NeutrosophicdeductivefiltersonBL-algebras
R.A.Borzooei∗ ,H.FarahaniandM.Moniri
FacultyofMathematicalSciences,ShahidBeheshtiUniversity,G.C.,Evin,Tehran,Iran
Abstract.Inthispaper,weintroducethenotionsofneutrosophicdeductivefilter,Booleanneutrosophicdeductivefilter(BNDF) andimplicativeneutrosophicdeductivefilter(INDF)on BL-algebrasasgeneralizationsofthefuzzycorrespondingversions.We alsoinvestigatesomepropertiesofthesefiltersanddrivesomecharacterizationsofthem.TherelationbetweenBNDFandINDFis investigatedanditisprovedthateveryBNDFisanINDF,buttheconverseistruewhencertainconditionissatisfied.Furthermore, weconstructaquotientstructurerelatedtotheneutrosophicdeductivefilterandprovecertainisomorphismtheorems.
Keywords: BL-algebra,neutrosophicdeductivefilter,quotientstructure
1.Introduction
FuzzysettheorywasintroducedbyZadehin1965 [11].Afuzzysubset A ofaset X isafunction µA : X → [0, 1],whereforeach x ∈ X, µA (x)represents thegradeofmembershipoftheelement x ∈ X to A. In[1],Atanassovintroducedtheintuitionisticfuzzy setsasageneralizationoffuzzysets.Theintuitionisticfuzzysetsconsiderbothmembershipdegreeand nonmembershipdegree.
In1998, neutrosophy hasbeenproposedbySmarandache[9]asanewbranchofphilosophyinorderto formallyrepresentneutralities.Thefundamentalthesis ofneutrosophyisthateveryideahasnotonlyacertaindegreeoftruthandacertaindegreeoffalsitybut alsoan indeterminacy degreethathavetobeconsideredindependentlyfromeachother.Inneutrosophicset theory,indeterminacyismeasuredexplicitlyandindependently.Thisassumptionisveryimportantinmany applicationssuchasinformationfusioninwhichwetry tocombinethedatafromdifferentsensors.Asanexample,supposethereare10votersduringavotingprocess. Onepossiblesituationisthattherearethree yes votes, two no votesandfive undecided ones.Wenotethatthis
∗
Correspondingauthor.R.A.Borzooei,FacultyofMathematicalSciences,ShahidBeheshtiUniversity,G.C.,Evin,Tehran,Iran. E-mail:borzooei@sbu.ac.ir.
exampleisbeyondthescopeofintuitionisticfuzzyset theory.
In1960AbrahamRobinsonintroduced non-standard analysis asaformalizationofanalysisandabranchof mathematicallogic.Innon-standardanalysisanonzero number ε issaidtobeinfinitelysmall,orinfinitesimalifandonlyifforallpositiveintegers n, |ε|≤ 1/n. Inthiscasethereciprocal δ = 1/ε willbeinfinitely large,orsimplyinfinite,meaningthatforallpositive integers n,wehave |δ| >n.Thesetofhyper-realnumbersisanextensionofthesetofrealnumberswhich includestheclassofinfinitenumbersandtheclassof infinitesimalnumbers.The non-standardunitinterval is]0 , 1+ [= 0 ∪ [0, 1] ∪ 1+ .Here0 isthesetofall hyper-realnumbers0 ε,and1+ isthesetofallhyperrealnumbers1 + λ,where ε and λ areinfinitesimal.
If U isaset,aneutrosophicsetdefinedonthe universe U assignstoeachelement x ∈ U ,atriple (T (x),I (x),F (x)),where T (x),I (x)and F (x)arestandardornon-standardelementsof]0 , 1+ [. T isthe degreeofmembershipintheset U , I isthedegreeof indeterminacy-membershipintheset U and F isthe degreeofnonmembershipintheset U .Inthispaper weworkwithspecialnetrosophicsetsthattheirneutrosophicelementsarestandardrealnumbersin[0,1].
Neutrosophyhaslaidthefoundationforawholefamilyofnewmathematicaltheories,suchasneutrosophic
settheory,neutrosophicprobability,neutrosophic statisticsandneutrosophiclogic.Inrecentyearsneutrosophicalgebraicstructureshavebeeninvestigated(see [3,5]).
BL-algebrasprovideanalgebraicsemanticsfor H ´ ajek’sBasicLogic[2].Themainexampleofa BLalgebraistheunitinterval[0,1]endowedwiththe structureinducedbyacontinuoust-norm.MV-algebras, GodelalgebrasandProductalgebrasarethemost knownclassesof BL-algebras.Filtertheoryplaysan importantruleinstudyingthesealgebras.Fromthe logicalpointofview,variousfilterscorrespondtovarioussetsofprovableformulas.In[4]and[7],the notionsoffuzzyprimefilter,fuzzyBooleanfilter,fuzzy implicativefilterandfuzzypositiveimplicativefilteron BL-algebraswereintroducedandsomeoftheirpropertiesandcharacterizationswereinvestigated.
Inthispaperwegeneralizetheconceptoffuzzy filerona BL-algebraanddefinetheconceptof neutrosophicdeductivefilter.WedefineBooleanneutrosophic deductivefilterandimplicativeneutrosophicdeductive filterandinvestigatesomeoftheirproperties.Wedrive severalcharacterizationsofthesefilters.Also,weinvestigaterelationbetweenBNDFandINDFandprovethat everyBNDFisanINDF,buttheconversemaynotbe true.Furthermore,theconditionunderwhichanINDF isBNDFisestablished.Finally,weconstructaquotient structurerelatedtotheneutrosophicdeductivefilterand provesomeisomorphismtheorems.
2.Preliminaries
Inthissection,wegivesomedefinitionsandresults fromtheliterature.
Definition2.1. [9]Let X beaset.A neutrosophicsubset A ofXisatriple(TA ,IA ,FA )where TA : X → [0, 1]isthemembershipfunction, IA : X → [0, 1]is theindeterminacyfunctionand FA : X → [0, 1]isthe nonmembershipfunction.Hereforeach x ∈ X, TA (x), IA (x)and FA (x)areallstandardrealnumbersin[0,1].
Notethatthereisnorestrictionsonthevaluesof TA (x),IA (x)and FA (x)andweonlyhavetheobvious condition
0 ≤ TA (x) + IA (x) + FA (x) ≤ 3
Definition2.2. [9]Let A and B betwoneutrosophic setson X.Define A ≤ B ifandonlyif
TA (x) ≤ TB (x),IA (x) ≥ IB (x),FA (x) ≥ FB (x)
forall x ∈ X.
Definition2.3. [9]Let A and B betwoneutrosophic setson X.Define
A ∧ B = (TA ∧ TB ,IA ∨ IB ,FA ∨ FB )
A ∨ B = (TA ∨ TB ,IA ∧ IB ,FA ∧ FB )
where, ∧ istheminimumand ∨ isthemaximum betweenrealnumbers.
Definition2.4. [2]TheaxiomsofpropositionalHa ´ jek BasicLogicintheHilbert-stylesystemareasthe following:
(A1)(ϕ → ψ ) → ((ψ → χ) → (ϕ → χ)), (A2)(ϕ &ψ ) → ϕ , (A3)(ϕ &ψ ) → (ψ &ϕ ), (A4)(ϕ &(ϕ → ψ )) → (ψ &(ψ → ϕ )), (A5)(ϕ → (ψ → χ)) → ((ϕ &ψ ) → χ (A6)((ϕ &ψ ) → χ) → (ϕ → (ψ → χ)), (A7)((ϕ → ψ ) → χ) → (((ψ → ϕ ) → χ) → χ) (A8)0 → ϕ ,
Theonlyinferenceruleismoduspones(MP).Weshow theconsequencerelationof BL intheHilbert-style axiomatizationby BL .Ifaformula ϕ isprovablein BL,wewrite BL ϕ
Proposition2.5. [2] In BL thefollowingstatements hold:
(1) BL (ϕ → (ψ → ϕ )), (2) BL (ϕ ∧ ψ → ψ ∧ ϕ ), (3) BL (ϕ ∧ ψ → ψ ), (4) BL ϕ →¬¬ϕ , (5) BL (ϕ → ψ ) → (¬ψ →¬ϕ ), (6) BL ¬ϕ → (ϕ → ψ ), (7) BL (ϕ → (ψ → χ)) ↔ (ψ → (ϕ → χ)), (8) BL (ϕ → (ψ → χ)) ↔ ((ϕ &ψ ) → χ)), (9) BL (ϕ → ψ ) → ((χ → ϕ ) → (χ → ψ )), (10) BL (ϕ → ψ ) → ((ϕ → χ) → (ψ → χ)), (11) BL ((ϕ →¬ϕ ) →¬ϕ ) ↔ (ϕ →¬¬ϕ ), (12) BL [(ϕ ∨¬ϕ ) → (ψ → ϕ )] ↔ [(ϕ → (ψ → ϕ )) ∧ (¬ϕ → (ψ → ϕ ))], (13) BL ψ ⇒ BL (ϕ → (ϕ → ψ )), (14) BL (ϕ → ψ ), BL (χ → θ ) ⇒ BL (ϕ &χ → ψ &θ ),
Definition2.6. [2]A BL-algebraisanalgebra (L, ∨, ∧, , →, 0, 1)oftype(2,2,2,2,0,0)suchthat
(BL1)(L, ∨, ∧, 0, 1)isaboundedlattice, (BL2)(L, , 1)isacommutativemonoid, (BL3) x y ≤ z ifandonlyif x ≤ y → z,forall x,y,z ∈ L, (BL4) x ∧ y = x (x → y ), (BL5)(x → y ) ∨ (y → x) = 1.
Ifa BL-algebra L satisfies ¬¬x = x,foreach x ∈ L, itiscalledan MV-algebra.
Proposition2.7. [2,6] Inany BL-algebra L,thefollowingpropertieshold:
(R1) x ≤ y ⇔ x → y = 1, (R2)1 → x = x,x → 1 = 1,x → x = 1, 0 → x = 1,x → (y → x) = 1, (R3) x ≤ y → z ⇔ y ≤ x → z, (R4) x → (y → z) = (x y ) → z = y → (x → z), (R5) x ≤ y ⇒ (z → x ≤ z → y and y → z ≤ x → z), (R6) z → y ≤ (x → z) → (x → y ), z → y ≤ (y → x) → (z → x), (R7)(x → y ) (y → z) ≤ x → z, (R8) ¬x =¬¬¬x,x ≤¬¬x,when ¬x = x → 0, (R9) ¬x ∧¬y =¬(x ∨ y ), (R10) x ∨¬x = 1 ⇒ x ∧¬x = 0, (M1) x y ≤ x ∧ y , (M2) x ≤ y ⇒ x z ≤ y z, (M3) y → z ≤ x ∨ y → x ∨ z, (M4) ¬x ∨¬y =¬(x ∧ y ), (M5)(x ∨ y ) → z = (x → z) ∧ (y → z), (M6) x ∨ y = ((x → y ) → y ) ∧ ((y → x) → x), (M7) x → (y ∨ z) = (x → y ) ∨ (x → z), (M8) x ∧ (y ∨ z) = (x ∧ y ) ∨ (x ∧ z), (B1) ¬¬(x ∧ y ) = (¬¬x ∧¬¬y ), ¬¬(x ∨ y ) = (¬¬x ∨¬¬y ), ¬¬(x y ) = (¬¬x ¬¬y ), (B2) ¬(¬¬x → x) = 0, ¬¬(x → y ) = (¬¬x → ¬¬y ).
Definition2.8. [2,10]Let F beanonemptysubsetof a BL -algebra L suchthat1 ∈ F . F iscalled:
(i)afilteron L,if (∀x,y ∈ L)(x,y ∈ F ⇒ x y ∈ F )and (∀x,y ∈ L)(x ∈ F,x ≤ y ⇒ y ∈ F ), (ii)a Booleanfilter on L,ifitisafilterandmoreover wehave (∀x ∈ L)(x ∨¬x ∈ F ),
(iii)an implicativefilter on L,ifitisafilterandmoreoverforall x,y,z ∈ L wehave
[x → y,x → (y → z) ∈ F ⇒ (x → z) ∈ F ].
Proposition2.9. [10] AnonemptysubsetFofa BLalgebra L isafilterifandonlyif (DS1)1 ∈ F , (DS2)(∀x,y ∈ L)(x ∈ F,x → y ∈ F ⇒ y ∈ F ). Theorem2.10. [2] LetFbeafilterona BL-algebra L. Definethebinaryrelation ∼F on L by
x ∼F y ⇔ (x → y ∈ F and y → x ∈ F )
Then ∼F isacongruenceon L,andthesetofallcongruenceclasses L/F ={[x]F : x ∈ L} withthefollowing operationsforma BL-algebra:
[x] • [y ] = [x y ], [x] [y ] = [x → y ], [x] [y ] = [x ∨ y ], [x] [y ] = [x ∧ y ]
Lemma2.11. [6] Let F1 and F2 betwofilterson BLalgebra L which F1 ⊆ F2 .Then F1 isafilteron F2 and F2 /F1 isafilteron L/F1
Definition2.12. [6]Theneutrosophicset F ofa BLalgebra L has Sup-InfProperty ifforanynonempty subset S of L,thereexist x0 ,x1 ,x2 ∈ S ,suchthat sup x∈S TF (x) = TF (x0 ), inf x∈S IF (x) = IF (x1 ), inf x∈S FF (x) = FF (x2 )
Fromnowon,weusethesamenotationsforcorrespondinglogicalandalgebraicnotions.Also,ifthere isnoconfusion,weuse ∧ and ∨ forminimumand maximumforrealnumbers.
3.Neutrosophicdeductivefilterson BL-algebras
Inthissection,wedefinetheneutrosophicdeductive filtersandprovesomepropertiesofthem.Furthermore, wecharacterizetheneurosophicdecuctivefiltergeneratedbyaneutrosophicdeductiveset.
Definition3.1. Supposethat and betwosubsetsof [0, 1]3 .Wedefinetherelation |= asfollows: |= ⇔∧ ≤∧
If =∅,thenwedefine ∧ = (1, 0, 0),andif =∅, thenwedefine ∧ = (0, 1, 1).
R.A.Borzooeietal./NeutrosophicdeductivefiltersonBL-algebras
Fromnowon,if |= and |= ,wewrite =
Definition3.2. Let L beaBL-algebraand bea consequencerelationonthesetofBL-formulas.A neutrosophicsubset F of L iscalledaneutrosophic filterwithrespectto ,ifforeachassignment v into L andforeveryset ∪{ϕ } of BL-formulas,if ϕ ,then {F (v( ))}|= F (v(ϕ )),where F (v( )) = {F (v(γ )): γ ∈ }.
Inparticular,if ispresentedbyaHilbert-stylesystem,forexampleif is BL ,thenitisenoughtocheck theaboveconditionfortheinferencerules( ,ϕ )and theaxioms(∅,ϕ )oftheproofsystem.
Definition3.3. AneutrosophicsubsetFofaBL-algebra L iscalleda neutrosophicdeductivefilter (briefly, NDF),if F isaneutrosophicfilterwithrespectto BL .
Lemma3.4. AneutrosophicsubsetFofaBL-algebra L isaNDFiffforallformulas ϕ,ψ andeachassignment v into L:
(NDF1) F (v(ϕ )) |= F (1), (NDF2) {F (v(ϕ )), F (v(ϕ → ψ ))}|= F (v(ψ )).
Proof. Thiscanbeeasilyobtainedfromthefactthatin aBL-algebra,allaxiomsofBLareevaluatedto1under allassignmentsand(MP)istheonlyinferencerule.
Corollary3.5. AneutrosophicsubsetFofaBL-algebra L isaNDFiff
(NDF1) (∀a ∈ L)( F (a) |= F (1)), (NDF2) (∀a,b ∈ L)( {F (a), F (a → b)}|= F (b))
Corollary3.6. Let F beaNDF.Thenwehave
(i) (∀a ∈ L)(F (a) ≤ F (1)), (ii) (∀a,b ∈ L)(F (a) ∧ F (a → b) ≤ F (b))
Example3.7. Let L ={0,a,b, 1}.Forall x,y ∈ L, wedefine x ∧ y = min{x,y }, x ∨ y = max{x,y } and and → asfollows: 0ab1 0 0000 a 00aa b 0abb 1 0ab1
→ 0ab1 0 1111 a a111 b 0a11 1 0ab1
Then(L, ∧, ∨, , →, 0, 1)isa BL-algebra.The neutrosophicsubset F of L definedby F (0) = F (a) = (t1 ,t3 ,t3 ), F (b) = (t2 ,t2 ,t2 ), F (1) = (t3 ,t1 ,t1 ),where 0 ≤ t1 <t2 <t3 ≤ 1arethreefixedrealnumbersin
[0,1],isaNDF. Theorem3.8. Let F beaneutrosophicsubsetof L Then F isaNDFifandonlyifforallformulas ϕ,ψ andallassignment v into L,if BL (ϕ → (ψ → χ)) then {F (v(ϕ )), F (v(ψ ))}|= F (v(χ))
Proof. Let F beaNDFon L, v beanassignment into L and BL (ϕ → (ψ → χ)),forsomeformulas ϕ,ψ .ByLemma3.4,wehave {F (v(ϕ → (ψ → χ)), F (v(ϕ ))}|= F (v(ψ → χ))and {F (v(ψ → χ)), F (v(ψ ))}|= F (v(χ)).Then {F (v(ϕ )), F (v(ϕ → (ψ → χ))), F (v((ψ ))}|= F (v(χ)).Now, since BL (ϕ → (ψ → χ)),so F (v(ϕ → (ψ → χ))) = F (1).Thus,weobtainthat {F (v(ϕ )), F (v(ψ ))}|= F (v(χ)).
Conversely,assumethattheconditionholds. Weknowif BL ψ ,then BL ϕ → (ϕ → ψ )and so {F (v((ϕ )), F (v((ϕ ))}|= F (v(ψ )) = F (1),therefore {F (v((ϕ ))}|= F (1).Now,since BL ϕ → ((ϕ → ψ ) → ψ ),bytheconditionweget {F (v(ϕ )), F (v(ϕ → ψ ))}|= F (v(ψ )),whichcompletestheproof.
Corollary3.9. Let F beaneutrosophicsubsetof L Then F isaNDFifandonlyifforallformulas ϕ,ψ andallassignments v into L,if BL (ϕ &ψ ) → χ,then {F (v(ϕ )), F (v(ψ ))}|= F (v(χ)).
Theorem3.10. Let F beaneutrosophicsubsetof L Then F isaNDFifandonlyifforallformulas ϕ,ψ andallassignments v into L
(i) BL (ϕ → ψ ) ⇒{F (v(ϕ ))}|= F (v(ψ )), (ii) {F (v(ϕ )), F (v(ψ ))}|= F (v(ϕ &ψ )).
Proof. Supposethat F isaNDF.Since BL (ϕ &ϕ → ϕ )and BL (ϕ → ψ ),wehave BL (ϕ &ϕ → ψ ).So,byCorollary3.9,itfollowsthatforall assignments v, {F (v(ϕ )), F (v(ϕ ))}|= F (v(ψ )),therefore {F (v(ϕ ))}|= F (v(ψ ))whichproves(i).Since BL (ϕ &ψ → ϕ &ψ ),byCorollary3.9wehave {F (v(ϕ )), F (v(ψ ))}|= F (v(ϕ &ψ )),proving(ii).
Conversely,assumethat(i)and(ii)holdand BL ((ϕ &ψ ) → χ),forsomeformulas ϕ,ψ,χ.Thenby(i) wehave {F (v(ϕ &ψ ))}|= F (v(χ))andsinceby(ii) wehave {F (v(ϕ )), F (v(ψ ))}|= F (v(ϕ &ψ )),weget {F (v(ϕ )), F (v(ψ ))}|= F (v(χ)).Therefore,theresult isobtainedbyCorollary3.9.
By(NDF2),Theorem3.10and(R1)-(R6),wegetthe followingcorollary.
Corollary3.11. ForaNDF F on L,wehave:
(1) BL (ϕ → ψ ) ⇒{F (v(ϕ ))}|= F (v(ϕ )),
(2) |= F (v(ϕ → ψ )) ⇒{F (v(ϕ ))}|= F (v(ψ )), (3) {F (v(ϕ &ψ ))}={F (v(ϕ )), F (v(ψ ))}, (4) {F (v(ϕ ∧ ψ ))}={F (v(ϕ )), F (v(ψ ))}, (5) {F (v(ϕ n ))}= F (v(ϕ )),whereϕ n = ϕ &...&ϕ (ntimes), (6) {F (v(ϕ )), F (v(¬ϕ ))}= (0, 1, 1), (7) {F (v(ϕ → ψ ))}|= F (v((χ → ϕ ) → (χ → ψ ))), (8) {F (v(ϕ → ψ ))}|= F (v((ψ → χ) → (ϕ → χ))), (9) {F (v(ϕ → ψ ))}|= F (v((ϕ &χ) → (ψ &χ))), (10) {F (v(ϕ → ψ )), F (v(ψ → χ))}|= F (v(ϕ → χ))), (11) {F (v(ϕ → (ψ → χ))), F (v(ϕ → ψ ))}|= {F (v(ϕ → (ϕ → χ)))}
Corollary3.12. ForaNDF F on L,thefollowinghold:
(1) x ≤ y ⇒ F (x) ≤ F (y ), (2) F (x → y ) = F (1) ⇒ F (x) ≤ F (y ), (3) F (x y ) = F (x) ∧ F (y ), (4) F (x ∧ y ) = F (x) ∧ F (y ), (5) F (xn ) = F (x),wherexn = x x... x (ntimes), (6) F (x) ∧ F (¬x) = F (0) = (0, 1, 1), (7) F (x → y ) ≤ F ((z → x) → (z → y )), (8) F (x → y ) ≤ F ((y → z) → (x → z)), (9) F (x → y ) ≤ F (x z → y z), (10) F (x → z) ≤ F (x → y ) ∧ F (y → z), (11) F (x → (y → z)) ∧ F (x → y ) ≤ F (x → (x → z)).
Proposition3.13. Let beanonemptysetand Fi be aNDF,foreach i ∈ .Then i∈ Fi isaNDF.
Definition3.14. Let F beaneutrosophicsubsetof L and G beaNDF. G issaidtobegeneratedby F ,if F ≤ G andforanyNDF H, F ≤ H impliesthat G ≤ H.The NDFgeneratedby F willbedenotedby F
Theorem3.15. Let F beaneutrosophicsubset of L.Thenforeachformula ψ andassignment v wehave F (v(ψ )) = {F (v(ϕ1 )) ∧ ∧ F (v(ϕn )) | BL ((ϕ1 & &ϕn ) → ψ ),forsome n ∈ N,ϕ1 ,...,ϕn ∈ Fm}
Proof. Wefirstprovethat F isaNDF.Obviously, F (v(ψ )) |= F (1),foreachformula ψ .Now,consideringtheformulas ϕ,ψ ,weobservethatifthereexist n,m ∈ N andformulas ϕ1 ,...,ϕn ,ψ1 ,...,ψm suchthat BL (ϕ1 &...&ϕn → ϕ ), BL (ψ1 &...&ψm → (ϕ → ψ )),thenweget BL (ϕ1 &...&ϕn &ψ1 &...&ψm → (ϕ &(ϕ → ψ ))).Hence,sincebyProposition2,
BL ((ϕ &(ϕ → ψ )) → ψ ),weget BL (ϕ1 &...&ϕn &ψ1 &...&ψm → ψ )
Thus
F (v(ϕ1 )) ∧ ∧ F (v(ϕn )) ∧ F (v(ψ1 )) ∧ ∧ F (v(ψm )) ≤ F (v(ψ )), andthen
F (v(ϕ )) ∧ F (v(ϕ → ψ )) = (∨{F (v(ϕ1 )) ∧ ... ∧ F (v(ϕn )) | BL ((ϕ1 & &ϕn ) → ϕ ), forsome n ∈ N, ϕ1 ,...,ϕn ∈ Fm}) ∧(∨{F (v(ψ1 )) ∧ ∧ F (v(ψm )) | BL ((ψ1 & &ψm ) → (ϕ → ψ )), forsome m ∈ N, ψ1 ,...,ψm ∈ Fm}) = (∨{F (v(ϕ1 )) ∧ ∧ F (v(ϕn )) ∧F (v(ψ1 )) ∧ ∧ F (v(ψm ))}| BL ((ϕ1 &...&ϕn ) → ϕ ), BL ((ψ1 &...&ψm ) → (ϕ → ψ )), forsome n,m ∈ N,ϕ1 ,...,ϕn ,ψ1 ,...,ψm ∈ Fm}) ≤ F (v(ψ ))
Therefore, F isaNDF. Now,Since BL (ϕ &ϕ ) → ϕ ,thenbydefinition of F ,itfollowsthat F (v(ϕ )) ≥ F (v(ϕ &ϕ )). Also, F (v(ϕ &ϕ )) ={F (v(ϕ )), F (v(ϕ ))}= F (v(ϕ )), byCorollary3.11.So, F (v(ϕ )) ≥ F (v(ϕ )).Therefore, F ≤ F .Finally,supposethat H isaNDFsuch that F ≤ H and ϕ isaformula.Then, F (v(ϕ )) = {F (v(ϕ1 )) ∧ ∧ F (v(ϕn )) |
BL ((ϕ1 & &ϕn ) → ϕ ),n ∈ N, ϕ1 ,...,ϕn ∈ Fm}
≤ {H(v(ϕ1 )) ∧ ∧ H(v(ϕn )) |
BL ((ϕ1 & &ϕn ) → ϕ ),n ∈ N, ϕ1 ,...,ϕn ∈ Fm}
≤ {H(v(ϕ ))}= H(v(ϕ )),
Therefore, F ≤ H,whichcompletestheproof.
Example3.16. Supposethat(L, ∧, ∨, , →, 0, 1)be the BL-algebradefinedinExample3.7.Definethe
R.A.Borzooeietal./NeutrosophicdeductivefiltersonBL-algebras
neutrosophicsubset F of L by F (0) = (t1 ,t1 ,t1 ), F (a) = F (b) = (t1 ,t2 ,t2 ), F (1) = (t2 ,t2 ,t2 )(0 ≤ t1 <t2 ≤ 1)andtheneutrosophicsubset G of L by G (0) = G (a) = G (b) = (t1 ,t1 ,t1 ), G (1) = (t2 ,t1 ,t1 ).Onecan easilycheckthat G = F .
4.Booleanneutrosophicdeductivefilters
Inthissectionwedefineandstudythenotionof Booleanneutrosophicdeductivefilteron BL-algebras. Definition4.1. Let F beaNDFon L F iscalleda Booleanneutrosophicdeductivefilter (briefly,BNDF) if F (1) |= F (v(ϕ ∨¬ϕ )),forallformula ϕ andall assignments v.
Example4.2. Let L ={0,a,b, 1} beachainwith cayleytablesasfollow: 0ab1 0 0000 a 0aaa b 0aab 1 0ab1
→ 0ab1 0 1111 a 0111 b 0b11 1 0ab1
Define ∧ and ∨ on L asminandmax,respectively.Then(L, ∧, ∨, , →, 0, 1)isa BL-algebra. Theneutrosophicsubset F of L definedby F (0) = (t1 ,t2 ,t2 ), F (a) = F (b) = F (1) = (t2 ,t1 ,t1 ),where 0 ≤ t1 <t2 ≤ 1aretwofixedrealnumbersin[0,1], isaBNDF.
Proposition4.3. Let F beaNDFon L. F isaBNDF ifandonlyif F (x ∨¬x) = F (1),forall x ∈ L.
Since ϕ ∨ ψ = ((ϕ → ψ ) → ψ ) ∧ ((ψ → ϕ ) → ϕ ), byCorollary3.11,wehavethefollowingproposition.
Proposition4.4. Let F beaNDFon L.Then, F isa BNDFifandonlyifforallformula ϕ andallassignments v: F (v((ϕ →¬ϕ ) →¬ϕ ))) = F (v((¬ϕ → ϕ ) → ϕ ))) = F (1)
Proposition4.5. Let F beaNDFon L.Then, F isa BNDFifandonlyifforall x ∈ L,
F ((x →¬x) →¬x) = F ((¬x → x) → x) = F (1) Definition4.6. Let F beaNDFon L.Then,foreach t ∈ [0, 1]wedefine Ft = (TFt ,IFt ,FFt ),where TF t = {x ∈ L : TF ≥ t }, IF t ={x ∈ L : IF ≤ t }, FF t = {x ∈ L : FF ≤ t }.
Theorem4.7. Let F beaNDFon L.Then, F isaBNDF ifandonlyifforeach t ∈ [0, 1], ∅ / = TF t , ∅ / = IF t , ∅ / = FF t andallofthembeBooleanfilterson L.
Proof. Supposethat F isaBNDF,and ∅ / = TF t , ∅ / = IF t and ∅ / = FF t ,forsome t ∈ [0, 1].Then,there exist x0 ∈ TF t , x1 ∈ IF t and x2 ∈ FF t .So,foreach x ∈ L, TF (x ∨¬x) = TF (1) ≥ TF (x0 ) ≥ t andhence x ∨¬x ∈ TF t .Similarly, x ∨¬x ∈ IF t and x ∨¬x ∈ FF t .Thus, TF t , IF t and FF t areBooleanfilterson L. Conversely,supposethat ∅ / = TF t , ∅ / = IF t and ∅ / = FF t areBooleanfilterson L,foreach t ∈ [0, 1]. Then, TTF (1) , IIF (1) and FFF (1) areBooleanfiltersandso x ∨¬x ∈ TTF (1) , x ∨¬x ∈ IIF (1) and x ∨¬x ∈ FFF (1) Thisimpliesthat F (x ∨¬x) = F (1),forall x ∈ L,and then F isaBNDF,byProposition4.3.
Corollary4.8. Let F beaNDFon L.Then F isaBNDF ifandonlyif TTF (1) , IIF (1) and FFF (1) areBoolean filters.
Theorem4.9. Let F and G betwoNDFson L,which F ≤ G , F (1) = G (1).If F isaBNDF,then G isaBNDF too.
Proof. UseDefinition4.1. Theorem4.10. Let F beaNDFon L, ϕ,ψ,χ beformulasand v beanassignmenton L.Thenthefollowing areequivalent:
(i) {F (v(ϕ → (¬χ → ψ ))), F (v(ψ → χ)}|= F (v(ϕ → χ) (ii) {F (v(ϕ → (¬χ → χ)))}|= F (v(ϕ → χ)) (iii) F (v(ϕ → (¬χ → χ))) = F (v(ϕ → χ)) (iv) {F (v(ψ → (ϕ → (¬χ → χ)))), F (v(ψ ))}|= F (v(ϕ → χ))
Proof. (i) ⇒ (ii)Itisenoughtolet ψ = χ in(i). (ii) ⇒ (iii)Itfollowsfrom BL (ϕ → χ) → (¬χ → (ϕ → χ)), BL (¬χ → (ϕ → χ)) ↔ (ϕ → (¬χ → χ))andTheorem3.10. (iii) ⇒ (iv)Since F isaNDF, {F (v(ψ → (ϕ → (¬χ → χ)))), F (v(ψ ))}|= F (v(ϕ → (¬χ → χ))). Thentheresultisobtainedby F (ϕ → (¬χ → χ)) = F (ϕ → χ).
(iv) ⇒ (i)ByCorollary3.11,wehave {F (v(ϕ → (¬χ → ψ ))), F (v(ψ → χ))}={F (v((ϕ ¬χ) → ψ )), F (v(ψ → χ))}|= F (v((ϕ ¬χ) → χ))and since F (v((ϕ ¬χ) → χ)) = F (v(ϕ → (¬χ → χ))), wehave {F (v(ϕ → (¬χ → ψ ))), F (v(ψ → χ))}|= F (v(ϕ → (¬χ → χ))).
OntheotherhandforeachBL-provableformula
ψ wehave F (v(ϕ → (¬χ → χ))) ={F (v(ψ → (ϕ → (¬χ → χ)))), F (v(ψ ))}|= F (v(ϕ → χ))by(iv).Thus weget {F (v(ϕ → (¬χ → ψ ))), F (v(ψ → χ))}|= F (v(ϕ → χ)),whichproves(i).
Definition4.11. Let F beaNDFon L.Wesay F has ImplicativeProperty,ifforallformulas ϕ,ψ,χ andall assignments v,itsatisfies: {F (v(ϕ → (¬χ → ψ ))), F (v(ψ → χ))} |= F (v(ϕ → χ))
Theorem4.12. ANDF F on L isaBNDFifandonly ifitsatisfiestheImplicativeProperty.
Proof. Supposethat F isaBNDFon L.From BL (χ ∨¬χ) → (ϕ → χ) ↔ (χ → (ϕ → χ)) ∧ (¬χ → (ϕ → χ)) ↔¬χ → (ϕ → χ) ↔ ϕ → (¬χ → χ),it followsthat F (v(ϕ → (¬χ → χ))) = F (v((χ ∨¬χ) → (ϕ → χ))) ={F (v((χ ∨¬χ) → (ϕ → χ))), F (v(χ ∨¬χ))} |= F (v(ϕ → χ)) whichprovesthat F satisfiestheImplicativeProperty, byTheorem4.10(i),(ii). Conversely,supposethat F satisfiestheImplicativeProperty.ByTheorem4.10(iii),replacing ϕ by ¬ϕ → ϕ and χ by ϕ ,wehave F (v((¬ϕ → ϕ ) → ϕ )) = F (v((¬ϕ → ϕ ) → (¬ϕ → ϕ ))) = F (1) andreplacing ϕ by ϕ →¬ϕ and χ by ¬ϕ ,weget F (v((ϕ →¬ϕ ) →¬ϕ )) = F (v((ϕ →¬ϕ ) → (¬¬ϕ →¬ϕ ))) = F (v((ϕ →¬ϕ ) → (ϕ →¬ϕ ))) = F (1).Then F (v(ϕ ∨¬ϕ )) ={F (v((ϕ →¬ϕ ) → ¬ϕ )), F (v((¬ϕ → ϕ ) → ϕ ))}= F (1).Thus F isa BNDFon L.
Theorem4.13. Let F beaNDFon L, ϕ,ψ,χ beformulasand v beanassignmenton L.Thenthefollowing areequivalent: (i) F isaBNDF, (ii) F (v(ϕ )) = F (v(¬ϕ → ϕ )), (iii) F (v((ϕ → ψ ) → ϕ )) |= F (v(ϕ )), (iv) F (v((ϕ → ψ ) → ϕ )) = F (v(ϕ )), (v) {F (v(χ → ((ϕ → ψ ) → ϕ ))),F (v(χ))}|= F (v(ϕ )).
Proof. (i) ⇒ (ii)Since BL ϕ → (¬ϕ → ϕ ),thenby Corollary3.11wehave F (v(ϕ )) |= F (v(¬ϕ → ϕ )). TheotherdirectionfollowsfromTheorem4.12,replacing ϕ byaBL-provableformulasand ψ,χ by ϕ in Definition4.11.
(ii) ⇒ (iii)From BL ¬ϕ → (ϕ → ψ ),weget BL ((ϕ → ψ ) → ϕ ) → (¬ϕ → ϕ ).Thus, F (v((ϕ → ψ ) → ϕ )) |= F (v(¬ϕ → ϕ )) = F (v(ϕ )).
(iii) ⇒ (iv)Itisenoughtoprovethat F (v(ϕ )) |= F (v((ϕ → ψ ) → ϕ ))andthisfollowsfrom BL ϕ → ((ϕ → ψ ) → ϕ ).
(iv) ⇒ (v)Since F isaNDF,then {F (v(χ → ((ϕ → ψ ) → ϕ ))), F (v(χ))}|= F (v((ϕ → ψ ) → ϕ )) = F (v(ϕ )).
(v) ⇒ (i)Let F beaNDF.Inordertoverifythat F isaBNDF,byTheorems4.10and4.12,itisenough toprovethat F (v(ϕ → (¬χ → χ))) |= F (v(ϕ → χ)), forallformulas ϕ,χ andallassignments v.Since, BL χ → (ϕ → χ)wehave BL (¬(ϕ → χ)) →¬χ andthen BL (¬χ → (ϕ → χ)) → (¬(ϕ → χ) → (ϕ → χ)).From(v),replacing ϕ by ϕ → χ, χ bya BL-provableformula θ and ψ byacontradiction(Inthis casetheformula ϕ → ψ isequivalentto ¬(ϕ → χ)),we get F (v(ϕ → (¬χ → χ))) = F (v(¬χ → (ϕ → χ))) |= F (v(¬(ϕ → χ) → (ϕ → χ))) ={F (v(θ → (¬(ϕ → χ) → (ϕ → χ)))), F (v(θ ))}|= F (v(ϕ → χ)). Therefore, F isaBNDFon L
5.Implicativeneutrosophicdeductivefilters
Inthissectionwedefineandstudythenotionof implicativedeductivefilteron BL-algebras.Also,we investigatesomerelationsbetweenBNDFsandINDFs. Definition5.1. Aneutrosophicsubset F of L iscalled an implicativeneutrosophicdeductivefilter (briefly, INDF)ifforallformulas ϕ,ψ,χ andallasignments v {F (v(ϕ → (ψ → χ))), F (v(ϕ → ψ ))}|= F (v(ϕ → χ)).
Asanimmediateresultwehave:
Theorem5.2. EveryINDFisaNDF. Proof. Let F beanINDFon L.Thenforeach BL-provableformula θ ,wehave {F (v(θ → (ψ → χ))), F (v(θ → ψ ))}|= F (v(θ → χ)),so {F (v(θ ) → v(ψ → χ))), F (v(θ ) → v(ψ ))}|= F (v(θ ) → v(χ)), then {F (1 → v(ψ → χ))), F (1 → v(ψ ))}|= F (1 → v(χ)),whichimpliesthat {F (v(ψ → χ))), F (v(ψ ))}|= F (v(χ))
Thus F isaNDFon L.
Proposition5.3. Aneutrosophicsubset F of L isa INDFifandonlyifforall x,y,z ∈ L
R.A.Borzooeietal./NeutrosophicdeductivefiltersonBL-algebras
F (x → z) ≥ F (x → (y → z)) ∧ F (x → y )
Proof. ItcanbeeasilyverifiedbyDefinition5.1. Theorem5.4. Let F beaNDFon L,thenthefollowing statementsareequivalent:
(i) F isanINDF, (ii) F (v(ϕ → (ϕ → ψ ))) |= F (v(ϕ → ψ )), (iii) F (v(ϕ → ψ )) = F (v(ϕ → (ϕ → ψ ))), (iv) F (v(ϕ → (ψ → χ))) |= F (v((ϕ → ψ ) → (ϕ → χ))), (v) F (v(ϕ → (ψ → χ))) = F (v((ϕ → ψ ) → (ϕ → χ))), (vi) F (v((ϕ ψ ) → χ)) = F (v((ϕ ∧ ψ ) → χ)).
Proof. (i) ⇒ (ii)Itisenoughtoput ψ = ϕ and χ = ψ , inthedefinition.
(ii) ⇒ (iii)Itfollowsfrom BL (ϕ → ψ ) → (ϕ → (ϕ → ψ )).
(iii) ⇒ (i)UsingCorollary3.11,wehave {F (v(ϕ → (ψ → χ))), F (v(ϕ → ψ ))}|={F (v(ϕ → (ϕ → χ)))}= F (v(ϕ → χ))
(i) ⇒ (iv)Supposethat F isanINDFon L Then, {F (v(ϕ → (ψ → χ))), F (v(ϕ → ((ψ → χ) → ((ϕ → ψ ) → χ))))}|={F (v(ϕ → ((ϕ → ψ ) → χ)))}={F (v((ϕ → ψ ) → (ϕ → χ)))}.Since BL (ψ → χ) → ((ϕ → ψ ) → (ϕ → χ)),then {F (v(ϕ → (ψ → χ))), F (1)}|={F (v((ϕ → ψ ) → (ϕ → χ)))}.Therefore, F (v(ϕ → (ψ → χ))) |= {F (v((ϕ → ψ ) → (ϕ → χ)))}. (iv) ⇒ (v)Itiseasy. (v) ⇒ (vi)Wehave F (v(ϕ &ψ → χ)) = F (v(ϕ → (ψ → χ))) = F (v((ϕ → ψ ) → (ϕ → χ))) = F (v(((ϕ → ψ )&ϕ ) → χ)) = F (v((ϕ &(ϕ → ψ )) → χ)) = F (v(ϕ ∧ ψ → χ)). (vi) ⇒ (i)ByCorollary3.11,wehave F (v(ϕ → χ)) = F (v((ϕ ∧ ϕ ) → χ)) = F (v((ϕ &ϕ ) → χ)) = F (v(ϕ → (ϕ → χ))),alsobyCorollary3.11 {F (v(ϕ → ψ )), F (v(ϕ → (ψ → χ)))}={F (v(ϕ → ψ )), F (v(ψ → (ϕ → χ)))}|= F (v(ϕ → (ϕ → χ))), therefore {F (v(ϕ → ψ )), F (v(ϕ → (ψ → χ)))}|= F (v(ϕ → (ϕ → χ))).Hence, F isanINDFon L. Theorem5.5. ANDF F on L isanINDFifandonlyif foreach t ∈ [0, 1], ∅ / = TF t , ∅ / = IF t , ∅ / = FF t andall ofthemareimplicativefilterson L. Proof. Supposethat F isanINDF, ∅ / = TF t , ∅ / = IF t and ∅ / = FF t ,forsome t ∈ [0, 1].Let x0 ∈ TF t , x1 ∈ IF t and x2 ∈ FF t .Then, TF (x0 ) ≥ t,IF (x1 ) ≤ t and FF (x1 ) ≤ t .
Since F isanINDF, TF (1) ≥ TF (x0 ) ≥ t , IF (1) ≤ IF (x1 ) ≤ t and FF (1) ≤ FF (x2 ) ≤ t ,i.e.1 ∈ TF t ,1 ∈ IF t and1 ∈ FF t .Now,supposethat x → y,x → (y → z) ∈ TF t , x → y,x → (y → z) ∈ IF t and x → y,x → (y → z) ∈ FF t ,forsome x,y,z ∈ L.Then, F (x → z) ≥ F (x → y ) ∧ F (x → (y → z)) ≥ t
Thus x → z ∈ TF t , x → z ∈ IF t and x → z ∈ FF t . Thisimpliesthat TF t , IF t and FF t areimplicativefilters on L.
Conversely,supposethatforeach t ∈ [0, 1], ∅ / = TF t , ∅ / = IF t and ∅ / = FF t areimplicative filterson L.Obviously, ∅ / = TTF (x) , ∅ / = IIF (x) and ∅ / = FFF (x) ,forany x ∈ L.Then, TTF (x) , IIF (x) and FFF (x) areimplicativefilterson L andso1 ∈ TTF (x) , 1 ∈ IIF (x) and1 ∈ FFF (x) ,i.e. F (1) ≥ F (x).Let t0 = TF (x → y ) ∧ TF (x → (y → z)), t1 = IF (x → y ) ∨ IF (x → (y → z))and t2 = FF (x → y ) ∨ FF (x → (y → z)),forsome x,y,z ∈ L.Then, x → y,x → (y → z) ∈ TF t0 , x → y,x → (y → z) ∈ IF t1 and x → y,x → (y → z) ∈ FF t2 andso x → z ∈ TF t0 , x → z ∈ IF t1 and x → z ∈ FF t2 .Hence, TF (x → z) ≥ t0 = TF (x → (y → z)) ∧ TF (x → y ), IF (x → z) ≤ t1 = IF (x → (y → z)) ∨ IF (x → y ), FF (x → z) ≤ t2 = FF (x → (y → z)) ∨ FF (x → y ).
Thus, F isanINDFon L Theorem5.6. EveryBNDFisanINDF.
Proof. Let F beaBNDFon L.From BL ((ϕ ∨¬ϕ ) → (ϕ → χ)) ↔ [((ϕ → (ϕ → χ)) ∧ (¬ϕ → (ϕ → χ)))] ↔ (ϕ → (ϕ → χ))itfollowsthat {F (v(ϕ → (ϕ → χ)))}={F (v((ϕ ∨¬ϕ ) → (ϕ → χ)))}={F (v((ϕ ∨¬ϕ ) → (ϕ → χ))), F (v(ϕ ∨¬ϕ ))} |= F (v(ϕ → χ)).Therefore, F isanINDFon L,by Theorem5.4(ii).
Example5.7. Let L ={0,a,b, 1} beachainwith cayleytablesasfollows: 0ab1 0 0000 a 0aaa b 0abb 1 0ab1
→ 0ab1 0 1111 a 0111 b 0a11 1 0ab1
Define ∧ and ∨ on L asminandmax,respectively. Then(L, ∧, ∨, , →, 0, 1)isa BL-algebra.Theneutrosophicsubset F of L definedby F (0) = F (a) = F (b) = (t1 ,t2 ,t2 ), F (1) = (t2 ,t1 ,t1 ),where0 ≤ t1 < t2 ≤ 1aretwofixedrealnumbersin[0,1],isanINDF, butitisnotaBNDF.
Theorem5.8. Let F beanINDFon L. F isaBNDF ifandonlyifforallformulas ϕ,ψ andallassignments v: F (v((ϕ → ψ ) → ψ )) = F (v((ψ → ϕ ) → ϕ ))(5.1)
Proof. Supposethat F isaBNDF.From BL ϕ → ((ψ → ϕ ) → ϕ ),itfollowsthat BL (¬((ψ → ϕ ) → ϕ )) →¬ϕ andsincebyProposition2, BL ¬ϕ → (ϕ → ψ )),thenwehave BL (¬((ψ → ϕ ) → ϕ )) → (ϕ → ψ )),so BL ((ϕ → ψ ) → ψ ) → (¬((ψ → ϕ ) → ϕ ) → ψ ).Inaddition, since BL ψ → (ϕ ∨ ψ )and BL (ϕ ∨ ψ ) → ((ψ → ϕ ) → ϕ )wehave BL ψ → ((ψ → ϕ ) → ϕ ).Thus BL (¬((ψ → ϕ ) → ϕ ) → ψ ) → (¬((ψ → ϕ ) → ϕ ) → ((ψ → ϕ ) → ϕ )).Therefore, BL ((ϕ → ψ ) → ψ ) → (¬((ψ → ϕ ) → ϕ ) → ((ψ → ϕ ) → ϕ ))and hence F (v((ϕ → ψ ) → ψ )) |= F (v(¬((ψ → ϕ ) → ϕ ) → ((ψ → ϕ ) → ϕ ))),byCorollary3.11. SincebyTheorem4.13(ii), F (v((ψ → ϕ ) → ϕ )) = F (v(¬((ψ → ϕ ) → ϕ ) → ((ψ → ϕ ) → ϕ ))),then F (v((ϕ → ψ ) → ψ )) |= F (v((ψ → ϕ ) → ϕ )) Similarly,wecanprovethat F (v((ψ → ϕ ) → ϕ )) |= F (v((ϕ → ψ ) → ψ )).Then, F (v((ϕ → ψ ) → ψ )) = F (v((ψ → ϕ ) → ϕ )).
Conversely,Let F beanINDFon L and(5.1)holds. Then,inordertoprovethat F isaBNDF,itisenough toshowthat F (v((ϕ →¬ϕ ) →¬ϕ )) = F (1),by Proposition2.SincebyProposition4.4, BL ((ϕ → ¬ϕ ) →¬ϕ ) ↔ (ϕ →¬¬ϕ )and BL ϕ →¬¬ϕ , thenitfollowsthat F (v((ϕ →¬ϕ ) →¬ϕ )) = F (v(ϕ →¬¬ϕ )) = F (1),whichcompletesthe proof.
Theorem5.9. Let F beanINDFon L F isaBNDFif andonlyif F (v(¬¬ϕ )) = F (v(ϕ )),foreachformula ϕ andeachassignment v
Proof. Assumethat F (v(¬¬ϕ )) = F (v(ϕ )).Then, F (v(¬ϕ →¬¬ϕ )) = F (v(¬¬ϕ )),byTheorem5.4 (iii).Alsowehave F (v(¬ϕ → ϕ )) |= F (v(¬ϕ → ¬¬ϕ )).Hence F (v(¬ϕ → ϕ )) |= F (v(ϕ )).Obviously, F (v(ϕ )) |= F (v(¬ϕ → ϕ )).Thus F isaBNDFon L, ,byTheorem4.13.Theconverseisobtainedbyusing Theorem5.8.
Corollary5.10. Let L beaMV-Algebraand F be aNDFon L.Then,thefollowingstatementsare equivalent:
(i) F isaBNDFon L, (ii) F isanINDF, (iii) F (v(ϕ → χ)) = F (v(ϕ → (ϕ → χ))), (iv) F (v(ϕ → χ)) = F (v(ϕ → (¬χ → χ))), (v) F (v((ϕ → ψ ) → ϕ )) = F (v(ϕ )), (vi) F (v(ϕ → (ψ → χ))) = F (v((ϕ → ψ ) → (ϕ → χ))), (vii) F (v((ϕ ψ ) → χ)) = F (v((ϕ ∧ ψ ) → χ)).
6.Quotientstructures
Inthissectionwedefinethequotientstructurefor neutrosophicdeductivefiltersandstudysomeofits properties.
Let F beaNDFon L and x ∈ L.Theneutrosophicset F x : L → [0, 1]3 whichisdefinedby F x (y ) = (TF x (y ),IF x (y ),FF x (y )),forany y ∈ L, where TF x (y ) = TF (x → y ) ∧ TF (y → x), IF x (y ) = IF (x → y ) ∨ IF (y → x)and FF x (y ) = FF (x → y ) ∨ FF (y → x),iscalledthe neutrosophiccoset (briefly, NC)of F .DenotethesetofallNCsof F by L/F .
Now,wehavethefollowinglemma.
Lemma6.1. Let F beaNDFon L.Then, F x = F y if andonlyif F (x → y ) = F (y → x) = F (1)
Proof. Assumethat F x = F y ,forsome x,y ∈ L Then(TF x (x),IF x (x),FF x (x)) = F x (x) = F y (x) = (TF y (x),IF y (x),FF y (x)),so, TF x (x) = TF y (x), IF x (x) = IF y (x)and FF x (x) = FF y (x).Hence, TF (1) = TF (x → x) = TF (y → x) ∧ TF (x → y ), IF (1) = IF (x → x) = IF (y → x) ∨ IF (x → y )and FF (1) = FF (x → x) = FF (y → x) ∨ FF (x → y )which impliesthat TF (y → x) = TF (x → y ) = TF (1), IF (y → x) = IF (x → y ) = IF (1)and FF (y → x) = FF (x → y ) = FF (1).Then, F (x → y ) = (TF (x → y ),IF (x → y ),FF (x → y )) = (TF (1),IF (1),FF (1)) = F (1)andsimilarly, F (y → x) = F (1). Conversely,assumethat F (x → y ) = F (y → x) = F (1).Then, TF (x → y ) = TF (y → x) = TF (1), IF (x → y ) = IF (y → x) = IF (1)and FF (x → y ) = FF (y → x) = FF (1).Thus,byCorollary3.12, foreach x,y,z ∈ L,weobtainsthat F (z → x) ≥ F (z → y ) ∧ F (y → x) = F (z → y )and F (x → z) ≥ F (x → y ) ∧ F (y → z) = F (y → z),hence F x (z)
R.A.Borzooeietal./NeutrosophicdeductivefiltersonBL-algebras
= F (z → x) ∧ F (x → z) ≥ F (z → y ) ∧ F (y → z) = F y (z) Similarly, F y (z) ≥ F x (z),therefore, F x = F y .
Supposethat F beaNDFon L.Let FF (1) ={x ∈ L : F (x) = F (1)},thenitiseasytoverifythat FF (1) = TT (1) ∩ II (1) ∩ FF (1) .Hence FF (1) isafilteron L. Corollary6.2. Let F beaNDFon L.Then, F x = F y ifandonlyif x ∼FF (1) y ,where x ∼FF (1) y ⇔ x → y ∈ FF (1) ,y → x ∈ FF (1) Let F beaNDFon L.Forany F x , F y ∈ L/F , define F x ∧ F y = F x∧y , F x ∨ F y = F x∨y , F x F y = F x y ,and F x → F y = F x→y
Now,wegetthefollowinglemma. Lemma6.3. Let F beaNDFon L, F x = F z and F y = F w ,forsome x,y,z,w ∈ L.Then, F x ∨ F y = F z ∨ F w , F x ∧ F y = F z ∧ F w , F x F y = F z F w ,and F x → F y = F z → F w
Proof. Assumethat F x = F z and F y = F w ,for some x,y,z,w ∈ L.ByCorollary6.2, x ∼FF (1) z and y ∼FF (1) w.Since,byTheorem2, ∼FF (1) isacongruenceon L,then x ∧ y ∼FF (1) z ∧ w,x ∨ y ∼FF (1) z ∨ w,x y ∼FF (1) z w and x → y ∼FF (1) z → w. Thisimpliesthat F x ∧ F y = F x∧y = F z∧w = F z ∧ F w andsimilarly, F x ∨ F y = F z ∨ F w , F x F y = F z F w and F x → F y = F z → F w . Wenotethatthelatticeorder ≤ on L/F isdefined by F x ≤ F y ifandonlyif F x ∨ F y = F y .
Lemma6.4. Let F beaNDFon L.Then, F x ≤ F y if andonlyif F (x → y ) = F (1).
Proof. Let x,y ∈ L.Then, F x ≤ F y ⇔ F x∨y = F x ∨ F y = F y ⇔ TF x∨y = TF y ,IF x∨y = IF y ,FF x∨y = FF y ⇔ TF (1) = TF (y → (x ∨ y )) = TF (x ∨ y → y ), IF (1) = IF (y → (x ∨ y )) = IF (x ∨ y → y ), FF (1) = FF (y → (x ∨ y )) = FF (x ∨ y → y ) ⇔ TF (x → y ) = TF (1),IF (x → y ) = IF (1), FF (x → y ) = FF (1) ⇔ F (x → y ) = (TF (x → y ),IF (x → y ),FF (x → y )) = (TF (1),IF (1),FF (1)) = F (1).
Theorem6.5. Let F beaNDFon L.Then (L/F , ∧, ∨, , →, F 1 , F 0 ) isa BL-algebra.
Proof. ByLemma6.3,theoperations ∧, ∨, and → on L/F arewell-defined.Weonlyneedtoprove L/F satisfiestheaxiomsof BL-algebras.Theaxioms (BL1),(BL2),(BL4)and(BL5)canbeeasilyproved. Let F x , F y , F z ∈ L/F ,thenbyCorollary6.4
7.Isomorphismtheorems
Inthissectionweprovethreeisomorphismtheorems concerningquotientsofneutrosophicdeductivefilters.
Wenotethat,since FF (1) isafilteron L,thenby Theorem2.10, L/FF (1) isa BL-algebra. Theorem7.1. Let F beaNDFon L.Then L/F L/FF (1)
Proof. Defineamap : L/F −→ L/FF (1) by (F x ) = [x]FF (1) Weprovethat isanisomorphism.Supposethat F x , F y ∈ L/F .Then F x = F y ifandonlyif x ∼FF (1) y ifandonlyif [x]FF (1) = [y ]FF (1) ,whichimpliesthat isanoneto-onefunction.Obviously, issurjective.Now, (F x F y ) = (F x y ) = [x y ]FF (1) = [x]FF (1) • [y ]FF (1) = (F x ) • (F y ).Similarly,itcanbe obtainedthat (F x → F y ) = (F x ) (F y ). Thus, isanisomorphismandhence L/F L/FF (1) . Corollary7.2. Let f : L −→ L beahomomorphism ofBL-algebrasand F beaNDFon L which kerf = FF (1) .Then, L/F f (L) Definition7.3. Let L1 and L2 be BL-algebrasand f be amapfrom L1 into L2 .Alsolet F beaneutrosophic subsetof L1 .Theneutrosophicsubset f (F )of L2 is definedby f (F )(l2 ) = (Tf (F ) (l2 ),If (F ) (l2 ),Ff (F ) (l2 )),f 1 (l2 ) / =∅ (0, 1, 1), otherwise
forany l2 ∈ L2 ,where
Tf (F ) (l2 ) =∨{TF (l1 ): l1 ∈ L1 ,f (l1 ) = l2 },
If (F ) (l2 ) =∧{IF (l1 ): l1 ∈ L1 ,f (l1 ) = l2 },
Ff (F ) (l2 ) =∧{FF (l1 ): l1 ∈ L1 ,f (l1 ) = l2 }
Let L1 and L2 be BL-algebrasand F1 and F2 beneutrosophicsetsof L1 and L2 ,respectively.A homomorphism f of L1 onto L2 iscalled weakhomomorphism of F1 into F2 ,if f (F1 ) ≤ F2 .Inthiscase, wesaythat F1 isweaklyhomomorphicto F2 andwe write F1 ∼f F2 orsimply F1 ∼ F2 .If f isbijective, wesaythat F1 isweaklyisomorphicto F2 andwewrite F1 f F2 orsimply F1 F2 .Ahomomorphism f of L1 onto L2 iscalleda homomorphism of F1 into F2 ,if f (F1 ) = F2 .Inthiscase,wesaythat F1 ishomomorphicto F2 andwewrite F1 ≈f F2 orsimply F1 ≈ F2 If f isbijective,wesaythat F1 isisomorphicto F2 and wewrite F1 ∼ =f F2 orsimply F1 ∼ = F2 (See[8]).
Foraneutrosophicsubset F of L,define F ∗ ={x ∈ L,TF (x) > 0,IF (x) < 1,FF (x) < 1}
Lemma7.4. Let F beaNDFon L, TF (1) > 0, IF (1) < 1 and FF (1) < 1.Then F ∗ isafilteron L
Proof. Theproofiseasy.
Recallthat F1 ≤ F2 meansthat TF1 (x) ≤ TF2 (x), IF1 (x) ≥ IF2 (x)and FF1 (x) ≥ FF2 (x),forany x ∈ L.Now,if F1 ≤ F2 and x ∈ F1 ∗ ,then 0 <TF1 (x) ≤ TF2 (x), 1 >IF1 (x) ≥ IF2 (x)and 1 >FF1 (x) ≥ FF2 (x)andso x ∈ F2 ∗ ,whichimplies that F ∗ 1 ⊆ F ∗ 2 .Hence F ∗ 1 isafilteron F ∗ 2 and F ∗ 2 /F ∗ 1 isa BL-algebra,byLemma2.11.
Theorem7.5. Let F1 and F2 betwoNDFson L, F1 ≤ F2 and F2 hasSup-Infproperty.Define ξ : F2 ∗ /F1 ∗ −→ [0, 1] by ξ ([x]F ∗ 1 ) = (Tξ ([x]F ∗ 1 ),Iξ ([x]F ∗ 1 ),Fξ ([x]F ∗ 1 )) where Tξ ([x]F ∗ 1 ) =∨{TF2 (y )| y ∈ [x]F ∗ 1 }, Iξ ([x]F ∗ 1 ) = ∧{IF2 (y )| y ∈ [x]F ∗ 1 } and Fξ ([x]F ∗ 1 ) =∧{FF2 (y )| y ∈ [x]F ∗ 1 },forall[x]F ∗ 1 ∈ F2 ∗ /F1 ∗ .Then ξ isaneutrosophicset.
Proof. Since, F2 hasSup-Infproperty,thereexist y0 ,y1 ,y2 ∈ [x]F ∗ 1 suchthat Tξ ([x]F ∗ 1 ) = TF2 (y0 ), Iξ ([x]F ∗ 1 ) = IF2 (y1 )and Fξ ([x]F ∗ 1 ) = FF2 (y2 ).Obviously, ξ ([x]F ∗ 1 ) = (TF2 (y0 ),IF2 (y1 ),FF2 (y2 ))isaneutrosophicset,whichcompletestheproof.Wecallthe
neutrosophicset ξ definedinTheorem7.5,the quotient neutrosophicset on F2 relativeto F1 anddenoteitby F2 /F1 .
Theorem7.6. Let F1 and F2 betwoNDFson L, F1 ≤ F2 and F2 hasSup-Infproperty.Then
F2 |F ∗ 2 ≈ F2 /F1
Proof. Let f : F ∗ 2 −→ F ∗ 2 /F ∗ 1 bethenaturalepimorphismand[x]F ∗ 1 ∈ F ∗ 2 /F ∗ 1 .Then f (F2 |F ∗ 2 )([x]F ∗ 1 ) = (∨{TF2 (z)|z ∈ F ∗ 2 ,f (z) = [x]F ∗ 1 }, ∧{IF2 (z)|z ∈ F ∗ 2 , f (z) = [x]F ∗ 1 }, ∧{FF2 (z)|z ∈ F ∗ 2 ,f (z) = [x]F ∗ 1 }) = (∨ {TF2 (y )|y ∈ [x]F ∗ 1 }, ∧{IF2 (y )|y ∈ [x]F ∗ 1 }, ∧{FF2 (y )|y ∈ [x]F ∗ 1 }) = (F2 /F1 )([x]F ∗ 1 ) Therefore, F2 |F ∗ 2 ≈ F2 /F1 .
Theorem7.7. Let F1 and F2 betwoNDFson BLalgebras L1 and L2 ,respectively, F1 ≈ F2 and F1 has Sup-Infproperty.ThenthereexistsaNDF F3 suchthat F3 ≤ F1 and F1 /F3 ∼ = F2 |F ∗ 2 .
Proof. Since F1 ≈ F2 ,thereisanhomomorphism f from L1 onto L2 suchthat f (F1 ) = F2 .Definethe neutrosophicset F3 asfollows: F3 (x) = (TF1 (x),IF1 (x),FF1 (x)),x ∈ ker (f ) (0, 1, 1), otherwise
forany x ∈ L1 .Itiseasytoshowthat F3 isaNDFon L1 .Since F1 ≈ F2 ,then f (F ∗ 1 ) = F ∗ 2 .Let g = f |F ∗ 1 , then g isahomomorphismfrom F ∗ 1 onto F ∗ 2 and ker (g) = F ∗ 3 .Thus,bythefirstisomorphismtheorem, thereexistsanisomorphism h : F1 ∗ /F3 ∗ −→ F ∗ 2 such that h([x]F ∗ 3 ) = g(x) = f (x),forany x ∈ F1 ∗ .Now, h(F1 /F3 )(z) = (∨{TF1 /F3 ([x]F ∗ 3 ) | x ∈ F1 ∗ , h([x]F ∗ 3 ) = z}, ∧{IF1 /F3 ([x]F ∗ 3 ) | x ∈ F1 ∗ ,h([x]F ∗ 3 ) = z}, ∧{FF1 /F3 ([x]F ∗ 3 ) | x ∈ F1 ∗ ,h([x]F ∗ 3 ) = z}) = (∨{∨ {TF1 (y ): y ∈ [x]F ∗ 3 }| x ∈ F1 ∗ ,g(x) = z}, ∧{∧{IF1 (y ): y ∈ [x]F ∗ 3 }| x ∈ F1 ∗ ,g(x) = z}, ∧{∧{FF1 (y ): y ∈ [x]F ∗ 3 }| x ∈ F1 ∗ ,g(x) = z}) = (∨{TF1 (y )|y ∈ F1 ∗ , g(y ) = z}, ∧{IF1 (y )|y ∈ F1 ∗ ,g(y ) = z}, ∧{FF1 (y )|y ∈ F1 ∗ ,g(y ) = z}) = (∨{TF1 (y )|y ∈ L1 ,f (y ) = z}, ∧ {IF1 (y )|y ∈ L1 ,f (y ) = z}, ∧{FF1 (y )|y ∈ L1 ,f (y ) = z}) = F2 (z)forany z ∈ F2 ∗ .Therefore F1 /F3 ∼ = F2 |F ∗ 2
Lemma7.8. Let F1 and F2 betwoNDFson L such that F1 ≤ F2 .Then F ∗ 2 /F ∗ 1 = (F2 /F1 )∗ .
Proof. Theproofiseasy.
R.A.Borzooeietal./NeutrosophicdeductivefiltersonBL-algebras
Lemma7.9. Let F1 , F2 and F3 beNDFson L, F1 ≤ F2 ≤ F3 and F2 , F3 haveSup-Infproperty.Then (F2 /F1 ) and (F3 /F1 ) areneurosophicsubsetsof L suchthat (F2 /F1 ) ≤ (F3 /F1 ).
Proof. UseTheorem7.5. Theorem7.10. Let F1 , F2 and F3 beNDFson L, F1 ≤ F2 ≤ F3 and F2 , F3 haveSup-Infproperty,suchthat F3 /F1 , F2 /F1 areNDFs.Then
(F3 /F1 )/(F2 /F1 ) ≈ F3 /F2 .
Proof. ItcanbeprovedbyusingTheorem2and Lemmas7.8,7.9.
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