FundamentalHomomorphismTheoremsfor NeutrosophicExtendedTripletGroups
MehmetÇelik,MogesMekonnenShallaandNecatiOlgun* ID DepartmentofMathematics,GaziantepUniversity,Gaziantep27310,Turkey;mathcelik@gmail.com(M.Ç.); ms09268@mail2.gantep.edu.tr(M.M.S.)
* Correspondence:olgun@gantep.edu.tr;Tel.:+90-536-321-4006
Received:17July2018;Accepted:31July2018;Published:3August2018
Abstract: Inclassicalgrouptheory,homomorphismandisomorphismaresignificanttostudythe relationbetweentwoalgebraicsystems.Throughthisarticle,weproposeneutro-homomorphismand neutro-isomorphismfortheneutrosophicextendedtripletgroup(NETG)whichplaysasignificant roleinthetheoryofneutrosophictripletalgebraicstructures.Then,wedefineneutro-monomorphism, neutro-epimorphism,andneutro-automorphism.Wegiveandprovesometheoremsrelatedtothese structures.Furthermore,theFundamentalhomomorphismtheoremfortheNETGisgivenand somespecialcasesarediscussed.Firstandsecondneutro-isomorphismtheoremsarestated.Finally, byapplyinghomomorphismtheoremstoneutrosophicextendedtripletalgebraicstructures,wehave examinedhowcloselydifferentsystemsarerelated.
Keywords: neutro-monomorphism;neutro-epimorphism;neutro-automorphism;fundamental neutro-homomorphismtheorem;firstneutro-isomorphismtheorem;andsecondneutro -isomorphismtheorem
1.Introduction
Groupsarefiniteorinfinitesetofelementswhicharevitaltomodernalgebraequippedwith anoperation(suchasmultiplication,addition,orcomposition)thatsatisfiesthefourbasicaxiomsof closure,associativity,theidentityproperty,andtheinverseproperty.Groupscanbefoundingeometry studiedby“Felixkleinin1872”[1],characterizingphenomenalitylikesymmetryandcertaintypes oftransformations.Grouptheory,firstlyintroducedby“Galois”[2],withthestudyofpolynomials hasapplicationsinphysics,chemistry,andcomputerscience,andalsopuzzlesliketheRubik’scube asitmaybeexpressedutilizinggrouptheory.Homomorphismisbothamonomorphismandan epimorphismmaintainingamapbetweentwoalgebraicstructuresofthesametype(suchastwo groups,tworings,twofields,twovectorspaces)andisomorphismisabijectivehomomorphism definedasamorphism,whichhasaninversethatisalsomorphism.Accordingly,homomorphisms areeffectiveinanalyzingandcalculatingalgebraicsystemsastheyenableonetorecognizehow intentlydistinctsystemsareassociated.Similartotheclassicalone,neuro-homomorphismisthe transformbetweentwoneutrosophictripletalgebraicobjectsNandH.Thatis,ifelementsinNsatisfy somealgebraicequationinvolvingbinaryoperation“*”,theirimagesinHsatisfythesamealgebraic equation.Aneutro-isomorphismidentifiestwoalgebraicobjectswithoneanother.Themostcommon useofneutro-homomorphismsandneutro-isomorphismsinthisstudyistodealwithhomomorphism theoremswhichallowfortheidentificationofsomeneutrosophictripletquotientobjectswithcertain otherneutrosophictripletsubgroups,andsoon.
Theneutrosophiclogicandaneutrosophicset,firstlymadeknownbyFlorentinSmarandache[3] in1995,hasbeenwidelyappliedtoseveralscientificfields.Thisstudyleadstoanew direction,exploration,pathofthinkingtomathematicians,engineers,computerscientists,and
Symmetry 2018, 10,321;doi:10.3390/sym10080321 www.mdpi.com/journal/symmetry
manyotherresearchers,sotheareaofstudygrewextremelyandapplicationswerefound inmanyareasofneutrosophiclogicandsetssuchascomputationalmodelling[4],artificial intelligence[5],datamining[6],decisionmakingproblems[7],practicalachievements[8],andsoforth. FlorentinSmarandacheandMumtaziAliinvestigatedtheneutrosophictripletgroupandneutrosophic tripletasexpansionofmatterplasma,nonmatterplasma,andantimatterplasma[9,10].Byusing theconceptofneutrosophictheoryVasanthaandSmarandacheintroducedneutrosophicalgebraic systemsand N-algebraicstructures[11]andthiswasthefirstneutrosoficationofalgerbraicstructures. Thecharacterizationofcancellableweakneutrosophicdupletsemi-groupsandcancellableNTGare investigated[12]in2017.FlorentinSmarandacheandMumtazAliexaminedtheapplicationsofthe neutrosophictripletfieldandneutrosophictripletring[13,14]in2017.¸SahinMehmetandAbdullah Kargındevelopedtheneutrosophictripletnormedspaceandneutrosophictripletinnerproduct[15,16]. TheneutrosophictripletG-moduleandfixedpointtheoremforNTpartialmetricspacearegiven inliterature[17,18].Similaritymeasuresofbipolarneutrosophicsetsandsinglevaluedtriangular neutrosophicnumbersandtheirappliancetomulti-attributegroupdecisionmakinginvestigated in[19,20].Byutilizingdistance-basedsimilaritymeasures,refinedneutrosophichierchicalclustering methodsareachievedin[21].Singlevaluedneutrosophicsetstodealwithpatternrecognitionproblems aregivenwiththeirapplicationin[22].Neutrosophicsoftlatticesandneutrosophicsoftexpertsetsare analyzedin[23,24].CentroidsinglevaluedneutrosophicnumbersandtheirapplicationsinMCDM isconsideredin[25].BalMikail,MogesMekonnenShalla,andNecatiOlgunreviewedneutrosophic tripletcosetsandquotientgroups[26]byusingtheconceptofNETin2018.Theconceptsconcerning neutrosophicsetsandneutrosophicmodulesaredescribedin[27,28],respectively.Amethodtohandle MCDMproblemsundertheSVNSsareintroducedin[29].Bipolarneutrosophicsoftexpertsettheory anditsbasicoperationsaredefinedin[30].
Theotherpartsofapaperiscoordinatedthusly.Subsequently,throughtheliteratureanalysis inthefirstsectionandpreliminariesinthesecondsection,weinvestigatedneutro-monomorphism, neutro-epimorphism,neutro-isomorphism,andneutro-automorphisminSection 3 andafundamental homomorphismtheoremforNETGinSection 4.Wegiveandprovethefirstneutro-isomorphism theoremforNETGinSection 5,andthenthesecondneutro-isomorphismtheoremforNETGisgiven inSection 6.Finally,resultsaregiveninSection 7
2.Preliminaries
Inthissection,weprovidebasicdefinitions,notationsandfactswhicharesignificanttodevelop thepaper.
2.1.NeutrosophicExtendedTriplet
LetUbeauniverseofdiscourse,and(N, ∗)asetincludedinit,endowedwithawell-defined binarylaw ∗. Definition1([3]). ThesetNiscalledaneutrosophicextendedtripletsetifforanyx ∈ Nthereexisteneut(x) ∈ Nandeanti(x) ∈ N.Thus,aneutrosophicextendedtripletisanobjectoftheform(x,eneut(x),eanti(x))where eneut(x) isextendedneutralofx,whichcanbeequalordifferentfromtheclassicalalgebraicunitaryelementif any,suchthat x ∗ eneut(x) = eneut(x) ∗ x = x andeanti(x) ∈ Nistheextendedoppositeofxsuchthat x ∗ eanti(x) = eanti(x) ∗ x = eneut(x) Ingeneral,foreachx ∈ Ntherearemanyexistingeneut(x) s andeanti(x) s
Theorem1([11]). Let(N, ∗)beacommutativeNETwithrespectto ∗ anda,b ∈ N;
(i) neut(a) ∗ neut(b) = neut(a ∗ b); (ii) anti(a) ∗ anti(b) = anti(a ∗ b);
Theorem2([11]). Let(N, ∗)beacommutativeNETwithrespectto ∗ anda ∈ N;
(i) neut(a) ∗ neut(a) = neut(a); (ii) anti(a) ∗ neut(a) = neut(a) ∗ anti(a) = anti(a)
2.2.NETG
Definition2([3]). Let(N, ∗)beaneutrosophicextendedtripletset.Then(N, ∗)iscalledaNETG,ifthe followingclassicalaxiomsaresatisfied.
(a) (N, ∗)iswelldefined,i.e.,foranyx, y ∈ Nonehasx ∗ y ∈ N. (b) (N, ∗)isassociative,i.e.,foranyx, y, z ∈ Nonehasx ∗ (y ∗ z) = (x ∗ y) ∗ z. Weconsider,thattheextendedneutralelementsreplacetheclassicalunitaryelementaswelltheextended oppositeelementsreplacetheinverseelementofclassicalgroup.Therefore,NETGsarenotagroupinclassical way.InthecasewhenNETGenrichesthestructureofaclassicalgroup,sincetheremaybeelementswithmore extendedopposites.
2.3.NeutrosophicExtendedTripletSubgroup
Definition3([26]). GivenaNETG(N, ∗),aneutrosophictripletsubsetHiscalledaneutrosophicextended tripletsubgroupofNifititselfformsaneutrosophicextendedtripletgroupunder ∗.Explicitythismeans (1) Theextendedneutralelementeneut(x) liesinH. (2) Foranyx, y ∈ H, x ∗ y ∈ H (3) Ifx ∈ Htheneanti(x) ∈ H. Ingeneral,wecanshowH ≤ Nasx ∈ Handtheneanti(x) ∈ H, i.ex ∗ eanti(x) = eneut(x) ∈ H.
Definition4. SupposethatNisNETGand H1, H2 ≤ N H1 andH2 arecalledneutrosophictripletconjugates ofNifn ∈ NtherebyH1 = nH2(anti(n))
2.4.Neutro-Homomorphism
Definition5([26]). Let(N1, ∗)and(N2, ◦)betwoNETGs.Amapping f : N1 → N2 iscalleda neutro-homomorphismif
(a) Foranyx, y ∈ N, wehave f (x ∗ y) = f (x) ◦ f (y) (b) If (x, neut(x), anti(x) isaneutrosophicextendedtripletfromN1,then f (neut(x)) = neut( f (x)) and f (anti(x)) = anti( f (x)).
Definition6([26]). Letf:N1→N2 beaneutro-homomorphismfromaNETG(N1, ∗)toaNETG(N2, ◦). Theneutrosophictripletimageoffis
Im( f ) = { f (g) : g ∈ N1, ∗}
Definition7([26]). Letf:N1→N2 beaneutro-homomorphismfromaNETG(N1, ∗)toaNETG(N2, ◦)andB ⊆ N2.Then
f 1(B) = {x ∈ N1 : f (x) ∈ B} istheneutrosophictripletinverseimageofBunderf.
Definition8([26]). Let f : N1 → N2 beaneutro-homomorphismfromaNETG(N1, ∗)toaNETG(N2, ◦). Theneutrosophictripletkerneloffisasubset
Ker( f ) = {x ∈ N1 : f (x) = neut(x)} ofN1,
whereneut(x)denotestheneutralelementofN2
Definition9. Theneutrosophictripletkernelof φ iscalledtheneutrosophictripletcenterofNETGNanditis denotedbyZ(N).Explicitly, Z(N) = {a ∈ N : ϕa = neutN } = {a ∈ N : ab(anti(a)) = b, ∀b ∈ N} = {a ∈ N : ab = ba, ∀b ∈ N}.
HenceZ(N)istheneutrosophictripletsetofelementsinNthatcommutewithallelementsinN.Notethat obviouslyZ(N)isaneutrosophictriplet.WehaveZ(N) = NinthecasethatNisabelian.
Definition10([26]). LetNbeaNETGand H ⊆ N.∀x ∈ N, theset xh/h ∈ H iscalledneutrosophictriplet cosetdenotedbyxH.Analogously, Hx = hx/h ∈ H and (xH)anti(x) = (xh)anti(x)/h ∈ H
When h ≤ N, xH iscalledtheleftneutrosophictripletcosetofHinNcontainingx,andHxiscalledthe rightneutrosophictripletcosetofHinNcontainingx. | xH | and | Hx | areusedtodenotethenumberof elementsinxHandHx, respectively.
2.5.NeutrosophicTripletNormalSubgroupandQuotientGroup
Definition11([26]). AneutrosophicextendedtripletsubgroupHofaNETGofNiscalledaneutrosophic tripletnormalsubgroupofNif aH(anti(a)) ⊆ H, ∀x ∈ N andwedenoteitas H NandH NifH = N
Example1. LetNbeNETG.{neut} NandN N.
Definition12([26]). IfNisaNETGandH Nisaneutrosophictripletnormalsubgroup,thenthe neutrosophictripletquotientgroupN/Hhaselements xH : x ∈ N, theneutrosophictripletcosetsofHinN, andoperation (xH)(yH) = (xy)H
3.Neutro-Monomorphism,Neutro-Epimorphism,Neutro-Isomorphism,Neutro-Automorphism
Inthissection,wedefineneutro-monomorphism,neutro-epimorphism,neutro-isomorphism,and neutro-automorphism.Then,wegiveandsomeimportanttheoremsrelatedtothem.
3.1.Neutro-Monomorphism
Definition13. Assumethat(N1, ∗)and(N2, ◦)betwoNETG’s.Ifamapping f : N1 → N2 ofNETGisonly onetoone(injective)fiscalledneutro-monomorphism.
Theorem3. Let(N1, ∗)and(N2, ◦)betwoNETG’s. ϕ : N1 → N2 isaneutro-monomorphismofNETGif andonlyifker ϕ = {neutN1}
Proof. Assume ϕ isinjective.If a ∈ kerϕ,then ϕ(a) = neutN2 = ϕ(neutN1), ∀a ∈ N1 andhencebyinjectivity a = neutN1.Conversely,assumekerϕ = ϕ(neutN1).Let a,b ∈ N1 suchthat ϕ(a)= ϕ(b).Weneedtoshowthat a = b.
neutH = ϕ(b)anti(ϕ(a)) = ϕ(b)ϕ(anti(a)) = ϕ(b(anti(a)))
Thus b(anti(a))) ∈ ker ϕ, andhence,byassumption ker ϕ = ϕ(neutN1) Weconcludethat b(anti(a))) = neutN1, i e., a = b
Definition14. Let(N1, ∗)and(N2, ◦)betwoNETG’s.Ifamapping f : N1 → N isonlyonto(surjective)fis calledneutro-epimorphism.
Theorem4. LetNandHbetwoNETG’s.If ϕ : N → H isaneutro-homomorphismofNETG,thensois ϕ 1: H → N.
Proof. Let x = ϕ(a), y = ϕ(b), ∀a, b ∈ N and ∀x, y ∈ H.So a = anti(ϕ(x)), b = anti(ϕ(y)).Now anti(xy) = ϕ(ϕ(a)ϕ(b)) = anti(ϕ(ab) = ab = anti(ϕ(x))anti(ϕ(y))
Theorem5. LetNbeNETGand a, b ∈ N Themap φ : N → AutN Then, a → ϕ a, isa neutro-homomorphism.
Proof. Foranyfixed n ∈ N,wehave
ϕab (N) = abn(anti(ab)) = abn(anti(a))anti(b) = ϕa (bn(anti(b)) = ϕa ϕb (n), So ϕab = ϕa ϕb, i e., φ(ab) = φ(a)φ(b)
Itisinfacthasanti-neutralelementi.e., ϕ(anti(n)) = anti(ϕn ). Since ϕn anti(ϕn (a)) = n(anti(n)an)anti(n) = a,andso ϕn isinjective.
Theorem6. Let f : N → H beaneutro-homomorphismofNETGNandH.For h ∈ H and x ∈ f 1(h), f 1(h) = x ∈ kerf .
Proof. (1) Let’sshowthat f 1(h) ⊆ xkerf If x ∈ f 1(h), then f (x) = h and b ∈ f 1(h), then f (b) = h.If f (x) = f (y),then: anti( f (x)) f (x) = anti( f (x)) f (b)(bytheorem1) neutH = f (anti(x)) f (b)(bydefinition1) ⇒ anti(x)b ∈ kerf
Foratleast k ∈ kerf , anti(x)b = k.If b = xk,then, b ∈ xkerf ⇒ f 1(h) ⊆ xkerf (1) (2) Let’sshowthat xkerf ⊆ f 1(h).Let b ∈ xkerf .Foratleast k ∈ kerf , b = xk ⇒ f (b) = f (xk) = f (x) f (k) = hneutH = h
If f 1(h) = b and b ∈ f 1(h),then xkerf ⊆ f 1(h) (2) by(1)and(2),weobtain xkerf = f 1(h)
Theorem7. Let ϕ : N1 → N2 beaneutro-homomorphismofNETGN1 andN2
(1) IfH2 N2, then ϕ 1(H2) N1. (2) IfH1 N1 and ϕ isaneutro epimorhismthen ϕ(H1) N2
Proof. (1) If x ∈ ϕ 1(H2) and a ∈ N1, then ϕ(x) ∈ H2 andso ϕ((ax)(anti(a)) = ϕ(a)ϕ(x)anti(ϕ(a)) ∈ H2 .Since H2 isneutrosophictripletnormalsubgroup. Weconclude ax(anti(a)) ∈ ϕ 1(H2)
(2) Since H1 isneutrosophictripletnormalsubgroup,wehave ϕ(a)ϕ(H1)anti(ϕ(a)) ⊆ ϕ(H1) Sinceweassume ϕ issurjective,every b ∈ N2 canbewrittenas b = ϕ(a), a ∈ N1 Therefore, b ϕ(H1)anti(b) ∈ ϕ(H1).
Theorem8([26]). Letf : N → Hbeaneutro-homomorphismfromaNETGNtoaNETGH.Kerf N
Theorem9. LetNbeNETGandH N.Themap ϕ : N → N/H, n → nH, isaneutro-homomorphism withneutrosophictripletkernelker ϕ = H.
Proof. Wehave ϕ(ab) = (ab)H = (aH)(bH) = ϕ(a)ϕ(a), so φ isaneutro-homomorphism.Asto theneutrosophictripletkernel, a ∈ ker ϕ ⇔ ϕ(a) = H (since H isneutralin N/H) ⇔ aH = H (by definitionof φ) ⇔ a ∈ H.
Theorem10. LetNbeNETGand H ⊆ N beanon-emptyneutrosophicextendedtripletsubset.Then H N, ifandonlyifthereexistsaneutro-homomorphism ϕ : N1 → N2 withH = ker ϕ.
Proof. Itsstraightforward.
3.2.Neutro-Isomorphism
Definition15. Let(N1, ∗)and(N2, ◦)betwoNETGs.Ifamapping f : N1 → N2 neutro-homomorphismis onetooneandontofiscalledneutro-isomorphism.Here,N1 andN2 arecalledneutro-isomorphicanddenoted asN1 ∼ = N2
Theorem11. Let(N1, ∗)and(N2, ◦)betwoNETG’s.If f : N1 → N2 isaneutro-isomorphismofNETG’s, thensoisf 1 : N2 → N1
Proof. Itisobvioustoshowthatfisonetooneandonto.Nowlet’sshowthatfis neutro-homomorphism.Let x = ϕ(a), y = ϕ(b), ∀a, b ∈ N1, ∀x, y ∈ N2 andso, a = anti(ϕ(x)), b = anti(ϕ(y)).Now anti(xy) = anti(ϕ(ϕ(a)ϕ(b))) = anti(ϕ(ϕ(ab))) = ab = anti(ϕ(x))anti(ϕ(y)) 3.3.Neutro-Automorphism.
Definition16. Let(N1, ∗)and(N2, ◦)betwoNETG’S.Ifamapping f : N1 → N2 isonetooneandontofis calledneutro-automorphism.
Definition17. LetNbeNETG. ϕ ∈ AutN iscalledaneutro-innerautomorphismifthereisa n ∈ N such that ϕ = ϕn Proposition1. LetNbeaNETG.For a ∈ N, fa : N → N suchthat x → ax(anti(a) isa neutro-automorphism (AutN)
Proof. (1) ∀x, y ∈ N,wehavetoshowthat
f (x) = f (y) ⇒ x = y ax(anti(a)) = ay(anti(a)) ⇒ ax(anti(a))a = ay(anti(a))a ⇒ ax(neut(a)) = ay(neut(a)) ⇒
Therefore, f isonetoone. (2) ∀x, y ∈ N,wehavetoshowthat
f (x) = ax(anti(a)) = y.ax(anti(a))a = ya ⇒ ax(neut(a)) = ya ⇒ ax = ya ⇒ anti(a)ax = anti(a)ya ⇒ neut(a)x =
So, f isonto.Therefore, fa isaneutro-automorphism.
Lemma1. LetabeanelementofNETGNsuchthata2 = a. Thena = neut(a).
Proof. Wehave
= (anti(a) ∗ a) ∗ a for anti(a) ∈ N (antiaxiom)
= anti(a) ∗ a2 (associativityaxiom)
= anti(a) ∗ a (byassumption)
= neut(a)(bydefinitionofanti)
Theorem12. LetNbeNETGand H1, H2 ≤ N Thentheneutrosophicextendedtripletset H1 H2 = {ab : a ∈ H1, b ∈ H2} isaneutrosophicextendedtripletsubgroupinthecasethatH1 H2 = H2 H1
Proof. Suppose H1H2 isaneutrosophicextendedtripletsubgroup.Then,forall a ∈ H1, b ∈ H2, we have anti(a)anti(b) ∈ H1 H2, i.e., H2 H1 ⊆ H1 H2. Butalsofor h ∈ H1 H2 wefind a ∈ H1, b ∈ H2 thereby anti(h) = ab, andthen h = anti(b)anti(a) ∈ H2 H1. So H1 H2 ⊆ H2 H1,that’s, H1 H2 = H2 H1. Onthe otherhand,assumethat H1 H2 = H2 H1. Then ∀a, a ∈ H1, b, b ∈ H2 wehave aba b ∈ aH2 H1b = aH1 H2b = H1 H2 Furthermore, ∀a ∈ H1, b ∈ H2 wehave anti(ab) = anti(b)anti(a) ∈ H2 H1 = H1 H2
Symmetry 2018, 10, x 9 of 15 12 , aHbH ∈∈ thereby () ,antihab = and then ()() 21 . hantibantiaHH ∈ = So 1221 HHHH ⊆ , that’s, 1221 . HHHH = On the other hand, assume that 1221 . HHHH = Then '' ,12 ,, aaHbbH ∀∈∈ we have '''' 211212 . ababaHHbaHHbHH ∈== Furthermore, 12 , aHbH∀∈∈ we have ()()() 2112 antiabantibantiaHHHH == ∈ □
4.FundamentalTheoremofNeutro-Homomorphism
4. Fundamental Theorem of Neutro-Homomorphism
The fundamental theorem of neutro-homomorphism relates the structure of two objects between which a neutrosophic kernel and image of the neutro-homomorphism is given. It is also significant to prove neutro-isomorphism theorems. In this section, we give and prove the fundamental theorem of neutro-homomorphism. Then, we discuss a few special cases. Finally, we give examples by using NETG.
Thefundamentaltheoremofneutro-homomorphismrelatesthestructureoftwoobjectsbetween whichaneutrosophickernelandimageoftheneutro-homomorphismisgiven.Itisalsosignificant toproveneutro-isomorphismtheorems.Inthissection,wegiveandprovethefundamentaltheorem ofneutro-homomorphism.Then,wediscussafewspecialcases.Finally,wegiveexamplesby usingNETG.
Theorem13. LetN1,N2 beNETG’sand φ : N1 → N2 beaneutro-homomorphism.Then, N1/ker(φ) ∼ = im(φ). Furthermoreif ϕ isneutro-epimorphism,then N1/kerφ ∼ = N2
Theorem 13. Let N1, N2 be NETG’s and :12 NN φ → be a neutro-homomorphism. Then, ()() 1 /.Nkerimφφ ≌ Furthermore if ϕ is neutro-epimorphism, then N1/kerϕ ≌ N2. N1 ϕ im(ϕ) φ i N1/ker(ϕ)
Proof. Wewillconstructanexplicitmap i : N1/ker(φ) → im(φ) andprovethatitisa neutro-isomorphismandwelldefined.Since ker(ϕ)isneutrosophictripletnormalsubgroupof N1 Let K = ker(φ), andrecallthat N1/K = {aK : a ∈ N1} Define i : N1/K → im(φ), i : nK → φ(n), n ∈ N1 Thus,weneedtocheckthefollowingconditions.
Proof. We will construct an explicit map ()() :1 / iNkerimφφ → and prove that it is a neutro-isomorphism and well defined. Since ker(ϕ) is neutrosophic triplet normal subgroup of N1. Let () , Kker φ = and recall that 11 /:{. } NKaKaN =∈ Define ()() :11 /,:,. iNKiminKnnN φφ →→∈ Thus, we need to check the following conditions.
(1) i iswelldefined
(1) i is well defined
(2) i is injective
(2) i isinjective
(3) i is surjective
(3) i issurjective (4) i isaneutro-homomorphism
(4) i is a neutro-homomorphism
(1) We must show that if , aKbK = then ()() iaKbK = Suppose . aKbK = We have ()() . aKbKantibaKKantibaK = = ∈
Here, () () ()() 2 ()( n neutantibaantiba ==φφφ () () ()()() . antibaab φφφφ = = Hence, ()()()() . iaKabibK φφ === Therefore, it is well defined. ()() ()()
(2) i is injective (3) i is surjective (4) i is a neutro-homomorphism
(1) We must show that if , aKbK = then ()() . iaKbK = Suppose aKbK = We have ()() aKbKantibaKKantibaK = = ∈
(1) Wemustshowthatif aK = bK, then i(aK) = (bK). Suppose aK = bK. Wehave aK = bK ⇒ anti(b)aK = K ⇒ anti(b)a ∈ K. Here, neut(n2) = φ(anti(b)a) = φ(anti(b)φ(a) = anti(φ(b))φ(a) ⇒ φ(a) = φ(b). Hence, i(aK) = φ(a) = φ(b) = i(bK). Therefore,itis welldefined.
Here, () () ()() 2 ()( n neutantibaantiba ==φφφ () () ()()() antibaab φφφφ = = Hence, ()()()() iaKabibK φφ ===
Therefore, it is well defined.
(2) Wemustshowthat i(aK) = i(bK) ⇒ aK = bK.Supposethat i(aK) = i(bK).Then
(2) We must show that ()() iaKibKaKbK = = Suppose that ()() iaKibK = Then ()() . iaKibKaKbK = = () () ()()() () ()() 22 nn antibaneutantibaneutantibaK φφφ = = ∈ () 222 () antibaKKaNNaN ==⇔∈
Thus, i isinjective.
Thus, i is injective.
(3) Wemustshowthatforanyelementinthedomain(N1/K)getsmappedtoitby i.let’spickany element φ(a) ∈ im(φ).Bydefinition, i(aK) = φ(a),hence i issurjective.
(3) We must show that for any element in the domain (N1/K) gets mapped to it by i. let’s pick any element ()() . aim φφ ∈ By definition, ()() , iaKa φ = hence i is surjective.
(4) Wemustshowthat i(aKbK) = i(aK)i(bK) i(aKbK) = i(abK)(aKbK = abK) = φ(ab) = φ(a)φ(b) = i(aKbK) = i(aK)i(bK).Thus, i isaneutro-homomorphism.
Insummary,since i : N1/K → im(φ) isawell-definedneutro-homomorphismthatisinjective andsurjective.Therefore,itisaneutro-isomorphism. i.e., N1/K ∼ = im(φ), andthefundamental theoremofneutro-homomorphismisproven.
Corollary1(AFewSpecialCasesofFundamentalTheoremofNeutro-homomorphism).
• LetN=(1,1,1)beatrivialneutrosophicextendedtriplet.If ϕ:N1→N2 isanembedding,thenneutrosophic ker(ϕ)={neut(1)=1N1}.TheTheorem12saysthatim(ϕ) ∼ = {N1/1N1} ∼ = N1
• If ϕ:N1→N2 isamap ϕ(n)=neut(1)=1N2 foralln2 ∈ N1,thenneutrosophicker(ϕ)=N1,soTheorem 13saysthat1N2 =im(ϕ) ∼ = N1/N1
Example2. TheneutrosophicextendedtripletalternatinggroupAn (theneutrosophicextendedtripletsubgroup ofevenpermutationinNETGSn)hasindex2inSn
Solution. Toprovethat[Sn:An]=2.Wewillconstructasurjectiveneutro-homomorphism φ:Sn→Z2 with neutrosophictriplet kerφ = An.HeretheneutrosophicextendedtripletsofZ2 are(0,0,0)and(1,1,1).Ifthis isachieved,itwouldfollowthat Sn /An ∼ = Z2, so|Sn/An|=|Z2|=2,andtherefore[Sn:An]=|Sn/An|=2, asdesired.Define φ:Sn→Z2 by φ(f)= [0] iffiseven [1] iffisodd
Byconstruction φ issurjective.Toprovethat φ isaneutro-homomorphismweneedtoshowthat φ(x)+ φ(y)= φ(xy), ∀x,y ∈ Sn.Hereifxandyarebothevenorbothodd,thenxyiseven.Ifxiseven andyisodd,orifxisoddandyiseven,thenxyisodd.Letusseethesefourdifferentcasesasfollows:
(1) xandyarebotheven.Thenxyisalsoeven.So, φ(x)= φ(y)= φ(xy)=[0].Since[0]+[0]= [0]holds.
(2) xiseven,andyisodd.Thenxyisodd.So, φ(x)+ φ(y)=[0]+[1]=[1]= φ(xy).
(3) xisodd,andyiseven.Thiscaseisanalogoustocase2.
(4) xandyarebothodd.Thenxyiseven,so φ(x)+ φ(y)=[1]+[1]=[0]= φ(xy).Thus,weverified that φ isaneutro-homomorphism.Finally,neutrosophictrpletkerφ ={x ∈ Sn: φ(x)=[0]2}isthe neutrosophicextendedtripletsetofallevenpermutations,soneutrosophictriet ker ϕ = An
5.FirstNeutro-IsomorphismTheorem
Thefirstneutro-isomorphismtheoremrelatestwoneutrosophictripletquotientgroupsinvolving productsandintersectionsofneutrosophicextendedtripletsubgroups.Inthissection,wegiveand provethefirstneutro-isomorphismtheorem.Finally,wegiveanexamplebyusingNETG.
Theorem14. LetNbeNETGandH,KbetwoneutrosophicextendedtripletsubgroupofNandHisa neutrosophictripletnormalinK.Then
(a) HKisneutrosophictripletsubgroupofN. (b) H KisneutrosophictripletnormalsubgroupinK. (c) HK H ∼ = K H K
Proof. (a) Let xy ∈ HK.If x = h1k1 and y = h2k2, h1h2 ∈ H and k1, k2 ∈ K.Consider
x(anti(y)) = (h1k1) anti(h2k2)
= (h1k1)anti(k2)anti(h2) = h1(k1(anti(k2)))anti(h2), (k3 = k1(anti(k2)) : k3 ∈ K = h1k3(anti(h2)) = h1k3(anti(h2))anti(k3)k3 = h1k3(anti(h2))anti(k3)k3 = h1h2k3 becauseH ksoh3 = k3(anti(h2))anti(k3) ∈ H ⇒ x(anti(y) = h4k3 ∈ HK, (h4 = h1h2) ⇒ HK isNETGof N
(b) Wehavetoprove H ∩ K isneutrosophictripletnormalsubgroupinkor H ∩ K k Let x ∈ H ∩ K and x ∈ K If x ∈ H and x ∈ K, then kx(anti(k)) ∈ H because H k and kx(anti(k)) ∈ K because xk ∈ K.Thus, kx(anti(k)) ∈ H ∩ K.Since H ∩ K k (c) HK H ∼ = K H K .Let H K = D,so K D = K H K .Nowlet’sdefineamapping ϕ: HK→ K D by φ(hk) = KD
1. ϕ iswelldefined
h1k1 = h2k2, h1h2 ∈ H and k1k2 ∈ K k1h1 = k2h2 ⇒ anti(k2)k1h1 = h2 ⇒ anti(k2)k1 = h2(anti(h1)), h2(anti(h1)) ∈ H ⇒ anti(k2)k1 ∈ H,but anti(k2)k1 ∈ K ⇒ anti(k2)k1 ∈ H ∩ K = D ⇒ anti(k2)k1 ∈ D ⇒ anti(k2)k1 D = D ⇒ k1 D = k2 D ⇒ φ(h1k1) = φ(h2k2)
2. ϕ isneutro-homomorphism.
Φ(h1k1.h2k2) = φ(h1(k1h2)k2 = φ(h1h2 k1k2) = K1k2 D = k1 Dk2 D = φ(h1k1).φ(h2k2)
3. ϕ isonto.
Sinceforevery KD ∈ K/D, neut.k ∈ HK under ϕ suchthat φ(neut.k) = KD.Hence, ϕ isonto. NowbyTheorem13, HK/Kerφ ∼ = K/D
Nowitisenoughtoprovethat kerφ = H.Let h ∈ H, h(neut) ∈ HK.Thus
φ(h) = φ(h neut) = neut D = D ⇒ φ(h) = D ⇒ h ∈ kerφ i eH ⊆ kerφ
Conversly, hk ∈ kerφ,where h ∈ H and k ∈ K.If φ(hk) = D,then KD = D ⇒ k ∈ D = H ∩ K ⇒ h ∈ H and k ∈ K ⇒ hk ⊆ H ⇒ kerφ ⊆ H.Thus H = kerφ by(1) HK H ∼ = K H K Example3. LetNbeNETG.Neutro-isomorphismtheoremsareforinstanceusefulinthecalculationofNETG orders,sinceneutro-isomorphicgroupshavethesameorder.IfH ≤ NandK NsothatHKisfinite,then Lagrange’stheorem[26]inneutrosophictripletwiththeorem13yield
|HK|/|K|=|HK : K| =|HK/K| =|H/H K| =|H : H K| =|H|/|H K|,thatis |HK|=|H||K|/|H K|
6.SecondNeutro-IsomorphismTheorem
Thesecondneutro-isomorphismtheoremisextremelyusefulinanalyzingtheneutrosophic extendednormalsubgroupsofaneutrosophictripletquotientgroup.Inthissection,wegiveand provethesecondneutro-homomorphismtheoremforNETG.
Theorem15. LetNbeaNETG.LetHandKbeneutrosophictripletnormalsubgroupofNwith K ⊆ H. Then H/K N/KandN/KH/K ∼ = N/H
Proof. Considerthenaturalmap Ψ:N→N/H.Theneutrosophictripletkernel, H contains K. Thus,by theuniversalpropertyof N/K,itfollowsthatthereisaneutro-homomorphism N/K → N/H. This mapisclearlysurjective.Infact,itsendstheneutrosophictripletleftcoset nK totheneutrosophic tripletleftcoset nH. NowsupposethatnKisintheneutrosophictripletkernel.Thentheneutrosophic tripletleftcoset nH istheneutralneutrosophictripletcoset,thatis, nH = H, sothat n ∈ H. Thusthe neutrosophictripletkernelconsistsofthoseneutrosophictripletleftcosetsoftheform nK,for n ∈ H, thatis, H/K.
1. Ψ iswelldefined.Let ak = bk.
anti(b)ak = k anti(b)a ∈ k ⇒ K H anti(b)a ∈ H aH = bH(anti(b)aH = H) Ψ(ak) = Ψ(bk)
2. Ψ isneutro-homomorphism ak, bk ∈ N/K Ψ(ak bk ) =Ψ(abk) = abH = aHbH =Ψ(ak)Ψ(bk) 3. Ψ isonto Forall y = aH ∈ N/H, x = ak ∈ N/K ⇒ Ψ(x) = y. 4. kerΨ = H/K Theneutralelementof N/H is H.Therefore
ker Ψ: {xk ∈ N/K : Ψ(xk) = H} = {xk ∈ N/K : Ψ(xk) = xH = H} = {xk ∈ N/K : x ∈ H} = {xk ∈ H/K} = H/K
ByTheorem13 N/KH/K ∼ = N/H.
7.Conclusions
Thispaperismainlyfocusedonfundamentalhomomorphismtheoremsforneutrosophicextended tripletgroups.Wegaveandprovedthefundamentaltheoremofneutro-homomorphism,aswell asfirstandsecondneutro-isomorphismtheoremsexplainedforNETG.Furthermore,wedefine neutro-monomorphism,neutro-epimorphism,neutro-automorphism,innerneutro-automorphism, andcenterforneutrosophicextendedtriplets.Finally,byapplyingthemtoneutrosophicalgebraic structures,wehaveexaminedhowcloselydifferentsystemsarerelated.Byusingtheconceptofa fundamentaltheoremofneutro-homomorphismandneutro-isomorphismtheorems,therelation betweenneutrosophicalgebraicstructures(neutrosophictripletring,neutrosophictripletfield, neutrosophictripletvectorspace,neutrosophictripletnormedspace,neutrosophicmodules,etc.)can bestudiedandthefieldofstudyinneutrosophicalgebraicstructureswillbeextended.
AuthorContributions: M.Ç.definedandstudiedpreliminariesandthefundamentaltheoremof neutro-homomorphismtheoremwithexamplesandwrotepartofthepaper.M.M.S.definedandstudiedfirstand secondneutro-isomorphismtheoremswithexample.Therevisionandsubmissionofthepaperwascompletedby N.O.
Funding: Thisresearchreceivednoexternalfunding. ConflictsofInterest: Theauthorsdeclarenoconflictofinterest.
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