NeutrosophicSetsandSystems,Vol.39,2021
UniversityofNewMexico
NeutroOrderedAlgebra:ApplicationstoSemigroups
MadeleineAl-Tahan1, ,F.Smarandache2 ,andBijanDavvaz3
1 DepartmentofMathematics,LebaneseInternationalUniversity,Bekaa,Lebanon;madeline.tahan@liu.edu.lb
2 MathematicsandScienceDivision,UniversityofNewMexico,GallupCampus,USA,smarand@unm.edu
3 DepartmentofMathematics,YazdUniversity,Yazd,Iran;davvaz@yazd.ac.ir
∗ Correspondence:madeline.tahan@liu.edu.lb
Abstract StartingwithapartialorderonaNeutroAlgebra,wegetaNeutroStructure.Thelatterifitsatisfies theconditionsofNeutroOrder,itbecomesaNeutroOrderedAlgebra.Inthispaper,weapplyournewdefined notiontosemigroupsbystudyingNeutroOrderedSemigroups.Moreprecisely,wedefinesomerelatedtermslike NeutrosOrderedSemigroup,NeutroOrderedIdeal,NeutroOrderedFilter,NeutroOrderedHomomorphism,etc.,illustratethemviasomeexamples,andstudysomeoftheirproperties.
Keywords: NeutroAlgebra,NeutroSemigroup,NeutroOrderedAlgebra,NeutrosOrderedSemigroup,NeutroOrderedIdeal,NeutroOrderedFilter,NeutroOrderedHomomorphism,NeutroOrderedStrongHomomorphism.
AMSMathematicsSubjectClassification: 08A99,06F05,06F25.
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1. Introduction
Neutrosophy,thestudyofneutralities,isanewbranchofPhilosophyinitiatedbySmarandachein1995.Ithasmanyapplicationsinalmosteveryfield.Manyalgebraistsworkedonthe connectionbetweenneutrosophyandalgebraicstructures.Foremoredetails,wereferto[1–3]. Unliketheidealisticorabstractalgebraicstructures,frompuremathematics,constructedon agivenperfectspace(set),wheretheaxioms(laws,rules,theorems,resultsetc.)aretotally (100%)trueforallspaceselements,ourworldandrealityconsistofapproximations,imperfections,vagueness,andpartialities.Startingfromthelatteridea,Smarandacheintroduced NeutroAlgebra.In2019and2020,he[11–13]generalizedtheclassicalAlgebraicStructuresto NeutroAlgebraicStructures(orNeutroAlgebras)whoseoperationsandaxiomsarepartially true,partiallyindeterminate,andpartiallyfalseasextensionsofPartialAlgebra,andtoAntiAlgebraicStructures(orAntiAlgebras)whoseoperationsandaxiomsaretotallyfalse.And ingeneral,heextendedanyclassicalStructure,innomatterwhatfieldofknowledge,toa
M.Al-Tahan,F.Smarandache,andB.Davvaz,NeutroOrderedAlgebra:ApplicationstoSemigroups
NeutroStructureandanAntiStructure.APartialAlgebraisanalgebrathathasatleastone PartialOperation,andallitsAxiomsareclassical.Throughatheorem,Smarandache[11] provedthataNeutroAlgebraisageneralizationofPartialAlgebraandgavesomeexamples ofNeutroAlgebrasthatarenotPartialAlgebras.Manyresearchersworkedonspecialtypes ofNeutroAlgebrasandAntiAlgebrasbyapplyingthemtodifferenttypesofalgebraicstructuressuchasgroups,rings, BE-Algebras, BCK-Algebras,etc.Formoredetails,werefer to[4–6,9,10,14,15].
InspiredbyNeutroAlgebraandorderedAlgebra,ourpaperintroducesandstudiesNeutroOrderedAlgebra.Anditisconstructedasfollows:AfteranIntroduction,inSection2, weintroduceNeutroOrderedAlgebraandsomerelatedtermssuchasNeutroOrderedSubAlgebraandNeutroOrderedHomomorphism.AndinSection3,weapplytheconceptofNeutroOrderedAlgebratosemigroupsandstudyNeutroOrderedSemigroupsbypresentingseveral examplesandstudyingsomeoftheirinterestingproperties.
2. NeutroOrderedAlgebra
Inthissection,wecombinethenotionsoforderedalgebraicstructuresandNeutroAlgebra tointroduce NeutroOrderedAlgebra.Somenewdefinitionsrelatedtothenewconceptare presented.Fordetailsaboutorderedalgebraicstructures,wereferto[7,8].
Definition2.1. [11]Anon-emptyset A endowedwith n operations“ i”for i =1,...,n,is called NeutroAlgebra ifithasatleastoneNeutroOperationoratleastoneNeutroAxiomwith noAntiOperationsnorAntiAxioms.
Definition2.2. [8]Let A beanAlgebrawith n operations“ i”and“≤”beapartialorder (reflexive,anti-symmetric,andtransitive)on A.Then(A, 1,..., n, ≤)isanOrderedAlgebra ifthefollowingconditionshold.
If x ≤ y ∈ A then z i x ≤ z i y and x i z ≤ y i z forall i =1,...,n and z ∈ A
Definition2.3. Let A beaNeutroAlgebrawith n (Neutro)operations“ i”and“≤”bea partialorder(reflexive,anti-symmetric,andtransitive)on A.Then(A, 1,..., n, ≤)isa NeutroOrderedAlgebra ifthefollowingconditionshold.
(1) Thereexist x ≤ y ∈ A with x = y suchthat z i x ≤ z i y and x i z ≤ y i z forall z ∈ A and i =1,...,n.(Thisconditioniscalleddegreeoftruth,“T ”.)
(2) Thereexist x ≤ y ∈ A and z ∈ A suchthat z i x z i y or x i z y i z forsome i =1,...,n.(Thisconditioniscalleddegreeoffalsity,“F ”.)
(3) Thereexist x ≤ y ∈ A and z ∈ A suchthat z i x or z i y or x i z or y i z are indeterminate,ortherelationbetween z i x and z i y,ortherelationbetween x i z M.Al-Tahan,F.Smarandache,andB.Davvaz,NeutroOrderedAlgebra:Applicationsto Semigroups
and y i z areindeterminateforsome i =1,...,n.(Thisconditioniscalleddegreeof indeterminacy,“I”.)
Where(T,I,F )isdifferentfrom(1, 0, 0)thatrepresentstheclassicalOrderedAlgebraaswell from(0, 0, 1)thatrepresentstheAntiOrderedAlgebra.
Definition2.4. Let(A, 1,..., n, ≤)beaNeutroOrderedAlgebra.If“≤”isatotalorderon A then A iscalled NeutroTotalOrderedAlgebra
Definition2.5. Let(A, 1,..., n, ≤A)beaNeutroOrderedAlgebraand ∅ = S ⊆ A.Then S isa NeutroOrderedSubAlgebra of A if(S, 1,..., n, ≤A)isaNeutroOrderedAlgebraandthere exists x ∈ S with(x]= {y ∈ A : y ≤A x}⊆ S.
Remark2.6. ANeutroOrderedAlgebrahasatleastoneNeutroOrderedSubAlgebrawhichis itself.
Definition2.7. Let(A, 1,..., n, ≤A)and(B, 1,..., n, ≤B )beNeutroOrderedAlgebras and φ : A → B beafunction.Then
(1) φ iscalled NeutroOrderedHomomorphism ifthereexist x,y ∈ A suchthatforall i = 1,...,n, φ(x i y)= φ(x) i φ(y),andthereexist a ≤A b ∈ A with a = b suchthat φ(a) ≤B φ(b).
(2) φ iscalled NeutroOrderedIsomomorphism if φ isabijectiveNeutroOrderedHomomorphism.Inthiscase,wewrite A ∼ =I B.
(3) φ iscalled NeutroOrderedStrongHomomorphism ifforall x,y ∈ A andforall i = 1,...,n,wehave φ(x i y)= φ(x) i φ(y)and a ≤A b ∈ A isequivalentto φ(a) ≤B φ(b) forall a,b ∈ A.
(4) φ iscalled NeutroOrderedStrongIsomomorphism if φ isabijectiveNeutroOrderedStrongHomomorphism.Inthiscase,wewrite A ∼ =SI B.
Example2.8. Let(A, 1,..., n, ≤A)beaNeutroOrderedAlgebra, B aNeutroOrderedSubAlgebraof A,and φ : B → A betheinclusionmap(φ(x)= x forall x ∈ B).Then φ isa NeutroOrderedStrongHomomorphism.
Example2.9. Let(A, 1,..., n, ≤A)beaNeutroOrderedAlgebraand φ : A → A bethe identitymap(φ(x)= x forall x ∈ A).Then φ isaNeutroOrderedStrongIsomomorphism.
Remark2.10. EveryNeutroOrderedStrongHomomorphism(NeutroOrderedStrongIsomorphism)isaNeutroOrderedHomomorphism(NeutroOrderedIsomorphism).
Theorem2.11. Therelation“∼ =SI ”isanequivalencerelationonthesetofNeutroOrderedAlgebras.
M.Al-Tahan,F.Smarandache,andB.Davvaz,NeutroOrderedAlgebra:Applicationsto Semigroups
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Proof. BytakingtheidentitymapandusingExample2.9,wecaneasilyprovethat“∼ =SI ” isareflexiverelation.Let A ∼ =SI B.ThenthereexistaNeutroOrderedStrongIsomorphism φ :(A, 1,..., n, ≤A) → (B, 1,..., n, ≤B ).Weprovethat φ 1 : B → A isaNeutroOrderedStrongIsomorphism.Forall b1,b2 ∈ B,thereexist a1,a2 ∈ A with φ(a1)= b1 and φ(a2)= b2.Forall i =1,...,n,wehave: φ 1(b1 i b2)= φ 1(φ(a1) i φ(a2))= φ 1(φ(a1 i a2))= a1 i a2 = φ 1(b1) i φ 1(b2) Moreover,having a1 ≤A a2 ∈ A equivalentto φ(a1) ≤B φ(a2) ∈ B and φ anontofunction impliesthat b1 = φ(a1) ≤B φ(a2)= b2 ∈ B isequivalentto a1 = φ 1(b1) ≤A a2 = φ 1(b2) ∈ A Thus, B ∼ =SI A andhence,“∼ =SI ”isasymmetricrelation.Let A ∼ =SI B and B ∼ =SI C.Then thereexistNeutroOrderedStrongIsomorphisms φ : A → B and ψ : B → C.Onecaneasilysee that ψ ◦ φ : A → C isaNeutroOrderedStrongIsomorphism.Thus, A ∼ =SI C andhence,“∼ =SI ” isatransitiverelation.
Remark2.12. Therelation“∼ =I ”isareflexiveandsymmetricrelationonthesetofNeutroOrderedAlgebras.Butitmayfailtobeatransitiverelation.
3. NeutroOrderedSemigroup
Inthissection,weusethedefinednotionofNeutroOrderedAlgebrainSection2andapply ittosemigroups.Asaresult,wedefineNeutroOrderedSemigroupandotherrelatedconcepts. Moreover,wepresentsomeexamplesoffiniteaswellasinfiniteNeutroOrderedSemigroups. Finally,westudysomepropertiesofNeutroOrderedSubSemigroups,NeutroOrderedIdeals,and NeutroOrderedFilters.
Definition3.1. [8]Let(S, ·)beasemigroup(“·”isanassociativeandabinaryclosed operation)and“≤”apartialorderon S.Then(S,, ≤)isan orderedsemigroup ifforevery x ≤ y ∈ S, z x ≤ z y and x z ≤ y z forall z ∈ S.
Definition3.2. [8]Let(S, · , ≤)beanorderedsemigroupand ∅ = M ⊆ S.Then
(1) M isan orderedsubsemigroup of S if(M,, ≤)isanorderedsemigroupand(x] ⊆ M forall x ∈ M .i.e.,if y ≤ x then y ∈ M
(2) M isan orderedleftideal of S if M isanorderedsubsemigroupof S andforall x ∈ M , r ∈ S,wehave rx ∈ M .
(3) M isan orderedrightideal of S if M isanorderedsubsemigroupof S andforall x ∈ M , r ∈ S,wehave xr ∈ M
(4) M isan orderedideal of S if M isboth:anorderedleftidealof S andanorderedright idealof S M.Al-Tahan,F.Smarandache,andB.Davvaz,NeutroOrderedAlgebra:Applicationsto Semigroups
(5) M isan orderedfilter of S if(M, )isasemigroupandforall x,y ∈ S with x y ∈ M , wehave x,y ∈ M and[y) ⊆ M forall y ∈ M .i.e.,if y ∈ M with y ≤ x then x ∈ M
Definition3.3. Let(S, ·)beaNeutroSemigroupand“≤”beapartialorder(reflexive,antisymmetric,andtransitive)on S.Then(S,, ≤)isa NeutroOrderedSemigroup ifthefollowing conditionshold.
(1) Thereexist x ≤ y ∈ S with x = y suchthat z x ≤ z y and x z ≤ y z forall z ∈ S. (Thisconditioniscalleddegreeoftruth,“T ”.)
(2) Thereexist x ≤ y ∈ S and z ∈ S suchthat z x z y or x z y z.(Thiscondition iscalleddegreeoffalsity,“F ”.)
(3) Thereexist x ≤ y ∈ S and z ∈ S suchthat z · x or z · y or x · z or y · z areindeterminate, ortherelationbetween z x and z y,ortherelationbetween x z and y z are indeterminate.(Thisconditioniscalleddegreeofindeterminacy,“I”.)
Where(T,I,F )isdifferentfrom(1, 0, 0)thatrepresentstheclassicalOrderedSemigroup,and from(0, 0, 1)thatrepresentstheAntiOrderedSemigroup.
Definition3.4. Let(S, · , ≤)beaNeutroOrderedSemigroup.If“≤”isatotalorderon A then A iscalled NeutroTotalOrderedSemigroup
Definition3.5. Let(S,, ≤)beaNeutroOrderedSemigroupand ∅ = M ⊆ S.Then
(1) M isa NeutroOrderedSubSemigroup of S if(M,, ≤)isaNeutroOrderedSemigroupand thereexist x ∈ M with(x]= {y ∈ S : y ≤ x}⊆ M .
(2) M isa NeutroOrderedLeftIdeal of S if M isaNeutroOrderedSubSemigroupof S and thereexists x ∈ M suchthat r x ∈ M forall r ∈ S
(3) M isa NeutroOrderedRightIdeal of S if M isaNeutroOrderedSubSemigroupof S and thereexists x ∈ M suchthat x · r ∈ M forall r ∈ S
(4) M isa NeutroOrderedIdeal of S if M isaNeutroOrderedSubSemigroupof S andthere exists x ∈ M suchthat r · x ∈ M and x · r ∈ M forall r ∈ S.
(5) M isa NeutroOrderedFilter of S if(M, · , ≤)isaNeutroOrderedSemigroupandthere exists x ∈ S suchthatforall y,z ∈ S with x y ∈ M and z x ∈ M ,wehave y,z ∈ M andthereexists y ∈ M [y)= {x ∈ S : y ≤ x}⊆ M .
Proposition3.6. Let (S, · , ≤) beaNeutroOrderedSemigroupand ∅ = M ⊆ S.Thenthe followingstatementsaretrue.
(1) If S containsaminimumelement(i.e.thereexists m ∈ S suchthat m ≤ x for all x ∈ S.)and M isaNeutroOrderedSubSemigroup(orNeutroOrderedRightIdealor NeutroOrderedLeftIdealorNeutroOrderedIdeal)of S thentheminimumelementisin M
M.Al-Tahan,F.Smarandache,andB.Davvaz,NeutroOrderedAlgebra:Applicationsto Semigroups
(2) IfIf S containsamaximumelement(i.e.thereexists n ∈ S suchthat x ≤ n forall x ∈ S.)and M isaNeutroOrderedFilterof S then M containsthemaximumelement of S
Proof. Theproofisstraightforward.
Remark3.7. Let(S, · , ≤)beaNeutroOrderedSemigroup.TheneveryNeutroOrderedIdeal of S isNeutroOrderedLeftIdealof S andaNeutroOrderedRightIdealof S.Buttheconverse maynothold.(SeeExample3.16.)
Definition3.8. Let(A, , ≤A)and(B, , ≤B )beNeutroOrderedSemigroupsand φ : A → B beafunction.Then
(1) φ iscalled NeutroOrderedHomomorphism if φ(x y)= φ(x) φ(y)forsome x,y ∈ A andthereexist a ≤A b ∈ A with a = b suchthat φ(a) ≤B φ(b).
(2) φ iscalled NeutroOrderedIsomomorphism if φ isabijectiveNeutroOrderedHomomorphism.
(3) φ iscalled NeutroOrderedStrongHomomorphism if φ(x y)= φ(x) φ(y)forall x,y ∈ A and a ≤A b ∈ A isequivalentto φ(a) ≤B φ(b) ∈ B.
(4) φ iscalled NeutroOrderedStrongIsomomorphism if φ isabijectiveNeutroOrderedStrongHomomorphism.
Example3.9. Let S1 = {s,a,m} and(S1, 1)bedefinedbythefollowingtable. 1 sam s sms a mam m mmm Since s ·1 (s ·1 s)= s =(s ·1 s) ·1 s and s ·1 (a ·1 m)= s = m =(s ·1 a) ·1 m,itfollowsthat (S1, 1)isaNeutroSemigroup. Bydefiningthetotalorder ≤1= {(m,m), (m,s), (m,a), (s,s), (s,a), (a,a)}
on S1,wegetthat(S1, ·1, ≤1)isaNeutroTotalOrderedSemigroup.Thisiseasilyseenas: m ≤1 s impliesthat m 1 x ≤1 s 1 x and x 1 m ≤1 x 1 s forall x ∈ S1.Andhaving s ≤1 a but s 1 s = s 1 m = a 1 s.
M.Al-Tahan,F.Smarandache,andB.Davvaz,NeutroOrderedAlgebra:Applicationsto Semigroups
Sets and Systems, Vol. 39, 2021
Example3.10. Let S2 = {0, 1, 2, 3} and(S2, 2)bedefinedbythefollowingtable. 2 0123 0 0003 1 0113 2 0322 3 3333
Since0 ·2 (0 ·2 0)=0=(0 ·2 0) ·2 0and1 ·2 (2 ·2 3)=1 =3=(1 ·2 2) ·2 3,itfollowsthat(S2, ·2) isaNeutroSemigroup. Bydefiningthetotalorder ≤2= {(0, 0), (0, 1), (0, 2), (0, 3), (1, 1), (1, 2), (1, 3), (2, 2), (2, 3), (3, 3)} on S2,wegetthat(S2, 2, ≤2)isaNeutroTotalOrderedSemigroup.Thisiseasilyseenas: 0 ≤2 3impliesthat0 ·2 x ≤2 3 ·2 x and x ·2 0 ≤2 x ·2 3forall x ∈ S2.Andhaving1 ≤2 2but 2 ·2 1=3 2 2=2 ·2 2.
WepresentexamplesonNeutroOrderedSemigroupsthatarenotNeutroTotalOrderedSemigroups.
Example3.11. Let S2 = {0, 1, 2, 3} and(S2, 2)bedefinedbythefollowingtable. ·2 0123 0 0000 1 0111 2 0132 3 0132
Since0 2 (0 2 0)=0=(0 2 0) 2 0and2 2 (2 2 3)=3 =2=(2 2 2) 2 3,itfollowsthat(S2, 2) isaNeutroSemigroup. Bydefiningthepartialorder(whichisnotatotalorder) ≤2= {(0, 0), (0, 1), (0, 2), (1, 1), (2, 2), (3, 3)}
on S2,wegetthat(S2, ·2, ≤2)isaNeutroOrderedSemigroup(thatisnotaNeutroTotalOrderedSemigroup).Thisiseasilyseenas:
0 ≤2 1impliesthat0 2 x = x 2 0=0 ≤2 1=1 2 x = x 2 1.Andhaving0 ≤2 2but 2 ·2 0=0 2 3=2 ·2 2.
M.Al-Tahan,F.Smarandache,andB.Davvaz,NeutroOrderedAlgebra:Applicationsto Semigroups
Example3.12. Let S3 = {0, 1, 2, 3, 4} and(S3, 3)bedefinedbythefollowingtable. 3 01234 0 00030 1 01211 2 04233 3 04233 4 00040
Since0 3 (0 3 0)=0=(0 3 0) 3 0and1 3 (2 3 1)=1 =4=(1 3 2) 3 1,itfollowsthat(S3, 3) isaNeutroSemigroup. Bydefiningthepartialorder ≤3= {(0, 0), (0, 1), (0, 3), (0, 4), (1, 1), (1, 3), (1, 4), (2, 2), (3, 3), (3, 4), (4, 4)} on S3,wegetthat(S3, 3, ≤3)isaNeutroOrderedSemigroupthatisnotNeutroTotalOrderedSemigroupas“≤3”isnotatotalorderon S3.Thisiseasilyseenas: 0 ≤3 4impliesthat0 ·3 x ≤3 4 ·3 x and x ·3 0 ≤3 x ·3 4forall x ∈ S3.Andhaving0 ≤3 1but 0 3 2=0 3 2=1 3 2.
Example3.13. Let Z bethesetofintegersanddefine“ ”on Z asfollows: x y = xy 1 forall x,y ∈ Z.Since0 (1 0)= 1=(0 1) 0and0 (1 2)= 1 = 3=(0 1) 2,it followsthat(Z, )isaNeutroSemigroup.Wedefinethepartialorder“≤Z”on Z as 1 ≤Z x forall x ∈ Z andfor a,b ≥ 0, a ≤Z b isequivalentto a ≤ b andfor a,b< 0, a ≤Z b isequivalent to a ≥ b.Inthisway,weget 1 ≤Z 0 ≤Z 1 ≤Z 2 ≤Z and 1 ≤Z 2 ≤Z 3 ≤Z .Having 0 ≤Z 1and x 0=0 x = 1 ≤Z x 1=1 x = x 1forall x ∈ Z and 1 ≤Z 0but ( 1) ( 1)=0 Z 1=0 ( 1)impliesthat(Z, , ≤Z)isaNeutroOrderedSemigroupwith 1asminimumelement.
Example3.14. Let“≤”betheusualorderon Z and“ ”betheoperationdefineon Z in Example3.13.Onecaneasilyseethat(Z, , ≤)isnotaNeutroTotalOrderedSemigroupas thereexistno x ≤ y ∈ Z (with x = y)suchthat z x ≤ z y forall z ∈ Z.Inthiscaseand accordingtoDefinition3.3,(T,I,F )=(0, 0, 1).
Example3.15. Let Z bethesetofintegersanddefine“⊗”on Z asfollows: x ⊗ y = xy +1for all x,y ∈ Z.Since0 ⊗ (1 ⊗ 0)=1=(0 ⊗ 1) ⊗ 0and0 ⊗ (1 ⊗ 2)=1 =3=(0 ⊗ 1) ⊗ 2,itfollows that(Z, ⊗)isaNeutroSemigroup.Wedefinethepartialorder“≤⊗”on Z as1 ≤⊗ x forall x ∈ Z andfor a,b ≥ 1, a ≤⊗ b isequivalentto a ≤ b andfor a,b ≤ 0, a ≤⊗ b isequivalent to a ≥ b.Inthisway,weget1 ≤⊗ 2 ≤⊗ 3 ≤⊗ 4 ≤⊗ ... and1 ≤⊗ 0 ≤⊗ 1 ≤⊗ 2 ≤⊗ .... Having0 ≤⊗ 1and x ⊗ 0=0 ⊗ x =1 ≤⊗ x +1= 1 ⊗ x = x ⊗ ( 1)forall x ∈ Z and M.Al-Tahan,F.Smarandache,andB.Davvaz,NeutroOrderedAlgebra:Applicationsto Semigroups
1 ≤⊗ 0but1 ⊗ 1=2 ⊗ 1=0 1impliesthat(Z, ⊗, ≤⊗)isaNeutroOrderedSemigroupwith 1asminimumelement.
WepresentsomeexamplesonNeutroOrderedSubSemigroups,NeutroOrderedRightIdeals, NeutroOrderedLeftIdeals,NeutroOrderedIdeals,andNeutroOrderedFilters.
Example3.16. Let(S3, 3, ≤3)betheNeutroOrderedSemigrouppresentedinExample3.12. Then I = {0, 1, 2} isaNeutroSubSemigroupof S3 as(I, ·3)isNeutroOperation(withno AntiAxiomas0 3 (0 3 0)=(0 3 0) 3 0)and0 ≤3 1 ∈ I but2 3 0=0 ≤3 4=2 3 1is indeterminateover I as4 / ∈ I.Moreover,(0]= {0}⊆ I.Since g 3 0=0 ∈ I forall g ∈ S3, itfollowsthat I isaNeutroOrderedLeftIdealof S3.Moreover,having1 ·3 g ∈{0, 1, 2}⊆ I impliesthat I isaNeutroOrderedRightIdealof S3.Sincethereisno g ∈ S satisfying g 3 i ∈ I and i 3 g ∈ I foraparticular i ∈ I,itfollowsthat I isnotaNeutroOrderedIdealof S3.
Remark3.17. UnlikethecaseinOrderedSemigroups,theintersectionofNeutroOrderedSubsemigroupsmaynotbeaNeutroOrderedSubsemigroup.(SeeExample3.18.)
Example3.18. Let(S3, ·3, ≤3)betheNeutroOrderedSemigrouppresentedinExample3.12. Onecaneasilyseethat J = {0, 1, 3} isaNeutroOrderedSubsemigroupof S3.FromExample 3.16,weknowthat I = {0, 1, 2} isaNeutroOrderedSubsemigroupof S3.Since({0, 1}, ·3)isa semigroupandnotaNeutroSemigroup,itfollowsthat(I ∩ J, ·3, ≤3)isnotaNeutroOrderedSubSemigroupof S3.Here, I ∩ J = {0, 1}.
Example3.19. Let(Z, , ≤Z)betheNeutroOrderedSemigrouppresentedinExample3.13. Then I = {−1, 0, 1, 2, 3, 4,...} isaNeutroOrderedIdealof Z.Thisisclearas:
(1) 0 (1 0)= 1=(0 1) 0and0 ( 1 −2)= 1 =1=(0 −1) −2; (2) g 0=0 g = 1 ∈ I forall g ∈ Z; (3) 1 ∈ I and( 1]= {−1}⊆ I; (4) 0 ≤Z 1 ∈ I impliesthat0 x = x 0= 1 ≤Z x 1= x 1=1 x forall x ∈ I and 1 ≤Z 0 ∈ I but 1 −1=0 Z 1=0 −1.
Example3.20. Let(Z, , ≤Z)betheNeutroOrderedSemigrouppresentedinExample3.13. Then F = {−1, 0, 1, 2, 3, 4,...} isaNeutroOrderedFilterof Z.Thisisclearas: (1) 0 (1 0)= 1=(0 1) 0and1 (2 3)=4 =2=(1 2) 3; (2) 1 ∈ F andforall x ∈ Z suchthat x 1=1 x = x 1 ∈ F ,wehave x ∈ F ; (3) 0 ∈ F and[0)= {0, 1, 2, 3, 4,...}⊆ F ; (4) 0 ≤Z 1 ∈ F and0 ⊗ ( 1)= 1 ≤−2=1 ⊗ ( 1)isindeterminatein F . Here, F isnotaNeutroOrderedSubSemigroupof Z asthereexistno x ∈ F with(x] ⊆ F
Example3.21. Let(S2, ·2, ≤2)betheNeutroTotalOrderedSemigrouppresentedinExample 3.10.Then F = {1, 2, 3} isaNeutroOrderedFilterof S2.Thisisclearas: M.Al-Tahan,F.Smarandache,andB.Davvaz,NeutroOrderedAlgebra:Applicationsto Semigroups
Neutrosophic
and Systems,
(1) 2 2 (2 2 2)=(2 2 2) 2 2and1 2 (2 2 1)=3 =1=(1 2 2) 2 1;
(2) 1 ·2 x ∈ F and z ·2 1 ∈ F impliesthat x,z ∈ F ; (3) 3 ∈ F and[3)= {3}⊆ F ; (4) 2 ≤2 3 ∈ F impliesthat2 ·2 x ≤2 3 ·2 x and x ·2 2 ≤2 x ·2 3forall x ∈ F and1 ≤2 2but 2 ·2 1=3 2 2=2 ·2 2.
Lemma3.22. Let (S, · , ≤S ) and (S, , ≤S ) beNeutroOrderedSemigroupsand φ : S → S be aNeutroOrderedStrongIsomorphism.Then S isaNeutroTotalOrderedSemigroupifandonly if S isaNeutroTotalOrderedSemigroup.
Proof. Theproofisstraightforward.
Remark3.23. Let(S, · , ≤S )and(S, , ≤S )beNeutroOrderedSemigroupsand φ : S → S beaNeutroOrderedIsomorphism.ThenLemma3.22maynothold.(SeeExample3.24.)
Example3.24. Let(S2, ·2, ≤2)betheNeutroTotalOrderedSemigrouppresentedinExample3.10,(S2, 2, ≤2)betheNeutroOrderedSemigrouppresentedinExample3.11,and φ :(S2, ·2, ≤2) → (S2, ·2, ≤2)bedefinedas φ(x)= x forall x ∈ S2.Onecaneasilysee that φ isaNeutroOrderedIsomorphismthatisnotNeutroOrderedStrongIsomorphismas: φ(0 2 0)= φ(0)=0= φ(0) 2 φ(0),0 ≤2 1and φ(0)=0 ≤2 1= φ(1),1 ≤2 3but φ(1)=1 2 3= φ(3).Having(S2, ·2, ≤2)aNeutroOrderedSemigroupthatisnotNeutroTotalOrderedSemigroupand(S2, ·2, ≤2)aNeutroTotalOrderedSemigroupillustratesouridea.
Lemma3.25. Let (S, · , ≤S ) and (S, , ≤S ) beNeutroOrderedSemigroupsand φ : S → S be aNeutroOrderedStrongIsomorphism.Then S containsaminimum(maximum)elementifand onlyif S containsaminimum(maximum)element.
Proof. Theproofisstraightforward.
Remark3.26. InLemma3.25,if φ : S → S isaNeutroOrderedIsomorphismthatisnota NeutroOrderedStrongIsomorphismthen S maycontainaminimum(maximum)elementand S doesnotcontain.(SeeExample3.27.)
Example3.27. WithreferencetoExample3.24,(S2, 2, ≤2)has0asitsminimumelement whereas(S2, ·2, ≤2)hasnominimumelement.
Proof. First,weprovethat(φ(M ), )isaNeutroSemigroup.Since(M, )isaNeutroSemigroup,itfollowsthat(M, ·)iseitherNeutroOperationorNeutroAssociative.
• Case(M, ·)isNeutroOperation.Thereexist x,y,a,b ∈ M suchthat x·y ∈ M and a·b/ ∈ M or x y isindeterminate.Thelatterimpliesthatthereexist φ(x),φ(y),φ(a),φ(b) ∈ φ(M )suchthat φ(x) φ(y)= φ(x y) ∈ φ(M )and φ(a) φ(b)= φ(a b) / ∈ φ(M )or φ(x) φ(y)= φ(x · y)isindeterminate.
• Case(M, )isNeutroAssociative.Thereexist x,y,z,a,b,c ∈ M suchthat(x y) z = x · (y · z)and(a · b) · c = a · (b · c).Thelatterimpliesthatthereexist φ(x),φ(y),φ(z),φ(a),φ(b),φ(c) ∈ φ(M )suchthat(φ(x) φ(y)) φ(z)= φ(x) (φ(y) φ(z))and(φ(a) φ(b)) φ(c) = φ(a) (φ(b) φ(c))(as φ isone-to-one.).
Since M isaNeutroOrderedSubsemigroupof S,itfollowsthatthereexist x ∈ M suchthat (x] ⊆ M .Itiseasytoseethat(φ(x)] ⊆ φ(M )asforall t ∈ S ,thereexist y ∈ S suchthat t = φ(y).For φ(y) ≤S φ(x),wehave y ≤S x.Thelatterimpliesthat y ∈ M andhence, t ∈ φ(M ).
Since M isaNeutroOrderedSubsemigroupof S,itfollowsthat:
(T) Thereexist x ≤S y ∈ M (with x = y)suchthat z · x ≤S z · y and x · z ≤S y · z forall z ∈ M ;
(F) Thereexist a ≤S b ∈ M and c ∈ M with a c S b c (or c a S c b);
(I) Thereexist x ≤S y ∈ M and z ∈ M with: z · x (or x · z or y · z or z · y)indeterminate or z x ≤S z y (or x z ≤S y z)indeterminatein M
Where(T,I,F ) =(1, 0, 0)and(T,I,F ) =(0, 0, 1).Thisimpliesthat
(T) Thereexist φ(x) ≤S φ(y) ∈ φ(M )(with φ(x) = φ(y)as x = y)suchthat φ(z) φ(x) ≤S φ(z) φ(y)and φ(x) φ(z) ≤S φ(y) φ(z)forall φ(z) ∈ φ(M );
(F) Thereexist φ(a) ≤S φ(b) ∈ φ(M )and φ(c) ∈ φ(M )with φ(a) φ(c) S φ(b) φ(c) (or φ(c) φ(a) S φ(c) φ(b));
(I) Thereexist φ(x) ≤S φ(y) ∈ φ(M )and φ(z) ∈ φ(M )with: φ(z) φ(x)(or φ(x) φ(z)or φ(y) φ(z)or φ(z) φ(y))indeterminateor φ(z) φ(x) ≤S φ(z) φ(y)(or φ(x) φ(z) ≤S φ(y) φ(z))indeterminatein φ(M ).
Where(T,I,F ) =(1, 0, 0)and(T,I,F ) =(0, 0, 1).Therefore, φ(M )isaNeutroOrderedSubsemigroupof S .
Lemma3.29. Let (S, · , ≤S ) and (S, , ≤S ) beNeutroOrderedSemigroupsand φ : S → S beaNeutroOrderedStrongIsomorphism.If M ⊆ S isaNeutroOrderedLeftIdeal(NeutroOrderedRightIdeal)of S then φ(M ) isaNeutroOrderedLeftIdeal(NeutroOrderedRightIdeal) of S M.Al-Tahan,F.Smarandache,andB.Davvaz,NeutroOrderedAlgebra:Applicationsto Semigroups
Proof. Weprovethatif M ⊆ S isaNeutroOrderedLeftIdealof S then φ(M )isaNeutroOrderedLeftIdealof T .ForNeutroOrderedRightIdeal,itisdonesimilarly.UsingLemma3.28, itsufficestoshowthatthereexist z ∈ φ(M )suchthatforall t ∈ S t z ∈ φ(M ).Since M is aNeutroOrderedLeftIdealof S,itfollowsthatthereexist m ∈ M suchthat s · m ∈ m forall s ∈ S.Having φ anontofunctionimpliesthatforall t ∈ S ,thereexist s ∈ S with t = φ(s). Bysetting z = φ(m),wegetthat t z = φ(s) φ(m)= φ(s m) ∈ φ(M ).
Lemma3.30. Let (S,, ≤S ) and (S, , ≤S ) beNeutroOrderedSemigroupsand φ : S → S be aNeutroOrderedStrongIsomorphism.If M ⊆ S isaNeutroOrderedIdealof S then φ(M ) isa NeutroOrderedIdealof S
Proof. TheproofissimilartothatofLemma3.29.
Example3.31. Let(Z, , ≤Z)and(Z, ⊗, ≤⊗)betheNeutroOrderedSemigroupspresentedin Example3.13andExample3.15respectively,and φ :(Z, , ≤Z) → (Z, ⊗, ≤⊗)bedefinedas φ(x)= x+2forall x ∈ Z.Onecaneasilyseethat φ isaNeutroOrderedStrongIsomorphism.By Example3.19,wehave I = {−1, 0, 1, 2, 3, 4,...} isaNeutroOrderedIdealof(Z, , ≤Z). ApplyingLemma3.30,wegetthat φ(I)= {1, 2, 3, 0, 1, 2,...} isaNeutroOrderedIdealof (Z, ⊗, ≤⊗).
Lemma3.32. Let (S, · , ≤S ) and (S, , ≤S ) beNeutroOrderedSemigroupsand φ : S → S be aNeutroOrderedStrongIsomorphism.If M ⊆ S isaNeutroOrderedFilterof S then φ(M ) isa NeutroOrderedFilterof S .
Proof. UsingLemma3.28,wegetthat(φ(M ), )isaNeutroSemigroupandthat ≤S isNeutroOrderon φ(M ).i.e.,Conditions(1),(2),and(3)ofDefinition3.3aresatisfied.
Since M isaNeutroOrderedFilterof S,itfollowsthatthereexist x ∈ M suchthat[x) ⊆ M Itiseasytoseethat[φ(x)) ⊆ φ(M )asforall t ∈ S ,thereexist y ∈ S suchthat t = φ(y).For φ(x) ≤S φ(y),wehave x ≤S y.Thelatterimpliesthat y ∈ M andhence, t ∈ φ(M ).
Since M isaNeutroOrderedFilterof S,itfollowsthatthereexist x ∈ M suchthatforall y,z ∈ S with x y ∈ M and z x ∈ M wehave y,z ∈ M .Thelatterandhaving φ ontoimplies thatthereexist t = φ(x) ∈ φ(M )suchthatforall φ(y),φ(z) ∈ S with φ(x) φ(y) ∈ φ(M ) and φ(z) φ(x) ∈ φ(M )wehave φ(y),φ(z) ∈ φ(M ).
Example3.33. Let(Z, , ≤Z)and(Z, ⊗, ≤⊗)betheNeutroOrderedSemigroupspresented inExample3.13andExample3.15respectively,and φ :(Z, , ≤Z) → (Z, ⊗, ≤⊗)bethe NeutroOrderedStrongIsomorphismdefinedas φ(x)= x +2forall x ∈ Z.ByExample3.20,we M.Al-Tahan,F.Smarandache,andB.Davvaz,NeutroOrderedAlgebra:Applicationsto Semigroups
Neutrosophic
Sets and Systems, Vol. 39, 2021
have F = {−1, 0, 1, 2, 3, 4,...} isaNeutroOrderedFilterof(Z, , ≤Z).ApplyingLemma3.32, wegetthat φ(F )= {1, 2, 3, 4, 5, 6,...} isaNeutroOrderedFilterof(Z, ⊗, ≤⊗).
Lemma3.34. Let (S, · , ≤S ) and (S, , ≤S ) beNeutroOrderedSemigroupsand φ : S → S be aNeutroOrderedStrongIsomorphism.If N ⊆ S isaNeutroOrderedSubsemigroupof S then φ 1(N ) isaNeutroOrderedSubsemigroupof S.
Proof. Theorem2.11assertsthat φ 1 : S → S isaNeutroOrderedStrongIsomorphism.Having N ⊆ S aNeutroOrderedSubsemigroupof S andbyusingLemma3.28,wegetthat φ 1(N ) isaNeutroOrderedSubsemigroupof S
Lemma3.35. Let (S,, ≤S ) and (S, , ≤S ) beNeutroOrderedSemigroupsand φ : S → S be aNeutroOrderedStrongIsomorphism.If N ⊆ S isaNeutroOrderedSubsemigroupof S then φ 1(N ) isaNeutroOrderedLeftIdeal(NeutroOrderedRightIdeal)of S.
Proof. Theorem2.11assertsthat φ 1 : S → S isaNeutroOrderedStrongIsomorphism.Having N ⊆ S aNeutroOrderedLeftIdeal(NeutroOrderedRightIdeal)of S andbyusingLemma3.29, wegetthat φ 1(N )isaNeutroOrderedLeftIdeal(NeutroOrderedRightIdeal)of S.
Lemma3.36. Let (S,, ≤S ) and (S, , ≤S ) beNeutroOrderedSemigroupsand φ : S → S be aNeutroOrderedStrongIsomorphism.If N ⊆ S isaNeutroOrderedSubsemigroupof S then φ 1(N ) isaNeutroOrderedIdealof S
Proof. Theorem2.11assertsthat φ 1 : S → S isaNeutroOrderedStrongIsomorphism.Having N ⊆ S aNeutroOrderedIdealof S andbyusingLemma3.35,wegetthat φ 1(N )isa NeutroOrderedIdealof S.
Lemma3.37. Let (S,, ≤S ) and (S, , ≤S ) beNeutroOrderedSemigroupsand φ : S → S be aNeutroOrderedStrongIsomorphism.If N ⊆ S isaNeutroOrderedFilterof S then φ 1(N ) is aNeutroOrderedFilterof S
Proof. Theorem2.11assertsthat φ 1 : S → S isaNeutroOrderedStrongIsomorphism.Having N ⊆ S aNeutroOrderedFilterof S andbyusingLemma3.32,wegetthat φ 1(N )isa NeutroOrderedFilterof S.
Wepresentourmaintheorems.
Neutrosophic
Proof. TheprooffollowsfromLemmas3.28and3.34.
Theorem3.39. Let (S,, ≤S ) and (S, , ≤S ) beNeutroOrderedSemigroupsand φ : S → S beaNeutroOrderedStrongIsomorphism.Then M ⊆ S isaNeutroOrderedLeftIdeal(NeutroOrderedRightIdeal)of S ifandonlyif φ(M ) isaNeutroOrderedLeftIdeal(NeutroOrderedRightIdeal)of S
Proof. TheprooffollowsfromLemmas3.29and3.35.
Theorem3.40. Let (S, · , ≤S ) and (S, , ≤S ) beNeutroOrderedSemigroupsand φ : S → S be aNeutroOrderedStrongIsomorphism.Then M ⊆ S isaNeutroOrderedIdealof S ifandonlyif φ(M ) isaNeutroOrderedIdealof S .
Proof. TheprooffollowsfromLemmas3.30and3.36.
Theorem3.41. Let (S, · , ≤S ) and (S, , ≤S ) beNeutroOrderedSemigroupsand φ : S → S be aNeutroOrderedStrongIsomorphism.Then M ⊆ S isaNeutroOrderedFilterof S ifandonly if φ(M ) isaNeutroOrderedFilterof S .
Proof. TheprooffollowsfromLemmas3.32and3.37.
4. Conclusion
ThispapercontributedtothestudyofNeutroAlgebrabyintroducing,forthefirsttime, NeutroOrderedAlgebra.ThenewdefinednotionwasappliedtosemigroupsandmanyinterestingpropertieswereprovedaswellillustrativeexamplesweregivenonNeutroOrderedSemigroups.
Forfutureresearch,itwillbeinterestingtoapplytheconceptofNeutroOrderedAlgebrato differentalgebraicstructuressuchasgroups,rings,modules,etc.andtostudyAntiOrderedAlgebra.
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Received: Sep 10, 2020. Accepted: Jan 7, 2021
M.Al-Tahan,F.Smarandache,andB.Davvaz,NeutroOrderedAlgebra:Applicationsto Semigroups