Neutro-BCK-Algebra

NEUTRO-BCK-ALGEBRA

1 MohammadHamidi, 2 FlorentinSmarandache

1 DepartmentofMathematics,UniversityofPayameNoor,Tehran,Iran, 2 Mathematics&Science,UniversityofNewMexico,GallupCampus,USA m.hamidi@pnu.ac.ir1 ,smarand@unm.edu2

Abstract

ThispaperintroducesthenovelconceptofNeutro-BCK-algebra.InNeutro-BCK-algebra,theoutcomeof anygiventwoelementsunderanunderlyingoperation(neutro-sophicationprocedure)hasthreecases,suchas: appurtenance,non-appurtenance,orindeterminate.Whileforanaxiom:equal,non-equal,orindeterminate. ThisstudyinvestigatestheNeutro-BCK-algebraandshowsthatNeutro-BCK-algebraaredifferentfromBCKalgebra.ThenotationofNeutro-BCK-algebrageneratesanewconceptofNeutroPosetandNeutro-HassdiagramforNeutroPosets.Finally,weconsideraninstanceofapplicationsoftheNeutro-BCK-algebra. Keywords: Neutro-BCK-algebra,NeutroPoset,Neutro-Hassdiagram.

1Introduction

Neutrosophy,asanewly-bornscience,isabranchofphilosophythatstudiestheorigin,natureandscope ofneutralities,aswellastheirinteractionswithdifferentideationalspectra.Itcanbedefinedastheincidenceoftheapplicationofalaw,anoperation,anaxiom,anidea,aconceptualaccreditedconstructionon anunclear,indeterminatephenomenon,contradictorytothepurposeofmakingitintelligible.Neutrosophic SetsandSystemsinternationaljournal(whichisinScopusandWebofScience)isatoolforpublicationsof advancedstudiesinneutrosophy,neutrosophicset,neutrosophiclogic,neutrosophicprobability,neutrosophic statistics,neutrosophicmeasure,neutrosophicintegral,andsoon,studiesthatstartedin1995andtheirapplicationsinanyfield,suchastheneutrosophicstructuresdevelopedinalgebra,geometry,topology,etc.Recently, FlorentinSmarandache[2019]generalizedtheclassicalAlgebraicStructurestoNeutroAlgebraicStructures NeutroAlgebras)andAntiAlgebraicStructures(AntiAlgebras)andheprovedthattheNeutroAlgebraisageneralizationofPartialAlgebra.7 Heconsidered <A> asanitem(concept,attribute,idea,proposition,theory, etc.).Throughtheprocessofneutrosphication,hesplitthenonemptyspaceandworkedontothreeregions twooppositeonescorrespondingto <A> and <antiA>,andonecorrespondingtoneutral(indeterminate) <neutA> (alsodenoted <neutroA>)betweentheopposites,regionsthatmayormaynotbedisjointdependingontheapplication,buttheyareexhaustive(theirunionequalsthewholespace).ANeutroAlgebra isanalgebrawhichhasatleastoneNeutroOperationoperationthatiswell-defined(alsocalledinner-defined) forsomeelements,indeterminateforothers,andouter-definedfortheothersoroneNeutroAxiom(axiomthat istrueforsomeelements,indeterminateforotherelements,andfalsefortheotherelements).APartialAlgebraisanalgebrathathasatleastonepartialoperation(well-definedforsomeelements,andindeterminatefor otherelements),andallitsaxiomsareclassical(i.e.,theaxiomsaretrueforallelements).Throughatheorem heprovedthatNeutroAlgebraisageneralizationofPartialAlgebra,andexamplesofNeutroAlgebrasthatare notpartialalgebrasweregiven.Also,theNeutroFunctionandNeutroOperationwereintroduced.7

Regardingthesepoints,wenowintroducetheconceptofNeutro-BCK-algebrasbasedonaxiomsof BCK-algebras,buthavingadifferentoutcome.Inthesystemof BCK-algebras,theoperationistotally well-definedforanytwogivenelements,butinNeutro-BCK-algebrasitsoutcomemaybewell-defined,outerdefined,orindeterminate.Any BCK-algebraisasystemwhichconsidersthatallitsaxiomsaretrue;butwe weakentheconditionsthattheaxiomsarenotnecessarilytotallytrue,butalsopartiallyfalse,andpartially indeterminate.So,oneofourmainmotivationisaweakcoverageoftheclassicalaxiomsof BCK-algebras. Thiscausesnewpartiallyorderedrelationsonanon-emptyset,suchasNeutroPosetsandNeutro-HassDiaDoi:10.5281/zenodo.3902754110

grams.IndeedNeutro-HassDiagramsofNeutroPosetscontainrelationsbetweenelementsinthesetthatare true,false,orindeterminate.

2Preliminaries

Inthissection,werecallsomedefinitionsandresultsfrom,7 whichareneededthroughoutthepaper. Let n ∈ N.Thenan n-aryoperation ◦ : X n → Y iscalledaNeutroOperationifithas x ∈ X n forwhich ◦(x) iswell-defined (degreeoftruth(T)), x ∈ X n forwhich ◦(x) isindeterminate (degreeofindeterminacy (I)),and x ∈ X n forwhich ◦(x) isouter-defined (degreeoffalsehood(F)),where T,I,F ∈ [0, 1],with (T,I,F ) =(1, 0, 0) thatrepresentsthe n-ary(total,orclassical)Operation,and (T,I,F ) =(0, 0, 1) that representsthe n-aryAntiOperation.Again,inthisdefinition“neutro”standsforneutrosophic,whichmeansthe existenceofouter-ness,orundefined-ness,orunknown-ness,orindeterminacyingeneral.Inthisregards,for anygivenset X,weclassify n-aryoperationon X n by (i);(classical) Operation isanoperationwell-defined forallset’selements, (ii); NeutroOperation isanoperationpartiallywell-defined,partiallyindeterminate, andpartiallyouter-definedonthegivensetand (iii); AntiOperation isanoperationouter-definedforallset’s elements.

Moreover,wehave (i); a (classical) Axiom definedonanon-emptysetisanaxiomthatistotallytrue (i.e.trueforallset’selements), (ii); NeutroAxiom(orneutrosophicaxiom) definedonanon-emptyset isanaxiomthatistrueforsomeelements(degreeoftrue=T),indeterminateforotherelements(degreeof indeterminacy=I),andfalsefortheotherelements(degreeoffalsehood=F),where T,I,F arein [0, 1] and (T,I,F ) isdifferentfrom (1, 0, 0) i.e.,differentfromtotallytrueaxiom,orclassicalAxiomand (T,I,F ) is differentfrom (0, 0, 1) i.e.,differentfromtotallyfalseaxiom,orAntiAxiom. (iii); an AntiAxiom oftype C definedonanon-emptysetisanaxiomthatisfalseforallset’selements.

Basedontheabovedefinitions,thereisaclassificationofalgebrasasfollows.Let X beanon-emptyset and O beafamilyofbinaryoperationson X.Then (A, O) iscalled

(i) a (classical) Algebra oftype C,if O isthesetofalltotalOperations(i.e.well-definedforallset’s elements)and (A, O) issatisfiedby (classical) Axiomsoftype C(trueforallset’selements).

(ii) a NeutroAlgebra(orneutro-algebraicstructure) oftype C,if O hasatleastoneNeutroOperation(or NeutroFunction),or (A, O) issatisfiedbyatleastoneNeutroAxiomoftype C thatisreferredtotheset’s (partial-,neutro-,ortotal-)operationsoraxioms;

(iii) an AntiAlgebra(oranti-algebraicstructure) oftype C,if O hasatleastoneAntiOperationor (A, O) issatisfiedbyatleastoneAntiAxiomoftype C

3.1ConceptofNeutro-BCK-algebra

Inthissection,weintroduceseveralconceptssuchas:Neutro-BCK-algebra,Neutro-BCK-algebraoftype 5,NeutroPosetandNeutro-HassDiagramandinvestigatethepropertiesoftheseconcepts.

Definition3.1. 2 Let X beanon-emptysetwithabinaryoperation “ ∗ ” andaconstant “0”.Then, (X, ∗, 0) iscalleda BCK -algebra ifitsatisfiesthefollowingconditions:

Now,wedefineNeutro-BCK-algebrasasfollows.

Definition3.2. Let X beanon-emptyset, 0 ∈ X beaconstantand “ ∗ ” beabinaryoperationon X.An algebra (X, ∗, 0) oftype (2, 0) issaidtobeaNeutro-BCK-algebra,ifitsatisfiesatleastoneofthefollowing NeutroAxioms(whiletheothersareclassicalBCK-axioms):

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(NBCI 1) ∃ x,y,z ∈ X suchthat ((x ∗ y) ∗ (x ∗ z)) ∗ (z ∗ y)=0) and ∃ x,y,z ∈ X suchthat ((x ∗ y) ∗ (x ∗ z)) ∗ (z ∗ y) =0 orindeterminate ;

(NBCI 2) ∃ x,y ∈ X suchthat (x ∗ (x ∗ y)) ∗ y =0 and ∃ x,y ∈ X suchthat (x ∗ (x ∗ y)) ∗ y =0 or indeterminate ;

(NBCI 3) ∃ x ∈ X suchthat x ∗ x =0 and ∃ x ∈ X suchthat x ∗ x =0 orindeterminate ;

(NBCI 4) ∃ x,y ∈ X,suchthatif x∗y = y∗x =0,wehave x = y and ∃ x,y ∈ X,suchthatif x∗y = y∗x =0, wehave x = y ;

(NBCK 5) ∃ x ∈ X suchthat 0 ∗ x =0 and ∃ x ∈ X suchthat 0 ∗ x =0 orindeterminate .Eachabove NeutroAxiomhasadegreeofequality(T ),degreeofnon-equality(F ),anddegreeofindeterminacy(I), where (T,I,F ) / ∈ (1, 0, 0), (0, 0, 1)

If (X, ∗, 0) isaNeutroAlgebraandsatisfiestheconditions (NBCI 1) to (NBCI 4) and (NBCK 5),then wewillcallitisaNeutro-BCK-algebraoftype5(i.e.itsatisfies5NeutroAxioms).

Example3.3. Let X = Z.Then (i)(X, ∗, 0) isaNeutro-BCK-algebra,whereforall x,y ∈ X,wehave x ∗ y = x y + xy (ii)(X, ∗, 1) isaNeutro-BCK-algebra,whereforall x,y ∈ X,wehave x ∗ y = xy.

(iii)(X, ∗, 1) isaNeutro-BCK-algebra,whereforall x,y ∈ X,wehave x ∗ y = 1 if x aneven xy if x anodd

Let X = ∅ beafiniteset.Wedenote NBCK (X ) and NNBCK (X ) bythesetofallNeutro-BCK-algebras andNeutro-BCK-algebrasoftype5thatareconstructedon X,respectively.

Theorem3.4. Let (X, ∗, 0) beaNeutro BCK -algebra.Then

(i) If |X| =1,then (X, ∗, 0) isatrivial BCK -algebra. (ii) If |X| =2,then |NBCK (X )| =2 and |NNBCK (X )| = ∞

(iii) If |X| =3,thenthereexists ∅ = Y ⊆ X ,suchthat (Y, ∗ , 0) isanontrivialortrivial BCK -algebra.

Proof. Weconsideronlythecases (ii), (iii),becausetheothersareimmediate. (ii) Let X = {0,x}.Thenwehave 2 trivialNeutro-BCK-algebras (X, ∗1 ), (X, ∗2 ) andaninfinite numberoftrivialNeutro-BCK-algebrasoftype 5(X, ∗, 0) inTable1,where w ∈ X (iii) Let X = {0,x,y}.Nowconsider Y = {0,x} anddefineaNeutro-BCK-algebra (X, ∗ , 0) inTable 1.Clearly (Y, ∗ , 0) isanon-trivial BCK-algebra.If Y = {0},itisatrivial BCK-algebra.

Table1:Neutro-BCK-algebrasoforder2 ∗1 0 x 0 0 x x 0 x , ∗2 0 x 0 x 0 x x 0 , ∗ 0 x 0 x 0 x w 0 and ∗ 0 xy 0 00 y x x 00 y 0 yx

Theorem3.5. Every BCK -algebra,canbeextendedtoaNeutro-BCK -algebra. Proof. Let (X, ∗, 0) bea BCK-algebraand α ∈ X,and U betheuniverseofdiscoursethatstrictlyincludes X ∪ α.Forall x,y ∈ X ∪{α},define ∗α on X ∪{α} by x ∗α y = x ∗ y where, x,y ∈ X andif α ∈{x,y}, define x ∗α y asindeterminateor x ∗α y/ ∈ X ∪ α.Then (X ∪{α}, ∗α, 0) isaNeutro-BCK-algebra. Example3.6. Let X = {0, 1, 2, 3, 4, 5}.ConsiderTable3. Then (i) If a =0,then (X, ∗1 , 0) isaNeutro-BCK-algebraandif a =1,then (X \{3, 4, 5}, ∗1 , 0) isa BCK-algebra. (ii)(X, ∗2 , 0) isaNeutro-BCK-algebraand (X \{4, 5}, ∗2 , 0) isa BCK-algebra. (iii) If s = t = y = z =0,w =3,then (X, ∗3 , 0) isaNeutro-BCK-algebraandfor s = t =1,y = 2,z =3, (X \{5}, ∗3 , 0) isa BCK-algebra.If s = t = y = z =0,w = √2,then (X, ∗3 , 0) isa Neutro-BCK-algebraoftype 5 where s,t ∈{0, 1},x ∈{4, 5},y ∈{2, 0},z ∈{3, 0} and w ∈{3, √2}

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Table2:Neutro-BCK-algebrasandNeutro-BCK-algebraoftype 5

∗1 012345 0 000 020 1 10 a 203 2 220 020 3 301205 4 040140 5 401023

,

∗2 012345 0 0000 20 1 1000 05 2 2100 50 3 3210 02 4 010412 5 50400 x

and

∗3 012345 0 00000 5 1 10 t 0 s 0 2 220 y 0 3 3 3130 z 0 4 44440 1 5 02020 w

,

Remark3.7. InNeutro-BCK-algebra (X, ∗3 , 0),whichisdefinedasinExample3.6,wehave (1, 5) ∈≤ and (5, 0) ∈≤,but (1, 0) ∈≤, where (x,y) ∈≤ means x ∗3 y =0.Thus ≤,necessarily,isnotatransitiverelation. Sowehavethefollowingdefinitionofneutro-partiallyorderedrelationonNeutro-BCK-algebra. Definition3.8. Let X beanon-emptysetand R beabinaryrelationon X.Then R iscalleda

(i) neutro-reflexive,if ∃ x ∈ X suchthat (x,x) ∈ R (degreeoftruth T ),and ∃ x ∈ X suchthat (x,x) is indeterminate(degreeofindeterminacy I),and ∃ x ∈ X suchthat (x,x) ∈ R (degreeoffalsehood F );

(ii) neutro-antisymmetric,if ∃ x,y ∈ X suchthat (x,y) ∈ R and (x,y) ∈ R implythat x = y (degreeof truth T ),and ∃ x,y ∈ X suchthat (x,y) or (y,x) areindeterminatein R (degreeofindeterminacy I), and ∃ x,y ∈ X suchthat (x,y) ∈ R and (y,x) ∈ R implythat x = y (degreeoffalsehood F );

(iii) neutro-transitive,if ∃ x,y,z ∈ X suchthat (x,y) ∈ R, (y,z) ∈ R implythat (x,z) ∈ R (degreeof truth T ),and ∃ x,y,z ∈ X suchthat (x,y) or (y,z) areindeterminatein R (degreeofindeterminacy I), and ∃ x,y,z ∈ X suchthat (x,y) ∈ R, (y,z) ∈ R,but (x,z) ∈ R (degreeoffalsehood F ).Inallabove neutro-axioms (T,I,F ) / ∈ (1, 0, 0), (0, 0, 1).

(iv) neutro-partiallyorderedbinaryrelation,iftherelationsatisfiesatleastoneoftheaboveneutro-axioms neutro-reflexivity,neutro-antisymmetry,neutro-transitivity,whiletheothers(ifany)areamongtheclassicalaxiomsreflexivity,antisymmetry,transitivity.

If R isaneutro-partiallyorderedrelationon X,wewillcall (X,R) byneutro-poset.Wewilldenote,the relateddiagramwithtoneutro-poset (X,R) byneutro-Hassdiagram.

Wedefinebinaryrelations ” ≤1 , ≤2 ” on X by x ≤1 y iforonlyif x ∗ y =0 or x ≤1 x and x ≤2 y if andonlyif(x ∗ y =0 orindeterminate)or x ≤2 x .Sowehavethefollowingtheorem.

Theorem3.9. Analgebra (X, ∗, 0) isaNeutro-BCK -algebraifandonlyifitsatisfiesthefollowingconditions:

(NBCI 1 ) ∃ x,y ∈ X suchthat ((x∗y)∗(x∗z)) ≤1 (z∗y) and ∃ x,y ∈ X suchthat ((x∗y)∗(x∗z)) ≤2 (z∗y) ,

(NBCI 2 ) ∃ x,y ∈ X suchthat (x ∗ (x ∗ y)) ≤1 y and ∃ x,y ∈ X suchthat (x ∗ (x ∗ y)) ≤2 y , (NBCI 3 ) ∃ x,y ∈ X suchthat x ≤1 x and ∃ x,y ∈ X suchthat x ≤2 x ,

(NBCI 4 ) ∀ x,y ∈ X ,if x ≤1 y and y ≤1 x,weget x = y and ∀ x,y ∈ X ,if x ≤2 y and y ≤2 x,weget x = y ,

(NBCK 5 ) ∃ x,y ∈ X suchthat 0 ≤1 x and ∃ x,y ∈ X suchthat 0 ≤2 x Proof. Let (X, ∗, 0) beaNeutro-BCK-algebra.Weproveonlytheitem (NBCI 1 ),otheritemsaresimilar to.Since (X, ∗, 0) beaNeutro-BCK-algebra, ∃ x,y ∈ X suchthat (x∗(x∗y))∗y =0 and ∃ x,y ∈ X such that (x ∗ (x ∗ y)) ∗ y =0 orindeterminate .Bydefinition, ∃ x,y ∈ X suchthat ((x ∗ y) ∗ (x ∗ z)) ≤1 (z ∗ y) and ∃ x,y ∈ X suchthat ((x ∗ y) ∗ (x ∗ z)) ≤2 (z ∗ y) .Conversely,lettheitems (NBCI 1 ) to (NBCI 4 ) and (NBCK 4 ).Justprove (NBCI 1) andotheritemsaresimilarto.Since ∃ x,y ∈ X suchthat ((x ∗ y) ∗ (x ∗ z)) ≤1 (z ∗ y) and ∃ x,y ∈ X suchthat ((x ∗ y) ∗ (x ∗ z)) ≤2 (z ∗ y) ,wegetthat ∃ x,y ∈ X suchthat ((x ∗ y) ∗ (x ∗ z)) ∗ (z ∗ y)=0) and ∃ x,y ∈ X suchthat ((x ∗ y) ∗ (x ∗ z)) ∗ (z ∗ y) =0 or indeterminate

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Let (X, ∗, 0) beaNeutro-BCK algebra.Definebinaryrelation ≤ on X,by x ≤ y ifandonly x ≤1 y and y ≤2 x.Sowehavethefollowingresults.

Theorem3.10. Let (X, ∗, 0) beaNeutro-BCK algebraand x,y,z ∈ X .Then (i) if x = y and x ≤ y,then y ≤ x; (ii) ≤ isareflexiveandsymmetricrelationon X ; (iii) ≤ isaneutro-transitivealgebrarelationon X .

Proof. (i) Let x = y ∈ X and x ≤ y.If y ≤ x,bydefinitionweobtain (x ∗ y = y ∗ x =0) and (x ∗ y = y ∗ x =0) andso x = y. (ii), (iii) Itisclearbyitem (i) andRemark3.7. (iii) Itisobtainedby (ii)

Corollary3.11. Let (X, ∗, 0) beaNeutro-BCK algebra.Then (X, ∗, 0, ≤1 ), (X, ∗, 0, ≤2 ) and (X, ∗, 0, ≤) areneutro-posets.

Let (X1 , ∗1 , 01 ) and (X2 , ∗2 , 02 ) be BCK-algebras,where X1 ∩ X2 = ∅.Forsome x,y ∈ X, definean



    

, where 01 ∗ 02 =02 and 02 ∗ 01 =01 .

Theorem3.12. Let (X1 , ∗1 , 01 ) and (X2 , ∗2 , 02 ) be BCK -algebras,where X1 ∩ X2 = ∅ and X = X1 ∪ X2 Then (i) (X, ∗, 01 ) isaNeutro-BCK -algebra; (ii) (X, ∗, 02 ) isaNeutro-BCK -algebra; Proof. (i) Weonlyprove (NBCI 4).Let x ∗ y =01 .Itfollowsthat x ∈ X1 and y ∈ X2 or x,y ∈ X1 . If x,y ∈ X1 ,because (X1 , ∗1 , 01 ) isa BCK-algebra, y ∗ x =01 impliesthat x = y.Butfor x ∈ X1 and y ∈ X2 ,wehave y ∗ x =01 so (NBCI 4) isvalidinanycases.Otheritemsareclear. (ii) Itissimilartoitem (i)

Example3.13. Let X1 = {a,b} and X2 = {w,x,y,z}.Then (X1 , ∗,a) and (X2 , ∗,w) are BCK-algebras. SobyTheorem3.12, (X1 ∪ X1 , ∗,a) and (X1 ∪ X1 , ∗,w) areNeutro-BCK-neutralgebrasinTable3.

Table3: BCK-algebrasandNeutro-BCK-algebra ∗ abwxyz a aa waaa b ba aaaa w aw wwww x ww xwww y ww yxww z ww zxxw

Corollary3.14. Let (X1 , ∗1 , 01 ) and (X2 , ∗2 , 02 ) be BCK -algebras.Then

(i) (X, ∗, 01 , ≤1 ), (X, ∗, 02 , ≤2 ) and (X, ∗, 02 , ≤2 ) areposets. (ii) (X, ∗, 01 , ≤2 ), (X, ∗, 02 , ≤1 ) areneutro-posets.

Example3.15. ConsidertheNeutro-BCK-algebrainExample3.13.Thenwehaveneutro-posets (X, ∗,w, ≤1 ), (X, ∗,a, ≤2 ) and (X, ∗, 02 , ≤) inTable4,where meansthatelementsarenotcomparableand I means thatareindeterminates.

Definition3.16. Let (X, ∗, 0) beaNeutro-BCK-algebra, θ ∈ X and Y ⊆ X.Then

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≤1 abwxyz a a axyz b bwxyz w awwwww x xxwxxx y yywxyy z zzwxyz

Table4:neutro-posets

,

≤2 abwxyz a abaxyz b bbwxyz w awwIII x xxIxII y yyIIyI z zzIIIz

,

≤ abwxyz a aawaaa b abbbbb w wbw x ab x y ab y z ab z

(i) Y iscalledaNeutro-BCK-subalgebra,if (1)0 ∈ Y , (2) forall x,y ∈ Y ,wehave x ∗ y ∈ Y , (3) satisfiesinconditions (NBCI 3), (NBCI 4) and (NBCK 5)

(ii) θ ∈ X iscalledasourceelement,ifitisaminimumormaximumelementinneutro-Hassdiagramof (X, ∗, 0)

Theorem3.17. Let (X, ∗, 0) beaNeutro-BCK -algebraand Y ⊆ X .If Y isaNeutro-BCK -subalgebraof X ,then

(i) (Y, ∗, 0) isaNeutro-BCK -algebra.

(ii) X isaNeutro-BCK -subalgebraof X

Proof. Theyareclear.

Corollary3.18. Let (X, ∗, 0) beaNeutro-BCK -algebraand |X| = n.Thenthereexist m ≤ n and x1 ,x2 ,...,xm ∈ X suchthat ({0,x1 ,x2 ,...,xm}, ∗, 0) isaNeutro-BCK -algebraof X .

Theorem3.19. Let X beanon-emptyset.Thenthereexistsabinaryoperation “ • ” on X and 0 ∈ X such that

(i) (X, •,x0 ) isaNeutro-BCK -algebra.

(ii) Forall ∅ = Y ⊆ X , Y ∪{x0 } isaNeutro-BCK -subalgebraof X

(iii) If X isacountableset,theninneutro-Hassdiagram (X, •,x0 ),wehave |Maximal(X )| =1 and Minimal(X )= |X|− 1(|X| iscardinalof X )

(iv) neutro-Hassdiagram (X, •,x0 ) hasasourceelement.

Proof. Let x,y ∈ X.Fixed x0 ∈ X anddefine x ∗ y = y.

(i) Somemodulationsshowthat (X, ∗,x0 ) isaNeutro-BCK-algebra.

(ii) ByTheorem3.4anddefinition,itisclear.

(iii) Let X = {x0 ,x1 ,x2 ,x3 ,...}.ThenbyCorollary3.11, (X, ≤,x0 ) isaneutro-posetandsohasa neutro-HassdiagramasFigure1. x0 • x2 • x1 • x3 • x4 • . . . ... ... ... xn •

. . . . . . . . . . . .

. . . . .

Figure1:neutro-Hassdiagram (X, ≤,x0 ) withsource x0 .

Theorem3.20. Let (X, ≤X ) beachain.Then

(i) thereexists ∗X on X and 0 ∈ X suchthat (X, ∗X , 0) isaNeutro-BCK -algebra.

(ii) forall x,y ∈ X ,wehave x ≤ y ifandonlyif y ≤X x

(iii) Inneutro-Hassdiagram (X, •,x0 ), 0 issourceelement.

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thereexists ∗X on X and 0 ∈ X suchthat (X, ∗X , 0) isaNeutro-BCK -algebra. Proof. Let 0,x,y ∈ X,where 0= Min(X ). (i) Define x ∗X y = x ∨ y if x ≤X y x ∧ y otherwise . Somemodulationsshowthat (X, ∗X , 0) isaNeutro-BCK algebra.

(ii) Let x,y ∈ X.Clearly x ∗ x = x,thenbydefinition x ≤ y ifandonlyif x ∗ y =0 and y ∗ x =0 ifand onlyif y =0 ifandonlyif y ≤X x (iii) Byitem (ii),wegettheneutro-Hassdiagram (X, ≤X , 0) inFigure1,so 0 issourceelement.

Let (X1 , ∗1 , 01 ) and (X2 , ∗2 , 02 ) betwoNeutro-BCK-algebras,where X1 ∩X2 = ∅.Define ∗ on X1 ∪X2 , by x ∗ y =      x ∗1 y if x,y ∈ X1 \ X2 x ∗2 y if x,y ∈ X2 \ X1 y otherwise

Theorem3.21. Let (X1 , ∗1 , 01 ) and (X2 , ∗2 , 02 ) betwoNeutro-BCK -algebras.Then (i) (X1 ∪ X2 , ∗, 01 ) isaNeutro-BCK -algebra. (ii) (X1 ∪ X2 , ∗, 02 ) isaNeutro-BCK -algebra.

Proof. Itisobvious. Let (X1 , ∗1 , 01 ) and (X2 , ∗2 , 02 ) betwoNeutro-BCK-algebras.Define ∗ on X1 ×X2 ,by (x,y)∗(x ,y )= (x ∗1 x ,y ∗2 y ),where (x,y), (x ,y ) ∈ X1 × X2 Theorem3.22. Let (X1 , ∗1 , 01 ) and (X2 , ∗2 , 02 ) betwoNeutro-BCK -algebras.Then (X1 × X2 , ∗, (01 , 02 )) isaNeutro-BCK -algebra. Proof. Weproveonlytheitem (NBCI 4).Let (x,y), (x ,y ) ∈ X1 × X2 .If (x,y) ∗ (x ,y )=(x ,y ) ∗ (x,y)=(01 , 02 ),then (x ∗1 x ,y ∗2 y )=(01 , 02 ) and (x ∗1 x,y ∗2 y)=(02 , 01 ).Itfollowsthat (x,y)= (x ,y ).Inasimilarway, (x,y) ∗ (x ,y )=(x ,y ) ∗ (x,y) =(01 , 02 ),wegetthat (x,y)=(x ,y ).Thus, (X1 × X2 , ∗, (01 , 02 )) isaNeutro-BCK-algebra.

3.2ApplicationofNeutro-BCK-algebra

Inthissubsection,wedescribesomeapplicationsofNeutro-BCK-algebra. Inthefollowingexample,wedescribesomeapplicationsofNeutro-BCK-algebra.WediscussapplicationsofNeutro-BCK-algebraforstudyingthecompetitionalongwithalgorithms.TheNeutro-BCK-algebra hasmanyutilizationsindifferentareas,whereweconnectNeutro-BCK-algebratoothersciencessuchaseconomics,computersciencesandotherengineeringsciences.WepresentanexampleofapplicationofNeutroBCK-algebrainCOVID-19.

Example3.23. (COVID-19) Let X = {a = China,b = Italy,c = USA,d = Spain,e = Germany,f = Iran} beasetoftopsixCOVID-19affectedcountries.Therearemanyrelationsbetweenthecountriesof theworld.Suppose ∗ isoneofrelationson X whichisdescribedinTable5.Thisrelationcanbeeconomic impact,politicalinfluence,scientificimpactorotherchasses.Forexample x ∗ y = z,meansthatthecountry z influencestherelationship ∗ fromcountry x tocountry y.Clearly (X, ∗,China) isaNeutro-BCK-algebra.

Table5:Neutro-BCK-algebra

Andsoweobtainneutro-HassdiagramasFigure2.ApplyingFigure2,weobtainthatChinaismainsource ofCOVID-19totopfiveaffectedcountriesandIran,Spain,ItalyareindeterminatedcountriesinCOVID-19 affectiontogether,USAeffectsSpainandGermanyeffectsIran.

Doi:10.5281/zenodo.3902754116

China • USA • Spain • Italy • Germany • Iran •

Figure2:neutro-Hassdiagram (X, ∗,China) associatedtoinfectedCOVID-19.

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