Annalsof Fuzzy Mathematicsand Informatics Volume14,No.1,(July2017),pp.87–97 ISSN:2093–9310(printversion) ISSN:2287–6235(electronicversion) http://www.afmi.or.kr
@FMI
c KyungMoonSaCo. http://www.kyungmoon.com
Neutrosophicsubalgebrasof BCK/BCI-algebras basedonneutrosophicpoints
A.BorumandSaeid,YoungBaeJun
Received27March2017; Revised28April2017; Accepted20May2017
Abstract. Propertiesonneutrosophic ∈∨ q-subsetsandneutrosophic q-subsetsareinvestigated.Relationsbetweenan(∈, ∈∨ q)-neutrosophic subalgebraanda(q, ∈∨ q)-neutrosophicsubalgebraareconsidered.Characterizationofan(∈, ∈∨ q)-neutrosophicsubalgebrabyusingneutrosophic ∈-subsetsarediscussed.Conditionsforan(∈, ∈∨ q)-neutrosophicsubalgebratobea(q, ∈∨ q)-neutrosophicsubalgebraareprovided.
2010AMSClassification: 06F35,03B60,03B52.
Keywords: Neutrosophicset,neutrosophic ∈-subset,neutrosophic q-subset,neutrosophic ∈∨ q-subset,neutrosophic TΦ-point,neutrosophic IΦ-point,neutrosophic FΦ-point.
CorrespondingAuthor: Y.B.Jun(skywine@gmail.com)
1. Introduction
Theconceptofneutrosophicset(NS)developedbySmarandache[17, 18, 19] isamoregeneralplatformwhichextendstheconceptsoftheclassicsetandfuzzy set(see[20],[21]),intuitionisticfuzzyset(see[1])andintervalvaluedintuitionistic fuzzyset(see[2]).Neutrosophicsettheoryisappliedtovariouspart(see[4],[5], [8],[9],[10],[11],[12],[13],[15],[16]).Forfurtherparticulars,wereferreadersto thesitehttp://fs.gallup.unm.edu/neutrosophy.htm.Barbhuiya[3]introducedand studiedtheconceptof(∈, ∈∨q)-intuitionisticfuzzyidealsof BCK/BCI-algebras. Jun[7]introducedthenotionofneutrosophicsubalgebrasin BCK/BCI-algebras withseveraltypes.Heprovidedcharacterizationsofan(∈, ∈)-neutrosophicsubalgebraandan(∈, ∈∨ q)-neutrosophicsubalgebra.Givenspecialsets,socalled neutrosophic ∈-subsets,neutrosophic q-subsetsandneutrosophic ∈∨ q-subsets,he consideredconditionsfortheneutrosophic ∈-subsets,neutrosophic q-subsetsand neutrosophic ∈∨ q-subsetstobesubalgebras.Hediscussedconditionsforaneutrosophicsettobea(q, ∈∨ q)-neutrosophicsubalgebra.
A.BorumandSaeidetal./Ann.FuzzyMath.Inform. 14 (2017),No.1,87–97
Inthispaper,wegiverelationsbetweenan(∈, ∈∨ q)-neutrosophicsubalgebraand a(q, ∈∨ q)-neutrosophicsubalgebra.Wediscusscharacterizationofan(∈, ∈∨ q)neutrosophicsubalgebrabyusingneutrosophic ∈-subsets.Weprovideconditions foran(∈, ∈∨ q)-neutrosophicsubalgebratobea(q, ∈∨ q)-neutrosophicsubalgebra. Weinvestigatepropertiesonneutrosophic q-subsetsandneutrosophic ∈∨ q-subsets.
2. Preliminaries
Bya BCI-algebra wemeananalgebra(X, ∗, 0)oftype(2, 0)satisfyingtheaxioms:
(a1) ((x ∗ y) ∗ (x ∗ z)) ∗ (z ∗ y)=0, (a2) (x ∗ (x ∗ y)) ∗ y =0, (a3) x ∗ x =0, (a4) x ∗ y = y ∗ x =0 ⇒ x = y, forall x,y,z ∈ X. Ifa BCI-algebra X satisfiestheaxiom (a5) 0 ∗ x =0forall x ∈ X, thenwesaythat X isa BCK-algebra.Anonemptysubset S ofa BCK/BCI-algebra X iscalleda subalgebra of X if x ∗ y ∈ S forall x,y ∈ S. Wereferthereadertothebooks[6]and[14]forfurtherinformationregarding BCK/BCI-algebras. Foranyfamily {ai | i ∈ Λ} ofrealnumbers,wedefine
{ai | i ∈ Λ} :=
max{ai | i ∈ Λ} ifΛisfinite, sup{ai | i ∈ Λ} otherwise. {ai | i ∈ Λ} := min{ai | i ∈ Λ} ifΛisfinite, inf{ai | i ∈ Λ} otherwise
IfΛ= {1, 2},wewillalsouse a1 ∨ a2 and a1 ∧ a2 insteadof {ai | i ∈ Λ} and {ai | i ∈ Λ},respectively.
Let X beanon-emptyset.Aneutrosophicset(NS)in X (see[18])isastructure oftheform:
A := { x; AT (x),AI (x),AF (x) | x ∈ X}
where AT : X → [0, 1]isatruthmembershipfunction, AI : X → [0, 1]isan indeterminatemembershipfunction,and AF : X → [0, 1]isafalsemembership function.Forthesakeofsimplicity,weshallusethesymbol A =(AT ,AI ,AF )for theneutrosophicset
A := { x; AT (x),AI (x),AF (x) | x ∈ X}. 88
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3. Neutrosophicsubalgebrasofseveraltypes
Givenaneutrosophicset A =(AT ,AI ,AF )inaset X, α,β ∈ (0, 1]and γ ∈ [0, 1), weconsiderthefollowingsets:
T∈(A; α):= {x ∈ X | AT (x) ≥ α},
I∈(A; β):= {x ∈ X | AI (x) ≥ β},
F∈(A; γ):= {x ∈ X | AF (x) ≤ γ},
Tq (A; α):= {x ∈ X | AT (x)+ α> 1},
Iq (A; β):= {x ∈ X | AI (x)+ β> 1},
Fq (A; γ):= {x ∈ X | AF (x)+ γ< 1},
T∈∨ q (A; α):= {x ∈ X | AT (x) ≥ α or AT (x)+ α> 1},
I∈∨ q (A; β):= {x ∈ X | AI (x) ≥ β or AI (x)+ β> 1},
F∈∨ q (A; γ):= {x ∈ X | AF (x) ≤ γ or AF (x)+ γ< 1}
Wesay T∈(A; α), I∈(A; β)and F∈(A; γ)are neutrosophic ∈-subsets; Tq (A; α), Iq (A; β) and Fq (A; γ)are neutrosophic q-subsets;and T∈∨ q (A; α), I∈∨ q (A; β)and F∈∨ q (A; γ) are neutrosophic ∈∨ q-subsets.ForΦ ∈{∈,q, ∈∨ q},theelementof TΦ(A; α)(resp., IΦ(A; β)and FΦ(A; γ))iscalleda neutrosophic TΦ-point (resp., neutrosophic IΦpoint and neutrosophic FΦ-point)withvalue α (resp., β and γ)(see[7]).
Itisclearthat
T∈∨ q (A; α)= T∈(A; α) ∪ Tq (A; α), (3.1)
I∈∨ q (A; β)= I∈(A; β) ∪ Iq (A; β), (3.2)
F∈∨ q (A; γ)= F∈(A; γ) ∪ Fq (A; γ) (3.3)
Definition3.1 ([7]). GivenΦ, Ψ ∈{∈,q, ∈∨ q},aneutrosophicset A =(AT ,AI ,AF ) ina BCK/BCI-algebra X iscalleda(Φ, Ψ)-neutrosophicsubalgebra of X ifthefollowingassertionsarevalid.
x ∈ TΦ(A; αx),y ∈ TΦ(A; αy ) ⇒ x ∗ y ∈ TΨ(A; αx ∧ αy ),
x ∈ IΦ(A; βx),y ∈ IΦ(A; βy ) ⇒ x ∗ y ∈ IΨ(A; βx ∧ βy ), x ∈ FΦ(A; γx),y ∈ FΦ(A; γy ) ⇒ x ∗ y ∈ FΨ(A; γx ∨ γy ) (3.4)
forall x,y ∈ X, αx,αy ,βx,βy ∈ (0, 1]and γx,γy ∈ [0, 1).
Lemma3.2 ([7]). Aneutrosophicset A =(AT ,AI ,AF ) ina BCK/BCI -algebra X isan (∈, ∈∨ q)-neutrosophicsubalgebraof X ifandonlyifitsatisfies: (∀x,y ∈ X) AT (x ∗ y) ≥ {AT (x),AT (y), 0 5} AI (x ∗ y) ≥ {AI (x),AI (y), 0.5} AF (x ∗ y) ≤ {AF (x),AF (y), 0 5} (3.5)
Theorem3.3. Aneutrosophicset A =(AT ,AI ,AF ) ina BCK/BCI -algebra X is an (∈, ∈∨ q)-neutrosophicsubalgebraof X ifandonlyiftheneutrosophic ∈-subsets T∈(A; α), I∈(A; β) and F∈(A; γ) aresubalgebrasof X forall α,β ∈ (0, 0 5] and γ ∈ [0 5, 1) 89
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Proof. Assumethat A =(AT ,AI ,AF )isan(∈, ∈∨ q)-neutrosophicsubalgebraof X.Forany x,y ∈ X,let α ∈ (0, 0.5]besuchthat x,y ∈ T∈(A; α).Then AT (x) ≥ α and AT (y) ≥ α.Itfollowsfrom(3.5)that
AT (x ∗ y) ≥ {AT (x),AT (y), 0.5}≥ α ∧ 0.5= α andsothat x ∗ y ∈ T∈(A; α).Thus T∈(A; α)isasubalgebraof X forall α ∈ (0, 0.5]. Similarly, I∈(A; β)isasubalgebraof X forall β ∈ (0, 0.5].Now,let γ ∈ [0.5, 1)be suchthat x,y ∈ F∈(A; γ).Then AF (x) ≤ γ and AF (y) ≤ γ.Hence
AF (x ∗ y) ≤ {AF x),AF (y), 0.5}≤ γ ∨ 0.5= γ by(3.5),andso x ∗ y ∈ F∈(A; γ).Thus F∈(A; γ)isasubalgebraof X forall γ ∈ [0.5, 1). Conversely,let α,β ∈ (0, 0.5]and γ ∈ [0.5, 1)besuchthat T∈(A; α), I∈(A; β)and F∈(A; γ)aresubalgebrasof X.Ifthereexist a,b ∈ X suchthat
AI (a ∗ b) < {AI (a),AI (b), 0.5},
thenwecantake β ∈ (0, 1)suchthat
AI (a ∗ b) <β< {AI (a),AI (b), 0 5} .(3.6) Thus a,b ∈ I∈(A; β)and β< 0 5,andso a ∗ b ∈ I∈(A; β).But,theleftinequalityin (3.6)induces a ∗ b/ ∈ I∈(A; β),acontradiction.Hence
AI (x ∗ y) ≥ {AI (x),AI (y), 0 5} forall x,y ∈ X.Similarly,wecanshowthat
AT (x ∗ y) ≥ {AT (x),AT (y), 0.5} forall x,y ∈ X.Nowsupposethat
AF (a ∗ b) > {AF (a),AF (b), 0 5} forsome a,b ∈ X.Thenthereexists γ ∈ (0, 1)suchthat
AF (a ∗ b) >γ> {AF (a),AF (b), 0.5}.
Itfollowsthat γ ∈ (0 5, 1)and a,b ∈ F∈(A; γ).Since F∈(A; γ)isasubalgebraof X, wehave a ∗ b ∈ F∈(A; γ)andso AF (a ∗ b) ≤ γ.Thisisacontradiction,andthus
AF (x ∗ y) ≤ {AF (x),AF (y), 0 5} forall x,y ∈ X.UsingLemma 3.2, A =(AT ,AI ,AF )isan(∈, ∈∨ q)-neutrosophic subalgebraof X
UsingTheorem 3.3 and[7,Theorem3.8],wehavethefollowingcorollary.
Corollary3.4. Foraneutrosophicset A =(AT ,AI ,AF ) ina BCK/BCI -algebra X,ifthenonemptyneutrosophic ∈∨ q-subsets T∈∨ q (A; α), I∈∨ q (A; β) and F∈∨ q (A; γ) aresubalgebrasof X forall α,β ∈ (0, 1] and γ ∈ [0, 1),thentheneutrosophic ∈subsets T∈(A; α), I∈(A; β) and F∈(A; γ) aresubalgebrasof X forall α,β ∈ (0, 0 5] and γ ∈ [0 5, 1) 90
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Theorem3.5. Givenneutrosophicset A =(AT ,AI ,AF ) ina BCK/BCI -algebra X,thenonemptyneutrosophic ∈-subsets T∈(A; α), I∈(A; β) and F∈(A; γ) aresubalgebrasof X forall α,β ∈ (0.5, 1] and γ ∈ [0, 0.5) ifandonlyifthefollowing assertionisvalid.
(∀x,y ∈ X)
AT (x ∗ y) ∨ 0 5 ≥ AT (x) ∧ AT (y) AI (x ∗ y) ∨ 0 5 ≥ AI (x) ∧ AI (y) AF (x ∗ y) ∧ 0.5 ≤ AF (x) ∨ AF (y)
(3.7)
Proof. Assumethatthenonemptyneutrosophic ∈-subsets T∈(A; α), I∈(A; β)and F∈(A; γ)aresubalgebrasof X forall α,β ∈ (0.5, 1]and γ ∈ [0, 0.5).Suppose thatthereare a,b ∈ X suchthat AT (a ∗ b) ∨ 0.5 <AT (a) ∧ AT (b):= α.Then α ∈ (0.5, 1]and a,b ∈ T∈(A; α).Since T∈(A; α)isasubalgebraof X,itfollowsthat a ∗ b ∈ T∈(A; α),thatis, AT (a ∗ b) ≥ α whichisacontradiction.Thus
AT (x ∗ y) ∨ 0 5 ≥ AT (x) ∧ AT (y)
forall x,y ∈ X.Similarly,weknowthat AI (x ∗ y) ∨ 0 5 ≥ AI (x) ∧ AI (y)for all x,y ∈ X.Now,if AF (x ∗ y) ∧ 0 5 >AF (x) ∨ AF (y)forsome x,y ∈ X,then x,y ∈ F∈(A; γ)and γ ∈ [0, 0 5)where γ = AF (x) ∨ AF (y).But, x ∗ y/ ∈ F∈(A; γ) whichisacontradiction.Hence AF (x ∗ y) ∧ 0 5 ≤ AF (x) ∨ AF (y)forall x,y ∈ X
Conversely,let A =(AT ,AI ,AF )beaneutrosophicsetin X satisfyingthecondition(3.7).Let x,y,a,b ∈ X and α,β ∈ (0 5, 1]besuchthat x,y ∈ T∈(A; α)and a,b ∈ I∈(A; β).Then
AT (x ∗ y) ∨ 0 5 ≥ AT (x) ∧ AT (y) ≥ α> 0 5, AI (a ∗ b) ∨ 0.5 ≥ AI (a) ∧ AI (b) ≥ β> 0.5.
Itfollowsthat AT (x ∗ y) ≥ α and AI (a ∗ b) ≥ β,thatis, x ∗ y ∈ T∈(A; α)and a ∗ b ∈ I∈(A; β).Now,let x,y ∈ X and γ ∈ [0, 0 5)besuchthat x,y ∈ F∈(A; γ). Then AF (x ∗ y) ∧ 0 5 ≤ AF (x) ∨ AF (y) ≤ γ< 0 5andso AF (x ∗ y) ≤ γ,i.e., x ∗ y ∈ F∈(A; γ).Thiscompletestheproof.
Weconsiderrelationsbetweena(q, ∈∨ q)-neutrosophicsubalgebraandan(∈, ∈∨ q)-neutrosophicsubalgebra.
Theorem3.6. Ina BCK/BCI -algebra,every (q, ∈∨ q)-neutrosophicsubalgebrais an (∈, ∈∨ q)-neutrosophicsubalgebra.
Proof. Let A =(AT ,AI ,AF )bea(q, ∈∨ q)-neutrosophicsubalgebraofa BCK/BCIalgebra X andlet x,y ∈ X.Let αx,αy ∈ (0, 1]besuchthat x ∈ T∈(A; αx)and y ∈ T∈(A; αy ).Then AT (x) ≥ αx and AT (y) ≥ αy .Suppose x∗y/ ∈ T∈∨ q (A; αx ∧αy ).
Then
AT (x ∗ y) <αx ∧ αy , (3.8)
AT (x ∗ y)+(αx ∧ αy ) ≤ 1. (3.9)
Itfollowsthat
AT (x ∗ y) < 0.5. (3.10) 91
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Combining(3.8)and(3.10),wehave
AT (x ∗ y) < {αx,αy , 0 5} andso
1 AT (x ∗ y) > 1 {αx,αy , 0.5} = {1 αx, 1 αy , 0 5} ≥ {1 AT (x), 1 AT (y), 0 5}
Hencethereexists α ∈ (0, 1]suchthat
1 AT (x ∗ y) ≥ α> {1 AT (x), 1 AT (y), 0 5} (3.11)
Therightinequalityin(3.11)induces AT (x)+ α> 1and AT (y)+ α> 1,thatis, x,y ∈ Tq (A; α).Since A =(AT ,AI ,AF )isa(q, ∈∨ q)-neutrosophicsubalgebra of X,wehave x ∗ y ∈ T∈∨ q (A; α).But,theleftinequalityin(3.11)impliesthat AT (x ∗ y)+ α ≤ 1,i.e., x ∗ y/ ∈ Tq (A; α),and AT (x ∗ y) ≤ 1 α< 1 0.5=0.5 <α, i.e., x ∗ y/ ∈ T∈(A; α).Hence x ∗ y/ ∈ T∈∨ q (A; α),acontradiction.Thus x ∗ y ∈ T∈∨ q (A; αx ∧ αy ).Similarly,wecanshowthatif x ∈ I∈(A; βx)and y ∈ I∈(A; βy ) for βx,βy ∈ (0, 1],then x ∗ y ∈ I∈∨ q (A; βx ∧ βy ).Now,let γx,γy ∈ [0, 1)besuch that x ∈ F∈(A; γx)and y ∈ F∈(A; γy ). AF (x) ≤ γx and AF (y) ≤ γy .If x ∗ y/ ∈ F∈∨ q (A; γx ∨ γy ),then
AF (x ∗ y) >γx ∨ γy , (3.12) AF (x ∗ y)+(γx ∨ γy ) ≥ 1. (3.13)
Itfollowsthat
AF (x ∗ y) > {γx,γy , 0.5} andsothat
1 AF (x ∗ y) < 1 {γx,γy , 0 5} = {1 γx, 1 γy , 0 5} ≤ {1 AF (x), 1 AF (y), 0.5}.
Thusthereexists γ ∈ [0, 1)suchthat
1 AF (x ∗ y) ≤ γ< {1 AF (x), 1 AF (y), 0 5} (3.14)
Itfollowsfromtherightinequalityin(3.14)that AF (x)+ γ< 1and AF (y)+ γ< 1, thatis, x,y ∈ Fq (A; γ),whichimpliesthat x ∗ y ∈ F∈∨ q (A; γ).But,wehave x ∗ y/ ∈ F∈∨ q (A; γ)bytheleftinequalityin(3.14).Thisisacontradiction,andso x ∗ y ∈ F∈∨ q (A; γx ∨ γy ).Therefore A =(AT ,AI ,AF )isan(∈, ∈∨ q)-neutrosophic subalgebraof X
ThefollowingexampleshowsthattheconverseofTheorem 3.6 isnottrue. 92
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Table1. Cayleytableoftheoperation ∗ ∗ 01234 000000 110011 221022 333303 444440
XAT (x) AI (x) AF (x) 00.60.80.3 10.20.30.6 20.20.30.6 30.70.10.7 40 40 40 9
Example3.7. Considera BCK-algebra X = {0, 1, 2, 3, 4} withthefollowingCayleytable. Let A =(AT ,AI ,AF )beaneutrosophicsetin X definedby Then
T∈(A; α)= {0, 3} if α ∈ (0 4, 0 5], {0, 3, 4} if α ∈ (0.2, 0.4], X if α ∈ (0, 0.2], I∈(A; β)= {0} if β ∈ (0 4, 0 5], {0, 4} if β ∈ (0 3, 0 4], {0, 1, 2, 4} if β ∈ (0 1, 0 3], X if β ∈ (0, 0 1], F∈(A; γ)= X if γ ∈ (0 9, 1), {0, 1, 2, 3} if γ ∈ [0 7, 0 9), {0, 1, 2} if γ ∈ [0 6, 0 7), {0} if γ ∈ [0 5, 0 6), whicharesubalgebrasof X forall α,β ∈ (0, 0 5]and γ ∈ [0 5, 1).UsingTheorem 3.3, A =(AT ,AI ,AF )isan(∈, ∈∨ q)-neutrosophicsubalgebraof X.Butitisnot a(q, ∈∨ q)-neutrosophicsubalgebraof X since2 ∈ Tq (A;0 83)and3 ∈ Tq (A;0 4), but2 ∗ 3=2 / ∈ T∈∨ q (A;0 4).
Weprovideconditionsforan(∈, ∈∨ q)-neutrosophicsubalgebratobea(q, ∈∨ q)neutrosophicsubalgebra.
Theorem3.8. Assumethatanyneutrosophic TΦ-pointandneutrosophic IΦ-point hasthevalue α and β in (0, 0.5],respectively,andanyneutrosophic FΦ-pointhas thevalue γ in [0.5, 1) for Φ ∈{∈,q, ∈∨ q}.Thenevery (∈, ∈∨ q)-neutrosophic subalgebraisa (q, ∈∨ q)-neutrosophicsubalgebra.
Proof. Let X bea BCK/BCI-algebraandlet A =(AT ,AI ,AF )bean(∈, ∈∨ q)neutrosophicsubalgebraof X.For x,y,a,b ∈ X,let αx,αy ,βa,βb ∈ (0, 0 5]be 93
A.BorumandSaeidetal./Ann.FuzzyMath.Inform. 14 (2017),No.1,87–97 suchthat x ∈ Tq (A; αx), y ∈ Tq (A; αy ), a ∈ Iq (A; βa)and b ∈ Tq (A; βb).Then AT (x)+ αx > 1, AT (y)+ αy > 1, AI (a)+ βa > 1and AI (b)+ βb > 1.Since αx,αy ,βa,βb ∈ (0, 0.5],itfollowsthat AT (x) > 1 αx ≥ αx, AT (y) > 1 αy ≥ αy , AI (a) > 1 βa ≥ βa and AI (b) > 1 βb ≥ βb,thatis, x ∈ T∈(A; αx), y ∈ T∈(A; αy ), a ∈ I∈(A; βa)and b ∈ I∈(A; βb).Also,let x ∈ Fq (A; γx)and y ∈ Fq (A; γy )for x,y ∈ X and γx,γy ∈ [0 5, 1).Then AF (x)+ γx < 1and AF (y)+ γy < 1,andso AF (x) < 1 γx ≤ γx and AF (y) < 1 γy ≤ γy since γx,γy ∈ [0 5, 1).Thisshowsthat x ∈ F∈(A; γx)and y ∈ F∈(A; γy ).Itfollowsfrom(3.4)that x ∗ y ∈ T∈∨ q (A; αx ∧ αy ), a∗b ∈ I∈∨ q (A; βa ∧βb),and x∗y ∈ F∈∨ q (A; γx ∨γy ).Consequently, A =(AT ,AI ,AF ) isa(q, ∈∨ q)-neutrosophicsubalgebraof X Theorem3.9. Both (∈, ∈)-neutrosophicsubalgebraand (∈∨ q, ∈∨ q)-neutrosophic subalgebraarean (∈, ∈∨ q)-neutrosophicsubalgebra.
Proof. Itisclearthat(∈, ∈)-neutrosophicsubalgebraisan(∈, ∈∨ q)-neutrosophic subalgebra.Let A =(AT ,AI ,AF )bean(∈∨ q, ∈∨ q)-neutrosophicsubalgebraof X.Forany x,y,a,b ∈ X,let αx,αy ,βa,βb ∈ (0, 1]besuchthat x ∈ T∈(A; αx), y ∈ T∈(A; αy ), a ∈ I∈(A; βa)and b ∈ I∈(A; βb).Then x ∈ T∈∨ q (A; αx), y ∈ T∈∨ q (A; αy ), a ∈ I∈∨ q (A; βa)and b ∈ I∈∨ q (A; βb)by(3.1)and(3.2).Itfollows that x ∗ y ∈ T∈∨ q (A; αx ∧ αy )and a ∗ b ∈ I∈∨ q (A; βa ∧ βb).Now,let x,y ∈ X and γx,γy ∈ [0, 1)besuchthat x ∈ F∈(A; γx)and y ∈ F∈(A; γy ).Then x ∈ F∈∨ q (A; γx) and y ∈ F∈∨ q (A; γy )by(3.3).Hence x ∗ y ∈ F∈∨ q (A; γx ∨ γy ).Therefore A = (AT ,AI ,AF )isan(∈, ∈∨ q)-neutrosophicsubalgebraof X
TheconverseofTheorem 3.9 isnottrueingeneral.Infact,the(∈, ∈∨ q)neutrosophicsubalgebra A =(AT ,AI ,AF )inExample 3.7 isneitheran(∈, ∈)neutrosophicsubalgebranoran(∈∨ q, ∈∨ q)-neutrosophicsubalgebra.
Theorem3.10. Foraneutrosophicset A =(AT ,AI ,AF ) ina BCK/BCI -algebra X,ifthenonemptyneutrosophic q-subsets Tq (A; α), Iq (A; β) and Fq (A; γ) aresubalgebrasof X forall α,β ∈ (0.5, 1] and γ ∈ (0, 0.5),then
x ∈ T∈(A; αx),y ∈ T∈(A; αy ) ⇒ x ∗ y ∈ Tq (A; αx ∨ αy ), x ∈ I∈(A; βx),y ∈ I∈(A; βy ) ⇒ x ∗ y ∈ Iq (A; βx ∨ βy ), x ∈ F∈(A; γx),y ∈ F∈(A; γy ) ⇒ x ∗ y ∈ Fq (A; γx ∧ γy ) (3.15)
forall x,y ∈ X, αx,αy ,βx,βy ∈ (0 5, 1] and γx,γy ∈ (0, 0 5)
Proof. Let x,y,a,b,u,v ∈ X and αx,αy ,βa,βb ∈ (0 5, 1]and γu,γv ∈ (0, 0 5)be suchthat x ∈ T∈(A; αx), y ∈ T∈(A; αy ), a ∈ I∈(A; βa), b ∈ I∈(A; βb), u ∈ F∈(A; γu) and v ∈ F∈(A; γv ).Then AT (x) ≥ αx > 1 αx, AT (y) ≥ αy > 1 αy , AI (a) ≥ βa > 1 βa, AI (b) ≥ βb > 1 βb, AF (u) ≤ γu < 1 γu and AF (v) ≤ γv < 1 γv . Itfollowsthat x,y ∈ Tq (A; αx ∨ αy ), a,b ∈ Iq (A; βa ∨ βb)and u,v ∈ Fq (A; γu ∧ γv ). Since αx ∨ αy ,βa ∨ βb ∈ (0.5, 1]and γu ∧ γv ∈ (0, 0.5),wehave x ∗ y ∈ Tq (A; αx ∨ αy ), a ∗ b ∈ Iq (A; βa ∨ βb)and u ∗ v ∈ Fq (A; γu ∧ γv )byhypothesis.Thiscompletesthe proof.
ThefollowingcorollaryisbyTheorem 3.10 and[7,Theorem3.7].
Corollary3.11. Every (∈, ∈∨ q)-neutrosophicsubalgebra A =(AT ,AI ,AF ) ina BCK/BCI -algebra X satisfiesthecondition (3.15) 94
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Corollary3.12. Every (q, ∈∨ q)-neutrosophicsubalgebra A =(AT ,AI ,AF ) ina BCK/BCI -algebra X satisfiesthecondition (3.15). Proof. ItisbyTheorem 3.6 andCorollary 3.11 Theorem3.13. Foraneutrosophicset A =(AT ,AI ,AF ) ina BCK/BCI -algebra X,ifthenonemptyneutrosophic q-subsets Tq (A; α), Iq (A; β) and Fq (A; γ) aresubalgebrasof X forall α,β ∈ (0, 0 5] and γ ∈ (0 5, 1),then x ∈ Tq (A; αx),y ∈ Tq (A; αy ) ⇒ x ∗ y ∈ T∈(A; αx ∨ αy ), x ∈ Iq (A; βx),y ∈ Iq (A; βy ) ⇒ x ∗ y ∈ I∈(A; βx ∨ βy ), x ∈ Fq (A; γx),y ∈ Fq (A; γy ) ⇒ x ∗ y ∈ F∈(A; γx ∧ γy )
(3.16)
forall x,y ∈ X, αx,αy ,βx,βy ∈ (0, 0 5] and γx,γy ∈ (0 5, 1)
Proof. Let x,y,a,b,u,v ∈ X and αx,αy ,βa,βb ∈ (0, 0.5]and γu,γv ∈ (0.5, 1) besuchthat x ∈ Tq (A; αx), y ∈ Tq (A; αy ), a ∈ Iq (A; βa), b ∈ Iq (A; βb), u ∈ Fq (A; γu)and v ∈ Fq (A; γv ).Then x,y ∈ Tq (A; αx ∨ αy ), a,b ∈ Iq (A; βa ∨ βb)and u,v ∈ Fq (A; γu ∧ γv ).Since αx ∨ αy ,βa ∨ βb ∈ (0, 0 5]and γu ∧ γv ∈ (0 5, 1),it followsfromthehypothesisthat x ∗ y ∈ Tq (A; αx ∨ αy ), a ∗ b ∈ Iq (A; βa ∨ βb)and u ∗ v ∈ Fq (A; γu ∧ γv ).Hence
AT (x ∗ y) > 1 (αx ∨ αy ) ≥ αx ∨ αy ,thatis, x ∗ y ∈ T∈(A; αx ∨ αy ),
AI (a ∗ b) > 1 (βa ∨ βb) ≥ βa ∨ βb,thatis, a ∗ b ∈ I∈(A; βa ∨ βb), AF (u ∗ v) < 1 (γu ∧ γv ) ≤ γu ∧ γv ,thatis, u ∗ v ∈ F∈(A; γu ∧ γv ). Consequently,thecondition(3.16)isvalidforall x,y ∈ X, αx,αy ,βx,βy ∈ (0, 0 5] and γx,γy ∈ (0 5, 1).
Theorem3.14. Givenaneutrosophicset A =(AT ,AI ,AF ) ina BCK/BCIalgebra X,ifthenonemptyneutrosophic ∈∨ q-subsets T∈∨ q (A; α), I∈∨ q (A; β) and F∈∨ q (A; γ) aresubalgebrasof X forall α,β ∈ (0, 0 5] and γ ∈ [0 5, 1),thenthe followingassertionsarevalid.
x ∈ Tq (A; αx),y ∈ Tq (A; αy ) ⇒ x ∗ y ∈ T∈∨ q (A; αx ∨ αy ), x ∈ Iq (A; βx),y ∈ Iq (A; βy ) ⇒ x ∗ y ∈ I∈∨ q (A; βx ∨ βy ), x ∈ Fq (A; γx),y ∈ Fq (A; γy ) ⇒ x ∗ y ∈ F∈∨ q (A; γx ∧ γy ) (3.17)
forall x,y ∈ X, αx,αy ,βx,βy ∈ (0, 0 5] and γx,γy ∈ [0 5, 1)
Proof. Let x,y,a,b,u,v ∈ X and αx,αy ,βa,βb ∈ (0, 0 5]and γu,γv ∈ [0 5, 1)be suchthat x ∈ Tq (A; αx), y ∈ Tq (A; αy ), a ∈ Iq (A; βa), b ∈ Iq (A; βb), u ∈ Fq (A; γu) and v ∈ Fq (A; γv ).Then x ∈ T∈∨ q (A; αx), y ∈ T∈∨ q (A; αy ), a ∈ I∈∨ q (A; βa), b ∈ I∈∨ q (A; βb), u ∈ F∈∨ q (A; γu)and v ∈ F∈∨ q (A; γv ).Itfollowsthat x,y ∈ T∈∨ q (A; αx ∨ αy ), a,b ∈ I∈∨ q (A; βa ∨ βb)and u,v ∈ F∈∨ q (A; γu ∧ γv )whichimply fromthehypothesisthat x ∗ y ∈ T∈∨ q (A; αx ∨ αy ), a ∗ b ∈ I∈∨ q (A; βa ∨ βb)and u ∗ v ∈ F∈∨ q (A; γu ∧ γv ).Thiscompletestheproof. Corollary3.15. Every (∈, ∈∨ q)-neutrosophicsubalgebra A =(AT ,AI ,AF ) ofa BCK/BCI -algebra X satisfiesthecondition (3.17) Proof. ItisbyTheorem 3.14 and[7,Theorem3.9]. 95
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Theorem3.16. Givenaneutrosophicset A =(AT ,AI ,AF ) ina BCK/BCIalgebra X,ifthenonemptyneutrosophic ∈∨ q-subsets T∈∨ q (A; α), I∈∨ q (A; β) and F∈∨ q (A; γ) aresubalgebrasof X forall α,β ∈ (0.5, 1] and γ ∈ [0, 0.5),thenthe followingassertionsarevalid.
x ∈ Tq (A; αx),y ∈ Tq (A; αy ) ⇒ x ∗ y ∈ T∈∨ q (A; αx ∨ αy ),
x ∈ Iq (A; βx),y ∈ Iq (A; βy ) ⇒ x ∗ y ∈ I∈∨ q (A; βx ∨ βy ),
x ∈ Fq (A; γx),y ∈ Fq (A; γy ) ⇒ x ∗ y ∈ F∈∨ q (A; γx ∧ γy ) (3.18) forall x,y ∈ X, αx,αy ,βx,βy ∈ (0.5, 1] and γx,γy ∈ [0, 0.5).
Proof. ItissimilartotheproofTheorem 3.14.
Corollary3.17. Every (q, ∈∨ q)-neutrosophicsubalgebra A =(AT ,AI ,AF ) ofa BCK/BCI -algebra X satisfiesthecondition (3.18)
Proof. ItisbyTheorem 3.16 and[7,Theorem3.10]. CombiningTheorems 3.14 and 3.16,wehavethefollowingcorollary.
Corollary3.18. Givenaneutrosophicset A =(AT ,AI ,AF ) ina BCK/BCIalgebra X,ifthenonemptyneutrosophic ∈∨ q-subsets T∈∨ q (A; α), I∈∨ q (A; β) and F∈∨ q (A; γ) aresubalgebrasof X forall α,β ∈ (0, 1] and γ ∈ [0, 1),thenthefollowing assertionsarevalid.
x ∈ Tq (A; αx),y ∈ Tq (A; αy ) ⇒ x ∗ y ∈ T∈∨ q (A; αx ∨ αy ), x ∈ Iq (A; βx),y ∈ Iq (A; βy ) ⇒ x ∗ y ∈ I∈∨ q (A; βx ∨ βy ),
x ∈ Fq (A; γx),y ∈ Fq (A; γy ) ⇒ x ∗ y ∈ F∈∨ q (A; γx ∧ γy ) forall x,y ∈ X, αx,αy ,βx,βy ∈ (0, 1] and γx,γy ∈ [0, 1)
Conclusions
Wehaveconsideredrelationsbetweenan(∈, ∈∨ q)-neutrosophicsubalgebraand a(q, ∈∨ q)-neutrosophicsubalgebra.Wehavediscussedcharacterizationofan(∈, ∈∨ q)-neutrosophicsubalgebrabyusingneutrosophic ∈-subsets,andhaveprovided conditionsforan(∈, ∈∨ q)-neutrosophicsubalgebratobea(q, ∈∨ q)-neutrosophic subalgebra.Wehaveinvestigatedpropertiesonneutrosophic q-subsetsandneutrosophic ∈∨ q-subsets.Ourfutureresearchwillbefocusedonthestudyofgeneralizationofthispaper.
4. Acknowledgements
Theauthorswishtothankstheanonymousreviewersfortheirvaluablesuggestions.
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ArshamBorumandSaeid (a b saeid@yahoo.com)
DepartmentofPureMathematics,FacultyofMathematicsandComputer,Shahid BahonarUniversityofKerman,Kerman,Iran
YoungBaeJun (skywine@gmail.com)
DepartmentofMathematicsEducation,GyeongsangNationalUniversity,Jinju52828, Korea
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