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

NeutrosophicPermeableValuesandEnergetic SubsetswithApplicationsin BCK/BCI-Algebras

YoungBaeJun 1,*,FlorentinSmarandache 2 ID ,Seok-ZunSong 3 ID andHashemBordbar 4 ID

1 DepartmentofMathematicsEducation,GyeongsangNationalUniversity,Jinju52828,Korea

2 Mathematics&ScienceDepartment,UniversityofNewMexico,705GurleyAve.,Gallup,NM87301,USA; fsmarandache@gmail.com

4

3 DepartmentofMathematics,JejuNationalUniversity,Jeju63243,Korea;szsong@jejunu.ac.kr

PostdoctoralResearchFellow,ShahidBeheshtiUniversity,Tehran,District1, DaneshjouBoulevard1983969411,Iran;bordbar.amirh@gmail.com

* Correspondence:skywine@gmail.com

Received:14March2018;Accepted:2May2018;Published:7May2018

Abstract: Theconceptofa (∈, ∈)-neutrosophicidealisintroduced,anditscharacterizationsare established.Thenotionsofneutrosophicpermeablevaluesareintroduced,andrelatedpropertiesare investigated.Conditionsfortheneutrosophiclevelsetstobeenergetic,rightstable,andrightvanished arediscussed.Relationsbetweenneutrosophicpermeable S-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.applieditto BCK/BCI-algebras(see[7–12]).Junetal.[8]introducedthenotionsofenergetic subsets,rightvanishedsubsets,rightstablesubsets,and(anti-)permeablevaluesin BCK/BCI-algebras andinvestigatedrelationsbetweenthesesets.

Inthispaper,weintroducethenotionsofneutrosophicpermeable S-values,neutrosophic permeable I-values, (∈, ∈)-neutrosophicideals,neutrosophicanti-permeable S-values, andneutrosophicanti-permeable I-values,whicharemotivatedbytheideaofsubalgebras

Mathematics 2018, 6,74;doi:10.3390/math6050074 www.mdpi.com/journal/mathematics

(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-)permeable S-or I-value.Weconsiderconditionsfortheupper(lower)neutrosophic ∈Φ-subsetsto berightstable(rightvanished)subsets.Weestablishrelationsbetweenneutrosophic(anti-)permeable S-and I-values.

2.Preliminaries

Analgebra (X; ∗,0) oftype (2,0) iscalleda BCI-algebra ifitsatisfiesthefollowingconditions:

(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)

Ifa BCI-algebra X satisfiesthefollowingidentity: (V) (∀x ∈ X)(0 ∗ x = 0), then X iscalleda BCK-algebra. Any BCK/BCI-algebra X satisfiesthefollowingconditions:

(∀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 ifandonlyif x ∗ y = 0. Anonemptysubset S ofa BCK/BCI-algebra X iscalleda subalgebra of X if x ∗ y ∈ S forall x, y ∈ S. Asubset I ofa BCK/BCI-algebra X iscalledan ideal of X if itsatisfiesthefollowing: 0 ∈ I,(5) (∀x, y ∈ X) (x ∗ y ∈ I, y ∈ I → x ∈ I) .(6)

Wereferthereadertothebooks[13]and[14]forfurtherinformationregarding BCK/BCI-algebras.

Foranyfamily {ai | i ∈ Λ} ofrealnumbers,wedefine

{ai | i ∈ Λ} = sup{ai | i ∈ Λ} and {ai | i ∈ Λ} = inf{ai | i ∈ Λ}.

If Λ = {1,2},wealsouse a1 ∨ a2 and a1 ∧ a2 insteadof {ai | i ∈{1,2}} and {ai | i ∈{1,2}}, respectively.

Welet X beanonemptyset.ANSin X (see[1])isastructureoftheform

A := { x; AT (x), AI (x), AF (x) | x ∈ X}, where AT : X → [0,1] isatruthmembershipfunction, AI : X → [0,1] isanindeterminatemembership function,and AF : X → [0,1] isafalsemembershipfunction.Forthesakeofsimplicity,weusethe symbol A =(AT , AI , AF ) fortheNS

A := { x; AT (x), AI (x), AF (x) | x ∈ X}

Asubset A ofa BCK/BCI-algebra X issaidtobe S-energetic(see[8])ifitsatisfies

(∀x, y ∈ X) (x ∗ y ∈ A ⇒{x, y}∩ A = ∅) .(7)

Asubset A ofa BCK/BCI-algebra X issaidtobe I-energetic (see[8])ifitsatisfies

(∀x, y ∈ X) (y ∈ A ⇒{x, y ∗ x}∩ A = ∅) .(8)

Asubset A ofa BCK/BCI-algebra X issaidtobe rightvanished (see[8])ifitsatisfies (∀x, y ∈ X) (x ∗ y ∈ A ⇒ x ∈ A) .(9)

Asubset A ofa BCK/BCI-algebra X issaidtobe rightstable (see[8])if A ∗ X := {a ∗ x | a ∈ A, x ∈ X}⊆ A

3.NeutrosophicPermeableValues

GivenaNS A =(AT , AI , AF ) inaset X, α, β ∈ (0,1] and γ ∈ [0,1),weconsiderthefollowingsets:

U∈ T (A; α)= {x ∈ X | AT (x) ≥ α}, U∈ T (A; α)∗ = {x ∈ X | AT (x) > α},

U∈ I (A; β)= {x ∈ X | AI (x) ≥ β}, U∈ I (A; β)∗ = {x ∈ X | AI (x) > β},

U∈ F (A; γ)= {x ∈ X | AF (x) ≤ γ}, U∈ F (A; γ)∗ = {x ∈ X | AF (x) < γ},

L∈ T (A; α)= {x ∈ X | AT (x) ≤ α}, L∈ T (A; α)∗ = {x ∈ X | AT (x) < α},

L∈ I (A; β)= {x ∈ X | AI (x) ≤ β}, L∈ I (A; β)∗ = {x ∈ X | AI (x) < β},

L∈ F (A; γ)= {x ∈ X | AF (x) ≥ γ}, L∈ F (A; γ)∗ = {x ∈ X | AF (x) > γ}

Wesay U∈ T (A; α), U∈ I (A; β),and U∈ F (A; γ) are upperneutrosophic ∈Φ-subsets of X,and L∈ T (A; α), L∈ I (A; β),and L∈ F (A; γ) are lowerneutrosophic ∈Φ-subsets of X,where Φ ∈{T, I, F}.Wesay U∈ T (A; α)∗ , U∈ I (A; β)∗,and U∈ F (A; γ)∗ are strongupperneutrosophic ∈Φ-subsets of X,and L∈ T (A; α)∗ , L∈ I (A; β)∗ , and L∈ F (A; γ)∗ are stronglowerneutrosophic ∈Φ-subsets of X,where Φ ∈{T, I, F}.

Definition1 ([7]). ANS A =(AT , AI , AF ) ina BCK/BCI-algebra X iscalledan (∈, ∈)neutrosophicsubalgebraofXifthefollowingassertionsarevalid:

x ∈ U∈ T (A; αx ), y ∈ U∈ T (A; αy ) ⇒ x ∗ y ∈ U∈ T (A; αx ∧ αy ), x ∈ U∈ I (A; βx ), y ∈ U∈ I (A; βy ) ⇒ x ∗ y ∈ U∈ I (A; βx ∧ βy ), x ∈ U∈ F (A; γx ), y ∈ U∈ F (A; γy ) ⇒ x ∗ y ∈ U∈ F (A; γx ∨ γy ),

(10) forallx, y ∈ X, αx, αy, βx, βy ∈ (0,1] and γx, γy ∈ [0,1).

Lemma1 ([7]). ANS A =(AT , AI , AF ) ina BCK/BCI-algebra X isan (∈, ∈)-neutrosophicsubalgebraof

Proof. Straightforward.

Theorem1. If A =(AT , AI , AF ) isan (∈, ∈)-neutrosophicsubalgebraofa BCK/BCI-algebra X,thenthe lowerneutrosophic ∈Φ-subsetsofXareS-energeticsubsetsofX,where Φ ∈{T, I, F} Proof. Let x, y ∈ X and α ∈ (0,1] besuchthat x ∗ y ∈ L∈ T (A; α).Then

α ≥ AT (x ∗ y) ≥ AT (x) ∧ AT (y), andthus AT (x) ≤ α or AT (y) ≤ α;thatis, x ∈ L∈ T (A; α) or y ∈ L∈ T (A; α).Thus {x, y}∩ L∈ T (A; α) = ∅ Therefore L∈ T (A; α) isan S-energeticsubsetof X.Similarly,wecanverifythat L∈ I (A; β) isan S-energetic subsetof X.Welet x, y ∈ X and γ ∈ [0,1) besuchthat x ∗ y ∈ L∈ F (A; γ).Then

γ ≤ AF (x ∗ y) ≤ AF (x) ∨ AF (y).

Itfollowsthat AF (x) ≥ γ or AF (y) ≥ γ;thatis, x ∈ L∈ F (A; γ) or y ∈ L∈ F (A; γ).Hence {x, y}∩ L∈ F (A; γ) = ∅,andtherefore L∈ F (A; γ) isan S-energeticsubsetof X Corollary1. If A =(AT , AI , AF ) isan (∈, ∈)-neutrosophicsubalgebraofa BCK/BCI-algebra X,thenthe stronglowerneutrosophic ∈Φ-subsetsofXareS-energeticsubsetsofX,where Φ ∈{T, I, F}

Proof. Straightforward. TheconverseofTheorem 1 isnottrue,asseeninthefollowingexample.

Example1. Considera BCK-algebra X = {0,1,2,3,4} withthebinaryoperation ∗ thatisgiveninTable 1 (see[14]). Table1. Cayleytableforthebinaryoperation“∗”. *01234 000000 110000 221001 332102 441110 LetA =(AT , AI , AF ) beaNSinXthatisgiveninTable 2. Table2. Tabulationrepresentationof A =(AT , AI , AF ) XAT (x) A I (x) AF (x) 00.60.80.2 10.40.50.7 20.40.50.6 30.40.50.5 40.70.80.2 If α ∈ [0.4,0.6), β ∈ [0.5,0.8),and γ ∈ (0.2,0.5],then L∈ T (A; α)= {1,2,3}, L∈ I (A; β)= {1,2,3}, andL∈ F (A; γ)= {1,2,3} areS-energeticsubsetsofX.Because AT (4 ∗ 4)= AT (0)= 0.6 0.7 = AT (4) ∧ AT (4)

   (13)

and/or AF (3 ∗ 2)= AF (1)= 0.7 0.6 = AF (3) ∨ AF (2), itfollowsfromLemma 1 thatA =(AT , AI , AF ) isnotan (∈, ∈)-neutrosophicsubalgebraofX. Definition2. Let A =(AT , AI , AF ) beaNSina BCK/BCI-algebra X and (α, β, γ) ∈ ΛT × ΛI × ΛF, where ΛT , ΛI ,and ΛF aresubsetsof [0,1].Then (α, β, γ) iscalledaneutrosophicpermeable S-valuefor A =(AT , AI , AF ) ifthefollowingassertionisvalid: (∀x, y ∈ X)    x ∗ y ∈ U∈ T (A; α) ⇒ AT (x) ∨ AT (y) ≥ α, x ∗ y ∈ U∈ I (A; β) ⇒ AI (x) ∨ AI (y) ≥ β, x ∗ y ∈ U∈ F (A; γ) ⇒ AF (x) ∧ AF (y) ≤ γ

Example2. LetX = {0,1,2,3,4} beasetwiththebinaryoperation ∗ thatisgiveninTable 3 Table3. Cayleytableforthebinaryoperation“∗”. *01234 000000 110110 222020 333303 444440

Then (X, ∗,0) isaBCK-algebra(see[14]).LetA =(AT, AI, AF) beaNSinXthatisgiveninTable 4. Table4. Tabulationrepresentationof A =(AT , AI , AF )

XAT (x) A I (x) AF (x) 00.20.30.7 10.60.40.6 20.50.30.4 30.40.80.5 40.70.60.2

Itisroutinetoverifythat (α, β, γ) ∈ (0,2,1] × (0.3,1] × [0,0.7) isaneutrosophicpermeable S-valuefor A =(AT , AI , AF ) Theorem2. Let A =(AT , AI , AF ) beaNSina BCK/BCI-algebra X and (α, β, γ) ∈ ΛT × ΛI × ΛF, where ΛT , ΛI ,and ΛF aresubsetsof [0,1].IfA =(AT , AI , AF ) satisfiesthefollowingcondition:

Similarly,if x ∗ y ∈ U∈ I (A; β) for x, y ∈ X,then AI (x) ∨ AI (y) ≥ β.Now,let a, b ∈ X besuchthat a ∗ b ∈ U∈ F (A; γ).Then

γ ≥ AF (a ∗ b) ≥ AF (a) ∧ AF (b)

Therefore (α, β, γ) isaneutrosophicpermeable S-valuefor A =(AT , AI , AF ).

Theorem3. Let A =(AT , AI , AF ) beaNSina BCK-algebra X and (α, β, γ) ∈ ΛT × ΛI × ΛF,where ΛT , ΛI ,and ΛF aresubsetsof [0,1].IfA =(AT , AI , AF ) satisfiesthefollowingconditions: (∀x ∈ X) (AT (0) ≤ AT (x), AI (0) ≤ AI (x), AF (0) ≥ AF (x)) (15) and (∀x, y ∈ X)   

AT (x) ≤ AT (x ∗ y) ∨ AT (y)

AI (x) ≤ AI (x ∗ y) ∨ AI (y) AF (x) ≥ AF (x ∗ y) ∧ AF (y)

   ,(16) then (α, β, γ) isaneutrosophicpermeableS-valueforA =(AT , AI , AF )

Proof. Let x, y, a, b, u, v ∈ X besuchthat x ∗ y ∈ U∈ T (A; α), a ∗ b ∈ U∈ I (A; β),and u ∗ v ∈ U∈ F (A; γ).Then

α ≤ AT (x ∗ y) ≤ AT ((x ∗ y) ∗ x) ∨ AT (x)

= AT ((x ∗ x) ∗ y) ∨ AT (x)= AT (0 ∗ y) ∨ AT (x)

= AT (0) ∨ AT (x)= AT (x),

β ≤ AI (a ∗ b) ≤ AI ((a ∗ b) ∗ a) ∨ AI (a)

= AI ((a ∗ a) ∗ b) ∨ AI (a)= AI (0 ∗ b) ∨ AI (a)

= AI (0) ∨ AI (a)= AI (a), and

γ ≥ AF (u ∗ v) ≥ AF ((u ∗ v) ∗ u) ∧ AF (u)

= AF ((u ∗ u) ∗ v) ∧ AF (u)= AF (0 ∗ v) ∧ AF (v) = AF (0) ∧ AF (v)= AF (v)

byEquations(3),(V),(15),and(16).Itfollowsthat

AT (x) ∨ AT (y) ≥ AT (x) ≥ α, AI (a) ∨ AI (b) ≥ AI (a) ≥ β, AF (u) ∧ AF (v) ≤ AF (u) ≤ γ

Therefore (α, β, γ) isaneutrosophicpermeable S-valuefor A =(AT , AI , AF ).

Theorem4. Let A =(AT , AI , AF ) beaNSina BCK/BCI-algebra X and (α, β, γ) ∈ ΛT × ΛI × ΛF, where ΛT , ΛI ,and ΛF aresubsetsof [0,1].If (α, β, γ) isaneutrosophicpermeable S-valuefor A =(AT , AI , AF ),thenupperneutrosophic ∈Φ-subsetsofXareS-energeticwhere Φ ∈{T, I, F}

Proof. Let x, y, a, b, u, v ∈ X besuchthat x ∗ y ∈ U∈ T (A; α), a ∗ b ∈ U∈ I (A; β),and u ∗ v ∈ U∈ F (A; γ). UsingEquation (13),wehave AT (x) ∨ AT (y) ≥ α, AI (a) ∨ AI (b) ≥ β,and AF (u) ∧ AF (v) ≤ γ. Itfollowsthat

AT (x) ≥ α or AT (y) ≥ α,thatis, x ∈ U∈ T (A; α) or y ∈ U∈ T (A; α);

AI (a) ≥ β or AI (b) ≥ β,thatis, a ∈ U∈ I (A; β) or b ∈ U∈ I (A; β); and

AF (u) ≤ γ or AF (v) ≤ γ,thatis, u ∈ U∈ F (A; γ) or v ∈ U∈ F (A; γ)

Hence {x, y}∩ U∈ T (A; α) = ∅, {a, b}∩ U∈ I (A; β) = ∅,and {u, v}∩ U∈ F (A; γ) = ∅ Therefore U∈ T (A; α), U∈ I (A; β),and U∈ F (A; γ) are S-energeticsubsetsof X.

Definition3. Let A =(AT , AI , AF ) beaNSina BCK/BCI-algebra X and (α, β, γ) ∈ ΛT × ΛI × ΛF, where ΛT , ΛI ,and ΛF aresubsetsof [0,1].Then (α, β, γ) iscalledaneutrosophicanti-permeable S-valuefor A =(AT , AI , AF ) ifthefollowingassertionisvalid: (∀x, y ∈ X)    x ∗ y ∈ L∈ T (A; α) ⇒ AT (x) ∧ AT (y) ≤ α, x ∗ y ∈ L∈ I (A; β) ⇒ AI (x) ∧ AI (y) ≤ β, x ∗ y ∈ L∈ F (A; γ) ⇒ AF (x) ∨ AF (y) ≥ γ

   .(17) Example3. LetX = {0,1,2,3,4} beasetwiththebinaryoperation ∗ thatisgiveninTable 5 Table5. Cayleytableforthebinaryoperation“∗”. *01234 000000 110010 221020 333303 444440 Then (X, ∗,0) isaBCK-algebra(see[14]).LetA =(AT, AI, AF) beaNSinXthatisgiveninTable 6. Table6. Tabulationrepresentationof A =(AT , AI , AF ) XAT (x) A I (x) AF (x) 00.70.60.4 10.40.50.6 20.40.50.6 30.50.20.7 40.30.30.9

Itisroutinetoverifythat (α, β, γ) ∈ (0.3,1] × (0.2,1] × [0,0.9) isaneutrosophicanti-permeable S-value forA =(AT , AI , AF )

Theorem5. Let A =(AT , AI , AF ) beaNSina BCK/BCI-algebra X and (α, β, γ) ∈ ΛT × ΛI × ΛF, where ΛT , ΛI ,and ΛF aresubsetsof [0,1].If A =(AT , AI , AF ) isan (∈, ∈)-neutrosophicsubalgebraof X, then (α, β, γ) isaneutrosophicanti-permeableS-valueforA =(AT , AI , AF )

Proof. Let x, y, a, b, u, v ∈ X besuchthat x ∗ y ∈ L∈ T (A; α), a ∗ b ∈ L∈ I (A; β),and u ∗ v ∈ L∈ F (A; γ). UsingLemma 1,wehave

AT (x) ∧ AT (y) ≤ AT (x ∗ y) ≤ α,

AI (a) ∧ AI (b) ≤ AI (a ∗ b) ≤ β,

AF (u) ∨ AF (v) ≥ AF (u ∗ v) ≥ γ, andthus (α, β, γ) isaneutrosophicanti-permeable S-valuefor A =(AT , AI , AF ).

Theorem6. Let A =(AT , AI , AF ) beaNSina BCK/BCI-algebra X and (α, β, γ) ∈ ΛT × ΛI × ΛF, where ΛT , ΛI ,and ΛF aresubsetsof [0,1].If (α, β, γ) isaneutrosophicanti-permeable S-valuefor A =(AT , AI , AF ),thenlowerneutrosophic ∈Φ-subsetsofXareS-energeticwhere Φ ∈{T, I, F}

Proof. Let x, y, a, b, u, v ∈ X besuchthat x ∗ y ∈ L∈ T (A; α), a ∗ b ∈ L∈ I (A; β),and u ∗ v ∈ L∈ F (A; γ) UsingEquation (17),wehave AT (x) ∧ AT (y) ≤ α, AI (a) ∧ AI (b) ≤ β,and AF (u) ∨ AF (v) ≥ γ, whichimplythat

AT (x) ≤ α or AT (y) ≤ α,thatis, x ∈ L∈ T (A; α) or y ∈ L∈ T (A; α);

AI (a) ≤ β or AI (b) ≤ β,thatis, a ∈ L∈ I (A; β) or b ∈ L∈ I (A; β); and

AF (u) ≥ γ or AF (v) ≥ γ,thatis, u ∈ L∈ F (A; γ) or v ∈ L∈ F (A; γ).

Hence {x, y}∩ L∈ T (A; α) = ∅, {a, b}∩ L∈ I (A; β) = ∅,and {u, v}∩ L∈ F (A; γ) = ∅ Therefore L∈ T (A; α), L∈ I (A; β),and L∈ F (A; γ) are S-energeticsubsetsof X Definition4. ANS A =(AT , AI , AF ) ina BCK/BCI-algebra X iscalledan (∈, ∈)-neutrosophicidealof Xifthefollowingassertionsarevalid: (∀x ∈ X)

(A;

.(20) Proof. AssumethatEquation (20) isvalid,andlet x ∈ U∈ T (A; α), a ∈ U∈ I (A; β),and u ∈ U∈ F (A; γ) forany x, a, u ∈ X, α, β ∈ (0,1] and γ ∈ [0,1).Then AT (0) ≥ AT (x) ≥ α, AI (0) ≥ AI (a) ≥ β, and AF (0) ≤ AF (u) ≤ γ.Hence 0 ∈ U∈ T (A; α), 0 ∈ U∈ I (A; β),and 0 ∈ U∈ F (A; γ),andthus Equation (18) isvalid.Let x, y, a, b, u, v ∈ X besuchthat x ∗ y ∈ U∈ T (A; αx ), y ∈ U∈ T (A; αy ), a ∗ b ∈ U∈ I (A; βa ), b ∈ U∈ I (A; βb ), u ∗ v ∈ U∈ F (A; γu ),and v ∈ U∈ F (A; γv ) forall αx, αy, βa, βb ∈ (0,1]

and γu, γv ∈ [0,1).Then AT (x ∗ y) ≥ αx, AT (y) ≥ αy, AI (a ∗ b) ≥ βa, AI (b) ≥ βb, AF (u ∗ v) ≤ γu, and AF (v) ≤ γv.ItfollowsfromEquation(20)that

AT (x) ≥ AT (x ∗ y) ∧ AT (y) ≥ αx ∧ αy,

AI (a) ≥ AI (a ∗ b) ∧ AI (b) ≥ βa ∧ βb, AF (u) ≤ AF (u ∗ v) ∨ AF (v) ≤ γu ∨ γv

Hence x ∈ U∈ T (A; αx ∧ αy ), a ∈ U∈ I (A; βa ∧ βb ),and u ∈ U∈ F (A; γu ∨ γv ).Therefore A =(AT , AI , AF ) isan (∈, ∈)-neutrosophicidealof X.

Conversely,let A =(AT , AI , AF ) bean (∈, ∈)-neutrosophicidealof X.Ifthereexists x0 ∈ X suchthat AT (0) < AT (x0),then x0 ∈ U∈ T (A; α) and 0 / ∈ U∈ T (A; α),where α = AT (x0).Thisisa contradiction,andthus AT (0) ≥ AT (x) forall x ∈ X.Assumethat AT (x0) < AT (x0 ∗ y0) ∧ AT (y0) for some x0, y0 ∈ X.Taking α := AT (x0 ∗ y0) ∧ AT (y0) impliesthat x0 ∗ y0 ∈ U∈ T (A; α) and y0 ∈ U∈ T (A; α); but x0 / ∈ U∈ T (A; α).Thisisacontradiction,andthus AT (x) ≥ AT (x ∗ y) ∧ AT (y) forall x, y ∈ X Similarly,wecanverifythat AI (0) ≥ AI (x) ≥ AI (x ∗ y) ∧ AI (y) forall x, y ∈ X.Now,suppose that AF (0) > AF (a) forsome a ∈ X.Then a ∈ U∈ F (A; γ) and 0 / ∈ U∈ F (A; γ) bytaking γ = AF (a) Thisisimpossible,andthus AF (0) ≤ AF (x) forall x ∈ X.Supposethereexist a0, b0 ∈ X such that AF (a0) > AF (a0 ∗ b0) ∨ AF (b0),andtake γ := AF (a0 ∗ b0) ∨ AF (b0).Then a0 ∗ b0 ∈ U∈ F (A; γ), b0 ∈ U∈ F (A; γ),and a0 / ∈ U∈ F (A; γ),whichisacontradiction.Thus AF (x) ≤ AF (x ∗ y) ∨ AF (y) forall x, y ∈ X.Therefore A =(AT , AI , AF ) satisfiesEquation(20).

Lemma2. Every (∈, ∈)-neutrosophicidealA =(AT , AI , AF ) ofaBCK/BCI-algebraXsatisfies

(∀x, y ∈ X) (x ≤ y ⇒ AT (x) ≥ AT (y), AI (x) ≥ AI (y), AF (x) ≤ AF (y)) .(21)

Proof. Let x, y ∈ X besuchthat x ≤ y.Then x ∗ y = 0,andthus

AT (x) ≥ AT (x ∗ y) ∧ AT (y)= AT (0) ∧ AT (y)= AT (y), AI (x) ≥ AI (x ∗ y) ∧ AI (y)= AI (0) ∧ AI (y)= AI (y), AF (x) ≤ AF (x ∗ y) ∨ AF (y)= AF (0) ∨ AF (y)= AF (y), byEquation(20).Thiscompletestheproof.

Theorem8. ANS A =(AT , AI , AF ) ina BCK-algebra X isan (∈, ∈)-neutrosophicidealof X ifandonlyif A =(AT , AI , AF ) satisfies (∀x,

Conversely,assumethat A =(AT , AI , AF ) satisfiesEquation (22).Because 0 ∗ x ≤ x forall x ∈ X, itfollowsfromEquation(22)that

AT (0) ≥ AT (x) ∧ AT (x)= AT (x),

AI (0) ≥ AI (x) ∧ AI (x)= AI (x),

AF (0) ≤ AF (x) ∨ AF (x)= AF (x),

forall x ∈ X.Because x ∗ (x ∗ y) ≤ y forall x, y ∈ X,wehave

AT (x) ≥ AT (x ∗ y) ∧ AT (y),

AI (x) ≥ AI (x ∗ y) ∧ AI (y),

AF (x) ≤ AF (x ∗ y) ∨ AF (y),

forall x, y ∈ X byEquation (22).ItfollowsfromTheorem 7 that A =(AT , AI , AF ) isan (∈, ∈)-neutrosophicidealof X

Theorem9. If A =(AT , AI , AF ) isan (∈, ∈)-neutrosophicidealofa BCK/BCI-algebra X,thenthelower neutrosophic ∈Φ-subsetsofXareI-energeticsubsetsofXwhere Φ ∈{T, I, F}

Proof. Let x, a, u ∈ X, α, β ∈ (0,1],and γ ∈ [0,1) besuchthat x ∈ L∈ T (A; α), a ∈ L∈ I (A; β), and u ∈ L∈ F (A; γ).UsingTheorem 7,wehave

α ≥ AT (x) ≥ AT (x ∗ y) ∧ AT (y),

β ≥ AI (a) ≥ AI (a ∗ b) ∧ AI (b),

γ ≤ AF (u) ≤ AF (u ∗ v) ∨ AF (v),

forall y, b, v ∈ X.Itfollowsthat

AT (x ∗ y) ≤ α or AT (y) ≤ α,thatis, x ∗ y ∈ L∈ T (A; α) or y ∈ L∈ T (A; α);

AI (a ∗ b) ≤ β or AI (b) ≤ β,thatis, a ∗ b ∈ L∈ T (A; β) or b ∈ L∈ T (A; β); and

AF (u ∗ v) ≥ γ or AF (v) ≥ γ,thatis, u ∗ v ∈ L∈ T (A; γ) or v ∈ L∈ T (A; γ).

Hence {y, x ∗ y}∩ L∈ T (A; α), {b, a ∗ b}∩ L∈ I (A; β),and {v, u ∗ v}∩ L∈ F (A; γ) arenonempty, andtherefore L∈ T (A; α), L∈ I (A; β) and L∈ F (A; γ) are I-energeticsubsetsof X

Corollary2. If A =(AT , AI , AF ) isan (∈, ∈)-neutrosophicidealofa BCK/BCI-algebra X,thenthestrong lowerneutrosophic ∈Φ-subsetsofXareI-energeticsubsetsofXwhere Φ ∈{T, I, F}

Proof. Straightforward.

Theorem10. Let (α, β, γ) ∈ ΛT × ΛI × ΛF,where ΛT , ΛI ,and ΛF aresubsetsof [0,1].If A =(AT , AI , AF ) isan (∈, ∈)-neutrosophicidealofaBCK-algebraX,then

(1) the(strong)upperneutrosophic ∈Φ-subsetsofXarerightstablewhere Φ ∈{T, I, F}; (2) the(strong)lowerneutrosophic ∈Φ-subsetsofXarerightvanishedwhere Φ ∈{T, I, F}

Proof. (1)Let x ∈ X, a ∈ U∈ T (A; α), b ∈ U∈ I (A; β),and c ∈ U∈ F (A; γ).Then AT (a) ≥ α, AI (b) ≥ β, and AF (c) ≤ γ.Because a ∗ x ≤ a, b ∗ x ≤ b,and c ∗ x ≤ c,itfollowsfromLemma 2 that AT (a ∗ x) ≥ AT (a) ≥ α, AI (b ∗ x) ≥ AI (b) ≥ β,and AF (c ∗ x) ≤ AF (c) ≤ γ;thatis, a ∗ x ∈ U∈ T (A; α),

b ∗ x ∈ U∈ I (A; β),and c ∗ x ∈ U∈ F (A; γ).Hencetheupperneutrosophic ∈Φ-subsetsof X arerightstable where Φ ∈{T, I, F}.Similarly,thestrongupperneutrosophic ∈Φ-subsetsof X arerightstablewhere Φ ∈{T, I, F}.

(2)Assumethat x ∗ y ∈ L∈ T (A; α), a ∗ b ∈ L∈ I (A; β),and c ∗ d ∈ L∈ F (A; γ) forany x, y, a, b, c, d ∈ X. Then AT (x ∗ y) ≤ α, AI (a ∗ b) ≤ β,and AF (c ∗ d) ≥ γ.Because x ∗ y ≤ x, a ∗ b ≤ a, and c ∗ d ≤ c,itfollowsfromLemma 2 that α ≥ AT (x ∗ y) ≥ AT (x), β ≥ AI (a ∗ b) ≥ AI (a), and γ ≤ AF (c ∗ d) ≤ AF (c);thatis, x ∈ L∈ T (A; α), a ∈ L∈ I (A; β),and c ∈ L∈ F (A; γ).Thereforethelower neutrosophic ∈Φ-subsetsof X arerightvanishedwhere Φ ∈{T, I, F}.Inasimilarway,weknowthat thestronglowerneutrosophic ∈Φ-subsetsof X arerightvanishedwhere Φ ∈{T, I, F}

Definition5. Let A =(AT , AI , AF ) beaNSina BCK/BCI-algebra X and (α, β, γ) ∈ ΛT × ΛI × ΛF, where ΛT , ΛI ,and ΛF aresubsetsof [0,1].Then (α, β, γ) iscalledaneutrosophicpermeable I-valuefor A =(AT , AI , AF ) ifthefollowingassertionisvalid: (∀x, y ∈ X)   

Example4. (1)InExample 2, (α, β, γ) isaneutrosophicpermeableI-valueforA =(AT , AI , AF ). (2)Considera BCI-algebra X = {0,1, a, b, c} withthebinaryoperation ∗ thatisgiveninTable 7 (see[14]). Table7. Cayleytableforthebinaryoperation“∗”. *01abc 000 abc 110 abc aaa 0 cb bbbc 0 a cccba 0

LetA =(AT , AI , AF ) beaNSinXthatisgiveninTable 8

Table8. Tabulationrepresentationof A =(AT , AI , AF ) XAT (x) A I (x) AF (x) 00.330.380.77 10.440.480.66 a 0.550.680.44 b 0.660.580.44 c 0.660.680.55

Itisroutinetocheckthat (α, β, γ) ∈ (0.33,1] × (0.38,1] × [0,0.77) isaneutrosophicpermeable I-value forA =(AT , AI , AF ) Lemma3. IfaNS A =(AT , AI , AF ) ina BCK/BCI-algebra X satisfiestheconditionofEquation (14),then (∀x ∈ X) (AT (0) ≤ AT (x), AI (0) ≤ AI (x), AF (0) ≥ AF (x)) .(24)

Proof. Straightforward. Theorem11. IfaNS A =(AT , AI , AF ) ina BCK-algebra X satisfiestheconditionofEquation (14), theneveryneutrosophicpermeable I-valuefor A =(AT , AI , AF ) isaneutrosophicpermeable S-valuefor A =(AT , AI , AF )

Proof. Let (α, β, γ) beaneutrosophicpermeable I-valuefor A =(AT , AI , AF ).Let x, y, a, b, u, v ∈ X besuchthat x ∗ y ∈ U∈ T (A; α), a ∗ b ∈ U∈ I (A; β),and u ∗ v ∈ U∈ F (A; γ).ItfollowsfromEquations(23), (3),(III),and(V)andLemma 3 that

α ≤ AT ((x ∗ y) ∗ x) ∨ AT (x)= AT ((x ∗ x) ∗ y) ∨ AT (x)

= AT (0 ∗ y) ∨ AT (x)= AT (0) ∨ AT (x)= AT (x),

β ≤ AI ((a ∗ b) ∗ a) ∨ AI (a)= AI ((a ∗ a) ∗ b) ∨ AI (a)

= AI (0 ∗ b) ∨ AI (a)= AI (0) ∨ AI (a)= AI (a), and

γ ≥ AF ((u ∗ v) ∗ u) ∧ AF (u)= AF ((u ∗ u) ∗ v) ∧ AF (u) = AF (0 ∗ v) ∧ AF (u)= AF (0) ∧ AF (u)= AF (u).

Hence AT (x) ∨ AT (y) ≥ AT (x) ≥ α, AI (a) ∨ AI (b) ≥ AI (a) ≥ β, and AF (u) ∧ AF (v) ≤ AF (u) ≤ γ Therefore (α, β, γ) isaneutrosophicpermeable S-valuefor A =(AT , AI , AF )

GivenaNS A =(AT , AI , AF ) ina BCK/BCI-algebra X,anyupperneutrosophic ∈Φ-subsetsof X maynotbe I-energeticwhere Φ ∈{T, I, F},asseeninthefollowingexample. Example5. Considera BCK-algebra X = {0,1,2,3,4} withthebinaryoperation ∗ thatisgiveninTable 9 (see[14]).

Table9. Cayleytableforthebinaryoperation“∗”. *01234 000000 110000 221010 331100 442120

LetA =(AT , AI , AF ) beaNSinXthatisgiveninTable 10. Table10. Tabulationrepresentationof A =(AT , AI , AF ) XAT (x) A I (x) AF (x) 00.750.730.34 10.530.450.58 20.670.860.34 30.530.560.58 40.460.560.66

Then U∈ T (A;0.6)= {0,2}, U∈ I (A;0.7)= {0,2},and U∈ F (A;0.4)= {0,2}.Because 2 ∈{0,2} and {1,2 ∗ 1}∩{0,2} = ∅,weknowthat {0,2} isnotanI-energeticsubsetofX.

Wenowprovideconditionsfortheupperneutrosophic ∈Φ-subsetstobe I-energeticwhere Φ ∈{T, I, F}

Theorem12. Let A =(AT , AI , AF ) beaNSina BCK/BCI-algebra X and (α, β, γ) ∈ ΛT × ΛI × ΛF, where ΛT , ΛI ,and ΛF aresubsetsof [0,1].If (α, β, γ) isaneutrosophicpermeable I-valuefor A =(AT , AI , AF ),thentheupperneutrosophic ∈Φ-subsetsofXareI-energeticsubsetsofXwhere Φ ∈{T, I, F}.

Proof. Let x, a, u ∈ X and (α, β, γ) ∈ ΛT × ΛI × ΛF,where ΛT , ΛI ,and ΛF aresubsetsof [0,1] such that x ∈ U∈ T (A; α), a ∈ U∈ I (A; β),and u ∈ U∈ F (A; γ).Because (α, β, γ) isaneutrosophicpermeable I-valuefor A =(AT , AI , AF ),itfollowsfromEquation(23)that

AT (x ∗ y) ∨ AT (y) ≥ α, AI (a ∗ b) ∨ AI (b) ≥ β,and AF (u ∗ v) ∧ AF (v) ≤ γ

forall y, b, v ∈ X.Hence

AT (x ∗ y) ≥ α or AT (y) ≥ α,thatis, x ∗ y ∈ U∈ T (A; α) or y ∈ U∈ T (A; α);

AI (a ∗ b) ≥ β or AI (b) ≥ β,thatis, a ∗ b ∈ U∈ I (A; β) or b ∈ U∈ I (A; β); and

AF (u ∗ v) ≤ γ or AF (v) ≤ γ,thatis, u ∗ v ∈ U∈ F (A; γ) or v ∈ U∈ F (A; γ)

Hence {y, x ∗ y}∩ U∈ T (A; α), {b, a ∗ b}∩ U∈ I (A; β),and {v, u ∗ v}∩ U∈ F (A; γ) arenonempty, andthereforetheupperneutrosophic ∈Φ-subsetsof X are I-energeticsubsetsof X where Φ ∈{T, I, F}

Theorem13. Let A =(AT , AI , AF ) beaNSina BCK/BCI-algebra X and (α, β, γ) ∈ ΛT × ΛI × ΛF, where ΛT , ΛI ,and ΛF aresubsetsof [0,1].IfA =(AT , AI , AF ) satisfiesthefollowingcondition: (∀x, y ∈ X)   

AT (x) ≤ AT (x ∗ y) ∨ AT (y)

AI (x) ≤ AI (x ∗ y) ∨ AI (y)

AF (x) ≥ AF (x ∗ y) ∧ AF (y)

   ,(25) then (α, β, γ) isaneutrosophicpermeableI-valueforA =(AT , AI , AF ) Proof. Let x, a, u ∈ X and (α, β, γ) ∈ ΛT × ΛI × ΛF,where ΛT , ΛI ,and ΛF aresubsetsof [0,1] such that x ∈ U∈ T (A; α), a ∈ U∈ I (A; β),and u ∈ U∈ F (A; γ).UsingEquation(25),weobtain

α ≤ AT (x) ≤ AT (x ∗ y) ∨ AT (y), β ≤ AI (a) ≤ AI (a ∗ b) ∨ AI (b), γ ≥ AF (u) ≥ AF (u ∗ v) ∧ AF (v), forall y, b, v ∈ X.Therefore (α, β, γ) isaneutrosophicpermeable I-valuefor A =(AT , AI , AF ). CombiningTheorems 12 and 13,wehavethefollowingcorollary.

Corollary3. Let A =(AT , AI , AF ) beaNSina BCK/BCI-algebra X and (α, β, γ) ∈ ΛT × ΛI × ΛF, where ΛT , ΛI ,and ΛF aresubsetsof [0,1].If A =(AT , AI , AF ) satisfiestheconditionofEquation (25), thentheupperneutrosophic ∈Φ-subsetsofXareI-energeticsubsetsofXwhere Φ ∈{T, I, F}

Definition6. Let A =(AT , AI , AF ) beaNSina BCK/BCI-algebra X and (α, β, γ) ∈ ΛT × ΛI × ΛF, where ΛT , ΛI ,and ΛF aresubsetsof [0,1].Then (α, β, γ) iscalledaneutrosophicanti-permeable I-valuefor A =(AT , AI , AF ) ifthefollowingassertionisvalid: (∀x, y ∈ X

Theorem14. Let A =(AT , AI , AF ) beaNSina BCK/BCI-algebra X and (α, β, γ) ∈ ΛT × ΛI × ΛF, where ΛT , ΛI ,and ΛF aresubsetsof [0,1].If A =(AT , AI , AF ) satisfiestheconditionofEquation (19), then (α, β, γ) isaneutrosophicanti-permeableI-valueforA =(AT , AI , AF ).

Proof. Let x, a, u ∈ X besuchthat x ∈ L∈ T (A; α), a ∈ L∈ I (A; β),and u ∈ L∈ F (A; γ).Then

AT (x ∗ y) ∧ AT (y) ≤ AT (x) ≤ α,

AI (a ∗ b) ∧ AI (b) ≤ AI (a) ≤ β,

AF (u ∗ v) ∨ AF (v) ≥ AF (u) ≥ γ,

forall y, b, v ∈ X byEquation (20).Hence (α, β, γ) isa neutrosophicanti-permeable I-valuefor A =(AT, AI, AF)

Theorem15. Let A =(AT , AI , AF ) beaNSina BCK/BCI-algebra X and (α, β, γ) ∈ ΛT × ΛI × ΛF, where ΛT , ΛI ,and ΛF aresubsetsof [0,1].If (α, β, γ) isaneutrosophicanti-permeable I-valuefor A =(AT , AI , AF ),thenthelowerneutrosophic ∈Φ-subsetsofXareI-energeticwhere Φ ∈{T, I, F}

Proof. Let x ∈ L∈ T (A; α), a ∈ L∈ I (A; β),and u ∈ L∈ F (A; γ).Then AT (x ∗ y) ∧ AT (y) ≤ α, AI (a ∗ b) ∧ AI (b) ≤ β,and AF (u ∗ v) ∨ AF (v) ≥ γ forall y, b, v ∈ X byEquation(26).Itfollowsthat

AT (x ∗ y) ≤ α or AT (y) ≤ α,thatis, x ∗ y ∈ L∈ T (A; α) or y ∈ L∈ T (A; α);

AI (a ∗ b) ≤ β or AI (b) ≤ β,thatis, a ∗ b ∈ L∈ I (A; β) or b ∈ L∈ I (A; β); and AF (u ∗ v) ≥ γ or AF (v) ≥ γ,thatis, u ∗ v ∈ L∈ F (A; γ) or v ∈ L∈ F (A; γ)

Hence {y, x ∗ y}∩ L∈ T (A; α), {b, a ∗ b}∩ L∈ I (A; β) and {v, u ∗ v}∩ L∈ F (A; γ) arenonempty, andthereforethelowerneutrosophic ∈Φ-subsetsof X are I-energeticwhere Φ ∈{T, I, F}

CombiningTheorems 14 and 15,weobtainthefollowingcorollary.

Corollary4. Let A =(AT , AI , AF ) beaNSina BCK/BCI-algebra X and (α, β, γ) ∈ ΛT × ΛI × ΛF, where ΛT , ΛI ,and ΛF aresubsetsof [0,1].If A =(AT , AI , AF ) satisfiestheconditionofEquation (19), thenthelowerneutrosophic ∈Φ-subsetsofXareI-energeticwhere Φ ∈{T, I, F}

Theorem16. If A =(AT , AI , AF ) isan (∈, ∈)-neutrosophicsubalgebraofa BCK-algebra X,thenevery neutrosophicanti-permeable I-valuefor A =(AT , AI , AF ) isaneutrosophicanti-permeable S-valuefor A =(AT , AI , AF )

Proof. Let (α, β, γ) beaneutrosophicanti-permeable I-valuefor A =(AT , AI , AF ). Let x, y, a, b, u, v ∈ X besuchthat x ∗ y ∈ L∈ T (A; α), a ∗ b ∈ L∈ I (A; β),and u ∗ v ∈ L∈ F (A; γ).Itfollows fromEquations(26),(3),(III),and(V)andProposition 1 that

α ≥ AT ((x ∗ y) ∗ x) ∧ AT (x)= AT ((x ∗ x) ∗ y) ∧ AT (x)

= AT (0 ∗ y) ∧ AT (x)= AT (0) ∧ AT (x)= AT (x),

β ≥ AI ((a ∗ b) ∗ a) ∧ AI (a)= AI ((a ∗ a) ∗ b) ∧ AI (a)

= AI (0 ∗ b) ∧ AI (a)= AI (0) ∧ AI (a)= AI (a), and

γ ≤ AF ((u ∗ v) ∗ u) ∨ AF (u)= AF ((u ∗ u) ∗ v) ∨ AF (u)

= AF (0 ∗ v) ∨ AF (u)= AF (0) ∨ AF (u)= AF (u).

Hence AT (x) ∧ AT (y) ≤ AT (x) ≤ α, AI (a) ∧ AI (b) ≤ AI (a) ≤ β,and AF(u) ∨ AF(v) ≥ AF(u) ≥ γ. Therefore (α, β, γ) isaneutrosophicanti-permeable S-valuefor A =(AT , AI , AF ).

4.Conclusions

Usingthenotionsofsubalgebrasandidealsin BCK/BCI-algebras,Junetal.[8]introducedthe notionsofenergeticsubsets,rightvanishedsubsets,rightstablesubsets,and(anti-)permeablevalues in BCK/BCI-algebras,aswellasinvestigatedrelationsbetweenthesesets.Asamoregeneralplatform thatextendstheconceptsofclassicandfuzzysets,intuitionisticfuzzysets,andinterval-valued (intuitionistic)fuzzysets,thenotionofNStheoryhasbeendevelopedbySmarandache(see[1,2])and hasbeenappliedtovariousparts:patternrecognition,medicaldiagnosis,decision-makingproblems, andsoon(see[3–6]).Inthisarticle,wehaveintroducedthenotionsofneutrosophicpermeable S-values, neutrosophicpermeable I-values, (∈, ∈)-neutrosophicideals,neutrosophicanti-permeable S-values, andneutrosophicanti-permeable I-values,whicharemotivatedbytheideaofsubalgebras(s-values) andideals(I-values),andhaveinvestigatedtheirproperties.Wehaveconsideredcharacterizations of (∈, ∈)-neutrosophicidealsandhavediscussedconditionsforthelower(upper)neutrosophic ∈Φ-subsetstobe S-and I-energetic.Wehaveprovidedconditionsforatriple (α, β, γ) ofnumbersto beaneutrosophic(anti-)permeable S-or I-value,andhaveconsideredconditionsfortheupper(lower) neutrosophic ∈Φ-subsetstoberightstable(rightvanished)subsets.Wehaveestablishedrelations betweenneutrosophic(anti-)permeable S-and I-values.

AuthorContributions: Y.B.J.andS.-Z.S.initiatedthemainideaofthisworkandwrotethepaper.F.S.andH.B. performedthefindingoftheexamplesandcheckingofthecontents.Allauthorsconceivedanddesignedthenew definitionsandresultsandreadandapprovedthefinalmanuscriptforsubmission.

Funding: Thisresearchreceivednoexternalfunding.

Acknowledgments: Theauthorswishtothanktheanonymousreviewersfortheirvaluablesuggestions. ConflictsofInterest: Theauthorsdeclarenoconflictofinterest.

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c 2018bytheauthors.LicenseeMDPI,Basel,Switzerland.Thisarticleisanopenaccess articledistributedunderthetermsandconditionsoftheCreativeCommonsAttribution (CCBY)license(http://creativecommons.org/licenses/by/4.0/).