof. Let N bethecollectionofallextendedneutrosophictripletsincluding(0,0,0)and(1, 1,1)andothertripletsassociatedwiththeneutral1.

Proofof(1):Inthecaseextendedtriplet N weseesumoftwoidempotentsneednotbe idempotentfor(1,1,1)+(1,1,1)=(2,2,2)isnotin N ,hence N istheNeutroAlgebra ofextendedneutrosophictripletswhichisnotapartialalgebraastheaxiomofneutrosophic tripletsisnotsatisfied.

Proofof(2):Consider N \{(0, 0, 0)}.Clearlyingeneraltheproductofanytwoidempotents isnotanidempotentin Zn,andseveraltripletsareundefinedanddonotingeneralsatisfy thetripletrelation[13].Hencetheclaim.

5. Discussions

ThestudyofNeutroAlgebraintroducedby[33]isverynew,heretheauthorsbuiltNeutroAlgebrausingidempotentsof {Zn, ×} asemigroupunder × modulo n forappropriate n whicharenotpartialalgebras.LikewiseNeutroAlgebrabuiltusingneutrosophictripletsset andextendedneutrosophictripletsset.Someopenproblemsbasedonourstudyisproposed inthesectiononconclusions. Kandasamy,V.andet.al.,NeutroAlgebraofNeutrosophicTriplets

6. Conclusions

ForthefirsttimeauthorshaveNeutroAlgebrausingidempotentsofasemigroup S = {Zn, ×};nacompositenumber5 <n< ∞,neutrosophictripletsandextendedneutrosophic triplets.WehaveobtainedNeutroAlgebrasofidempotentswhicharenotpartialalgebrasundertheclassicaloperationof+and × onlyusing S = {Zn, ×},thesemigroupunderproduct forappropriate n.Wehaveobtainedbothexistenceandnon-existencetheoremforNeutroAlgebrasofidempotentsinS.Wesuggestcertainopenproblemsforresearchersaswellasthese problemswillbetakenbytheauthorsforfuturestudy.

Problem1:Doesthereexista n (n acompositenumber)suchthatusing {Zn, ×} there arenonontrivialNeutroAlgebraofneutrosophictripletsetandNeutroAlgebrainextended neutrosophictripletset?

Problem2.Doesthereexista n, n acompositenumbersuchthat {Zn, ×} hasitscollection oftrivialandnontrivialidempotentsdenotedby N tobesuchthat;

• (N, +)isnotNeutroAlgebraofidempotents?

• (N, ×)isnotaNeutroAlgebraofidempotents?

Problem3:Proveincaseof {Z3p, ×} and {Z4p, ×},theidempotentsareonlyoftheform mentionedinTables10and11respectively.

Funding: Thisresearchreceivednoexternalfunding.

ConflictsofInterest:”Theauthorsdeclarenoconflictofinterest.

References

1. F.Smarandache,“Aunifyingfieldinlogics.neutrosophy:Neutrosophicprobability,setandlogic,”1999.

2. F.Smarandache, NeutrosophicPerspectives:Triplets,Duplets,Multisets,HybridOperators,ModalLogic, HedgeAlgebras.AndApplications. EuropaNova,2017.

3. W.B.Vasantha,I.Kandasamy,andF.Smarandache,“Aclassicalgroupofneutrosophictripletgroups using {Z2p , ×},” Symmetry,vol.10,no.6,2018.

4. W.B.Vasantha,I.Kandasamy,andF.Smarandache,“Neutrosophicdupletsof {Zpn , ×} and {Zpq , ×} and theirproperties,” Symmetry,vol.10,no.8,2018.

5. W.B.Vasantha,I.Kandasamy,andF.Smarandache,“Algebraicstructureofneutrosophicdupletsin neutrosophicrings,” NeutrsophicSetsandSystems,vol.23,pp.85–95,2018.

6. W.B.Vasantha,I.Kandasamy,andF.Smarandache,“Neutrosophictripletsinneutrosophicrings,” Mathematics,vol.7,no.6,2019.

7. W.B.Vasantha,I.Kandasamy,andF.Smarandache,“Neutrosophicquadruplevectorspacesandtheir properties,” Mathematics,vol.7,no.8,2019.

8. M.Ali,F.Smarandache,andM.Khan,“Studyonthedevelopmentofneutrosophictripletringandneutrosophictripletfield,” Mathematics,vol.6,no.4,2018.

9. W.B.Vasantha,I.Kandasamy,andF.Smarandache,“Semi-idempotentsinneutrosophicrings,” Mathematics,vol.7,no.6,2019.

10. W.B.Vasantha, Smarandachesemigroups.AmericanResearchPress,2002.

Kandasamy,V.andet.al.,NeutroAlgebraofNeutrosophicTriplets

11. F.SmarandacheandM.Ali,“Neutrosophictripletgroup,” NeuralComputandApplic,vol.29,p.595–601, 2018.

12. A.RezaeiandF.Smarandache,“Theneutrosophictripletofbi-algebras,” NeutrosophicSetsandSystems, vol.33,no.1,p.20,2020.

13. W.B.Vasantha,I.Kandasamy,andF.Smarandache, NeutrosophicTripletGroupsandTheirApplications toMathematicalModelling.EuropaNova:Brussels,Belgium,2017,2017.

14. A.A.A.Agboola,A.D.Akwu,andY.Oyebo,“Neutrosophicgroupsandsubgroups,” Smarandache Multispace&Multistructure,vol.28,p.105,2013.

15. W.B.Vasantha,I.Kandasamy,andF.Smarandache,“Neutrosophiccomponentssemigroupsandmultiset neutrosophiccomponentssemigroups,” Symmetry,vol.12,no.5,p.818,2020.

16. A.A.A.Agboola,B.Davvaz,andF.Smarandache,“Neutrosophicquadruplealgebraichyperstructures,” AnnalsofFuzzyMathematicsandInformatics,vol.14,pp.29–42,2017.

17. M.Abobala,“Ah-subspacesinneutrosophicvectorspaces,” InternationalJournalofNeutrosophicScience, vol.6,pp.80–86,2020.

18. M.Abdel-Basset,A.Gamal,L.H.Son,F.Smarandache, etal.,“Abipolarneutrosophicmulticriteria decisionmakingframeworkforprofessionalselection,” AppliedSciences,vol.10,no.4,p.1202,2020.

19. M.Abdel-Basset,R.Mohamed,A.E.-N.H.Zaied,A.Gamal,andF.Smarandache,“Solvingthesupply chainproblemusingthebest-worstmethodbasedonanovelplithogenicmodel,”in OptimizationTheory BasedonNeutrosophicandPlithogenicSets,pp.1–19,Elsevier,2020.

20. M.Abdel-Basset,W.Ding,R.Mohamed,andN.Metawa,“Anintegratedplithogenicmcdmapproachfor financialperformanceevaluationofmanufacturingindustries,” RiskManagement,vol.22,no.3,pp.192–218,2020.

21. W.B.Vasantha,I.Kandasamy,andF.Smarandache,“Neutrosophicquadruplealgebraiccodesoverz2and theirproperties,” NeutrosophicSetsandSystems,vol.33,no.1,p.12,2020.

22. M.Abdel-Basst,R.Mohamed,andM.Elhoseny,“Anovelframeworktoevaluateinnovationvaluepropositionforsmartproduct-servicesystems,” EnvironmentalTechnology&Innovation,p.101036,2020.

23. M.Abdel-Basst,R.Mohamed,andM.Elhoseny,“¡?covid19?¿amodelfortheeffectivecovid-19identificationinuncertaintyenvironmentusingprimarysymptomsandctscans,” HealthInformaticsJournal, p.1460458220952918,2020.

24. E.Adeleke,A.Agboola,andF.Smarandache, RefinedneutrosophicringsIi.InfiniteStudy,2020.

25. X.Zhang,X.Wang,F.Smarandache,T.G.Ja´ıy´eol´a,andT.Lian,“Singularneutrosophicextendedtriplet groupsandgeneralizedgroups,” CognitiveSystemsResearch,vol.57,pp.32–40,2019.

26. X.Zhou,P.Li,F.Smarandache,andA.M.Khalil,“Newresultsonneutrosophicextendedtripletgroups equippedwithapartialorder,” Symmetry,vol.11,no.12,p.1514,2019.

27. X.Zhang,F.Smarandache,andX.Liang,“Neutrosophicdupletsemi-groupandcancellableneutrosophic tripletgroups,” Symmetry,vol.9,no.11,p.275,2017.

28. X.Zhang,X.Wu,X.Mao,F.Smarandache,andC.Park,“Onneutrosophicextendedtripletgroups(loops) andabel-grassmann’sgroupoids(ag-groupoids),” JournalofIntelligent&FuzzySystems,no.Preprint, pp.1–11,2019.

29. X.Zhang,X.Wu,F.Smarandache,andM.Hu,“Left(right)-quasineutrosophictripletloops(groups)and generalizedbe-algebras,” Symmetry,vol.10,no.7,2018.

30. X.Zhang,Q.Hu,F.Smarandache,andX.An,“Onneutrosophictripletgroups:Basicproperties,ntsubgroups,andsomenotes,” Symmetry,vol.10,no.7,2018.

31. Y.Ma,X.Zhang,X.Yang,andX.Zhou,“Generalizedneutrosophicextendedtripletgroup,” Symmetry, vol.11,no.3,2019.

Kandasamy,V.andet.al.,NeutroAlgebraofNeutrosophicTriplets

32. Q.Li,Y.Ma,X.Zhang,andJ.Zhang,“Neutrosophicextendedtripletgroupbasedonneutrosophic quadruplenumbers,” Symmetry,vol.11,no.5,2019.

33. F.Smarandache, IntroductiontoNeutroAlgebraicStructuresandAntiAlgebraicStructures(revisited).InfiniteStudy,2019.

34. F.Smarandache, NeutroAlgebraisaGeneralizationofPartialAlgebra.InfiniteStudy,2020.

35. F.SmarandacheandA.Rezaei,“Onneutro-be-algebrasandanti-be-algebras,” InternationalJournalof NeutrosophicScience,vol.4,no.1,p.8,2020.

36. F.SmarandacheandM.Hamidi,“Neutro-bck-algebra,” InternationalJournalofNeutrosophicScience, vol.8,no.2,p.110,2020.

37. A.Agboola,M.Ibrahim,andE.Adeleke,“Elementaryexaminationofneutroalgebrasandantialgebrasviza-viztheclassicalnumbersystems,” InternationalJournalofNeutrosophicScience(IJNS),vol.4,pp.16–19, 2020.

Received: June 1, 2020. Accepted: Nov 23, 2020

Kandasamy,V.andet.al.,NeutroAlgebraofNeutrosophicTriplets