Neutrosophic Permeable Values and Energetic Subsets with Applications in BCK/BCI-Algebras

2017·0.3 MB·English

by  Young Bae Jun, Florentin Smarandache, Seok-Zun Song, Hashem Bordbar;

mathematics
Article
Neutrosophic Permeable Values and Energetic
Subsets with Applications in BCK/BCI-Algebras
YoungBaeJun1,*,FlorentinSmarandache2 ID,Seok-ZunSong3 ID andHashemBordbar4 ID
1 DepartmentofMathematicsEducation,GyeongsangNationalUniversity,Jinju52828,Korea
2 Mathematics&ScienceDepartment,UniversityofNewMexico,705GurleyAve.,Gallup,NM87301,USA;
[email protected]
3 DepartmentofMathematics,JejuNationalUniversity,Jeju63243,Korea;[email protected]
4 PostdoctoralResearchFellow,ShahidBeheshtiUniversity,Tehran,District1,
DaneshjouBoulevard1983969411,Iran;[email protected]
* Correspondence:[email protected]
(cid:1)(cid:2)(cid:3)(cid:1)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8)(cid:1)
(cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)
Received:14March2018;Accepted:2May2018;Published:7May2018
Abstract: The concept of a (∈, ∈)-neutrosophic ideal is introduced, and its characterizations are
established. Thenotionsofneutrosophicpermeablevaluesareintroduced,andrelatedpropertiesare
investigated.Conditionsfortheneutrosophiclevelsetstobeenergetic,rightstable,andrightvanished
arediscussed. RelationsbetweenneutrosophicpermeableS-and I-valuesareconsidered.
Keywords:(∈,∈)-neutrosophicsubalgebra;(∈,∈)-neutrosophicideal;neutrosophic(anti-)permeable
S-value;neutrosophic(anti-)permeable I-value;S-energeticset; I-energeticset
MSC:06F35;03G25;08A72
1. Introduction
Thenotionofneutrosophicset(NS)theorydevelopedbySmarandache(see[1,2])isamoregeneral
platformthatextendstheconceptsofclassicandfuzzysets,intuitionisticfuzzysets,andinterval-valued
(intuitionistic)fuzzysetsandthatisappliedtovariousparts: patternrecognition,medicaldiagnosis,
decision-makingproblems, andsoon(see[3–6]). Smarandache[2]mentionedthatacloudisaNS
becauseitsbordersareambiguousandbecauseeachelement(waterdrop)belongswithaneutrosophic
probabilitytotheset(e.g.,therearetypesofseparatedwaterdropsaroundacompactmassofwater
drops,suchthatwedonotknowhowtoconsiderthem: inoroutofthecloud). Additionally,weare
notsurewherethecloudendsnorwhereitbegins,andneitherwhethersomeelementsareorarenot
intheset. Thisiswhythepercentageofindeterminacyisrequiredandtheneutrosophicprobability
(usingsubsets—notnumbers—ascomponents)shouldbeusedforbettermodeling: itisamoreorganic,
smooth,andparticularlyaccurateestimation. Indeterminacyisthezoneofignoranceofaproposition’s
value,betweentruthandfalsehood.
Algebraicstructuresplayanimportantroleinmathematicswithwide-rangingapplicationsin
severaldisciplinessuchascodingtheory,informationsciences,computersciences,controlengineering,
theoreticalphysics,andsoon. NStheoryisalsoappliedtoseveralalgebraicstructures. Inparticular,
Junetal. appliedittoBCK/BCI-algebras(see[7–12]). Junetal. [8]introducedthenotionsofenergetic
subsets,rightvanishedsubsets,rightstablesubsets,and(anti-)permeablevaluesinBCK/BCI-algebras
andinvestigatedrelationsbetweenthesesets.
In this paper, we introduce the notions of neutrosophic permeable S-values, neutrosophic
permeable I-values, (∈, ∈)-neutrosophic ideals, neutrosophic anti-permeable S-values,
and neutrosophic anti-permeable I-values, which are motivated by the idea of subalgebras
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Mathematics2018,6,74 2of16
(i.e.,S-values)andideals(i.e.,I-values),andinvestigatetheirproperties. Weconsidercharacterizations
of(∈,∈)-neutrosophicideals. Wediscussconditionsforthelower(upper)neutrosophic∈Φ-subsetsto
be S-and I-energetic. Weprovideconditionsforatriple (α,β,γ) ofnumberstobeaneutrosophic
(anti-)permeableS-orI-value.Weconsiderconditionsfortheupper(lower)neutrosophic∈Φ-subsetsto
berightstable(rightvanished)subsets. Weestablishrelationsbetweenneutrosophic(anti-)permeable
S-and I-values.
2. Preliminaries
Analgebra(X;∗,0)oftype(2,0)iscalledaBCI-algebraifitsatisfiesthefollowingconditions:
(I) (∀x,y,z ∈ X)(((x∗y)∗(x∗z))∗(z∗y) =0);
(II) (∀x,y ∈ X)((x∗(x∗y))∗y =0);
(III) (∀x ∈ X)(x∗x =0);
(IV) (∀x,y ∈ X)(x∗y =0,y∗x =0 ⇒ x = y).
IfaBCI-algebraXsatisfiesthefollowingidentity:
(V) (∀x ∈ X)(0∗x =0),
thenXiscalledaBCK-algebra. AnyBCK/BCI-algebraXsatisfiesthefollowingconditions:
(∀x ∈ X)(x∗0= x), (1)
(∀x,y,z ∈ X)(x ≤ y ⇒ x∗z ≤ y∗z, z∗y ≤ z∗x), (2)
(∀x,y,z ∈ X)((x∗y)∗z = (x∗z)∗y), (3)
(∀x,y,z ∈ X)((x∗z)∗(y∗z) ≤ x∗y), (4)
where x ≤ y if and only if x∗y = 0. A nonempty subset S of a BCK/BCI-algebra X is called a
subalgebraofXifx∗y ∈ Sforallx,y ∈ S.Asubset I ofaBCK/BCI-algebraXiscalledanidealofXif
itsatisfiesthefollowing:
0∈ I, (5)
(∀x,y ∈ X)(x∗y ∈ I, y ∈ I → x ∈ I). (6)
We refer the reader to the books [13] and [14] for further information regarding
BCK/BCI-algebras.
Foranyfamily{a |i ∈ Λ}ofrealnumbers,wedefine
i
(cid:95){a |i ∈ Λ} =sup{a |i ∈ Λ}
i i
and
(cid:94){a |i ∈ Λ} =inf{a |i ∈ Λ}.
i i
IfΛ = {1,2},wealsousea ∨a anda ∧a insteadof(cid:87){a |i ∈ {1,2}}and(cid:86){a |i ∈ {1,2}},
1 2 1 2 i i
respectively.
WeletXbeanonemptyset. ANSinX(see[1])isastructureoftheform
A := {(cid:104)x;A (x),A (x),A (x)(cid:105) | x ∈ X},
T I F
where A : X → [0,1]isatruthmembershipfunction, A : X → [0,1]isanindeterminatemembership
T I
function,and A : X → [0,1]isafalsemembershipfunction. Forthesakeofsimplicity,weusethe
F
symbol A = (A , A , A )fortheNS
T I F
A := {(cid:104)x;A (x),A (x),A (x)(cid:105) | x ∈ X}.
T I F
Mathematics2018,6,74 3of16
Asubset AofaBCK/BCI-algebraXissaidtobeS-energetic(see[8])ifitsatisfies
(∀x,y ∈ X)(x∗y ∈ A ⇒ {x,y}∩A (cid:54)= ∅). (7)
Asubset AofaBCK/BCI-algebraXissaidtobe I-energetic(see[8])ifitsatisfies
(∀x,y ∈ X)(y ∈ A ⇒ {x,y∗x}∩A (cid:54)= ∅). (8)
Asubset AofaBCK/BCI-algebraXissaidtoberightvanished(see[8])ifitsatisfies
(∀x,y ∈ X)(x∗y ∈ A ⇒ x ∈ A). (9)
A subset A of a BCK/BCI-algebra X is said to be right stable (see [8]) if A∗X := {a∗x | a ∈
A, x ∈ X} ⊆ A.
3. NeutrosophicPermeableValues
GivenaNSA = (A ,A ,A )inasetX,α,β ∈ (0,1]andγ ∈ [0,1),weconsiderthefollowingsets:
T I F
U∈(A;α) = {x ∈ X | A (x) ≥ α}, U∈(A;α)∗ = {x ∈ X | A (x) > α},
T T T T
U∈(A;β) = {x ∈ X | A (x) ≥ β}, U∈(A;β)∗ = {x ∈ X | A (x) > β},
I I I I
U∈(A;γ) = {x ∈ X | A (x) ≤ γ}, U∈(A;γ)∗ = {x ∈ X | A (x) < γ},
F F F F
L∈(A;α) = {x ∈ X | A (x) ≤ α}, L∈(A;α)∗ = {x ∈ X | A (x) < α},
T T T T
L∈(A;β) = {x ∈ X | A (x) ≤ β}, L∈(A;β)∗ = {x ∈ X | A (x) < β},
I I I I
L∈(A;γ) = {x ∈ X | A (x) ≥ γ}, L∈(A;γ)∗ = {x ∈ X | A (x) > γ}.
F F F F
WesayU∈(A;α),U∈(A;β),andU∈(A;γ)areupperneutrosophic∈Φ-subsetsofX,and L∈(A;α),
T I F T
L∈(A;β),andL∈(A;γ)arelowerneutrosophic∈Φ-subsetsofX,whereΦ ∈ {T,I,F}. WesayU∈(A;α)∗,
I F T
U∈(A;β)∗, and U∈(A;γ)∗ are strong upper neutrosophic ∈Φ-subsets of X, and L∈(A;α)∗, L∈(A;β)∗,
I F T I
and L∈(A;γ)∗ arestronglowerneutrosophic∈Φ-subsetsofX,whereΦ ∈ {T,I,F}.
F
Definition 1 ([7]). A NS A = (A , A , A ) in a BCK/BCI-algebra X is called an (∈, ∈)-
T I F
neutrosophicsubalgebraofXifthefollowingassertionsarevalid:
x ∈U∈(A;α ), y ∈U∈(A;α ) ⇒ x∗y ∈U∈(A;α ∧α ),
T x T y T x y
x ∈U∈(A;β ), y ∈U∈(A;β ) ⇒ x∗y ∈U∈(A;β ∧β ), (10)
I x I y I x y
x ∈U∈(A;γ ), y ∈U∈(A;γ ) ⇒ x∗y ∈U∈(A;γ ∨γ ),
F x F y F x y
forallx,y ∈ X,α ,α ,β ,β ∈ (0,1]andγ ,γ ∈ [0,1).
x y x y x y
Lemma1([7]). ANS A = (A , A , A )inaBCK/BCI-algebraXisan(∈,∈)-neutrosophicsubalgebraof
T I F
Xifandonlyif A = (A , A , A )satisfies
T I F
 A (x∗y) ≥ A (x)∧A (y) 
T T T
(∀x,y ∈ X) A (x∗y) ≥ A (x)∧A (y) . (11)
 I I I 
A (x∗y) ≤ A (x)∨A (y)
F F F
Proposition1. Every(∈,∈)-neutrosophicsubalgebra A = (A , A , A )ofaBCK/BCI-algebraXsatisfies
T I F
(∀x ∈ X)(A (0) ≥ A (x), A (0) ≥ A (x), A (0) ≤ A (x)). (12)
T T I I F F
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Proof. Straightforward.
Theorem1. If A = (A , A , A )isan(∈,∈)-neutrosophicsubalgebraofaBCK/BCI-algebraX,thenthe
T I F
lowerneutrosophic∈Φ-subsetsofXareS-energeticsubsetsofX,whereΦ ∈ {T,I,F}.
Proof. Letx,y ∈ Xandα ∈ (0,1]besuchthatx∗y ∈ L∈(A;α). Then
T
α ≥ A (x∗y) ≥ A (x)∧A (y),
T T T
andthus A (x) ≤ αor A (y) ≤ α;thatis,x ∈ L∈(A;α)ory ∈ L∈(A;α). Thus{x,y}∩L∈(A;α) (cid:54)= ∅.
T T T T T
ThereforeL∈(A;α)isanS-energeticsubsetofX.Similarly,wecanverifythatL∈(A;β)isanS-energetic
T I
subsetofX. Weletx,y ∈ Xandγ ∈ [0,1)besuchthatx∗y ∈ L∈(A;γ). Then
F
γ ≤ A (x∗y) ≤ A (x)∨A (y).
F F F
Itfollowsthat A (x) ≥ γor A (y) ≥ γ;thatis, x ∈ L∈(A;γ)ory ∈ L∈(A;γ). Hence{x,y}∩
F F F F
L∈(A;γ) (cid:54)= ∅,andthereforeL∈(A;γ)isanS-energeticsubsetofX.
F F
Corollary1. If A = (A , A , A )isan(∈,∈)-neutrosophicsubalgebraofaBCK/BCI-algebraX,thenthe
T I F
stronglowerneutrosophic∈Φ-subsetsofXareS-energeticsubsetsofX,whereΦ ∈ {T,I,F}.
Proof. Straightforward.
TheconverseofTheorem1isnottrue,asseeninthefollowingexample.
Example1. ConsideraBCK-algebraX = {0,1,2,3,4}withthebinaryoperation∗thatisgiveninTable1
(see[14]).
Table1.Cayleytableforthebinaryoperation“∗”.
* 0 1 2 3 4
0 0 0 0 0 0
1 1 0 0 0 0
2 2 1 0 0 1
3 3 2 1 0 2
4 4 1 1 1 0
Let A = (A , A , A )beaNSinXthatisgiveninTable2.
T I F
Table2.TabulationrepresentationofA=(A ,A ,A ).
T I F
X A (x) A (x) A (x)
T I F
0 0.6 0.8 0.2
1 0.4 0.5 0.7
2 0.4 0.5 0.6
3 0.4 0.5 0.5
4 0.7 0.8 0.2
If α ∈ [0.4,0.6), β ∈ [0.5,0.8), and γ ∈ (0.2,0.5], then L∈(A;α) = {1,2,3}, L∈(A;β) = {1,2,3},
T I
and L∈(A;γ) = {1,2,3}areS-energeticsubsetsofX. Because
F
A (4∗4) = A (0) =0.6(cid:3)0.7= A (4)∧A (4)
T T T T
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and/or
A (3∗2) = A (1) =0.7(cid:2)0.6= A (3)∨A (2),
F F F F
itfollowsfromLemma1that A = (A , A , A )isnotan(∈,∈)-neutrosophicsubalgebraofX.
T I F
Definition 2. Let A = (A , A , A ) be aNS ina BCK/BCI-algebra X and (α,β,γ) ∈ Λ ×Λ ×Λ ,
T I F T I F
where Λ , Λ , and Λ are subsets of [0,1]. Then (α,β,γ) is called a neutrosophic permeable S-value for
T I F
A = (A , A , A )ifthefollowingassertionisvalid:
T I F
 x∗y ∈U∈(A;α) ⇒ A (x)∨A (y) ≥ α, 
T T T
(∀x,y ∈ X) x∗y ∈U∈(A;β) ⇒ A (x)∨A (y) ≥ β,  (13)
 I I I 
x∗y ∈U∈(A;γ) ⇒ A (x)∧A (y) ≤ γ
F F F
Example2. LetX = {0,1,2,3,4}beasetwiththebinaryoperation∗thatisgiveninTable3.
Table3.Cayleytableforthebinaryoperation“∗”.
* 0 1 2 3 4
0 0 0 0 0 0
1 1 0 1 1 0
2 2 2 0 2 0
3 3 3 3 0 3
4 4 4 4 4 0
Then(X,∗,0)isaBCK-algebra(see[14]).LetA=(A ,A ,A )beaNSinXthatisgiveninTable4.
T I F
Table4.TabulationrepresentationofA=(A ,A ,A ).
T I F
X A (x) A (x) A (x)
T I F
0 0.2 0.3 0.7
1 0.6 0.4 0.6
2 0.5 0.3 0.4
3 0.4 0.8 0.5
4 0.7 0.6 0.2
Itisroutinetoverifythat(α,β,γ) ∈ (0,2,1]×(0.3,1]×[0,0.7)isaneutrosophicpermeableS-valuefor
A = (A , A , A ).
T I F
Theorem 2. Let A = (A , A , A ) be a NS in a BCK/BCI-algebra X and (α,β,γ) ∈ Λ ×Λ ×Λ ,
T I F T I F
whereΛ ,Λ ,andΛ aresubsetsof[0,1]. If A = (A , A , A )satisfiesthefollowingcondition:
T I F T I F
 A (x∗y) ≤ A (x)∨A (y) 
T T T
(∀x,y ∈ X) A (x∗y) ≤ A (x)∨A (y) , (14)
 I I I 
A (x∗y) ≥ A (x)∧A (y)
F F F
then(α,β,γ)isaneutrosophicpermeableS-valuefor A = (A , A , A ).
T I F
Proof. Letx,y ∈ Xbesuchthatx∗y ∈U∈(A;α). Then
T
α ≤ A (x∗y) ≤ A (x)∨A (y).
T T T
Mathematics2018,6,74 6of16
Similarly,ifx∗y ∈U∈(A;β)forx,y ∈ X,then A (x)∨A (y) ≥ β. Now,leta,b ∈ Xbesuchthat
I I I
a∗b ∈U∈(A;γ). Then
F
γ ≥ A (a∗b) ≥ A (a)∧A (b).
F F F
Therefore(α,β,γ)isaneutrosophicpermeableS-valuefor A = (A , A , A ).
T I F
Theorem3. Let A = (A , A , A )beaNSinaBCK-algebraXand(α,β,γ) ∈ Λ ×Λ ×Λ ,whereΛ ,
T I F T I F T
Λ ,andΛ aresubsetsof[0,1]. If A = (A , A , A )satisfiesthefollowingconditions:
I F T I F
(∀x ∈ X)(A (0) ≤ A (x),A (0) ≤ A (x),A (0) ≥ A (x)) (15)
T T I I F F
and
 A (x) ≤ A (x∗y)∨A (y) 
T T T
(∀x,y ∈ X) A (x) ≤ A (x∗y)∨A (y) , (16)
 I I I 
A (x) ≥ A (x∗y)∧A (y)
F F F
then(α,β,γ)isaneutrosophicpermeableS-valuefor A = (A , A , A ).
T I F
Proof. Letx,y,a,b,u,v∈ Xbesuchthatx∗y∈U∈(A;α),a∗b∈U∈(A;β),andu∗v∈U∈(A;γ).Then
T I F
α ≤ A (x∗y) ≤ A ((x∗y)∗x)∨A (x)
T T T
= A ((x∗x)∗y)∨A (x) = A (0∗y)∨A (x)
T T T T
= A (0)∨A (x) = A (x),
T T T
β ≤ A (a∗b) ≤ A ((a∗b)∗a)∨A (a)
I I I
= A ((a∗a)∗b)∨A (a) = A (0∗b)∨A (a)
I I I I
= A (0)∨A (a) = A (a),
I I I
and
γ ≥ A (u∗v) ≥ A ((u∗v)∗u)∧A (u)
F F F
= A ((u∗u)∗v)∧A (u) = A (0∗v)∧A (v)
F F F F
= A (0)∧A (v) = A (v)
F F F
byEquations(3),(V),(15),and(16). Itfollowsthat
A (x)∨A (y) ≥ A (x) ≥ α,
T T T
A (a)∨A (b) ≥ A (a) ≥ β,
I I I
A (u)∧A (v) ≤ A (u) ≤ γ.
F F F
Therefore(α,β,γ)isaneutrosophicpermeableS-valuefor A = (A , A , A ).
T I F
Theorem 4. Let A = (A , A , A ) be a NS in a BCK/BCI-algebra X and (α,β,γ) ∈ Λ ×Λ ×Λ ,
T I F T I F
whereΛ ,Λ ,andΛ aresubsetsof[0,1]. If(α,β,γ)isaneutrosophicpermeableS-valuefor A = (A , A ,
T I F T I
AF),thenupperneutrosophic∈Φ-subsetsofXareS-energeticwhereΦ ∈ {T,I,F}.
Mathematics2018,6,74 7of16
Proof. Let x,y,a,b,u,v ∈ X besuchthat x∗y ∈ U∈(A;α), a∗b ∈ U∈(A;β), and u∗v ∈ U∈(A;γ).
T I F
Using Equation (13), we have A (x)∨ A (y) ≥ α, A (a)∨ A (b) ≥ β, and A (u)∧ A (v) ≤ γ.
T T I I F F
Itfollowsthat
A (x) ≥ αor A (y) ≥ α,thatis,x ∈U∈(A;α)ory ∈U∈(A;α);
T T T T
A (a) ≥ βor A (b) ≥ β,thatis,a ∈U∈(A;β)orb ∈U∈(A;β);
I I I I
and
A (u) ≤ γor A (v) ≤ γ,thatis,u ∈U∈(A;γ)orv ∈U∈(A;γ).
F F F F
Hence {x,y} ∩ U∈(A;α) (cid:54)= ∅, {a,b} ∩ U∈(A;β) (cid:54)= ∅, and {u,v} ∩ U∈(A;γ) (cid:54)= ∅.
T I F
Therefore U∈(A;α),U∈(A;β),andU∈(A;γ)areS-energeticsubsetsofX.
T I F
Definition 3. Let A = (A , A , A ) be aNS ina BCK/BCI-algebra X and (α,β,γ) ∈ Λ ×Λ ×Λ ,
T I F T I F
whereΛ ,Λ ,andΛ aresubsetsof[0,1]. Then(α,β,γ)iscalledaneutrosophicanti-permeableS-valuefor
T I F
A = (A , A , A )ifthefollowingassertionisvalid:
T I F
 x∗y ∈ L∈(A;α) ⇒ A (x)∧A (y) ≤ α, 
T T T
(∀x,y ∈ X) x∗y ∈ L∈(A;β) ⇒ A (x)∧A (y) ≤ β, . (17)
 I I I 
x∗y ∈ L∈(A;γ) ⇒ A (x)∨A (y) ≥ γ
F F F
Example3. LetX = {0,1,2,3,4}beasetwiththebinaryoperation∗thatisgiveninTable5.
Table5.Cayleytableforthebinaryoperation“∗”.
* 0 1 2 3 4
0 0 0 0 0 0
1 1 0 0 1 0
2 2 1 0 2 0
3 3 3 3 0 3
4 4 4 4 4 0
Then(X,∗,0)isaBCK-algebra(see[14]).LetA=(A ,A ,A )beaNSinXthatisgiveninTable6.
T I F
Table6.TabulationrepresentationofA=(A ,A ,A ).
T I F
X A (x) A (x) A (x)
T I F
0 0.7 0.6 0.4
1 0.4 0.5 0.6
2 0.4 0.5 0.6
3 0.5 0.2 0.7
4 0.3 0.3 0.9
Itisroutinetoverifythat(α,β,γ) ∈ (0.3,1]×(0.2,1]×[0,0.9)isaneutrosophicanti-permeableS-value
for A = (A , A , A ).
T I F
Theorem 5. Let A = (A , A , A ) be a NS in a BCK/BCI-algebra X and (α,β,γ) ∈ Λ ×Λ ×Λ ,
T I F T I F
whereΛ ,Λ ,andΛ aresubsetsof[0,1]. If A = (A , A , A )isan(∈,∈)-neutrosophicsubalgebraofX,
T I F T I F
then(α,β,γ)isaneutrosophicanti-permeableS-valuefor A = (A , A , A ).
T I F
Mathematics2018,6,74 8of16
Proof. Let x,y,a,b,u,v ∈ X be such that x∗y ∈ L∈(A;α), a∗b ∈ L∈(A;β), and u∗v ∈ L∈(A;γ).
T I F
UsingLemma1,wehave
A (x)∧A (y) ≤ A (x∗y) ≤ α,
T T T
A (a)∧A (b) ≤ A (a∗b) ≤ β,
I I I
A (u)∨A (v) ≥ A (u∗v) ≥ γ,
F F F
andthus(α,β,γ)isaneutrosophicanti-permeableS-valuefor A = (A , A , A ).
T I F
Theorem 6. Let A = (A , A , A ) be a NS in a BCK/BCI-algebra X and (α,β,γ) ∈ Λ ×Λ ×Λ ,
T I F T I F
whereΛ ,Λ ,andΛ aresubsetsof[0,1]. If(α,β,γ)isaneutrosophicanti-permeableS-valuefor A = (A ,
T I F T
AI, AF),thenlowerneutrosophic∈Φ-subsetsofXareS-energeticwhereΦ ∈ {T,I,F}.
Proof. Let x,y,a,b,u,v ∈ X be such that x∗y ∈ L∈(A;α), a∗b ∈ L∈(A;β), and u∗v ∈ L∈(A;γ).
T I F
Using Equation (17), we have A (x)∧ A (y) ≤ α, A (a)∧ A (b) ≤ β, and A (u)∨ A (v) ≥ γ,
T T I I F F
whichimplythat
A (x) ≤ αor A (y) ≤ α,thatis,x ∈ L∈(A;α)ory ∈ L∈(A;α);
T T T T
A (a) ≤ βor A (b) ≤ β,thatis,a ∈ L∈(A;β)orb ∈ L∈(A;β);
I I I I
and
A (u) ≥ γor A (v) ≥ γ,thatis,u ∈ L∈(A;γ)orv ∈ L∈(A;γ).
F F F F
Hence {x,y} ∩ L∈(A;α) (cid:54)= ∅, {a,b} ∩ L∈(A;β) (cid:54)= ∅, and {u,v} ∩ L∈(A;γ) (cid:54)= ∅.
T I F
Therefore L∈(A;α),L∈(A;β),andL∈(A;γ)areS-energeticsubsetsofX.
T I F
Definition4. ANS A = (A , A , A )inaBCK/BCI-algebraXiscalledan (∈,∈)-neutrosophicidealof
T I F
Xifthefollowingassertionsarevalid:
 x ∈U∈(A;α) ⇒ 0∈U∈(A;α) 
T T
(∀x ∈ X) x ∈U∈(A;β) ⇒ 0∈U∈(A;β) , (18)
 I I 
x ∈U∈(A;γ) ⇒ 0∈U∈(A;γ)
F F
 x∗y ∈U∈(A;α ), y ∈U∈(A;α ) ⇒ x ∈U∈(A;α ∧α ) 
T x T y T x y
(∀x,y ∈ X) x∗y ∈U∈(A;β ), y ∈U∈(A;β ) ⇒ x ∈U∈(A;β ∧β ) , (19)
 I x I y I x y 
x∗y ∈U∈(A;γ ), y ∈U∈(A;γ ) ⇒ x ∈U∈(A;γ ∨γ )
F x F y F x y
forallα,β,α ,α ,β ,β ∈ (0,1]andγ,γ ,γ ∈ [0,1).
x y x y x y
Theorem7. ANS A = (A , A , A )inaBCK/BCI-algebraXisan(∈,∈)-neutrosophicidealofXifand
T I F
onlyif A = (A , A , A )satisfies
T I F
 A (0) ≥ A (x) ≥ A (x∗y)∧A (y) 
T T T T
(∀x,y ∈ X) A (0) ≥ A (x) ≥ A (x∗y)∧A (y) . (20)
 I I I I 
A (0) ≤ A (x) ≤ A (x∗y)∨A (y)
F F F F
Proof. Assume that Equation (20) is valid, and let x ∈ U∈(A;α), a ∈ U∈(A;β), and u ∈ U∈(A;γ)
T I F
for any x,a,u ∈ X, α,β ∈ (0,1] and γ ∈ [0,1). Then A (0) ≥ A (x) ≥ α, A (0) ≥ A (a) ≥ β,
T T I I
and A (0) ≤ A (u) ≤ γ. Hence 0 ∈ U∈(A;α), 0 ∈ U∈(A;β), and 0 ∈ U∈(A;γ), and thus
F F T I F
Equation (18) is valid. Let x,y,a,b,u,v ∈ X be such that x∗y ∈ U∈(A;α ), y ∈ U∈(A;α ),
T x T y
a∗b ∈ U∈(A;β ),b ∈ U∈(A;β ),u∗v ∈ U∈(A;γ ),andv ∈ U∈(A;γ )forallα ,α ,β ,β ∈ (0,1]
I a I b F u F v x y a b
Mathematics2018,6,74 9of16
andγ ,γ ∈ [0,1). Then A (x∗y) ≥ α , A (y) ≥ α , A (a∗b) ≥ β , A (b) ≥ β , A (u∗v) ≤ γ ,
u v T x T y I a I b F u
and A (v) ≤ γ . ItfollowsfromEquation(20)that
F v
A (x) ≥ A (x∗y)∧A (y) ≥ α ∧α ,
T T T x y
A (a) ≥ A (a∗b)∧A (b) ≥ β ∧β ,
I I I a b
A (u) ≤ A (u∗v)∨A (v) ≤ γ ∨γ .
F F F u v
Hencex ∈U∈(A;α ∧α ),a ∈U∈(A;β ∧β ),andu ∈U∈(A;γ ∨γ ). Therefore A = (A , A ,
T x y I a b F u v T I
A )isan(∈,∈)-neutrosophicidealofX.
F
Conversely,let A = (A , A , A ) bean (∈, ∈)-neutrosophicidealof X. Ifthereexists x ∈ X
T I F 0
such that A (0) < A (x ), then x ∈ U∈(A;α) and 0 ∈/ U∈(A;α), where α = A (x ). This is a
T T 0 0 T T T 0
contradiction,andthusA (0) ≥ A (x)forallx ∈ X. AssumethatA (x ) < A (x ∗y )∧A (y )for
T T T 0 T 0 0 T 0
somex ,y ∈ X. Takingα := A (x ∗y )∧A (y )impliesthatx ∗y ∈U∈(A;α)andy ∈U∈(A;α);
0 0 T 0 0 T 0 0 0 T 0 T
but x ∈/ U∈(A;α). This is a contradiction, and thus A (x) ≥ A (x∗y)∧A (y) for all x,y ∈ X.
0 T T T T
Similarly, we can verify that A (0) ≥ A (x) ≥ A (x∗y)∧A (y) for all x,y ∈ X. Now, suppose
I I I I
that A (0) > A (a) for some a ∈ X. Then a ∈ U∈(A;γ) and 0 ∈/ U∈(A;γ) by taking γ = A (a).
F F F F F
This is impossible, and thus A (0) ≤ A (x) for all x ∈ X. Suppose there exist a ,b ∈ X such
F F 0 0
that A (a ) > A (a ∗b )∨A (b ), and take γ := A (a ∗b )∨A (b ). Then a ∗b ∈ U∈(A;γ),
F 0 F 0 0 F 0 F 0 0 F 0 0 0 F
b ∈U∈(A;γ),anda ∈/ U∈(A;γ),whichisacontradiction. Thus A (x) ≤ A (x∗y)∨A (y)forall
0 F 0 F F F F
x,y ∈ X. Therefore A = (A , A , A )satisfiesEquation(20).
T I F
Lemma2. Every(∈,∈)-neutrosophicideal A = (A , A , A )ofaBCK/BCI-algebraXsatisfies
T I F
(∀x,y ∈ X)(x ≤ y ⇒ A (x) ≥ A (y), A (x) ≥ A (y), A (x) ≤ A (y)). (21)
T T I I F F
Proof. Letx,y ∈ Xbesuchthatx ≤ y. Thenx∗y =0,andthus
A (x) ≥ A (x∗y)∧A (y) = A (0)∧A (y) = A (y),
T T T T T T
A (x) ≥ A (x∗y)∧A (y) = A (0)∧A (y) = A (y),
I I I I I I
A (x) ≤ A (x∗y)∨A (y) = A (0)∨A (y) = A (y),
F F F F F F
byEquation(20). Thiscompletestheproof.
Theorem8. ANS A = (A , A , A )inaBCK-algebraXisan(∈,∈)-neutrosophicidealofXifandonlyif
T I F
A = (A , A , A )satisfies
T I F
  A (x) ≥ A (y)∧A (z) 
 T T T
(∀x,y,z ∈ X) x∗y ≤ z ⇒ A (x) ≥ A (y)∧A (z)  (22)
 I I I 
 A (x) ≤ A (y)∨A (z)
F F F
Proof. Let A = (A , A , A ) bean (∈, ∈)-neutrosophicidealof X,andlet x,y,z ∈ X besuchthat
T I F
x∗y ≤ z. UsingTheorem7andLemma2,wehave
A (x) ≥ A (x∗y)∧A (y) ≥ A (y)∧A (z),
T T T T T
A (x) ≥ A (x∗y)∧A (y) ≥ A (y)∧A (z),
I I I I I
A (x) ≤ A (x∗y)∨A (y) ≤ A (y)∨A (z).
F F F F F
Mathematics2018,6,74 10of16
Conversely,assumethat A = (A , A , A )satisfiesEquation(22). Because0∗x ≤ xforallx ∈ X,
T I F
itfollowsfromEquation(22)that
A (0) ≥ A (x)∧A (x) = A (x),
T T T T
A (0) ≥ A (x)∧A (x) = A (x),
I I I I
A (0) ≤ A (x)∨A (x) = A (x),
F F F F
forallx ∈ X. Becausex∗(x∗y) ≤ yforallx,y ∈ X,wehave
A (x) ≥ A (x∗y)∧A (y),
T T T
A (x) ≥ A (x∗y)∧A (y),
I I I
A (x) ≤ A (x∗y)∨A (y),
F F F
for all x,y ∈ X by Equation (22). It follows from Theorem 7 that A = (A , A , A ) is an
T I F
(∈,∈)-neutrosophicidealofX.
Theorem9. If A = (A , A , A )isan(∈,∈)-neutrosophicidealofaBCK/BCI-algebraX,thenthelower
T I F
neutrosophic∈Φ-subsetsofXare I-energeticsubsetsofXwhereΦ ∈ {T,I,F}.
Proof. Let x,a,u ∈ X, α,β ∈ (0,1], and γ ∈ [0,1) be such that x ∈ L∈(A;α), a ∈ L∈(A;β),
T I
andu ∈ L∈(A;γ). UsingTheorem7,wehave
F
α ≥ A (x) ≥ A (x∗y)∧A (y),
T T T
β ≥ A (a) ≥ A (a∗b)∧A (b),
I I I
γ ≤ A (u) ≤ A (u∗v)∨A (v),
F F F
forally,b,v ∈ X. Itfollowsthat
A (x∗y) ≤ αor A (y) ≤ α,thatis,x∗y ∈ L∈(A;α)ory ∈ L∈(A;α);
T T T T
A (a∗b) ≤ βor A (b) ≤ β,thatis,a∗b ∈ L∈(A;β)orb ∈ L∈(A;β);
I I T T
and
A (u∗v) ≥ γor A (v) ≥ γ,thatis,u∗v ∈ L∈(A;γ)orv ∈ L∈(A;γ).
F F T T
Hence {y,x ∗ y} ∩ L∈(A;α), {b,a ∗ b} ∩ L∈(A;β), and {v,u ∗ v} ∩ L∈(A;γ) are nonempty,
T I F
andthereforeL∈(A;α),L∈(A;β)andL∈(A;γ)are I-energeticsubsetsofX.
T I F
Corollary2. If A = (A , A , A )isan(∈,∈)-neutrosophicidealofaBCK/BCI-algebraX,thenthestrong
T I F
lowerneutrosophic∈Φ-subsetsofXare I-energeticsubsetsofXwhereΦ ∈ {T,I,F}.
Proof. Straightforward.
Theorem10. Let(α,β,γ) ∈ Λ ×Λ ×Λ ,whereΛ ,Λ ,andΛ aresubsetsof[0,1]. If A = (A , A ,
T I F T I F T I
A )isan(∈,∈)-neutrosophicidealofaBCK-algebraX,then
F
(1) the(strong)upperneutrosophic∈Φ-subsetsofXarerightstablewhereΦ ∈ {T,I,F};
(2) the(strong)lowerneutrosophic∈Φ-subsetsofXarerightvanishedwhereΦ ∈ {T,I,F}.
Proof. (1) Let x ∈ X, a ∈ U∈(A;α), b ∈ U∈(A;β), and c ∈ U∈(A;γ). Then A (a) ≥ α, A (b) ≥ β,
T I F T I
and A (c) ≤ γ. Because a∗x ≤ a, b∗x ≤ b, and c∗x ≤ c, it follows from Lemma 2 that A (a∗
F T
x) ≥ A (a) ≥ α, A (b∗x) ≥ A (b) ≥ β, and A (c∗x) ≤ A (c) ≤ γ; that is, a∗x ∈ U∈(A;α),
T I I F F T