1/54 NeutroOrderedAlgebra:TheoryandExamples MadeleineAl-Tahan LebaneseInternationalUniversity madeline.tahan@liu.edu.lb “3rdInternationalWorkshoponAdvancedTopicsinDynamicalSystems(IWATDS2021)" UniversityofKufa,Iraq March1st ,2021 MadeleineAl-Tahan(LIU) NeutroOrderedAlgebra:TheoryandExamples March1st ,20211/54
2/54 Abstract Inthistalk,wefirstlyreviewsomebasicconceptsrelatedto neutrosophy.Also,wediscussNeutroAlgebra.Next,we presentsomeofourresultsrelatedtoournewdefined concept“NeutroOrderedAlgebra"andcompareittothe wellknownconceptof“OrderedAlgebra".Finally,weleave withsomequestionsthatopennewresearchoptionsinthis field. MadeleineAl-Tahan(LIU) NeutroOrderedAlgebra:TheoryandExamples March1st ,20212/54
3/54 1 Neutrosophy Definition Formalization 2 NeutroAlgebra Motivation Definitions 3 NeutroOrderedAlgebra Definitions NeutroOrderedSemigroups DefinitionsandExamplesofNeutroOrderedSemigroups SubsetsofNeutroOrderedSemigroups NeutroOrderedIsomorphisms ProductionalNeutroOrderedSemigroups 4 Conclusion 5 Collaborativeteam 6 References 7 Thankyou MadeleineAl-Tahan(LIU) NeutroOrderedAlgebra:TheoryandExamples March1st ,20213/54
4/54 Neutrosophy Definition 1 Neutrosophy Definition Formalization 2 NeutroAlgebra Motivation Definitions 3 NeutroOrderedAlgebra Definitions NeutroOrderedSemigroups DefinitionsandExamplesofNeutroOrderedSemigroups SubsetsofNeutroOrderedSemigroups NeutroOrderedIsomorphisms ProductionalNeutroOrderedSemigroups 4 Conclusion 5 Collaborativeteam 6 References 7 Thankyou MadeleineAl-Tahan(LIU) NeutroOrderedAlgebra:TheoryandExamples March1st ,20214/54
5/54 Neutrosophy Definition Neutrosophy[4]isanewbranchofphilosophywhichstudiestheorigin, nature,andscopeofneutralities,aswellastheirinteractionswithdifferent ideationalspectra.WhileHegel’sandMarx’sDialecticsdealsonlywiththe dynamicsofopposites,Neutrosophydealswiththedynamicsofopposites andtheirneutralsalltogether. Thismodeofthinking 1 proposesnewphilosophicaltheses,principles,laws,methods,formulas, movements; 2 interpretstheuninterpretable; 3 regards,frommanydifferentangles,oldconcepts,systems:showing thatanidea,whichistrueinagivenreferentialsystem,maybefalsein anotherone,andviceversa; 4 attemptstomakepeaceinthewarofideas,andtomakewarinthe peacefulideas; 5 measuresthestabilityofunstablesystems,andinstabilityofstable systems. MadeleineAl-Tahan(LIU) NeutroOrderedAlgebra:TheoryandExamples March1st ,20215/54
6/54 Neutrosophy Formalization 1 Neutrosophy Definition Formalization 2 NeutroAlgebra Motivation Definitions 3 NeutroOrderedAlgebra Definitions NeutroOrderedSemigroups DefinitionsandExamplesofNeutroOrderedSemigroups SubsetsofNeutroOrderedSemigroups NeutroOrderedIsomorphisms ProductionalNeutroOrderedSemigroups 4 Conclusion 5 Collaborativeteam 6 References 7 Thankyou MadeleineAl-Tahan(LIU) NeutroOrderedAlgebra:TheoryandExamples March1st ,20216/54
7/54 Neutrosophy Formalization Let’snoteby<A>anideaortheoryorconcept,by<Non-A>whatisnot <A>,andby<Anti-A>theoppositeof<A>.Also,<Neut-A>means whatisneither<A>,nor<Anti-A>,i.e.neutralityinbetweenthetwo extremes.And<A’>aversionof<A>.<Non-A>isdifferentfrom <Anti-A>. Example1. [4]If<A>=whitethen<Anti-A>=black(antonym).But <Non-A>=green,red,blue,yellow,black,etc.(anycolor,exceptwhite), while<Neut-A>=green,red,blue,yellow,etc.(anycolor,exceptwhite andblack).And<A’>=darkwhite,etc.(anyshadeofwhite). Example2. If<A>=lovethen<Anti-A>=hatred(antonym).But <Non-A>=anyfeelingexceptlove,while<Neut-A>=anyfeelingexcept loveandhatred.And<A’>=anytypeoflove. MadeleineAl-Tahan(LIU) NeutroOrderedAlgebra:TheoryandExamples March1st ,20217/54
8/54 Neutrosophy Formalization Let<A>beanidea.Thenthefollowingaretrue. 1 <Neut-A> ≡ <Neut-(Anti-A)>; 2 <Anti-A> ⊆ <Non-A>; 3 <A>, <Neut-A>,and <Anti-A> arepairwisedisjoint; 4 <Non-A> isthecompletitudeof<A>withrespecttotheuniversal set. Everyidea<A>tendstobeneutralized,diminished,balancedby<Non-A> ideasasastateofequilibrium.Inbetween<A>and<Anti-A>thereare infinitelymany<Neut-A>ideas,whichmaybalance<A>without necessarilyany<Anti-A>version.Toneuteranidea,weneedtodiscover allitsthreesides:ofsense(truth),ofnonsense(falsity),andof undecidability(indeterminacy)-thenreverse/combinethem.Afterwards, theideawillbeclassifiedasneutrality. MadeleineAl-Tahan(LIU) NeutroOrderedAlgebra:TheoryandExamples March1st ,20218/54
9/54 NeutroAlgebra Motivation 1 Neutrosophy Definition Formalization 2 NeutroAlgebra Motivation Definitions 3 NeutroOrderedAlgebra Definitions NeutroOrderedSemigroups DefinitionsandExamplesofNeutroOrderedSemigroups SubsetsofNeutroOrderedSemigroups NeutroOrderedIsomorphisms ProductionalNeutroOrderedSemigroups 4 Conclusion 5 Collaborativeteam 6 References 7 Thankyou MadeleineAl-Tahan(LIU) NeutroOrderedAlgebra:TheoryandExamples March1st ,20219/54
10/54 NeutroAlgebra Motivation Insoftsciencesthelawsareinterpretedandre-interpreted;insocialand politicallegislationthelawsareflexible;thesamelawmaybetruefroma pointofview,andfalsefromanotherpointofview.Thus,thelawis partiallytrueandpartiallyfalse(itisaNeutrosophicLaw).Forexample, “Wearingclothesfromanimals’leatherorfur".Therearepeoplesupporting itbecauseitpreventsfeelingcoldduringthewinter(andtheyareright),and people(supportingAnimals’rights)opposingitbecausetheyareagainst killinganimalstogettheirleatherorfur(andtheyarerighttoo).Wehave twooppositepropositions,bothofthemaretruebutfromdifferentpointsof view(fromdifferentcriteria/parameters;plithogeniclogic).Howcanwe solvethis?Goingtothemiddle,inbetweenopposites(asinneutrosophy): Leather/Furloverscanweareitherleather/furclothesthatarenotfrom animalsorfromdeadanimals. MadeleineAl-Tahan(LIU) NeutroOrderedAlgebra:TheoryandExamples March1st ,202110/54
11/54 NeutroAlgebra Motivation Anotherexampleis“Druglegislation".ManypeopleinLebanonsupportit formedicalandeconomicalpurposes(andtheyareright),andotherpeople inLebanonopposeitbecausetheywanttosavetheirfamiliesfromdrug addiction(andtheyarerighttoo).Wehavetwooppositepropositions,both ofthemaretruebutfromdifferentpointsofview(fromdifferent criteria/parameters;plithogeniclogic).Howcanwesolvethis?Goingtothe middle,inbetweenopposites(asinneutrosophy):TheLebanese governmenthastosupporttheusageofdrugsinmedicineandprohibits tradersfromsellingdrugstoindividualsbymonitoringtheprocessstarting fromplantingtodistributingandmanufacturing. MadeleineAl-Tahan(LIU) NeutroOrderedAlgebra:TheoryandExamples March1st ,202111/54
12/54 NeutroAlgebra Motivation
MadeleineAl-Tahan(LIU) NeutroOrderedAlgebra:TheoryandExamples March1st ,202112/54
“Inallclassicalalgebraicstructures,thelawsofcompositionsonagivenset arewell-defined.Butthisisarestrictivecase,becausetherearemanymore situationsinscienceandinanydomainofknowledgewhenalawof compositiondefinedonasetmaybeonlypartially-defined(orpartiallytrue) andpartially-undefined(orpartiallyfalse),thatwecallNeutroDefined,or totallyundefined(totallyfalse)thatwecallAntiDefined.Again,inall classicalalgebraicstructures,theAxioms(Associativity,Commutativity, etc.)definedonasetaretotallytrue,butitisagainarestrictivecase, becausesimilarlytherearenumeroussituationsinscienceandinanydomain ofknowledgewhenanAxiomdefinedonasetmaybeonlypartially-true (andpartially-false),thatwecallNeutroAxiom,ortotallyfalsethatwecall AntiAxiom."Basedonthis,FlorentinSmarandachein2019openednew fieldsofresearchcalledNeutroStructuresandAntiStructuresrespectively.
13/54 NeutroAlgebra Definitions 1 Neutrosophy Definition Formalization 2 NeutroAlgebra Motivation Definitions 3 NeutroOrderedAlgebra Definitions NeutroOrderedSemigroups DefinitionsandExamplesofNeutroOrderedSemigroups SubsetsofNeutroOrderedSemigroups NeutroOrderedIsomorphisms ProductionalNeutroOrderedSemigroups 4 Conclusion 5 Collaborativeteam 6 References 7 Thankyou MadeleineAl-Tahan(LIU) NeutroOrderedAlgebra:TheoryandExamples March1st ,202113/54
14/54 NeutroAlgebra Definitions Definition(F.Smarandache,2019) Let A beanynon-emptysetand“·"beanoperationon A.Then“·"is calleda NeutroOperation on A ifthefollowingconditionshold. 1 Thereexist x, y ∈ A with x · y ∈ A.(Thisconditioniscalleddegree oftruth,“T ".) 2 Thereexist x, y ∈ A with x · y / ∈ A.(Thisconditioniscalleddegree offalsity,“F ".) 3 Thereexist x, y ∈ A with x · y isindeterminatein A.(Thiscondition iscalleddegreeofindeterminacy,“I".) Where (T , I, F ) isdifferentfrom (1, 0, 0) thatrepresentstheclassical binaryclosedoperation,andfrom (0, 0, 1) thatrepresentsthe AntiOperation. MadeleineAl-Tahan(LIU) NeutroOrderedAlgebra:TheoryandExamples March1st ,202114/54
15/54 NeutroAlgebra Definitions IllustrativeExamples 1 Wecanviewthestandarddivision“÷"on R,thesetofrealnumbers asNeutroOperation.Thisiseasilyseenas x ÷ y is ∈ R if y = 0; undefined otherwise. 2 Wecanviewthestandarddivision“÷"on N,thesetofpositive integersasNeutroOperation.Thisiseasilyseenas x ÷ y is ∈ N if y divides x; / ∈ N otherwise. 3 Wecanviewthestandarddivision“÷"on Z,thesetofintegersas NeutroOperation.Thisiseasilyseenas x ÷ y is ∈ Z if y divides x and y = 0; / ∈ Z if y doesnotdivide x and y = 0; undefined if y = 0. MadeleineAl-Tahan(LIU) NeutroOrderedAlgebra:TheoryandExamples March1st ,202115/54
16/54 NeutroAlgebra Definitions Definition(F.Smarandache,2019) Let U beauniverseofdiscourse,endowedwithsomewell-definedlaws,a non-emptyset S ⊆ U andanAxiom α,definedon S,usingtheselaws. Then 1 Ifallelementsof S verifytheaxiom α,wehaveaClassicalAxiom,or simplywesayAxiom. 2 Ifsomeelementsof S verifytheaxiom α andothersdonot,wehavea NeutroAxiom(whichisalsocalledNeutAxiom). 3 Ifnoelementsof S verifytheaxiom α,thenwehaveanAntiAxiom. TheNeutrosophicTripletAxiomsare:(Axiom,NeutroAxiom,AntiAxiom) satisfyingthefollowing. NeutroAxiom ∪ AntiAxiom=NonAxiom, NeutroAxiom ∩ AntiAxiom = ∅. MadeleineAl-Tahan(LIU) NeutroOrderedAlgebra:TheoryandExamples March1st ,202116/54
17/54 NeutroAlgebra Definitions Illustration Let A beanynon-emptysetand“·"beanoperationon A.Then“·"is calleda NeutroAssociative on A ifthereexist x, y , z, a, b, c, e, f , g ∈ A withthefollowingconditions. 1 x · (y · z)=(x · y) · z;(Thisconditioniscalleddegreeoftruth,“T ".) 2 a · (b · c) =(a · b) · c;(Thisconditioniscalleddegreeoffalsity,“F ".) 3 e · (f · g ) isindeterminateor (e · f ) · g isindeterminateorwecannot findif e · (f · g ) and (e · f ) · g areequal.(Thisconditioniscalled degreeofindeterminacy,“I".) Where (T , I, F ) isdifferentfrom (1, 0, 0) thatrepresentstheclassical associativeaxiom,andfrom (0, 0, 1) thatrepresentsthe AntiAssociativeAxiom. MadeleineAl-Tahan(LIU) NeutroOrderedAlgebra:TheoryandExamples March1st ,202117/54
18/54 NeutroAlgebra Definitions Definition(F.Smarandache,2019) Anon-emptyset A endowedwith n operations“ i "for i = 1,..., n,is called NeutroAlgebra ifithasatleastoneNeutroOperationoratleastone NeutroAxiomwithnoAntiOperationsnorAntiAxioms. Remark(F.Smarandache,2019) NeutroAlgebraisageneralizationofPartialAlgebra. IncomparisonbetweenthePartialAlgebraandtheNeutroAlgebra. WhentheNeutroAlgebrahasnoNeutroAxiom,andnoouter-defined operation,itcoincideswiththePartialAlgebra. ThereareNeutroAlgebrasthathavenoNeutroOperations,buthave NeutroAxioms.ThesearedifferentfromPartialAlgebras. ThereareNeutroAlgebrasthathaveboth,NeutroOperationsand NeutroAxioms. MadeleineAl-Tahan(LIU) NeutroOrderedAlgebra:TheoryandExamples March1st ,202118/54
19/54 NeutroAlgebra Definitions IllustrativeExamples (1) Let S =]0, ∞[.Then (S, ÷) isaNeutroSemigroup[4].Since“÷"is well-definedthenweneedtoshowthat (S, ÷) isaNeutroAssociative. Thisiseasilyseenas x ÷ (y ÷ z) is =(x ÷ y) ÷ z if z = 1; =(x ÷ y) ÷ z otherwise. (2) Let S = {a, b}.Then (S, ) definedbythefollowingtableisa NeutroSemigroup[6]. a b a b a b a undefined Thisisclearas“ "isaNeutroOperationand a (a a)=(a a) a = a. MadeleineAl-Tahan(LIU) NeutroOrderedAlgebra:TheoryandExamples March1st ,202119/54
20/54 NeutroAlgebra Definitions
Examples(Cont’d)
(3) Let S = {a, b}.Then (S, ·) definedbythefollowingtableisa NeutroSemigroup[6]. · a b a b a b a c / ∈ S
MadeleineAl-Tahan(LIU) NeutroOrderedAlgebra:TheoryandExamples March1st ,202120/54
(4) Let S = {a, b}.Then (S, +) definedbythefollowingtableisa NeutroSemigroup[6]. + a b a b a b a aorb
21/54 NeutroOrderedAlgebra Definitions 1 Neutrosophy Definition Formalization 2 NeutroAlgebra Motivation Definitions 3 NeutroOrderedAlgebra Definitions NeutroOrderedSemigroups DefinitionsandExamplesofNeutroOrderedSemigroups SubsetsofNeutroOrderedSemigroups NeutroOrderedIsomorphisms ProductionalNeutroOrderedSemigroups 4 Conclusion 5 Collaborativeteam 6 References 7 Thankyou MadeleineAl-Tahan(LIU) NeutroOrderedAlgebra:TheoryandExamples March1st ,202121/54
22/54 NeutroOrderedAlgebra Definitions Anorderedalgebraicstructure A consistsofanalgebratogetherwitha partialorderingontheunderlyingsetofthealgebra.Werequirethatthe operationsofthealgebraarecompatiblewiththepartialorderinginthat theypreserveorreverseorderineachcoordinate.Orderedalgebraic structuresoccurinawidevarietyofareassuchaspartiallyorderedvector spaces,latticeorderedgroups,Booleanalgebras,Heytingalgebras,modal algebras,cylindricalgebras,relationalgebras,etc. Definition [3]Let A beanAlgebrawith n operations“ i "and“≤"beapartialorder (reflexive,anti-symmetric,andtransitive)on A.Then (A, 1,..., n , ≤) is anOrderedAlgebraifthefollowingconditionshold. 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. MadeleineAl-Tahan(LIU) NeutroOrderedAlgebra:TheoryandExamples March1st ,202122/54
23/54 NeutroOrderedAlgebra Definitions InspiredbyNeutroAlgebraandorderedAlgebra,weintroduced NeutroOrderedAlgebra in2021[1]. StartingwithapartialorderonaNeutroAlgebra,wegetaNeutroStructure. Thelatterifitsatisfiestheconditionsof NeutroOrder,itbecomesa NeutroOrderedAlgebra. MadeleineAl-Tahan(LIU) NeutroOrderedAlgebra:TheoryandExamples March1st ,202123/54
24/54 NeutroOrderedAlgebra Definitions Definition(M.Al-Tahan,F.Smarandache,andB.Davvaz,[1],2021) Let A beaNeutroAlgebrawith n (Neutro)operations“ i "and“≤"bea partialorder(reflexive,anti-symmetric,andtransitive)on A.Then (A, 1,..., n , ≤) isa NeutroOrderedAlgebra ifthefollowingconditions hold. 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.(Thisconditionis calleddegreeoftruth,“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.(Thisconditioniscalleddegree offalsity,“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 areindeterminate,ortherelationbetween z i x and z i y, ortherelationbetween x i z and y i z areindeterminateforsome i = 1,..., n.(Thisconditioniscalleddegreeofindeterminacy,“I".) Where (T , I, F ) isdifferentfrom (1, 0, 0) aswellfrom (0, 0, 1). MadeleineAl-Tahan(LIU) NeutroOrderedAlgebra:TheoryandExamples March1st ,202124/54
25/54 NeutroOrderedAlgebra Definitions Definition Let (A, 1,..., n , ≤) beaNeutroOrderedAlgebra.If“≤"isatotalorder on A then A iscalled NeutroTotalOrderedAlgebra [1]. Definition Let (A, 1,..., n , ≤A) beaNeutroOrderedAlgebraand ∅ = S ⊆ A. Then S isa NeutroOrderedSubAlgebra of A if (S, 1,..., n , ≤A) isa NeutroOrderedAlgebraandthereexists x ∈ S with (x]= {y ∈ A : y ≤A x}⊆ S [1]. Remark ANeutroOrderedAlgebrahasatleastoneNeutroOrderedSubAlgebrawhich isitself. MadeleineAl-Tahan(LIU) NeutroOrderedAlgebra:TheoryandExamples March1st ,202125/54
26/54 NeutroOrderedAlgebra NeutroOrderedSemigroups 1 Neutrosophy Definition Formalization 2 NeutroAlgebra Motivation Definitions 3 NeutroOrderedAlgebra Definitions NeutroOrderedSemigroups DefinitionsandExamplesofNeutroOrderedSemigroups SubsetsofNeutroOrderedSemigroups NeutroOrderedIsomorphisms ProductionalNeutroOrderedSemigroups 4 Conclusion 5 Collaborativeteam 6 References 7 Thankyou MadeleineAl-Tahan(LIU) NeutroOrderedAlgebra:TheoryandExamples March1st ,202126/54
27/54 NeutroOrderedAlgebra NeutroOrderedSemigroups 1 Neutrosophy Definition Formalization 2 NeutroAlgebra Motivation Definitions 3 NeutroOrderedAlgebra Definitions NeutroOrderedSemigroups DefinitionsandExamplesofNeutroOrderedSemigroups SubsetsofNeutroOrderedSemigroups NeutroOrderedIsomorphisms ProductionalNeutroOrderedSemigroups 4 Conclusion 5 Collaborativeteam 6 References 7 Thankyou MadeleineAl-Tahan(LIU) NeutroOrderedAlgebra:TheoryandExamples March1st ,202127/54
28/54 NeutroOrderedAlgebra NeutroOrderedSemigroups Definition(M.Al-Tahan,F.Smarandache,andB.Davvaz,[1],2021) Let (S, ·) beaNeutroSemigroupand“≤"beapartialorder(reflexive, anti-symmetric,andtransitive)on S.Then (S, · , ≤) isa NeutroOrderedSemigroup ifthefollowingconditionshold. 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.(Thisconditioniscalleddegreeoffalsity,“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,orthe relationbetween x · z and y · z areindeterminate.(Thisconditionis calleddegreeofindeterminacy,“I".) Where (T , I, F ) isdifferentfrom (1, 0, 0) thatrepresentstheclassical OrderedSemigroup,andfrom (0, 0, 1) thatrepresentsthe AntiOrderedSemigroup. MadeleineAl-Tahan(LIU) NeutroOrderedAlgebra:TheoryandExamples March1st ,202128/54
29/54 NeutroOrderedAlgebra NeutroOrderedSemigroups Example3. [1]Let S1 = {s, a, m} and (S1, ·1) bedefinedbythe followingtable. ·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) isa NeutroSemigroup. Bydefiningthetotalorder ≤1= {(m, m), (m, s), (m, a), (s, s), (s, a), (a, a)} on S1,wegetthat (S1, ·1, ≤1) isaNeutroTotalOrderedSemigroup.Thisis easilyseenas: 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. MadeleineAl-Tahan(LIU) NeutroOrderedAlgebra:TheoryandExamples March1st ,202129/54
30/54 NeutroOrderedAlgebra NeutroOrderedSemigroups Example4. [1]Let S2 = {0, 1, 2, 3} and (S2, ·2) bedefinedbythe followingtable. ·2 0123 0 0003 1 0113 2 0322 3 3333 Since 0 ·2 (0 ·2 0)= 0 =(0 ·2 0) ·2 0 and 1 ·2 (2 ·2 3)= 1 = 3 =(1 ·2 2) ·2 3,itfollowsthat (S2, ·2) isa NeutroSemigroup.Bydefiningthetotalorder“≤2"on S2 asfollows: {(0, 0), (0, 1), (0, 2), (0, 3), (1, 1), (1, 2), (1, 3), (2, 2), (2, 3), (3, 3)}, wegetthat (S2, ·2, ≤2) isaNeutroTotalOrderedSemigroup.Thisiseasily seenas: 0 ≤2 3 impliesthat 0 ·2 x ≤2 3 ·2 x and x ·2 0 ≤2 x ·2 3 forall x ∈ S2. Andhaving 1 ≤2 2 but 2 ·2 1 = 3 2 2 = 2 ·2 2. MadeleineAl-Tahan(LIU) NeutroOrderedAlgebra:TheoryandExamples March1st ,202130/54
31/54 NeutroOrderedAlgebra NeutroOrderedSemigroups Example5. [1]Let S3 = {0, 1, 2, 3, 4} and (S3, ·3) bedefinedbythe followingtable. ·3 01234 0 00030 1 01211 2 04233 3 04233 4 00040 Since 0 ·3 (0 ·3 0)= 0 =(0 ·3 0) ·3 0 and 1 ·3 (2 ·3 1)= 1 = 4 =(1 ·3 2) ·3 1,itfollowsthat (S3, ·3) isa NeutroSemigroup. Bydefiningthepartialorder“≤3"on S3 asfollows {(0, 0), (0, 1), (0, 3), (0, 4), (1, 1), (1, 3), (1, 4), (2, 2), (3, 3), (3, 4), (4, 4) wegetthat (S3, ·3, ≤3) isaNeutroOrderedSemigroupthatisnot NeutroTotalOrderedSemigroupas“≤3"isnotatotalorderon S3. MadeleineAl-Tahan(LIU) NeutroOrderedAlgebra:TheoryandExamples March1st ,202131/54
32/54 NeutroOrderedAlgebra NeutroOrderedSemigroups Example5.(Cont’d) Thisiseasilyseenas: 0 ≤3 4 impliesthat 0 ·3 x ≤3 4 ·3 x and x ·3 0 ≤3 x ·3 4 forall x ∈ S3. Andhaving 0 ≤3 1 but 0 ·3 2 = 0 3 2 = 1 ·3 2. Example6. [1]Let Z bethesetofintegersanddefine“ "on Z asfollows: x y = xy − 1 forall x, y ∈ Z.Since 0 (1 0)= −1 =(0 1) 0 and 0 (1 2)= −1 = −3 =(0 1) 2,itfollowsthat (Z, ) isa NeutroSemigroup.Wedefinethepartialorder“≤Z"on Z as −1 ≤Z x for all x ∈ Z andfor a, b ≥−1, a ≤Z b isequivalentto a ≤ b andfor a, b < −1, a ≤Z b isequivalentto a ≥ b.Inthisway,weget −1 ≤Z 0 ≤Z 1 ≤Z 2 ≤Z ... and −1 ≤Z −2 ≤Z −3 ≤Z ....Having 0 ≤Z 1 and x 0 = 0 x = −1 ≤Z x − 1 = 1 x = x 1 forall x ∈ Z and −1 ≤Z 0 but (−1) (−1)= 0 Z −1 = 0 (−1) implies that (Z, , ≤Z) isaNeutroOrderedSemigroupwith −1 asminimum element. MadeleineAl-Tahan(LIU) NeutroOrderedAlgebra:TheoryandExamples March1st ,202132/54
33/54 NeutroOrderedAlgebra NeutroOrderedSemigroups 1 Neutrosophy Definition Formalization 2 NeutroAlgebra Motivation Definitions 3 NeutroOrderedAlgebra Definitions NeutroOrderedSemigroups DefinitionsandExamplesofNeutroOrderedSemigroups SubsetsofNeutroOrderedSemigroups NeutroOrderedIsomorphisms ProductionalNeutroOrderedSemigroups 4 Conclusion 5 Collaborativeteam 6 References 7 Thankyou MadeleineAl-Tahan(LIU) NeutroOrderedAlgebra:TheoryandExamples March1st ,202133/54
34/54 NeutroOrderedAlgebra NeutroOrderedSemigroups Definition(M.Al-Tahan,F.Smarandache,andB.Davvaz,[1],2021) Let (S, · , ≤) beaNeutroOrderedSemigroupand ∅ = M ⊆ S.Then (1) M isa NeutroOrderedSubSemigroup of S if (M, · , ≤) isa NeutroOrderedSemigroupandthereexist x ∈ M with (x]= {y ∈ S : y ≤ x}⊆ M. (2) M isa NeutroOrderedLeftIdeal of S if M isa NeutroOrderedSubSemigroupof S andthereexists x ∈ M suchthat r · x ∈ M forall r ∈ S. (3) M isa NeutroOrderedRightIdeal of S if M isa NeutroOrderedSubSemigroupof S andthereexists x ∈ M suchthat x · r ∈ M forall r ∈ S (4) M isa NeutroOrderedIdeal of S if M isaNeutroOrderedSubSemigroup of S andthereexists x ∈ M suchthat r · x ∈ M and x · r ∈ M forall r ∈ S. MadeleineAl-Tahan(LIU) NeutroOrderedAlgebra:TheoryandExamples March1st ,202134/54
35/54 NeutroOrderedAlgebra NeutroOrderedSemigroups Remark UnlikethecaseinOrderedSemigroups,thenon-emptyintersectionof NeutroOrderedSubsemigroupsmaynotbeaNeutroOrderedSubsemigroup. Example7. [1]Let (S3, ·3, ≤3) betheNeutroOrderedSemigrouppresented inExample5.Onecaneasilyseethat I = {0, 1, 2}, J = {0, 1, 3} are NeutroOrderedSubsemigroupsof S3.Since ({0, 1}, ·3) isasemigroupand notaNeutroSemigroup,itfollowsthat (I ∩ J, ·3, ≤3) isnota NeutroOrderedSubSemigroupof S3.Here, I ∩ J = {0, 1}. MadeleineAl-Tahan(LIU) NeutroOrderedAlgebra:TheoryandExamples March1st ,202135/54
36/54 NeutroOrderedAlgebra NeutroOrderedSemigroups WepresentanexampleonNeutroOrderedIdealofaninfinite NeutroOrderedSemigroup. Example8. [1]Let (Z, , ≤Z) betheNeutroOrderedSemigrouppresented inExample6.Then I = {−1, 0, 1, −2, −3, −4,...} isa NeutroOrderedIdealof Z.Thisisclearas: (1) 0 (1 0)= −1 =(0 1) 0 and 0 (−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 impliesthat 0 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. MadeleineAl-Tahan(LIU) NeutroOrderedAlgebra:TheoryandExamples March1st ,202136/54
37/54 NeutroOrderedAlgebra NeutroOrderedSemigroups 1 Neutrosophy Definition Formalization 2 NeutroAlgebra Motivation Definitions 3 NeutroOrderedAlgebra Definitions NeutroOrderedSemigroups DefinitionsandExamplesofNeutroOrderedSemigroups SubsetsofNeutroOrderedSemigroups NeutroOrderedIsomorphisms ProductionalNeutroOrderedSemigroups 4 Conclusion 5 Collaborativeteam 6 References 7 Thankyou MadeleineAl-Tahan(LIU) NeutroOrderedAlgebra:TheoryandExamples March1st ,202137/54
38/54 NeutroOrderedAlgebra NeutroOrderedSemigroups 1 Neutrosophy Definition Formalization 2 NeutroAlgebra Motivation Definitions 3 NeutroOrderedAlgebra Definitions NeutroOrderedSemigroups DefinitionsandExamplesofNeutroOrderedSemigroups SubsetsofNeutroOrderedSemigroups NeutroOrderedIsomorphisms ProductionalNeutroOrderedSemigroups 4 Conclusion 5 Collaborativeteam 6 References 7 Thankyou MadeleineAl-Tahan(LIU) NeutroOrderedAlgebra:TheoryandExamples March1st ,202138/54
39/54 NeutroOrderedAlgebra NeutroOrderedSemigroups Definition(M.Al-Tahan,F.Smarandache,andB.Davvaz,[1],2021) 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 isabijective NeutroOrderedHomomorphism. (3) φ iscalled NeutroOrderedStrongHomomorphism if φ(x y)= φ(x) φ(y) forall x, y ∈ A and a ≤A b ∈ A is equivalentto φ(a) ≤B φ(b) ∈ B. (4) φ iscalled NeutroOrderedStrongIsomomorphism if φ isabijective NeutroOrderedStrongHomomorphism. MadeleineAl-Tahan(LIU) NeutroOrderedAlgebra:TheoryandExamples March1st ,202139/54
40/54 NeutroOrderedAlgebra NeutroOrderedSemigroups Lemma(M.Al-Tahan,F.Smarandache,andB.Davvaz,[1],2021) Let (S, · , ≤S ) and (S , , ≤S ) beNeutroOrderedSemigroupsand φ : S → S beaNeutroOrderedStrongIsomorphism.If M ⊆ S isa NeutroOrderedSubsemigroup(NeutroOrderedIdeal)of S then φ(M) isa (NeutroOrderedIdeal)of S . Example9. Let (Z, , ≤Z) betheNeutroOrderedSemigroupspresentedin Example6and (Z, ⊗, ≤⊗) betheNeutroOrderedSemigroupdefinedas follows: x ⊗ y = xy + 1 forall x, y ∈ Z and“≤⊗"on Z as 1 ≤⊗ x forall x ∈ Z andfor a, b ≥ 1, a ≤⊗ b isequivalentto a ≤ b andfor a, b ≤ 0, a ≤⊗ b isequivalentto a ≥ b.Let φ :(Z, , ≤Z) → (Z, ⊗, ≤⊗) be definedas φ(x)= x + 2 forall x ∈ Z.Onecaneasilyseethat φ isa NeutroOrderedStrongIsomorphism.Having I = {−1, 0, 1, −2, −3, −4,...} isaNeutroOrderedIdealof (Z, , ≤Z) andapplyingthepreviouslemma,wegetthat φ(I)= {1, 2, 3, 0, −1, −2,...} isaNeutroOrderedIdealof (Z, ⊗, ≤⊗). MadeleineAl-Tahan(LIU) NeutroOrderedAlgebra:TheoryandExamples March1st ,202140/54
41/54 NeutroOrderedAlgebra NeutroOrderedSemigroups Lemma(M.Al-Tahan,F.Smarandache,andB.Davvaz,[1],2021) (S, · , ≤S ) and (S , , ≤S ) beNeutroOrderedSemigroupsand φ : S → S beaNeutroOrderedStrongIsomorphism.If N ⊆ S isa NeutroOrderedSubsemigroup(NeutroOrderedIdeal)of S then φ−1(N) isa NeutroOrderedSubsemigroup(NeutroOrderedIdeal)of S. Theorem(M.Al-Tahan,F.Smarandache,andB.Davvaz,[1],2021) Let (S, · , ≤S ) and (S , , ≤S ) beNeutroOrderedSemigroupsand φ : S → S beaNeutroOrderedStrongIsomorphism.Then M ⊆ S isa NeutroOrderedSubsemigroup(NeutroOrderedIdeal)of S ifandonlyif φ(M) isaNeutroOrderedSubsemigroup(NeutroOrderedIdeal)of S . MadeleineAl-Tahan(LIU) NeutroOrderedAlgebra:TheoryandExamples March1st ,202141/54
42/54 NeutroOrderedAlgebra NeutroOrderedSemigroups 1 Neutrosophy Definition Formalization 2 NeutroAlgebra Motivation Definitions 3 NeutroOrderedAlgebra Definitions NeutroOrderedSemigroups DefinitionsandExamplesofNeutroOrderedSemigroups SubsetsofNeutroOrderedSemigroups NeutroOrderedIsomorphisms ProductionalNeutroOrderedSemigroups 4 Conclusion 5 Collaborativeteam 6 References 7 Thankyou MadeleineAl-Tahan(LIU) NeutroOrderedAlgebra:TheoryandExamples March1st ,202142/54
43/54 NeutroOrderedAlgebra NeutroOrderedSemigroups Let (Aα, ≤α) beapartialorderedsetforall α ∈ Γ.Wedefine“≤"on α∈Γ Aα asfollows:Forall (xα), (yα) ∈ α∈Γ Aα, (xα) ≤ (yα) ⇐⇒ xα ≤α yα forall α ∈ Γ. Onecaneasilyseethat ( α∈Γ Aα, ≤) isapartialorderedset. Let Aα beanynon-emptysetforall α ∈ Γ and“· α"beanoperationon Aα.Wedefine“·"on α∈Γ Aα asfollows:Forall (xα), (yα) ∈ α∈Γ Aα, (xα) · (yα)=(xα · α yα). Theorem(M.Al-Tahan,B.Davvaz,F.Smarandache,andO.Anis,[2],2021) Let (G1, ≤1), (G2, ≤2) bepartiallyorderedsetswithoperations ·1, ·2 respectively.Then (G1 × G2, · , ≤) isaNeutroOrderedSemigroup(NOS)if oneofthefollowingstatementsistrue. 1 G1 and G2 areNeutroSemigroupswithatleastoneofthemisanNOS. 2 Oneof G1, G2 isanNOSandtheotherisasemigroup. MadeleineAl-Tahan(LIU) NeutroOrderedAlgebra:TheoryandExamples March1st ,202143/54
44/54 NeutroOrderedAlgebra NeutroOrderedSemigroups Discussion Itisknowninorderedalgebraicstructuresthattheproduct G1 × G2 of semigroups G1, G2 isanorderedsemigroupifandonlyif G1, G2 areordered semigroups. Thisresultinclassicalalgebraicstructuresisnotvalidin NeutroAlgebraicStructures. Aswehave G1 × G2 isanNOSifeither G1, G2 arebothNOS, G1 isanNOSand G2 isaNeutroSemigroup, G1 isanNOS and G2 isasemigroup(oroderedsemigroup), G1 isaNeutroSemigroupand G2 isanNOS,or G1 isasemigroup(ororderedsemigroup)and G2 isan NOS. MadeleineAl-Tahan(LIU) NeutroOrderedAlgebra:TheoryandExamples March1st ,202144/54
45/54 NeutroOrderedAlgebra NeutroOrderedSemigroups Wepresentsomeapplicationsofourprevioustheorem. Example9. [2]Let S1 = {s, a, m}, (S1, ·1, ≤1) betheNOSpresentedin Example3,and“≤1"bethetrivialorderon S1.Theoremassertsthat Cartesianproduct (S1 × S1, · , ≤) resultingfrom (S1, ·1, ≤1) and (S1, ·1, ≤1) isanNOSoforder 9. Example10. [2]Let S1 = {s, a, m}, (S1, ·1, ≤1) betheNOSpresented inExample3,and (R, ·s , ≤u ) bethesemigroupofrealnumbersunder standardmultiplicationandusualorder.TheoremassertsthatCartesian product (R × S1, · , ≤) isanNOSofinfiniteorder. MadeleineAl-Tahan(LIU) NeutroOrderedAlgebra:TheoryandExamples March1st ,202145/54
46/54 Conclusion 1 Neutrosophy Definition Formalization 2 NeutroAlgebra Motivation Definitions 3 NeutroOrderedAlgebra Definitions
4 Conclusion 5 Collaborativeteam 6 References 7 Thankyou MadeleineAl-Tahan(LIU) NeutroOrderedAlgebra:TheoryandExamples March1st ,202146/54
NeutroOrderedSemigroups DefinitionsandExamplesofNeutroOrderedSemigroups SubsetsofNeutroOrderedSemigroups NeutroOrderedIsomorphisms ProductionalNeutroOrderedSemigroups
47/54 Conclusion Mostlyinidealisticorimaginaryorabstractorperfectspaces,wehaverigid lawsandrigidaxiomsthattotallyapply(thatare100 % true).Butthelaws andtheaxiomsshouldbemoreflexibleinordertodealwithourimperfect world.BecauseofthattheintroducingofNeutroAlgebrasisveryimportant. SuchaclassofNeutroAlgebrasisverylarge.Inthistalk,wepresented NeutroOrderedAlgebrasandwewereconcernedonlyabout NeutroOrderedSemigroupsasaspecialtypeofNeutroOrderedAlgebra. ManyotherdifferenttypesofNeutroOrderedAlgebrascanbedefined. Weleavewithsomerelatedopenquestions. 1 Canwefindanecessaryandasufficientconditionforthe productofNOStobeanNOS? 2 Givenafiniteset A.CanweclassifythedistinctNOSon A (up toNeutroStrongIsomorphism)? 3 CanweapplytheconceptofNeutroOrderedAlgebratoother structures? MadeleineAl-Tahan(LIU) NeutroOrderedAlgebra:TheoryandExamples March1st ,202147/54
48/54 Collaborativeteam 1 Neutrosophy Definition Formalization 2 NeutroAlgebra Motivation Definitions 3 NeutroOrderedAlgebra Definitions
4 Conclusion 5 Collaborativeteam 6 References 7 Thankyou MadeleineAl-Tahan(LIU) NeutroOrderedAlgebra:TheoryandExamples March1st ,202148/54
NeutroOrderedSemigroups DefinitionsandExamplesofNeutroOrderedSemigroups SubsetsofNeutroOrderedSemigroups NeutroOrderedIsomorphisms ProductionalNeutroOrderedSemigroups
BijanDavvaz fromYazdUniversity,Iran,
49/54
Collaborativeteam Theresultsinthispresentationarefromjointwork[1,2]with FlorentinSmarandache fromNewMexicoUniversity,USA,
MadeleineAl-Tahan(LIU) NeutroOrderedAlgebra:TheoryandExamples March1st ,202149/54
and OsmanAnis fromLebaneseInternationalUniversity,Lebanon.
50/54 Collaborativeteam
MadeleineAl-Tahan(LIU) NeutroOrderedAlgebra:TheoryandExamples March1st ,202150/54
51/54 References 1 Neutrosophy Definition Formalization 2 NeutroAlgebra Motivation Definitions 3 NeutroOrderedAlgebra Definitions
4 Conclusion 5 Collaborativeteam 6 References 7 Thankyou MadeleineAl-Tahan(LIU) NeutroOrderedAlgebra:TheoryandExamples March1st ,202151/54
NeutroOrderedSemigroups DefinitionsandExamplesofNeutroOrderedSemigroups SubsetsofNeutroOrderedSemigroups NeutroOrderedIsomorphisms ProductionalNeutroOrderedSemigroups
M.Al-Tahan,F.Smarandache,B.Davvaz,NeutroOrderedAlgebra: ApplicationstoSemigroups,NeutrosophicSetsandSystems,Vol.39, pp.133-147,2021.
M.Al-Tahan,B.Davvaz,F.Smarandache,O.Osman,OnSome PropertiesofProductionalNeutroOrderedSemigroups,Submitted.
L.Fuchs, PartiallyOrderedAlgebraicSystems,Int.Ser.ofMonographs onPureandAppl.Math.28,PergamonPress,Oxford,1963.
F.Smarandache, Neutrosophy/NeutrosophicProbability,Set,and Logic,ProQuestInformation&Learning,AnnArbor,Michigan,USA, 105p.,1998.
52/54 References
MadeleineAl-Tahan(LIU) NeutroOrderedAlgebra:TheoryandExamples March1st ,202152/54
F.Smarandache,NeutroAlgebraisaGeneralizationofPartialAlgebra, InternationalJournalofNeutrosophicScience(IJNS),Vol.2,No.1,pp. 08-17,2020,http://fs.unm.edu/NeutroAlgebra.pdf.
F.Smarandache,IntroductiontoNeutroAlgebraicStructuresand AntiAlgebraicStructures,inAdvancesofStandardandNonstandard NeutrosophicTheories,PonsPublishingHouseBrussels,Belgium,Ch.6, pp.240-265,2019; http://fs.unm.edu/AdvancesOfStandardAndNonstandard.pdf
F.Smarandache,IntroductiontoNeutroAlgebraicStructuresand AntiAlgebraicStructures(revisited),NeutrosophicSetsandSystems, Vol.31,pp.1-16,2020.DOI:10.5281/zenodo.3638232, http://fs.unm.edu/NSS/NeutroAlgebraic-AntiAlgebraic-Structures.pdf
53/54 References
MadeleineAl-Tahan(LIU) NeutroOrderedAlgebra:TheoryandExamples March1st ,202153/54
54/54 Thankyou MadeleineAl-Tahan(LIU) NeutroOrderedAlgebra:TheoryandExamples March1st ,202154/54