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SS symmetry Article Fundamental Homomorphism Theorems for Neutrosophic Extended Triplet Groups MehmetÇelik,MogesMekonnenShallaandNecatiOlgun* ID DepartmentofMathematics,GaziantepUniversity,Gaziantep27310,Turkey;[email protected](M.Ç.); [email protected](M.M.S.) * Correspondence:[email protected];Tel.:+90-536-321-4006 (cid:1)(cid:2)(cid:3)(cid:1)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8)(cid:1) (cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7) 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, the Fundamental homomorphism theorem for the NETG is given and somespecialcasesarediscussed. Firstandsecondneutro-isomorphismtheoremsarestated. Finally, byapplyinghomomorphismtheoremstoneutrosophicextendedtripletalgebraicstructures,wehave examinedhowcloselydifferentsystemsarerelated. Keywords: neutro-monomorphism; neutro-epimorphism; neutro-automorphism; fundamental neutro-homomorphism theorem; first neutro-isomorphism theorem; and second neutro -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 as it may be expressed utilizing group theory. Homomorphism is both a monomorphism and an epimorphism maintaining a map between two algebraic structures of the same type (such as two groups, two rings, two fields, two vector spaces) and isomorphism is a bijective homomorphism definedasamorphism,whichhasaninversethatisalsomorphism. Accordingly,homomorphisms are effective in analyzing and calculating algebraic systems as they enable one to recognize how intently distinct systems are associated. Similar to the classical one, neuro-homomorphism is the transformbetweentwoneutrosophictripletalgebraicobjectsNandH.Thatis,ifelementsinNsatisfy somealgebraicequationinvolvingbinaryoperation“*”,theirimagesinHsatisfythesamealgebraic equation. Aneutro-isomorphismidentifiestwoalgebraicobjectswithoneanother. Themostcommon useofneutro-homomorphismsandneutro-isomorphismsinthisstudyistodealwithhomomorphism theoremswhichallowfortheidentificationofsomeneutrosophictripletquotientobjectswithcertain otherneutrosophictripletsubgroups,andsoon. Theneutrosophiclogicandaneutrosophicset,firstlymadeknownbyFlorentinSmarandache[3] in 1995, has been widely applied to several scientific fields. This study leads to a new direction, exploration, path of thinking to mathematicians, engineers, computer scientists, and Symmetry2018,10,321;doi:10.3390/sym10080321 www.mdpi.com/journal/symmetry Symmetry2018,10,321 2of14 many other researchers, so the area of study grew extremely and applications were found in many areas of neutrosophic logic and sets such as computational modelling [4], artificial intelligence[5],datamining[6],decisionmakingproblems[7],practicalachievements[8],andsoforth. FlorentinSmarandacheandMumtaziAliinvestigatedtheneutrosophictripletgroupandneutrosophic triplet as expansion of matter plasma, nonmatter plasma, and antimatter plasma [9,10]. By using theconceptofneutrosophictheoryVasanthaandSmarandacheintroducedneutrosophicalgebraic systemsandN-algebraicstructures[11]andthiswasthefirstneutrosoficationofalgerbraicstructures. Thecharacterizationofcancellableweakneutrosophicdupletsemi-groupsandcancellableNTGare investigated[12]in2017. FlorentinSmarandacheandMumtazAliexaminedtheapplicationsofthe neutrosophictripletfieldandneutrosophictripletring[13,14]in2017. S¸ahinMehmetandAbdullah Kargındevelopedtheneutrosophictripletnormedspaceandneutrosophictripletinnerproduct[15,16]. The neutrosophic triplet G-module and fixed point theorem for NT partial metric space are given inliterature[17,18]. Similaritymeasuresofbipolarneutrosophicsetsandsinglevaluedtriangular neutrosophic numbers and their appliance to multi-attribute group decision making investigated 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-automorphisminSection3andafundamental homomorphismtheoremforNETGinSection4. Wegiveandprovethefirstneutro-isomorphism theoremforNETGinSection5,andthenthesecondneutro-isomorphismtheoremforNETGisgiven inSection6. Finally,resultsaregiveninSection7. 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)(cid:48)s andeanti(x)(cid:48)s. Theorem1([11]). Let(N,∗)beacommutativeNETwithrespectto∗anda,b∈N; Symmetry2018,10,321 3of14 (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 Definition 2 ([3]). Let (N, ∗) be a neutrosophic extended triplet set. Then (N, ∗) is called a NETG, if the 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 ≤ N asx ∈ Handtheneanti(x) ∈ H, i.ex∗eanti(x) = eneut(x) ∈ H. Definition4. SupposethatNisNETGand H ,H ≤ N.H and H arecalledneutrosophictripletconjugates 1 2 1 2 ofNifn ∈ NtherebyH = nH (anti(n)). 1 2 2.4. Neutro-Homomorphism Definition 5 ([26]). Let (N , ∗) and (N , ◦) be two NETGs. A mapping f : N → N is called a 1 2 1 2 neutro-homomorphismif (a) Foranyx,y ∈ N,wehave f(x∗y) = f(x)◦ f(y) (b) If(x,neut(x),anti(x) isaneutrosophicextendedtripletfromN ,then 1 f(neut(x)) = neut(f(x)) and f(anti(x)) = anti(f(x)). Symmetry2018,10,321 4of14 Definition 6 ([26]). Let f: N →N be a neutro-homomorphism from a NETG (N , ∗) to a NETG (N , ◦). 1 2 1 2 Theneutrosophictripletimageoffis Im(f) = {f(g) : g ∈ N ,∗}. 1 Definition7([26]). Letf: N →N beaneutro-homomorphismfromaNETG(N ,∗)toaNETG(N ,◦)andB 1 2 1 2 ⊆N . Then 2 f−1(B) = {x ∈ N : f(x) ∈ B} 1 istheneutrosophictripletinverseimageofBunderf. Definition8([26]). Let f : N → N beaneutro-homomorphismfromaNETG(N ,∗)toaNETG(N ,◦). 1 2 1 2 Theneutrosophictripletkerneloffisasubset Ker(f) = {x ∈ N : f(x) = neut(x)}of N , 1 1 whereneut(x)denotestheneutralelementofN . 2 Definition9. TheneutrosophictripletkernelofφiscalledtheneutrosophictripletcenterofNETGNanditis denotedbyZ(N).Explicitly, Z(N) = {a ∈ N : ϕ = neut } a N = {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]). LetNbeaNETGandH ⊆ N.∀x ∈ N,thesetxh/h ∈ Hiscalledneutrosophictriplet cosetdenotedbyxH.Analogously, Hx = hx/h ∈ H and (xH)anti(x) = (xh)anti(x)/h ∈ H. Whenh ≤ N,xHiscalledtheleftneutrosophictripletcosetofHinNcontainingx,andHxiscalledthe rightneutrosophictripletcosetofHinNcontainingx. | xH | and | Hx |areusedtodenotethenumberof elementsinxH and Hx,respectively. 2.5. NeutrosophicTripletNormalSubgroupandQuotientGroup Definition11([26]). AneutrosophicextendedtripletsubgroupHofaNETGofNiscalledaneutrosophic tripletnormalsubgroupofNifaH(anti(a)) ⊆ H,∀x ∈ NandwedenoteitasH (cid:69) N andH (cid:67) Nif H (cid:54)= N. Example1. LetNbeNETG.{neut} NandN N. (cid:1) (cid:2) Definition 12 ([26]). If N is a NETG and H N is a neutrosophic triplet normal subgroup, then the neutrosophictripletquotientgroupN/Hhaselem(cid:2)entsxH : x ∈ N,theneutrosophictripletcosetsofHinN, andoperation(xH)(yH) = (xy)H. Symmetry2018,10,321 5of14 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(N ,∗)and(N ,◦)betwoNETG’s. Ifamapping f : N → N ofNETGisonly 1 2 1 2 onetoone(injective)fiscalledneutro-monomorphism. Theorem3. Let(N ,∗)and(N ,◦)betwoNETG’s. ϕ : N → N isaneutro-monomorphismofNETGif 1 2 1 2 andonlyifkerϕ = {neut }. N1 Proof. Assume ϕisinjective. Ifa∈kerϕ,then ϕ(a) = neut = ϕ(neut ),∀a ∈ N N2 N1 1 andhencebyinjectivitya=neut . Conversely,assumekerϕ= ϕ(neut ). Leta,b∈N suchthat ϕ(a)= N1 N1 1 ϕ(b). Weneedtoshowthata=b. neut = ϕ(b)anti(ϕ(a)) H = ϕ(b)ϕ(anti(a)) = ϕ(b(anti(a))). Thus b(anti(a))) ∈ kerϕ, and hence, by assumption kerϕ = ϕ(neut ). We conclude that N1 b(anti(a)))= neut ,i.e., a = b. N1 Definition14. Let(N ,∗)and(N ,◦)betwoNETG’s. Ifamapping f : N → N isonlyonto(surjective)fis 1 2 1 calledneutro-epimorphism. Theorem4. LetNandHbetwoNETG’s. If ϕ : N → H isaneutro-homomorphismofNETG,thensois ϕ−1: H→N. Proof. Letx = ϕ(a),y = ϕ(b),∀a,b ∈ Nand∀x,y ∈ H.Soa = 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)). Theorem 5. Let N be NETG and a,b ∈ N. The map φ : N → AutN.Then,a → ϕ , is a a neutro-homomorphism. Proof. Foranyfixedn ∈ N,wehave ϕ (N) = abn(anti(ab)) = abn(anti(a))anti(b) ab = ϕ (bn(anti(b)) = ϕ ϕ (n), a a b So ϕ = ϕ ϕ ,i.e.,φ(ab) = φ(a)φ(b). ab a b Symmetry2018,10,321 6of14 It is in fact has anti-neutral element i.e., ϕ(anti(n)) = anti(ϕ ). Since ϕ anti(ϕ (a)) = n n n n(anti(n)an)anti(n) = a,andso ϕ isinjective. n Theorem 6. Let f : N → H be a neutro-homomorphism of NETG N and H. For h ∈ H and x ∈ f−1(h), f−1(h) = x ∈ kerf. Proof. (1) Let’s show that f−1(h) ⊆ x kerf. 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) neut = f(anti(x))f(b)(bydefinition1) H ⇒ anti(x)b ∈ kerf. Foratleastk ∈ kerf,anti(x)b = k.Ifb = xk,then, b ∈ xkerf ⇒ f−1(h) ⊆ xkerf (1) (2) Let’sshowthatxkerf ⊆ f−1(h).Letb ∈ xkerf.Foratleastk ∈ kerf,b = xk ⇒ f(b) = f(xk) = f(x)f(k) = hneut = h H If f−1(h) = bandb ∈ f−1(h),then xkerf ⊆ f−1(h) (2) by(1)and(2),weobtainxkerf = f−1(h). Theorem7. Let ϕ : N → N beaneutro-homomorphismofNETGN andN . 1 2 1 2 (1) If H (cid:69) N , then ϕ−1(H )(cid:69) N . 2 2 2 1 (2) IfH (cid:69) N and ϕisaneutro−epimorhismthen ϕ(H )(cid:69) N . 1 1 1 2 Proof. (1) If x ∈ ϕ−1(H ) and a ∈ N , then ϕ(x) ∈ H and so 2 1 2 ϕ((ax)(anti(a)) = ϕ(a)ϕ(x)anti(ϕ(a)) ∈ H . SinceH isneutrosophictripletnormalsubgroup. 2 2 Weconcludeax(anti(a)) ∈ ϕ−1(H ). 2 (2) Since H is neutrosophic triplet normal subgroup, we have ϕ(a)ϕ(H )anti(ϕ(a)) ⊆ ϕ(H ). 1 1 1 Since we assume ϕ is surjective, every b ∈ N can be written as b = ϕ(a),a ∈ N . Therefore, 2 1 bϕ(H )anti(b) ∈ ϕ(H ). 1 1 Theorem8([26]). Let f : N → H beaneutro-homomorphismfromaNETGNtoaNETGH.Kerf (cid:67) N. Theorem9. LetNbeNETGandH N.Themap ϕ : N → N/H, n → nH, isaneutro-homomorphism withneutrosophictripletkernelker ϕ(cid:2)= H. Symmetry2018,10,321 7of14 Proof. We have ϕ(ab) = (ab)H = (aH)(bH) = ϕ(a)ϕ(a), so φ is a neutro-homomorphism. As to the neutrosophic triplet kernel, a ∈ kerϕ ⇔ ϕ(a) = H (since H is neutral in N/H) ⇔ aH = H (by definitionofφ) ⇔ a ∈ H. Theorem10. LetNbeNETGandH ⊆ Nbeanon-emptyneutrosophicextendedtripletsubset. Then H (cid:69) N, ifandonlyifthereexistsaneutro-homomorphism ϕ : N → N with H = kerϕ. 1 2 Proof. Itsstraightforward. 3.2. Neutro-Isomorphism Definition15. Let(N ,∗)and(N ,◦)betwoNETGs. Ifamapping f : N → N neutro-homomorphismis 1 2 1 2 onetooneandontofiscalledneutro-isomorphism. Here,N andN arecalledneutro-isomorphicanddenoted 1 2 ∼ asN = N . 1 2 Theorem11. Let(N ,∗)and(N ,◦)betwoNETG’s. If f : N → N isaneutro-isomorphismofNETG’s, 1 2 1 2 thensois f−1 : N → N . 2 1 Proof. It is obvious to show that f is one to one and onto. Now let’s show that f is neutro-homomorphism. Let x = ϕ(a),y = ϕ(b),∀a,b ∈ N ,∀x,y ∈ N andso, a = anti(ϕ(x)),b = 1 2 anti(ϕ(y)).Nowanti(xy) = anti(ϕ(ϕ(a)ϕ(b)))= anti(ϕ(ϕ(ab))) = ab = anti(ϕ(x))anti(ϕ(y)). 3.3. Neutro-Automorphism. Definition16. Let(N ,∗)and(N ,◦)betwoNETG’S.Ifamapping f : N → N isonetooneandontofis 1 2 1 2 calledneutro-automorphism. Definition17. LetNbeNETG. ϕ ∈ AutN iscalledaneutro-innerautomorphismifthereisan ∈ N such that ϕ = ϕ . n Proposition 1. Let N be a NETG. For a ∈ N, f : N → N such that x → ax(anti(a) is a a 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)) ⇒ ax = ay ⇒ anti(a)ax = anti(a)ay⇒ neut(a)x = neut(a)y ⇒ x = y. 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 = anti(a)ya ⇒ x = anti(a)ya. So,f isonto. Therefore,f isaneutro-automorphism. a Symmetry2018,10,321 8of14 Lemma1. LetabeanelementofNETGNsuchthata2 = a.Thena = neut(a). Proof. Wehave = (anti(a)∗a)∗aforanti(a) ∈ N (antiaxiom) = anti(a)∗a2 (associativityaxiom) = anti(a)∗a(byassumption) = neut(a)(bydefinitionofanti) Theorem 12. Let N be NETG and H ,H ≤ N. Then the neutrosophic extended triplet set H H = 1 2 1 2 {ab : a ∈ H ,b ∈ H }isaneutrosophicextendedtripletsubgroupinthecasethatH H = H H . 1 2 1 2 2 1 Symmetry 2018, 10, x 9 of 15 Proof. SupposeH H isaneutrosophicextendedtripletsubgroup. Then,foralla ∈ H ,b ∈ H ,we 1 2 1 2 haveaa∈ntHi(a,) abn∈ti(Hb) ∈ Hth1eHre2,biy.e .,Ha2nHti1(⊆h)H 1=H 2a.bB,ut aalsnodf orthhe∈n H1hH 2=w eafinntid(ba)∈aHnt1i,(ba∈)∈H2HthHere.by So anti(h) =1ab,andt2henh = anti(b)anti(a) ∈ H H .So H H ⊆ H H ,that’s, H H = H H .O2n1the otheHr1hHan2d⊆, aHss2uHm1e, tthhaatt’sH, 1HH21H=2H=2 HH12.HTh1.e nO2∀na 1,thae(cid:48) ∈othH1e1r, 2bh,abn(cid:48)d∈,2 aHs12suwmeeh tahvaet 1aHb2a1(cid:48)Hb(cid:48)2∈=2a HH122HH11b.(cid:48) =Then aH1∀Ha2b,(cid:48) a='∈HH1H,2 b.,F ubr't∈heHrmor ew,e∀ ah∈avHe 1,bab∈a'Hb'2∈waeHhaHveba' n=ti( aabH) =Habn't i=(b) HantHi(a.) ∈ FuHr2thHe1rm=ore, 1 2 2 1 1 2 1 2 H1H∀2a. ∈H , b∈H we have anti(ab) = anti(b)anti(a)∈H H = H H . □ 1 2 2 1 1 2 4. Fundamental Theorem of Neutro-Homomorphism 4. FundamentalTheoremofNeutro-Homomorphism The fundamental theorem of neutro-homomorphism relates the structure of two objects Thefundamentaltheoremofneutro-homomorphismrelatesthestructureoftwoobjectsbetween between which a neutrosophic kernel and image of the neutro-homomorphism is given. It is also whichaneutrosophickernelandimageoftheneutro-homomorphismisgiven. Itisalsosignificant significant to prove neutro-isomorphism theorems. In this section, we give and prove the toproveneutro-isomorphismtheorems. Inthissection,wegiveandprovethefundamentaltheorem fundamental theorem of neutro-homomorphism. Then, we discuss a few special cases. Finally, we of neutro-homomorphism. Then, we discuss a few special cases. Finally, we give examples by give examples by using NETG. usingNETG. TheTohreemore1m3. L1e3t. NL1,etN 2Nb1e, NNE2T Gb’es aNndETφG:’sN 1a→nd Nφ2 b:e aNn1e→utroN-h2om boem oar phniesumtr.oT-hhoemno,mNo1r/pkheirsm(φ. ) T∼=hen, im(φN).F/ukretrhe(rφm)o≌reiifmϕ(iφs)n.e uFturor-tehpeirmmoorrpeh iifs mϕ ,isth neenutro-epimorphism, then 1 ∼ NN1//kkeerrϕφ ≌= NN.2 . 1 2 N ϕ im(ϕ) 1 φ i N /ker(ϕ) 1 Proof. We will construct an explicit map i : N /ker(φ)→im(φ) and prove that it is a 1 Pronoef.utWroe-isowmillorpchonissmtr uacntd awnelle dxpefliicnietd.m Sainpce ik:erN(ϕ1/) kise rn(φeu)t→rosiomp(hφi)c trainpdlet pnroorvmeal tshuabtgritouips ofa N1. neuLtreot -isomoKrp=hiskmera(φnd),well d aenfidn ed. Srinecceallk er(ϕ)tihsant eutrosoNph/icKt ri=pl e{tanKor m: aal∈suNbg}ro.up of DNe1f.ine 1 1 LetiK := Nker/(φK),→andimre(cφal)l,th ia t:N n1/KK→=φ{a(Kn):,a n∈∈N1N}..D Tefhinues, iw:eN n1/eeKd→ to icmh(eφck), tih:en fKol→lowφi(nng) ,cnon∈dNit1io.ns. Thus,wen1eedtocheckthefollowingconditions. 1 (1) i is well defined (1) iiswelldefined (2) i is injective (2) iisinjective (3) i is surjective (3) iissurjective (4) i is a neutro-homomorphism (4) iisaneutro-homomorphism (1) We must show that if aK = bK, then i(aK) = (bK). Suppose aK = bK. We have aK = bK anti(b)aK = K anti(b)a∈K. Here, neut = φ(anti(b)a) = φ(anti(b) φ(a) (n2) = anti(φ(b)) φ(a)φ(a) = φ(b). Hence, i(aK) = φ(a) = φ(b) = i(bK). Therefore, it is well defined. i(aK) = i(bK)aK = bK. i(aK) = i(bK). (2) We must show that Suppose that Then i(aK) = i(bK)aK = bK. φ(anti(b)) φ(a) = neut φ(anti(b)a) = neut anti(b)a∈K (n2) (n2) anti(b)aK = K (aN = N ⇔a∈N ). 2 2 2 Thus, i is injective. (3) We must show that for any element in the domain (N1/K) gets mapped to it by i. let’s pick any element φ(a)∈im(φ). By definition, i(aK) = φ(a), hence i is surjective. Symmetry 2018, 10, x 9 of 15 a∈H , b∈H thereby anti(h) = ab, and then h = anti(b)anti(a)∈H H . So 1 2 2 1 H H ⊆ H H , that’s, H H = H H . On the other hand, assume that H H = H H . Then 1 2 2 1 1 2 2 1 1 2 2 1 ∀a, a'∈H , b, b'∈H we have aba'b'∈aH H b' = aH H b' = H H . Furthermore, 1 2 2 1 1 2 1 2 ∀a∈H , b∈H we have anti(ab) = anti(b)anti(a)∈H H = H H . □ 1 2 2 1 1 2 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. Theorem 13. Let N1, N2 be NETG’s and φ : N1 → N2 be a neutro-homomorphism. Then, N /ker(φ)≌im(φ). Furthermore if ϕ is neutro-epimorphism, then 1 N /kerϕ ≌ N . 1 2 N ϕ im(ϕ) 1 φ i N /ker(ϕ) 1 Proof. We will construct an explicit map i : N /ker(φ)→im(φ) and prove that it is a 1 neutro-isomorphism and well defined. Since ker(ϕ) is neutrosophic triplet normal subgroup of N1. Let K =ker(φ), and recall that N /K = {aK : a∈N }. Define 1 1 i : N /K →im(φ), i : nK →φ(n), n∈N . Thus, we need to check the following conditions. 1 1 (1) i is well defined (2) i is injective (3) i is surjective (4) i is a neutro-homomorphism (1) We must show that if aK = bK, then i(aK) = (bK). Suppose aK = bK. We have Symmetry2018,10,321 9of14 aK = bK anti(b)aK = K anti(b)a∈K. (1) We mHuestre,s how that if aK = bK, then i(aK) =ne(ubtK). =Su pφp(oasnetia(Kb)a=) =bK .φ(Waentih(abv)e φ(a) (n2) aK = b=K a⇒ntain(φti((bb))a)K φ=(aK)⇒aφnt(ia(b)) a=∈ Kφ.(bH)e.re, Hneeuntc(en,2 ) =i(aφK(a)n ti=(b )φa)(a=) =φ( aφn(tbi()b )φ=(a i)(bK). = anti(φ(b))φ(a) ⇒ φ(a) = φ(b). Hence, i(aK) = φ(a) = φ(b) = i(bK). Therefore, it is welldTefihnereedf.ore, it is well defined. (2) Wemustshowthat i(aK) =i(ia(bKK)) ⇒= aiK(b=Kb)K. SaupKp o=se bthKat.i(aK) =i(bK).Tih(aenK) = i(bK). (2) We must show that Suppose that Then i(aK) = i(bK)aK = bK. φ(anti(b)) φ(a) = neut φ(anti(b)a) = neut anti(b)a∈K (n2) (n2) anti(b)aK = K (aN = N ⇔a∈N ). 2 2 2 Thus,Tiihsuisn,j ei cisti vinej.ective. (3) W(3e)m uWste smhouwst tshhaotwfo trhaant yfoerl eamnye netleinmtehnet dino mthaei ndo(Nma/inK )(Nge1/tKs)m gaeptsp emdatpopietdb ytoi .itl ebty’s ip. ilcekt’sa npyick any elemenelteφm(ean)t∈ φim(a(φ))∈.Bimyd(φefi)n.i tBioyn d,ei(fainKi)tio=n,φ i((aa),Khe)n 1c=e iφis(sau)r,je chteivnec.e i is surjective. (4) We must show that 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,iisaneutro-homomorphism. Insummary,since i : N /K →im(φ) isawell-definedneutro-homomorphismthatisinjective 1 and surjective. Therefore, it is a neutro-isomorphism. i.e.,N /K ∼= im(φ), and the fundamental 1 theoremofneutro-homomorphismisproven. Corollary1(AFewSpecialCasesofFundamentalTheoremofNeutro-homomorphism). • LetN=(1,1,1)beatrivialneutrosophicextendedtriplet.Ifϕ:N →N isanembedding,thenneutrosophic 1 2 ∼ ∼ ker(ϕ)={neut(1)=1N }. TheTheorem12saysthatim(ϕ)={N /1N }=N . 1 1 1 1 • If ϕ: N →N isamap ϕ(n)=neut(1)=1N foralln ∈N ,thenneutrosophicker(ϕ)=N ,soTheorem 1 2 2 2 1 1 ∼ 13saysthat1N =im(ϕ)=N /N . 2 1 1 Example2. TheneutrosophicextendedtripletalternatinggroupA (theneutrosophicextendedtripletsubgroup n ofevenpermutationinNETGS )hasindex2inS . n n Solution. Toprovethat[S :A ]=2. Wewillconstructasurjectiveneutro-homomorphismφ: S →Z with n n n 2 neutrosophictripletkerφ = A . HeretheneutrosophicextendedtripletsofZ are(0,0,0)and(1,1,1). Ifthis n 2 ∼ isachieved,itwouldfollowthatS /A = Z ,so|S /A |=|Z |=2,andtherefore[S :A ]=|S /A |=2, n n 2 n n 2 n n n n (cid:40) [0]if f iseven asdesired. Defineφ: S →Z byφ(f)= n 2 [1]if f isodd Byconstructionφissurjective. Toprovethatφisaneutro-homomorphismweneedtoshowthat φ(x)+φ(y)=φ(xy),∀x,y∈S . Hereifxandyarebothevenorbothodd,thenxyiseven. Ifxiseven n andyisodd,orifxisoddandyiseven,thenxyisodd. Letusseethesefourdifferentcasesasfollows: (1) x and y are both even. Then xy is also even. 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,soφ(x)+φ(y)=[1]+[1]=[0]=φ(xy). Thus,weverified thatφisaneutro-homomorphism. Finally,neutrosophictrpletkerφ={x∈S : φ(x)=[0] }isthe n 2 neutrosophicextendedtripletsetofallevenpermutations,soneutrosophictrietkerϕ = A . n Symmetry2018,10,321 10of14 5. FirstNeutro-IsomorphismTheorem Thefirstneutro-isomorphismtheoremrelatestwoneutrosophictripletquotientgroupsinvolving productsandintersectionsofneutrosophicextendedtripletsubgroups. Inthissection,wegiveand provethefirstneutro-isomorphismtheorem. Finally,wegiveanexamplebyusingNETG. Theorem 14. Let N be NETG and H, K be two neutrosophic extended triplet subgroup of N and H is a neutrosophictripletnormalinK.Then (a) HKisneutrosophictripletsubgroupofN. (cid:84) (b) H KisneutrosophictripletnormalsubgroupinK. (c) HHK ∼= HK(cid:84)K Proof. (a) Letxy ∈ HK.Ifx = h k andy = h k ,h h ∈ Handk ,k ∈ K.Consider 1 1 2 2 1 2 1 2 x(anti(y)) = (h k ) anti(h k ) 1 1 2 2 = (h k )anti(k )anti(h ) 1 1 2 2 = h (k (anti(k )))anti(h ),(k = k (anti(k2)) : k ∈ K 1 1 2 2 3 1 3 = h k (anti(h )) 1 3 2 = h k (anti(h ))anti(k )k 1 3 2 3 3 = h k (anti(h ))anti(k )k 1 3 2 3 3 = h h k because H (cid:67)ksoh = k (anti(h ))anti(k ) ∈ H 1 2 3 3 3 2 3 ⇒ x(anti(y) = h k ∈ HK,(h = h h ) 4 3 4 1 2 ⇒ HKisNETGofN. (b) WehavetoproveH∩KisneutrosophictripletnormalsubgroupinkorH∩K (cid:67)k.Letx ∈ H∩K andx ∈ K.Ifx ∈ Handx ∈ K,thenkx(anti(k)) ∈ HbecauseH (cid:67)kandkx(anti(k)) ∈ Kbecause xk ∈ K.Thus,kx(anti(k)) ∈ H∩K.SinceH∩K (cid:67)k. (c) HHK ∼= HK(cid:84)K. LetH(cid:84)K=D,so DK = HK(cid:84)K. Nowlet’sdefineamappingϕ: HK→DK byφ(hk) = KD. 1. ϕiswelldefined h k = h k ,h h ∈ Handk k ∈ K 1 1 2 2 1 2 1 2 k h(cid:48) = k h(cid:48) 1 1 2 2 ⇒ anti(k )k h(cid:48) = h(cid:48) 2 1 1 2 ⇒ anti(k )k = h(cid:48)(anti(h )),h(cid:48)(anti(h )) ∈ H 2 1 2 1 2 1 ⇒ anti(k )k ∈ H, butanti(k )k ∈ K 2 1 2 1 ⇒ anti(k )k ∈ H∩K = D 2 1 ⇒ anti(k )k ∈ D 2 1 ⇒ anti(k )k D = D 2 1 ⇒ k D = k D 1 2 ⇒ φ(h k ) = φ(h k ). 1 1 2 2 2. ϕisneutro-homomorphism. Φ(h k .h k ) = φ(h (k h )k 1 1 2 2 1 1 2 2 = φ(h h2(cid:48)k k ) 1 1 2 = K k D 1 2 = k Dk D 1 2 = φ(h k ).φ(h k ) 1 1 2 2 3. ϕisonto.