KYUNGPOOKMath.J.46(2006),409-416
SmarandacheFreshandCleanIdealsofSmarandache BCIalgebras
YoungBaeJun
DepartmentofMathematicsEducation,GyeongsangNationalUniversity,Chinju 660-701,Korea
e-mail : ybjun@gsnu.ac.kr
Abstract. ThenotionofSmarandachefreshandcleanidealsisintroduced,examplesare given,andrelatedpropertiesareinvestigated.Relationsbetween Q-Smarandachefreshidealsand Q-Smarandachecleanidealsaregiven.Extensionpropertiesfor Q-Smarandache freshidealsand Q-Smarandachecleanidealsareestablished.
1.Introduction
Generally,inanyhumanfield,a SmarandacheStructure onaset A meansa weakstructure W on A suchthatthereexistsapropersubset B of A whichisembeddedwithastrongstructure S.In[4],W.B.VasanthaKandasamystudiedthe conceptofSmarandachegroupoids,subgroupoids,idealofgroupoids,semi-normal subgroupoids,SmarandacheBolgroupoidsandstrongBolgroupoidsandobtained manyinterestingresultsaboutthem.Smarandachesemigroupsareveryimportantforthestudyofcongruences,anditwasstudiedbyR.Padilla[3].Itwillbe veryinterestingtostudytheSmarandachestructurein BCK/BCI-algebras.In [1],Y.B.JundiscussedtheSmarandachestructurein BCI-algebras.HeintroducedthenotionofSmarandache(positiveimplicative,commutative,implicative) BCI-algebras,SmarandachesubalgebrasandSmarandacheideals,andinvestigated somerelatedproperties.InthispaperwedealwithSmarandacheidealstructures inSmarandache BCI-algebras.WeintroducethenotionofSmarandachefresh idealsandSmarandachecleanidealsinSmarandache BCI-algebras,andinvestigateitsusefulproperties.Wegiverelationsbetween Q-Smarandachefreshideals and Q-Smarandachecleanideals.Wealsoestablishextensionpropertiesfor QSmarandachefreshidealsand Q-Smarandachecleanideals.
2.Preliminaries
Analgebra(X; ∗, 0)oftype(2, 0)iscalleda BCI-algebra ifitsatisfiesthe
ReceivedMarch24,2005.
2000MathematicsSubjectClassification:06F35,03G25.
Keywordsandphrases:Smarandache BCI-algebra,Smarandache(fresh,clean)ideal. ThispaperwassupportedbyKoreaResearchFoundationGrant(KRF-2003-005C00013). 409
followingconditions:
(a1) (∀x,y,z ∈ X)(((x ∗ y) ∗ (x ∗ z)) ∗ (z ∗ y)=0), (a2) (∀x,y ∈ X)((x ∗ (x ∗ y)) ∗ y =0), (a3) (∀x ∈ X)(x ∗ x =0), (a4) (∀x,y ∈ X)(x ∗ y =0,y ∗ x =0 ⇒ x = y)
Ifa BCI-algebra X satisfiesthefollowingidentity: (a5) (∀x ∈ X)(0 ∗ x =0), then X iscalleda BCK-algebra.Wecandefineapartialorder‘≤’on X by x ≤ y ifandonlyif x ∗ y =0 Every BCI-algebra X hasthefollowingproperties:
(b1) (∀x ∈ X)(x ∗ 0= x) (b2) (∀x,y,z ∈ X)((x ∗ y) ∗ z =(x ∗ z) ∗ y) (b3) (∀x,y,z ∈ X)(x ≤ y ⇒ x ∗ z ≤ y ∗ z,z ∗ y ≤ z ∗ x) (b4) (∀x,y ∈ X)(x ∗ (x ∗ (x ∗ y))= x ∗ y)
A Smarandache BCI-algebra [1]isdefinedtobea BCI-algebra X inwhich thereexistsapropersubset Q of X suchthat (s1) 0 ∈ Q and |Q|≥ 2, (s2) Q isa BCK-algebraundertheoperationof X.
3.Smarandachefreshideals
Inwhatfollows,let X and Q denoteaSmarandache BCI-algebraanda BCKalgebrawhichisproperlycontainedin X, respectively.
Definition3.1. Anonemptysubset I of X iscalleda Smarandacheideal of X relatedto Q (orbriefly, Q-Smarandacheideal of X)[1]ifitsatisfies: (c1) 0 ∈ I, (c2) (∀x ∈ Q)(∀y ∈ I)(x ∗ y ∈ I ⇒ x ∈ I)
If I isaSmarandacheidealof X relatedtoevery BCK-algebracontainedin X, we simplysaythat I isa Smarandacheideal of X.
Proposition3.2. If Q satisfies Q ∗ X ⊂ Q, thenevery Q-Smarandacheideal I of X satisfiesthefollowingimplication.
(1)(∀x,y ∈ I)(∀z ∈ Q)((z ∗ y) ∗ x =0 ⇒ z ∈ I)
SmarandacheFreshandCleanIdealsofSmarandache BCI -algebras 411
Proof. Assumethat Q ∗ X ⊂ Q andlet I bea Q-Smarandacheidealof X. Suppose that(z ∗ y) ∗ x =0forall x,y ∈ I and z ∈ Q. Then z ∗ y ∈ Q byassumption,and (z ∗ y) ∗ x ∈ I, andso z ∗ y ∈ I by(c2).Since y ∈ I, itfollowsfrom(c2)againthat z ∈ I. Thiscompletestheproof.
Problem3.3. Ifanonemptysubset I of X satisfiesthecondition(1),thenis I a Q-Smarandacheidealof X? Definition3.4. Anonemptysubset I of X iscalleda Smarandachefreshideal of X relatedto Q (orbriefly, Q-Smarandachefreshideal of X)ifitsatisfiesthe condition(c1)and (c3) (∀x,y,z ∈ Q)((x ∗ y) ∗ z ∈ I,y ∗ z ∈ I ⇒ x ∗ z ∈ I)
Example3.5. Let X = {0, 1, 2, 3, 4, 5} beasetwiththefollowingCayleytable: ∗ 012345 0 000005 1 101015 2 220205 3 313035 4 444405 5 555550
Table3.1
Then(X; ∗, 0)isaSmarandache BCI-algebra.Notethat Q = {0, 1, 2, 3, 4} is a BCK-algebrawhichisproperlycontainedin X. Itiseasilycheckedthatsubsets I1 = {0, 1, 3} and I2 = {0, 1, 2, 3} are Q-Smarandachefreshidealsof X.
Theorem3.6. Let Q1 and Q2 be BCK-algebraswhichareproperlycontainedin X suchthat Q1 ⊂ Q2 . Thenevery Q2 -Smarandachefreshidealisa Q1 -Smarandache freshideal. Proof. Straightforward.
ThefollowingexampleshowsthattheconverseofTheorem3.6isnottruein general.
Example3.7. Let X = {0, 1, 2, 3, 4, 5} beasetwiththefollowingCayleytable:
∗ 012345 0 000044 1 100144 2 220244 3 333044 4 444400 5 544510
Table3.2
Then(X; ∗, 0)isaSmarandache BCI-algebra.Notethat Q1 = {0, 1} and Q2 = {0, 1, 2, 3} are BCK-algebraswhichareproperlycontainedin X. Itiseasily checkedthatasubset I = {0, 2, 3} isa Q1 -Smarandachefreshidealof X, butit isnota Q2 -Smarandachefreshidealof X since(1 ∗ 2) ∗ 3=0 ∗ 3=0 ∈ I and 2 ∗ 3=2 ∈ I but1 ∗ 3=1 / ∈ I. Proposition3.8. If I isa Q-Smarandachefreshidealof X, then (i) (∀x,y ∈ Q)((x ∗ y) ∗ y ∈ I ⇒ x ∗ y ∈ I) (ii) (∀x,y,z ∈ Q)((x ∗ y) ∗ z ∈ I ⇒ (x ∗ z) ∗ (y ∗ z) ∈ I)
Proof. (i).Assumethat(x ∗ y) ∗ y ∈ I forall x,y ∈ Q. Since y ∗ y =0 ∈ I by(a3) and(c1),itfollowsfrom(c3)that x ∗ y ∈ I whichisthedesiredresult.
(ii).Supposethat(x ∗ y) ∗ z ∈ I forall x,y,z ∈ Q. Since (((x ∗ z) ∗ (y ∗ z)) ∗ z) ∗ ((x ∗ y) ∗ z)=0 ∈ I, wehave((x ∗ (y ∗ z)) ∗ z) ∗ z =((x ∗ z) ∗ (y ∗ z)) ∗ z ∈ I. Applying(i)and(b2),we concludethat(x ∗ z) ∗ (y ∗ z)=(x ∗ (y ∗ z)) ∗ z ∈ I.
Theorem3.9. Every Q-Smarandachefreshidealwhichiscontainedin Q isa QSmarandacheideal.
Proof. Let I bea Q-Smarandachefreshidealof X whichiscontainedin Q andlet x ∈ Q and y ∈ I besuchthat x ∗ y ∈ I. Then(x ∗ y) ∗ 0= x ∗ y ∈ I and y ∗ 0= y ∈ I. Since x ∈ Q and y ∈ I ⊂ Q, itfollowsfrom(c3)and(b1)that x = x ∗ 0 ∈ I sothat I isa Q-Smarandacheidealof X.
ThefollowingexampleshowsthattheconverseofTheorem3.9isnottruein general.
Example3.10. Let X = {0, 1, 2, 3, 4, 5} beasetwiththefollowingCayleytable:
SmarandacheFreshandCleanIdealsofSmarandache BCI -algebras 413 ∗ 012345 0 000005 1 100015 2 210125 3 311035 4 444405 5 555550
Table3.3
Then(X; ∗, 0)isaSmarandache BCI-algebra.Notethat Q = {0, 1, 2, 3, 4} isa BCK-algebrawhichisproperlycontainedin X. Itiseasilycheckedthatasubset I = {0, 4} isa Q-Smarandacheidealof X whichisnota Q-Smarandachefreshideal of X since(2 ∗ 1) ∗ 3=0 ∈ I and1 ∗ 3=0 ∈ I, but2 ∗ 3=1 / ∈ I. Weprovideconditionsfora Q-Smarandacheidealtobea Q-Smarandachefresh ideal.
Theorem3.11. If I isa Q-Smarandacheidealof X suchthat (2)(∀x,y,z ∈ Q)((x ∗ y) ∗ z ∈ I ⇒ (x ∗ z) ∗ (y ∗ z) ∈ I), then I isa Q-Smarandachefreshidealof X. Proof. Assumethat(x∗y)∗z ∈ I and y∗z ∈ I forall x,y,z ∈ Q. Then(x∗z)∗(y∗z) ∈ I by(2),andso x ∗ z ∈ I by(c2).Therefore I isa Q-Smarandachefreshidealof X.
Corollary3.12. If I isa Q-Smarandacheidealof X thatsatisfiesthefollowing implication: (3)(∀x,y ∈ Q)((x ∗ y) ∗ y ∈ I ⇒ x ∗ y ∈ I), then I isa Q-Smarandachefreshidealof X. Proof. Let x,y,z ∈ Q besuchthat(x ∗ y) ∗ z ∈ I. Since (((x ∗ z) ∗ (y ∗ z)) ∗ z) ∗ ((x ∗ y) ∗ z)=0 ∈ I, itfollowsfrom(b2)and(c2)that((x ∗ (y ∗ z)) ∗ z) ∗ z =((x ∗ z) ∗ (y ∗ z)) ∗ z ∈ I sofrom(b2)and(3)that(x ∗ z) ∗ (y ∗ z)=(x ∗ (y ∗ z)) ∗ z ∈ I. Hence I isa Q-Smarandachefreshidealof X byTheorem3.11.
Proposition3.13. If I isa Q-Smarandachefreshidealof X whichiscontained in Q, then (4)(∀x,y ∈ Q)(∀z ∈ I)(((x ∗ y) ∗ y) ∗ z ∈ I ⇒ x ∗ y ∈ I)
Proof. Assumethat((x ∗ y) ∗ y) ∗ z ∈ I forall x,y ∈ Q and z ∈ I. If I isa QSmarandachefreshidealof X whichiscontainedin Q, then I isa Q-Smarandache idealof X (seeTheorem3.9).Using(c2),weknowthat(x ∗ y) ∗ y ∈ I andso x ∗ y ∈ I byProposition3.8(i).Thiscompletestheproof.
Theorem3.14 (ExtensionProperty). Let I and J be Q-Smarandacheidealsof X and I ⊂ J. If I isa Q-Smarandachefreshidealof X, thensois J. Proof. Let x,y,z ∈ Q besuchthat(x ∗ y) ∗ z ∈ J. Using(a3)and(b2),wehave ((x ∗ ((x ∗ y) ∗ z)) ∗ y) ∗ z =((x ∗ y) ∗ z) ∗ ((x ∗ y) ∗ z)=0 ∈ I. Since I isa Q-Smarandachefreshidealof X, itfollowsfromProposition3.8(ii)and (b2)that ((x ∗ z) ∗ (y ∗ z)) ∗ ((x ∗ y) ∗ z)=((x ∗ ((x ∗ y) ∗ z)) ∗ z) ∗ (y ∗ z) ∈ I ⊂ J sofrom(c2)that(x ∗ z) ∗ (y ∗ z) ∈ J. Thisprovesthat J isa Q-Smarandachefresh idealof X.
4.Smarandachecleanideals
Definition4.1. Anonemptysubset I of X iscalleda Smarandachecleanideal of X relatedto Q (orbriefly, Q-Smarandachecleanideal of X)ifitsatisfiesthe condition(c1)and (c4) (∀x,y ∈ Q)(∀z ∈ I)((x ∗ (y ∗ x)) ∗ z ∈ I ⇒ x ∈ I) Example4.2. Let X = {0, 1, 2, 3, 4, 5} beasetwiththefollowingCayleytable: ∗ 012345 0 000005 1 100005 2 210105 3 333005 4 444405 5 555550
Table4.1 Then(X; ∗, 0)isaSmarandache BCI-algebra.Notethat Q = {0, 1, 2, 3, 4} isa BCK-algebrawhichisproperlycontainedin X. Itiseasilycheckedthatasubset I = {0, 1, 2, 3} isa Q-Smarandachecleanidealof X.
Theorem4.3. Every Q-Smarandachecleanidealisa Q-Smarandacheideal. Proof. Let I bea Q-Smarandachecleanidealof X andlet x ∈ Q and z ∈ I besuch that x ∗ z ∈ I. Taking y = x in(c4),wehave (x ∗ (x ∗ x)) ∗ z =(x ∗ 0) ∗ z = x ∗ z ∈ I
SmarandacheFreshandCleanIdealsofSmarandache BCI -algebras 415 andso x ∈ I by(c4).Hence I isa Q-Smarandacheidealof X.
ThefollowingexampleshowsthattheconverseofTheorem4.3isnottruein general.
Example4.4. Let X = {0, 1, 2, 3, 4, 5} beaSmarandache BCI-algebradescribed inExample3.5.Itiseasilycheckedthatasubset I = {0, 2} isa Q-Smarandache idealof X, butitisnota Q-Smarandachecleanidealof X since(1 ∗ (3 ∗ 1)) ∗ 2= (1 ∗ 1) ∗ 2=0 ∗ 2=0 ∈ I and1 / ∈ I. Notethatif Q isimplicative,thatis, Q satisfiesthecondition: (∀x,y ∈ Q)(x = x ∗ (y ∗ x)),
thenevery Q-Smarandacheidealisa Q-Smarandachecleanideal.
Theorem4.5. Every Q-Smarandachecleanidealisa Q-Smarandachefreshideal.
Proof. Let I bea Q-Smarandachecleanidealof X. Assumethat(x ∗ y) ∗ z ∈ I and y ∗ z ∈ I forall x,y,z ∈ Q. Notethat (((x ∗ z) ∗ z) ∗ (y ∗ z)) ∗ ((x ∗ y) ∗ z)=(((x ∗ z) ∗ z) ∗ (y ∗ z)) ∗ ((x ∗ z) ∗ y)=0 ∈ I. Since I isa Q-Smarandacheideal,itfollowsfrom(b4)and(c2)that (x ∗ z) ∗ (x ∗ (x ∗ z))=(x ∗ (x ∗ (x ∗ z))) ∗ z =(x ∗ z) ∗ z ∈ I sofrom(b1)that((x ∗ z) ∗ (x ∗ (x ∗ z))) ∗ 0=(x ∗ z) ∗ (x ∗ (x ∗ z)) ∈ I. Thus x ∗ z ∈ I by(c4).Thiscompletestheproof.
TheconverseofTheorem4.5isnottrueingeneralasseeninthefollowing example. Example4.6. Let X = {0, 1, 2, 3, 4, 5} beaSmarandache BCI-algebradescribed inExample3.5.Then I := {0, 1, 3} isa Q-Smarandachefreshideal,butnota Q-Smarandachecleanidealsince(2 ∗ (4 ∗ 2)) ∗ 3=0
Weprovideaconditionfora Q-Smarandachefreshidealtobea Q-Smarandache cleanideal.
Theorem4.7. Ifa Q-Smarandachefreshideal I of X whichiscontainedin Q satisfiesthefollowingimplication: (5)(∀x,y ∈ Q)(y ∗ (y ∗ x) ∈ I ⇒ x ∗ (x ∗ y) ∈ I), then I isa Q-Smarandachecleanidealof X.
Proof. Let x,y ∈ Q and z ∈ I besuchthat(x ∗ (y ∗ x)) ∗ z ∈ I. If I isa QSmarandachefreshidealwhichiscontainedin Q, then I isa Q-Smarandacheideal (seeTheorem3.9).Thus x ∗ (y ∗ x) ∈ I by(c2).Since ((y ∗ (y ∗ x)) ∗ (y ∗ x)) ∗ (x ∗ (y ∗ x))=0 ∈ I,
itfollowsfrom(c2)that(y ∗ (y ∗ x)) ∗ (y ∗ x) ∈ I sofromProposition3.8(i)that y ∗ (y ∗ x) ∈ I. Hence,by(5), x ∗ (x ∗ y) ∈ I. Ontheotherhand,notethat ((x ∗ y) ∗ z) ∗ (x ∗ (y ∗ x))=0 ∈ I.
Using(c2),weget(x ∗ y) ∗ z ∈ I, andso x ∗ y ∈ I. Therefore x ∈ I. Thiscompletes theproof.
Proposition4.8. Every Q-Smarandachecleanideal I of X satisfiesthecondition (5) Proof. Let I bea Q-Smarandachecleanidealof X. Then I isa Q-Smarandache idealof X (seeTheorem4.3).Assumethat y ∗ (y ∗ x) ∈ I forall x,y ∈ Q. Since ((x ∗ (x ∗ y)) ∗ (y ∗ (x ∗ (x ∗ y)))) ∗ (y ∗ (y ∗ x))=0 ∈ I, itfollowsfrom(b1)and(c2)that ((x ∗ (x ∗ y)) ∗ (y ∗ (x ∗ (x ∗ y)))) ∗ 0=(x ∗ (x ∗ y)) ∗ (y ∗ (x ∗ (x ∗ y))) ∈ I sofrom(c4)that x ∗ (x ∗ y) ∈ I. Thiscompletestheproof.
Theorem4.9 (ExtensionProperty). Let I and J be Q-Smarandacheidealsof X and I ⊂ J ⊂ Q. If I isa Q-Smarandachecleanidealof X, thensois J. Proof. If I isa Q-Smarandachecleanidealof X, then I isa Q-Smarandachefresh idealof X (seeTheorem4.5),andso J isa Q-Smarandachefreshidealof X by Theorem3.14.Let x,y ∈ Q besuchthat y∗(y∗x) ∈ J. Since(y∗(y∗x))∗(y∗(y∗x))= 0 ∈ I, itfollowsfrom(b2)andProposition3.8(ii)that (y ∗ (y ∗ (y ∗ x))) ∗ ((y ∗ (y ∗ (y ∗ x))) ∗ x)=(y ∗ (y ∗ (y ∗ x))) ∗ ((y ∗ x) ∗ (y ∗ (y ∗ x))) ∈ I sofrom(b4)andProposition4.8that x ∗ (x ∗ (y ∗ x))= x ∗ (x ∗ (y ∗ (y ∗ (y ∗ x)))) ∈ I ⊂ J. Since((x ∗ (x ∗ y)) ∗ (x ∗ (x ∗ (y ∗ x)))) ∗ (y ∗ (y ∗ x))=0 ∈ J, weget x ∗ (x ∗ y) ∈ J by (c2).UsingTheorem4.7,weconcludethat J isa Q-Smarandachecleanidealof X.
References
[1] Y.B.Jun, SmarandacheBCI-algebras,Sci.Math.Jpn., 62(1)(2005),137-142. [2] J.MengandY.B.Jun, BCK-algebras,KyungmoonsaCo.Seoul,Korea(1994).
[3] R.Padilla, Smarandachealgebraicstructures,Bull.PureAppl.Sci.,Delhi, 17E(1)(1998),119-121;http://www.gallup.unm.edu/smarandache/alg-s-tx.txt.
[4] W.B.VasanthaKandasamy,Smarandachegroupoids;http://www.gallup.unm.edu/ smarandache/Groupoids.pdf.
