Some Results on Neutrosophic Triplet Group and Their Applications

SomeResultsonNeutrosophicTripletGroupand TheirApplications

Tèmító . pé . Gbó . láhànJaíyéo . lá 1,* ID andFlorentinSmarandache 2 ID

1 DepartmentofMathematics,ObafemiAwolowoUniversity,IleIfe220005,Nigeria

2 DepartmentofMathematicsandScience,UniversityofNewMexico,705GurleyAve., Gallup,NM87301,USA;smarand@unm.edu

* Correspondence:tjayeola@oauife.edu.ng;Tel.:+234-813-961-1718

Received:23April2018;Accepted:15May2018;Published:6June2018

Abstract: Thisarticleisbasedonnewdevelopmentsonaneutrosophictripletgroup(NTG)and applicationsearlierintroducedin2016bySmarandacheandAli.NTGsprangupfromneutrosophic tripletset X:acollectionoftriplets (b, neut(b), anti(b)) foran b ∈ X thatobeyscertainaxioms (existenceofneutral(s)andopposite(s)).Someresultsthataretrueinclassicalgroupswere investigatedinNTGandwereshowntobeeitheruniversallytrueinNTGortrueinsomepeculiar typesofNTG.DistinguishingfeaturesbetweenanNTGandsomeotheralgebraicstructuressuchas: generalizedgroup(GG),quasigroup,loopandgroupwereinvestigated.Someneutrosophictriplet subgroups(NTSGs)ofaneutrosophictripletgroupwerestudied.Inparticular,foranyarbitrarily fixed a ∈ X,thesubsets Xa = {b ∈ X : neut(b)= neut(a)} and ker fa = {b ∈ X| f (b)= neut( f (a))} of X, where f : X → Y is aneutrosophictripletgrouphomomorphism,wereshowntobeNTSGand normalNTSG,respectively.Both Xa and ker fa wereshowntobe a-normalNTSGsandfoundto partition X.Consequently,aLagrange-likeformulawasfoundforafiniteNTG X; |X| = ∑ a∈X [Xa : ker fa ]| ker fa | basedonthefactthat | ker fa | |Xa |.Thefirstisomorphismtheorem X/ker f ∼ = Im f wasestablishedforNTGs.Usinganarbitrarynon-abelianNTG X anditsNTSG Xa,aBolstructure wasconstructed.Applicationsoftheneutrosophictripletset,andourresultsonNTGinrelationto managementandsports,arehighlightedanddiscussed.

Keywords: generalizedgroup;neutrosophictripletset;neutrosophictripletgroup;group

MSC: Primary20N02;Secondary20N05

1.Introduction

1.1.GeneralizedGroup

Unifiedgaugetheoryhasthealgebraicstructureofageneralizedgroupabstrusely,initsphysical background.Ithasbeenachallengeforphysicistsandmathematicianstofindadesirableunified theoryfortwistortheory,isotopiestheory,andsoon.Generalizedgroupsareinstrumentsfor constructionsinunifiedgeometrictheoryandelectroweaktheory.Completelysimplesemigroups arepreciselygeneralizedgroups(Araujoetal.[1]).AsrecordedinAdeniranetal.[2],studieson thepropertiesandstructuresofgeneralizedgroupshavebeencarriedoutinthepast,andthesehave beenextendedtosmoothgeneralizedgroupsandsmoothgeneralizedsubgroupsbyAgboola[3,4], topologicalgeneralizedgroupsbyMolaei[5],MolaeiandTahmoresi[6],andquotientspaceof generalizedgroupsbyMalekiandMolaei[7].

Symmetry 2018, 10,202;doi:10.3390/sym10060202 www.mdpi.com/journal/symmetry

Definition1 (GeneralizedGroup(GG)). Ageneralizedgroup X isanon-voidsetwithabinaryoperation calledmultiplicationobeyingthesetofrulesgivenbelow.

(i) (ab)c = a(bc) foralla, b, c ∈ X.

(ii) Foreach a ∈ X thereisaunique e(a) ∈ X suchthat ae(a)= e(a)a = a (existenceanduniquenessof identityelement).

(iii) Foreacha ∈ X,thereisa 1 ∈ Xsuchthataa 1 = a 1 a = e(a) (existenceofinverseelement).

Definition2. Let X beanon-voidset.Let(·)beabinaryoperationon X.Whenever a · b ∈ X forall a, b ∈ X, then (X, ·) iscalledagroupoid.

Whenevertheequation c · x = d (or y · c = d)haveuniquesolutionwithrespectto x (or y)i.e.,satisfies theleft(orright)cancellationlaw,then (X, ·) iscalledaleft(orright)quasigroup.Ifagroupoid (X, ·) isboth aleftquasigroupandrightquasigroup,thenitiscalledaquasigroup.Ifthereisanelement e ∈ X calledthe identityelementsuchthatforalla ∈ X,a · e = e · a = a,thenaquasigroup (X, ·) iscalledaloop.

Definition3. AloopiscalledaBolloopwheneveritsatisfiestheidentity

((ab)c)b = a((bc)b)

Remark1. OneofthemoststudiedclassesofloopsistheBolloop.

Formoreonquasigroupsandloops,interestedreaderscancheck[8–15]. Ageneralizedgroup X hasthefollowingproperties:

(i) Foreach a ∈ X,thereisaunique a 1 ∈ X (ii) e(e(a))= e(a) and e(a 1)= e(a) if a ∈ X (iii) If X iscommutative,then X isagroup.

1.2.NeutrosophicTripletGroup

Neutrosophyisanovelsubdivisionofphilosophythatstudiesthenature,origination,andambitof neutralities,includingtheirinteractionwithideationalspectra.FlorentinSmarandache[16]introduced thenotionofneutrosophiclogicandneutrosophicsetsforthefirsttimein1995.Asamatteroffact, theneutrosophicsetisthegeneralizationofclassicalsets[17],fuzzysets[18],intuitionisticfuzzy sets[17,19],andintervalvaluedfuzzysets[17],tociteafew.Thegrowthprocessofneutrosophic sets,fuzzysets,andintuitionisticfuzzysetsarestillevolving,withdiverseapplications.Somerecent researchfindingsinthesedirectionsare[20–27].

SmarandacheandAli[28]werethefirsttointroducethenotionoftheneutrosophictriplet,which theyhadearliertalkedaboutataconference.Theseneutrosophictripletswereusedbythemto introducetheneutrosophictripletgroup,whichdiffersfromtheclassicalgroupbothinfundamental andstructuralproperties.Thedistinctionandcomparisonoftheneutrosophictripletgroupwiththe classicalgeneralizedgroupweregiven.Theyalsodrewabriefoutlineofthepotentialapplications oftheneutrosophictripletgroupinotherresearchfields.Fordiscussionsofresultsonneutrosophic tripletgroups,neutrosophicquadruples,andneutrosophicdupletsofalgebraicstructures,aswellas newapplicationsofneutrosophy,seeJaiyéo . láandSmarandache[29].Jaiyéo . láandSmarandache[29] werethefirsttointroduceandstudyinversepropertyneutrosophictripletloopswithapplicationsto cryptographyforthefirsttime.

Definition4 (NeutrosophicTripletSet-NTS). Let X beanon-voidsettogetherwithabinaryoperation definedonit.Then X iscalledaneutrosophictripletsetif,forany a ∈ X,thereisaneutralof‘a’denotedby neut(a) (notnecessarilytheidentityelement)andanoppositeof‘a’denotedby anti(a),with neut(a), anti(a) ∈ Xsuchthat

a neut(a)= neut(a) a = aanda anti(a)= anti(a) a = neut(a)

Theelements a, neut(a) and anti(a) aretogethercalledneutrosophictriplet,andrepresentedby (a, neut(a), anti(a)).

Remark2. Foran a ∈ X,eachof neut(a) and anti(a) maynotbeunique.Inaneutrosophictripletset (X, ), anelement b (or c)isthesecond(orthird)componentofaneutrosophictripletif a, c ∈ X (a, b ∈ X)suchthat a b = b a = aanda c = c a = b.Thus, (a, b, c) isaneutrosophictriplet.

Example1 (SmarandacheandAli[28]). Consider (Z6, ×6) suchthat Z6 = {0,1,2,3,4,5} and ×6 is multiplicationinmodulo 6 (2,4,2), (4,4,4),and (0,0,0) areneutrosophictriplets,but 3 willnotgiverisetoa neutrosophictriplet.

Definition5 (NeutrosophicTripletGroup—NTG). Let (X, ) beaneutrosophictripletset.Then (X, ) is referredtoasaneutrosophictripletgroupif (X, ) isasemigroup.Furthermore,if (X, ) obeysthecommutativity law,then (X, ) isreferredtoasacommutativeneutrosophictripletgroup.

Let (X, ) beaneutrosophictripletgroup.Whenever neut(ab)= neut(a)neut(b) forall a, b ∈ X, thenXisreferredtoasanormalneutrosophictripletgroup.

Let (X, ) beaneutrosophictripletgroupandlet H ⊆ X H isreferredtoasaneutrosophictriplet subgroup(NTSG)of X if (H, ) isaneutrosophictripletgroup.Whence,foranyfixed a ∈ X, H iscalled a-normalNTSGofX,writtenH a Xifayanti(a) ∈ Hforally ∈ H.

Remark3. AnNTGisnotnecessarilyagroup.However,agroupisanNTGwhere neut(a)= e,thegeneral identityelementforalla ∈ X,andanti(a) isuniqueforeacha ∈ X.

Example2 (SmarandacheandAli[28]). Consider (Z10, ⊗) suchthat c ⊗ d = 3cd mod10. (Z10, ⊗) isa commutativeNTGbutneitheraGGnoraclassicalgroup.

Example3 (SmarandacheandAli[28]). Consider (Z10, ) suchthat c d = 5c + d mod10. (Z10, ) isa non-commutativeNTGbutnotaclassicalgroup.

Definition6 (NeutrosophicTripletGroupHomomorphism). Let f : X → Y beamappingsuchthat X and Y aretwoneutrosophictripletgroups.Then f isreferredtoasaneutrosophictripletgrouphomomorphism iff (cd)= f (c) f (d) forallc, d ∈ X.Thekerneloffata ∈ Xisdefinedby ker fa = {x ∈ X : f (x)= neut( f (a))}.

TheKerneloffisdefinedby ker f = a∈X ker fa suchthatfa = f |Xa ,whereXa = {x ∈ X : neut(x)= neut(a)}

Remark4. Thedefinitionofneutrosophictripletgrouphomomorphismaboveismoregeneralthanthat inSmarandacheandAli[28].InTheorem 5,itisshownthat,foranNTGhomomorphism f : X → Y, f (neut(a))= neut( f (a)) andf (anti(a))= anti( f (a)) foralla ∈ X.

Thepresentworkisacontinuationofthestudyofaneutrosophictripletgroup(NTG)andits applications,whichwasintroducedbySmarandacheandAli[28].Someresultsthataretrueinclassical groupswereinvestigatedinNTGandwillbeprovedtobeeithergenerallytrueinNTGortruein someclassesofNTG.Someapplicationsoftheneutrosophictripletset,andourresultsonNTGin relationtomanagementandsportswillbediscussed.

ThefirstsectionintroducesGGandNTGandhighlightsexistingresultsthatarerelevantto thepresentstudy.Section 2 establishesnewresultsonalgebraicpropertiesofNTGsandNTG homomorphisms,amongwhichareLagrange’sTheoremandthefirstisomorphismtheorem, andpresentsamethodoftheconstructionofBolalgebraicstructuresusinganNTG.Thethirdsection describesapplicationsofNTGstohumanmanagementandsports.

2.MainResults

Weshallfirstestablishtherelationshipamonggeneralizedgroups,quasigroups,andloopswith aneutrosophictripletgroupassumed.

Lemma1. LetXbeaneutrosophictripletgroup.

1. X isageneralizedgroupifitsatisfiestheleft(orright)cancellationlawor X isaleft(orright)quasigroup.

2. Xisageneralizedgroupifandonlyifeachelementx ∈ Xhasauniqueneut(x) ∈ X.

3. WheneverXhasthecancellationlaws(orisaquasigroup),thenXisaloopandgroup.

Proof. 1. Let x haveatleasttwoneutralelements,say neut(x), neut(x) ∈ X.Then xx = xx ⇒ xxanti(x)= xxanti(x) ⇒ xneut(x)= xneut(x) leftquasigroup =⇒ leftcancellationlaw neut(x)= neut(x) .Therefore, X isageneralizedgroup.Similarly, X isageneralizedgroupifitishastherightcancellationlaw orifitisarightquasigroup.

2. Thisfollowsbydefinition.

3. Thisisstraightforwardbecauseeveryassociativequasigroupisaloopandgroup.

2.1.AlgebraicPropertiesofNeutrosophicTripletGroup

WenowestablishsomenewalgebraicpropertiesofNTGs.

Theorem1. LetXbeaneutrosophictripletgroup.Foranya ∈ X,anti anti(a) = a.

Proof. anti anti(a) anti(a)= neut anti(a) = neut(a) byTheorem1([29]).Aftermultiplyingby a, weobtain anti anti(a) anti(a) a = neut(a)a = a.(1)

LHS = anti anti(a) anti(a)a = anti anti(a) neut(a) = anti anti(a) neut anti(a) = anti anti(a) neut anti anti(a) = anti anti(a) (2)

Hence,basedonEquations(1)and(2), anti anti(a) = a

Theorem2. Let X beaneutrosophictripletgroupsuchthattheleftcancellationlawissatisfied, and neut(a)= neut aanti(b) ifandonlyif aanti(b)= a.Then X isanidempotentneutrosophictriplet groupifandonlyifneut(a)anti(b)= anti(b)neut(a) ∀ a, b ∈ X.

Proof. neut(a)anti(b)= anti(b)neut(a) ⇔ (aneut(a))anti(b)= aanti(b)neut(a) ⇔ aanti(b)= aanti(b)neut(a) ⇔ neut(a)= neut aanti(b) ⇔ aanti(b)= a ⇔ aanti(b)b = ab ⇔ aneut(b)= ab ⇔ anti(a)aneut(b)= anti(a)ab ⇔ neut(a)neut(b)= neut(a)b ⇔ neut(b)= b ⇔ b = bb.

Theorem3. Let X beanormalneutrosophictripletgroupinwhich neut(a)anti(b)= anti(b)neut(a) ∀ a, b ∈ X. Then,anti(ab)= anti(b)anti(a) ∀ a, b ∈ X.

Proof. Since anti(ab)(ab)= neut(ab),thenbymultiplyingbothsidesoftheequationontherightby anti(b)anti(a),weobtain

anti(ab)ab anti(b)anti(a)= neut(ab)anti(b)anti(a).(3)

GoingbyTheorem1([29]),

anti(ab)ab anti(b)anti(a)= anti(ab)a banti(b) anti(a)= anti(ab)a(neut(b)anti(a))

= anti(ab)(aanti(a))neut(b)= anti(ab) neut(a)neut(b)

= anti(ab)neut(ab)= anti(ab)neut anti(ab))= anti(ab)

UsingEquations(3)and(4),weobtain

anti(ab)ab anti(b)anti(a)= anti(ab) ⇒

neut(ab) anti(b)anti(a) = anti(ab) ⇒ anti(ab)= anti(b)anti(a)

(4)

Itisworthcharacterizingtheneutrosophictripletsubgroupofagivenneutrosophictripletgroup toseehowanewNTGcanbeobtainedfromexistingNTGs.

Lemma2. LetHbeanon-voidsubsetofaneutrosophictripletgroupX.Thefollowingareequivalent.

(i) HisaneutrosophictripletsubgroupofX. (ii) Foralla, b ∈ H,aanti(b) ∈ H. (iii) Foralla, b ∈ H,ab ∈ H,andanti(a) ∈ H.

Proof. (i)⇒ (ii) If H isanNTSGof X and a, b ∈ H,then anti(b) ∈ H.Therefore,byclosureproperty, aanti(b) ∈ H ∀ a, b ∈ H. (ii)⇒ (iii) If H = ∅,and a, b ∈ H,thenwehave banti(b)= neut(b) ∈ H, neut(b)anti(b)= anti(b) ∈ H, and ab = aanti(anti(b)) ∈ H,i.e., ab ∈ H (iii)⇒ (i) H ⊆ X,so H isassociativesince X isassociative.Obviously,forany a ∈ H, anti(a) ∈ H Let a ∈ H,then anti(a) ∈ H.Therefore, aanti(a)= anti(a)a = neut(a) ∈ H.Thus, H isan NTSGof X

Theorem4. LetGandHbeneutrosophictripletgroups.ThedirectproductofGandHdefinedby G × H = {(g, h) : g ∈ Gandh ∈ H} isaneutrosophictripletgroupunderthebinaryoperation ◦ definedby (g1, h1) ◦ (g2, h2)=(g1 g2, h1h2) Proof. Thisissimplydonebycheckingtheaxiomsofneutrosophictripletgroupforthepair (G × H, ◦), inwhichcase neut(g, h)= neut(g), neut(h) and anti(g, h)= anti(g), anti(h) Lemma3. Let H = {Hi }i∈Ω beafamilyofneutrosophictripletsubgroupsofaneutrosophictripletgroup Xsuchthat i∈Ω Hi = ∅.Then i∈Ω Hi isaneutrosophictripletsubgroupofX.

Proof. ThisisaroutineverificationusingLemma 2.

2.2.NeutrosophicTripletGroupHomomorphism

LetusnowestablishresultsonNTGhomomorphisms,itskernels,andimages,aswellas aLagrange-likeformulaandtheFirstIsomorphismTheoremforNTGs.

Theorem5. Letf : X → YbeahomomorphismwhereXandYaretwoneutrosophictripletgroups.

1. f (neut(a))= neut( f (a)) foralla ∈ X.

2. f (anti(a))= anti( f (a)) foralla ∈ X.

3.

4.

IfHisaneutrosophictripletsubgroupofX,thenf (H) isaneutrosophictripletsubgroupofY.

IfKisaneutrosophictripletsubgroupofY,then ∅ = f 1(K) isaneutrosophictripletsubgroupofX.

5. IfXisanormalneutrosophictripletgroupandthesetX f = {(neut(a), f (a)) : a ∈ X} withtheproduct (neut(a), f (a))(neut(b), f (b)) :=(neut(ab), f (ab)), then X f isaneutrosophictripletgroup.

Proof. Since f isanhomomorphism, f (ab)= f (a) f (b) forall a, b ∈ X

1. Place b = neut(a) in f (ab)= f (a) f (b) toobtain f aneut(a) = f (a) f (neut(a)) ⇒ f (a)= f (a) f (neut(a)).Additionally,place b = neut(a) in f (ba)= f (b) f (a) toobtain f neut(a)a = f (neut(a)) f (a) ⇒ f (a)= f (neut(a)) f (a).Thus, f (neut(a))= neut( f (a)) forall a ∈ X.

2. Place b = anti(a) in f (ab)= f (a) f (b) toobtain f aanti(a) = f (a) f (anti(a)) ⇒ f (neut(a))= f (a) f (anti(a)) ⇒ neut( f (a))= f (a) f (anti(a)).Additionally,place b = anti(a) in f (ba)= f (b) f (a) toobtain f anti(a)a = f (anti(a)) f (a) ⇒ f (neut(a))= f (a) f (anti(a)) ⇒ neut( f (a))= f (anti(a)) f (a).Thus, f (anti(a))= anti( f (a)) forall a ∈ X

3. If H isanNTSGof G,then f (H)= { f (h) ∈ Y : h ∈ H}.Weshallprovethat f (H) isanNTSGof Y byLemma 2

Since f (neut(a))= neut( f (a)) ∈ f (H) for a ∈ H, f (H) = ∅.Let a , b ∈ f (H).Then a = f (a) and b = f (b).Thus, a anti(b )= f (a)anti( f (b))= f (a) f (anti(b))= f (aanti(b)) ∈ f (H). Therefore, f (H) isanNTSGof Y.

4. If K isaneutrosophictripletsubgroupof Y,then ∅ = f 1(K)= {a ∈ X : f (a) ∈ K}.Weshall provethat f (H) isanNTSGof Y byLemma 2 Let a, b ∈ f 1(K).Then a , b ∈ K suchthat a = f (a) and b = f (b).Thus, a anti(b )= f (a)anti( f (b))= f (a) f (anti(b))= f (aanti(b)) ∈ K ⇒ aanti(b) ∈ f 1(K).Therefore, f 1(K) is anNTSGof X

5. Giventheneutrosophictripletgroup X andtheset X f = {(neut(a), f (a)) : a ∈ X} withthe product (neut(a), f (a))(neut(b), f (b)) :=(neut(ab), f (ab)) X f isagroupoid. (neut(a), f (a))(neut(b), f (b)) (neut(z), f (z))=(neut(ab), f (ab))(neut(z), f (z))= (neut(abz), f (abz)) =(neut(a), f (a))(neut(bz), f (bz))=(neut(a), f (a)) (neut(b), f (b))(neut(z), f (z)) Therefore, X f isasemigroup.

For (neut(a), f (a)) ∈ X f , let neut(neut(a), f (a))= neut(neut(a)), neut( f (a)) .Then neut(neut(a), f (a))= neut(a), ( f (neut(a)) ∈ X f .Additionally,let anti(neut(a), f (a))= anti(neut(a)), anti( f (a)) .Then anti(neut(a), f (a))= neut(a), f (anti(a)) ∈ X f . Thus, (neut(a), f (a))neut(neut(a), f (a))=(neut(a), f (a)) neut(a), ( f (neut(a)) = (neut(a), f (a)) neut(anti(a)), ( f (neut(a)) = neut(aanti(a)), f (aneut(a)) = neut(neut(a)), f (aneut(a)) =(neut(a), f (a)) ⇒ (neut(a), f (a))neut(neut(a), f (a))= (neut(a), f (a)) andsimilarly, neut(neut(a), f (a))(neut(a), f (a))=(neut(a), f (a)).

Ontheotherhand, (neut(a), f (a))anti(neut(a), f (a))=(neut(a), f (a)) neut(a), f (anti(a)) = (neut(a), f (a)) neut(anti(a)), ( f (anti(a)) = neut(aanti(a)), f (aanti(a)) = neut(neut(a)), f (neut(a)) = neut(a), ( f (neut(a)) = neut(neut(a), f (a)) ⇒ (neut(a), f (a)) anti(neut(a), f (a))= neut(neut(a), f (a)) andsimilarly, anti(neut(a), f (a)) (neut(a), f (a))= neut(neut(a), f (a)) Therefore, X f isaneutrosophictripletgroup.

Theorem6. Letf : X → Ybeaneutrosophictripletgrouphomomorphism.

1. ker fa a X.

2. Xa a X.

3. Xa isanormalneutrosophictripletgroup.

4. anti(cd)= anti(d)anti(c) ∀ c, d ∈ Xa.

5. Xa = c∈Xa

c ker fa foralla ∈ X.

6. If X isfinite, |Xa | = ∑c∈Xa |c ker fa | =[Xa :ker fa ]| ker fa | forall a ∈ X where [Xa :ker fa ] isthe indexof ker fa inXa,i.e.,thenumberofdistinctleftcosetsof ker fa inXa.

7. X = a∈X Xa 8. IfXisfinite, |X| = ∑ a∈X [Xa :ker fa ]| ker fa |

Proof. 1. f (neut(a))= neut( f (a))= neut(neut( f (a)))= neut( f (neut(a))) ⇒ neut(a) ∈ ker fa ⇒ ker fa = ∅.Let c, d ∈ ker fa,then f (c)= f (d)= neut( f (a)).WeshalluseLemma 2 f (canti(d))= f (c) f (anti(d))= f (c)anti( f (d))= neut( f (a))anti(neut( f (a)))= neut( f (a))neut( f (a))= neut( f (a)) ⇒ canti(d) ∈ ker fa Thus, ker fa isaneutrosophictripletsubgroupof X.Forthe a-normality,let d ∈ ker fa,then f (d)= neut( f (a)).Therefore, f (adanti(a))= f (a) f (d) f (anti(a))= f (a)neut( f (a))anti( f (a))= f (a)anti( f (a))= neut( f (a)) ⇒ adanti(a) ∈ ker fa forall d ∈ ker fa Therefore,ker fa a X 2. Xa = {c ∈ X : neut(c)= neut(a)}. neut(neut(a))= neut(a) ⇒ neut(a) ∈ Xa.Therefore, Xa = ∅. Let c, d ∈ Xa.Then neut(c)= neut(a)= neut(d). (cd)neut(a)= c(dneut(a))= c(dneut(d))= cd,and neut(a)(cd)=(neut(a)c)d =(neut(c)c)d = cd.Therefore, neut(cd)= neut(a). neut(anti(c))= anti(neut(c))= anti(neut(a))= neut(a) ⇒ anti(c) ∈ Xa.Thus, Xa is aneutrosophictripletsubgroupof X. neut(anti(a))= neut(a) ⇒ anti(a) ∈ Xa.Therefore, (acanti(a))neut(a)= (ac) anti(a)neut(a) = acanti(a),and neut(a)(acanti(a))= neut(a)a(canti(a))= acanti(a) Thus, neut(acanti(a))= neut(a) ⇒ acanti(a) ∈ Xa.Therefore, Xa a X

3. Let c, d ∈ Xa.Then neut(c)= neut(a)= neut(d).Therefore, neut(cd)= neut(a)= neut(a)neut(a)= neut(c)neut(d).Thus, Xa isanormalNTG.

4. Forall c, d ∈ Xa, neut(c)anti(d)= neut(a)anti(d)= neut(d)anti(d)= anti(d)= anti(d)neut(d)= anti(d)neut(a)= anti(d).Therefore,basedonPoint3andTheorem 3, anti(cd)= anti(d)anti(c) ∀ c, d ∈ Xa.

5. Definearelation on Xa asfollows: c d if anti(c)d ∈ ker fa forall c, d ∈ Xa anti(c)c = neut(c)= neut(a) ⇒ anti(c)c ∈ ker fa ⇒ c c.Therefore, isreflexive.

c d ⇒ anti(c)d ∈ ker fa by4. ⇒ anti(anti(c)d) ∈ ker fa ⇒ anti(d)c ∈ ker fa ⇒ d c.Therefore, issymmetric.

c d, d z ⇒ anti(c)d, anti(d)z ∈ ker fa ⇒ anti(c)danti(d)z = anti(c)neut(d)z = anti(c)neut(a)z = anti(c)z ∈ ker fa ⇒ c z.Therefore, istransitiveand isan

c ker fa forall a ∈ X

equivalencerelation. Theequivalenceclass [c] fa = {d : anti(c)d ∈ ker fa } = {d : canti(c)d ∈ c ker fa } = {d : neut(c)d ∈ c ker fa } = {d : neut(a)d ∈ c ker fa } = {d : d ∈ c ker fa } = c ker fa.Therefore, Xa / = {[c] fa }c∈Xa = {c ker fa }c∈Xa . Thus, Xa = c∈Xa

6. If X isfinite,then | ker fa | = |c ker fa | forall c ∈ Xa.Thus, |Xa | = ∑c∈Xa |c ker fa | =[Xa : ker fa ]| ker fa | forall a ∈ X where [Xa :ker fa ] istheindexof ker fa in Xa,i.e.,thenumberof distinctleftcosetsofker fa in Xa. 7. Definearelation ∼ on X: c ∼ d if neut(c)= neut(d). ∼ isanequivalencerelationon X,so X/ ∼= {Xc }c∈X and,therefore, X = a∈X Xa. 8. Hence,basedonPoint7,if X isfinite,then |X| = ∑ a∈X |Xa | = ∑ a∈X [Xa :ker fa ]| ker fa |

Theorem7. Leta ∈ Xandf : X → Ybeaneutrosophictripletgrouphomomorphism.Then

1. fisamonomorphismifandonlyif ker fa = {neut(a)} foralla ∈ X; 2. thefactorset X/ker f = a∈X Xa /ker fa isaneutrosophictripletgroup(neutrosophictripletfactorgroup) undertheoperationdefinedby c ker fa d ker fb =(cd) ker fab

Proof. 1. Let ker fa = {neut(a)} andlet c, d ∈ X.If f (c)= f (d),thisimpliesthat f (canti(d))= f (d)anti( f (d))= f (danti( f (d))) ⇒ f (canti(d))= neut( f (d)) ⇒ canti(d) ∈ ker fd ⇒ canti(d)= neut(d)= neut(anti(d)).(5)

Similarly, f (anti(d)c)= neut( f (d)) ⇒ anti(d)c ∈ ker fd ⇒ anti(d)c = neut(anti(d)).(6)

UsingEquations(5)and(6), c = anti(anti(d))= d.Therefore, f isamonomorphism.

Conversely,if f ismono,then f (d)= f (c) ⇒ d = c.Let k ∈ ker fa, a ∈ X Then f (k)= neut( f (a))= f (neut(a)) ⇒ k = neut(a).Therefore, ker fa = {neut(a)} forall a ∈ X.

2. Let c ker fa, d ker fb, z ker fc ∈ X/ker f = a∈X Xa /ker fa.

Groupoid: Basedonthemultiplication c ker fa d ker fb =(cd) ker fab,thefactorset X/ker f is agroupoid.

Semigroup: c ker fa d ker fb z ker fc =(cdz) ker fabc = c ker fa d ker fb z ker fc

Neutrality: Let neut(c ker fa )= neut(c) ker fneut(a).Then c ker fa · neut(c ker fa )= c ker fa · neut(c) ker fneut(a) =(cneut(c)) ker faneut(a) = c ker fa andsimilarly, neut(c ker fa ) · c ker fa = c ker fa

Opposite: Let anti(c ker fa )= anti(c) ker fanti(a).Then c ker fa anti(c ker fa )= c ker fa anti(c) ker fanti(a) =(canti(c)) ker faanti(a) = neut(c) ker fneut(a).Similarly, anti(c ker fa )) c ker fa = neut(c) ker fneut(a).

∴ X/ker f , isanNTG.

Theorem8. Let φ : X → Ybeaneutrosophictripletgrouphomomorphism.ThenX/ker φ ∼ = Im φ.

Proof. BasedonTheorem 6(7), X = a∈X Xa.Similarly,definearelation ≈ on φ(X)= Im φ: φ(c) ≈ φ(d) if neut(φ(c))= neut(φ(d)). ≈ isanequivalencerelationon φ(X),so φ(X)/ ≈= {φ(Xc )}c∈X and Im φ = c∈X φ(Xc ).Itshouldbenotedthat Xa a X inTheorem 6(2).

Let φa : Xa/ker φa → φ(Xa) givenby φa(c ker φa)= φ(c).Itshouldbenotedthat,byTheorem 6(1), ker φa a X.Therefore, c ker φa = d ker φa ⇒ anti(d)c ker φa = anti(d)d ker φa = neut(d) ker φa = ker φa ⇒ anti(d)c ker φa = ker φa ⇒ φ(anti(d)c)= neut(φ(a)) ⇒ anti(φ(d))φ(c)= neut(φ(a)) ⇒ φ(d)anti(φ(d))φ(c)= φ(d)neut(φ(a)) ⇒ neut(φ(d))φ(c)= φ(d)neut(φ(a)) ⇒ φ(neut(d))φ(c)= φ(d)φ(neut(a)) ⇒ φ(neut(d) c))= φ(dneut(a)) ⇒ φ(neut(a) c))= φ(dneut(a)) ⇒ φ(neut(c) c))= φ(dneut(c)) ⇒ φ(c)= φ(d) ⇒ φa (c ker φa )= φa (d ker φa ).Thus, φa iswelldefined. φa (c ker φa )= φa (d ker φa ) ⇒ φ(c)= φ(d) ⇒ anti(φ(d))φ(c)= anti(φ(d))φ(d)= neut(φ(d)) ⇒ φ(anti(d))φ(c)= neut(φ(d))= φ(neut(d))= φ(neut(a))= neut(φ(a)) ⇒ φ(anti(d) c)= neut(φ(a)) ⇒ anti(d) c ∈ ker φa ⇒ danti(d) c ∈ d ker φa ⇒ neut(d) c ∈ d ker φa ⇒ neut(a) c ∈ d ker φa ⇒ c ∈ d ker φa Theorem 6(1) =⇒ c ker φa = d ker φa.Thismeansthat φa is1-1. φa isobviouslyonto. Thus, φa isbijective. Now,basedontheaboveandTheorem 7(2),wehaveabijection Φ = a∈X φa : X/ker φ = a∈X Xa /ker φa → Im φ = φ(X)= a∈X φ(Xa ) definedby Φ(c ker φa )= φ(c).Thus,if c ker φa, d ker φb ∈ X/ker φ,then Φ c ker φa · d ker φb = Φ cd ker φa b = φ(cd)= φ(c)φ(d)= Φ c ker φa Φ d ker φb ∴ X/ker φ ∼ = Im φ

2.3.ConstructionofBolAlgebraicStructures

WenowpresentamethodofconstructingBolalgebraicstructuresusinganNTG. Theorem9. Let X beanon-abelianneutrosophictripletgroupandlet A = Xa × X foranyfixed a ∈ X For (h1, g1), (h2, g2) ∈ A,define ◦ onAasfollows: (h1, g1) ◦ (h2, g2)=(h1h2, h2 g1 anti(h2)g2).

Then (A, ◦) isaBolgroupoid.

Proof. Let a, b, c ∈ A.Bychecking,itistruethat a ◦ (b ◦ c) =(a ◦ b) ◦ c.Therefore, (A, ◦) is non-associative. Xa isanormalneutrosophictripletgroupbyTheorem 6(3). A isagroupoid.

LetusnowverifytheBolidentity: ((a ◦ b) ◦ c) ◦ b = a ◦ ((b ◦ c) ◦ b) LHS =((a ◦ b) ◦ c) ◦ b = h1h2h3h2, h2h3h2 g1 anti(h2)g2 anti(h3)g3 anti(h2)g2 .

FollowingTheorem 6(4), RHS = a ◦ ((b ◦ c) ◦ b)=

h1h2h3h2, h2h3h2 g1 anti h2h3h2 h2h3 g2 anti(h3)g3 anti(h2)g2 = h1h2h3h2, h2h3h2 g1 anti(h2) anti(h3) anti(h2)h2h3 g2 anti(h3)g3 anti(h2)g2 = h1h2h3h2, h2h3h2 g1 anti(h2) anti(h3) neut(h2)h3 g2 anti(h3)g3 anti(h2)g2 = h1h2h3h2, h2h3h2 g1 anti(h2) anti(h3) neut(a)h3 g2 anti(h3)g3 anti(h2)g2 =

h1h2h3h2, h2h3h2 g1 anti(h2)anti(h3)h3 g2 anti(h3)g3 anti(h2)g2 = h1h2h3h2, h2h3h2 g1 anti(h2)neut(h3)g2 anti(h3)g3 anti(h2)g2 = h1h2h3h2, h2h3h2 g1 anti(h2)neut(a)g2 anti(h3)g3 anti(h2)g2 = h1h2h3h2, h2h3h2 g1 anti(h2)g2 anti(h3)g3 anti(h2)g2

Therefore,LHS=RHS.Hence, (A, ◦) isaBolgroupoid.

Corollary1. Let H beasubgroupofanon-abelianneutrosophictripletgroup X,andlet A = H × X. For (h1, g1), (h2, g2) ∈ A,define ◦ onAasfollows:

(h1, g1) ◦ (h2, g2)=(h1h2, h2 g1 anti(h2)g2)

Then (A, ◦) isaBolgroupoid.

Proof. Asubgroup H isanormalneutrosophictripletgroup.Therestoftheclaimfollowsfrom Theorem 9.

Corollary2. Let H beaneutrosophictripletsubgroup(whichobeysthecancellationlaw)ofanon-abelian neutrosophictripletgroupX,andletA = H × X.For (h1, g1), (h2, g2) ∈ A,define ◦ onAasfollows: (h1, g1) ◦ (h2, g2)=(h1h2, h2 g1 anti(h2)g2)

Then (A, ◦) isaBolgroupoid.

Proof. ByTheorem 1(3), H isasubgroupof X.Hence,followingCorollary 1, (A, ◦) isaBolgroupoid.

Corollary3. Let H beaneutrosophictripletsubgroupofanon-abelianneutrosophictripletgroup X thathas thecancellationlawandletA = H × X.For (h1, g1), (h2, g2) ∈ A,define ◦ onAasfollows: (h1, g1) ◦ (h2, g2)=(h1h2, h2 g1 anti(h2)g2)

Then (A, ◦) isaBolloop.

Proof. ByTheorem 1(3), X isanon-abeliangroupand H isasubgroupof X.Hence, (A, ◦) isaloop andaBolloopbyTheorem 9.

3.ApplicationsinManagementandSports

3.1.One-WayManagementandDivisionofLabor

Consideracompanyorworkplaceconsistingofasetofpeople X with |X| numberofpeople. Aworkingunitorsubgroupwithaleader‘a’isdenotedby Xa neut(x) forany x ∈ X representsaco-worker(orco-workers)whohas(have)agood(non-critical) workingrelationshipwith x,while anti(x) representsaco-worker(orco-workers)whom x considers ashis/herpersonalcritic(s)atwork.

Hence, Xa canbesaidtoincludebothcriticsandnon-criticsofeachworker x.Itshouldbe notedthatin Xa, neut(a)= neut(x) forall x ∈ Xa.Thismeansthateveryworkerin Xa hasagood relationshipwiththeleader‘a’.

Thus,byTheorem 6(7)—X = a∈X Xa and |X| = Σa∈X |Xa |—thecompanyorworkplace X canbe saidtohaveagooddivisionoflaborforeffectiveperformanceandmaximumoutputbasedonthe compositionofitsvariousunits(Xa).SeeFigure 1 X (company) Xa1

WorkingUnit WorkingUnit Xa2 a2

WorkingUnit

Xa3 ...

Leader neut(x) x anti(x)

Figure1. One-waymanagementanddivisionoflabor.

3.2.Two-WayManagementDivisionofLabor

Consideracompanyorworkplaceconsistingofasetofpeople X with |X| numberofpeopleat alocation A andanothercompanyorworkplaceconsistingofpeople Y with |Y| numberofpeopleat anotherlocation B.Assumethatbothcompaniesareownedbythesameperson f .Hence, f : X → Y canbeconsideredasamovement(transfer)orworkinginteractionbetweenworkersat A andat B. Thefactthat f isaneutrosophictripletgrouphomomorphismindicatesthattheworkinginteraction between X and Y ispreserved. Let‘a’beaunitleaderat A whoseworkcorrelatestoanotherleader f (a) at B Then Kerfa representsthesetofworkers x inaunitat A undertheleadershipof‘a’suchthatthereare other,correspondingworkers f (x) at B undertheleadershipof f (a).Here, f (x)= neut( f (a)) means thatworkers f (x) at B undertheleadershipof f (a) areloyalandinagoodworkingrelationship. Themapping fa showsthattheoperationofasubgroupleader(theoperationisdenotedby‘a’)is subjecttothemodusoperandioftheownerofthetwocompanies,wheretheownerisdenotedby f Thefinalformula |X| = ∑ x∈X [Xa :ker fa ]|kerfa | inTheorem 6(8)showsthattheoverallperformance ofthesetofpeople X isdeterminedbyhowtheunitleaders‘a’at A properlyharmonizewiththeunit leadersat B intheeffectiveadministrationofker fa and Xa (Figure 2).

Figure2. Two-waymanagementdivisionoflabor.

3.3.Sports

Inthecompositionofateam,acoachcantake Xa asthesetofplayerswhoplayinaparticular department(e.g.,forward,middlefield,ordefence),where a istheleaderofthatdepartment. Let neut(x) representplayer(s)whoseperformanceisthesameasthatofplayer x,andlet anti(x) representplayer(s)thatcanperformbetterthanplayer x.Itshouldbenotedthatthecondition neut(x)= neut(a) forall x ∈ Xa meansthatthedepartment Xa hasplayer(s)whoareequalin performance;i.e.,thosewhoseperformanceareequaltothatofthedepartmentalleader a.Hence, aneutrosophictriplet (x, neut(x), anti(x)) isatriplefromwhichacoachcanmakeachoiceofhis/her startingplayerandmakeasubstitution.Theneutrosophictripletcanalsohelpacoachtomake thebestalternativechoicewheninjuriesarise.Forinstance,inthegoalkeepingdepartment(for soccer/football),threegoalkeepersoftenmakeuptheteamforanyinternationalcompetition.Imagine anincompletetriplet (x, neut(x), ?),i.e.,noplayerisfoundtobebetterthan x,whichreducesto aduplet.

Xa canalsobeusedforgroupingteamsincompetitionsinthepreliminaries.If x = team, then anti(x)= teams thatcanbeat x and neut(x)= teams thatcanplaydrawwith x.Therefore, neutrosophictriplet (x, neut(x), anti(x)) isatripletwithwhichcompetitionorganizerscandrawteams intogroupsforabalancedcompetition.TheFédérationInternationaledeFootballAssociation(FIFA) oftenusesthistemplateindrawingnationalteamsintogroupsforitscompetitions.Clubteamsfrom variousnationalleagues,toqualifyforcontinentalcompetitions(e.g.,UnionofEuropeanFootball Associations(UEFA)ChampionsLeagueandConfederationofAfricanFootball(CAF)Champions League),havetobeamongthefive.Thisimpliestheapplicationofduplets,triplets,quadruples, etc.(Figure 3).

anti(x)

x neut(x)

Figure3. Sports.

AuthorContributions: T.G.J.establishednewpropertiesofneutrosophictripletgroups.Hefurtherpresented applicationsoftheneutrosophictripletsetsandgroupstomanagementandsports.F.S.cointroducedthe neutrosophictripletsetandgroup,aswellastheirproperties.Heconfirmedtherelevanceoftheneutrosophic dupletandthequadrupleintheapplicationsofneutrosophictripletset.

ConflictsofInterest: Theauthorsdeclarenoconflictofinterest.

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