BCK-Algebras and Related Algebraic Systems

90 Pages·2012·2.92 MB·English

International Journal of Mathematics and Mathematical Sciences
BCK-Algebras and Related Algebraic Systems
Guest Editors: Young Bae Jun, Ivan Chajda, Hee Sik Kim, Eun Hwan Roh,
Jianming Zhan, and Afrodita Iorgulescu
BCK-Algebras and Related
Algebraic Systems
International Journal of Mathematics and
Mathematical Sciences
BCK-Algebras and Related
Algebraic Systems
Guest Editors: Young Bae Jun, Ivan Chajda,
Hee Sik Kim, Eun Hwan Roh, Jianming Zhan,
and Afrodita Iorgulescu
Copyrightq 2011HindawiPublishingCorporation.Allrightsreserved.
Thisisaspecialissuepublishedin“InternationalJournalofMathematicsandMathematicalSciences.”Allarticlesareopen
accessarticlesdistributedundertheCreativeCommonsAttributionLicense,whichpermitsunrestricteduse,distribution,
andreproductioninanymedium,providedtheoriginalworkisproperlycited.
Editorial Board
AsaoArai,Japan AttilaGila´nyi,Hungary HernandoQuevedo,Mexico
ErikJ.Balder,TheNetherlands JeromeA.Goldstein,USA JeanMichelRakotoson,France
A.Ballester-Bolinches,Spain SiegfriedGottwald,Germany RobertH.Redfield,USA
MartinoBardi,Italy N.K.Govil,USA B.E.Rhoades,USA
P.Basarab-Horwath,Sweden R.Grimshaw,UK PaoloE.Ricci,Italy
PeterW.Bates,USA HeinzPeterGumm,Germany Fre´de´ricRobert,France
HeinrichBegehr,Germany S.M.Gusein-Zade,Russia AlexanderRosa,Canada
HowardE.Bell,Canada SeppoHassi,Finland AndrewRosalsky,USA
KennethS.Berenhaut,USA PenttiHaukkanen,Finland MishaRudnev,UK
O´scarBlasco,Spain JosephHilbe,USA StefanSamko,Portugal
MartinBohner,USA HelgeHolden,Norway GideonSchechtman,Israel
HuseyinBor,Turkey HenrykHudzik,Poland NaseerShahzad,SaudiArabia
TomaszBrzezinski,UK PetruJebelean,Romania N.Shanmugalingam,USA
TeodorBulboaca˘,Romania PalleE.Jorgensen,USA ZhongminShen,USA
StefaanCaenepeel,Belgium ShyamL.Kalla,Kuwait MariannaA.Shubov,USA
W.zuCastell,Germany V.R.Khalilov,Russia H.S.Sidhu,Australia
AlbertoCavicchioli,Italy H.M.Kim,RepublicofKorea TheodoreE.Simos,Greece
DerChenChang,USA TaekyunKim,RepublicofKorea AndrzejSkowron,Poland
ShihSenChang,China EvgenyKorotyaev,Germany FrankSommen,Belgium
CharlesE.Chidume,Italy AloysKrieg,Germany LindaR.Sons,USA
HiJunChoe,RepublicofKorea WolfgangKu¨hnel,Germany F.C.R.Spieksma,Belgium
ColinChristopher,UK IrenaLasiecka,USA IlyaM.Spitkovsky,USA
ChristianCorda,Italy YuriLatushkin,USA MarcoSquassina,Italy
RodicaD.Costin,USA BaoQinLi,USA H.M.Srivastava,Canada
M.-E.Craioveanu,Romania SongxiaoLi,China YucaiSu,China
RalE.Curto,USA NoelG.Lloyd,UK PeterTakac,Germany
PrabirDaripa,USA R.Lowen,Belgium Chun-LeiTang,China
H.DeSnoo,TheNetherlands AnilMaheshwari,Canada MichaelM.Tom,USA
LokenathDebnath,USA RaulF.Manasevich,Chile RamU.Verma,USA
AndreasDefant,Germany B.N.Mandal,India AndreiI.Volodin,Canada
DavidE.Dobbs,USA EnzoLuigiMitidieri,Italy LucVrancken,France
S.S.Dragomir,Australia VladimirMityushev,Poland DorothyI.Wallace,USA
JewgeniDshalalow,USA ManfredMoller,SouthAfrica FrankWerner,Germany
J.Dydak,USA V.Nistor,USA RichardG.Wilson,Mexico
M.A.Efendiev,Germany EnricoObrecht,Italy IngoWitt,Germany
HansEngler,USA Chia-venPao,USA PeiYuanWu,Taiwan
RicardoEstrada,USA WenL.Pearn,Taiwan SiamakYassemi,Iran
B.Forster-Heinlein,Germany GeluPopescu,USA A.Zayed,USA
DaliborFroncek,USA MihaiPutinar,USA KaimingZhao,Canada
XianguoGeng,China FengQi,China YuxiZheng,USA
Contents
BCK-AlgebrasandRelatedAlgebraicSystems,YoungBaeJun,IvanChajda,HeeSikKim,
EunHwanRoh,JianmingZhan,andAfroditaIorgulescu
Volume2011,ArticleID268683,3pages
RoughFiltersinBL-Algebras,LidaTorkzadehandShokoofehGhorbani
Volume2011,ArticleID474375,13pages
TwoNewTypesofRingsConstructedfromQuasiprimeIdeals,ManalGhanemand
HassanAl-Ezeh
Volume2011,ArticleID473413,9pages
OnBE-Semigroups,SunShinAhnandYoungHeeKim
Volume2011,ArticleID676020,8pages
OnAlgebraicApproachinQuadraticSystems,MatejMencinger
Volume2011,ArticleID230939,12pages
GeneralizedDerivationsandBilocalJordanDerivationsofNestAlgebras,DanguiYanand
ChengchangZhang
Volume2011,ArticleID748159,6pages
FuzzyFilterSpectrumofaBCKAlgebra,XiaoLongXin,WeiJi,andXiuJuanHua
Volume2011,ArticleID795934,13pages
GraphsBasedonBCK/BCI-Algebras,YoungBaeJunandKyoungJaLee
Volume2011,ArticleID616981,8pages
AConstructionofMirrorQ-Algebras,KeumSookSo
Volume2011,ArticleID219496,6pages
CommutativePseudoValuationsonBCK-Algebras,MyungImDohandMinSuKang
Volume2011,ArticleID754047,6pages
HindawiPublishingCorporation
InternationalJournalofMathematicsandMathematicalSciences
Volume2011,ArticleID268683,3pages
doi:10.1155/2011/268683
Editorial
BCK-Algebras and Related Algebraic Systems
Young Bae Jun,1 Ivan Chajda,2 Hee Sik Kim,3 Eun Hwan Roh,4
Jianming Zhan,5 and Afrodita Iorgulescu6
1DepartmentofMathematicsEducation,GyeongsangNationalUniversity,
Chinju660-701,RepublicofKorea
2DepartmentofAlgebraandGeometry,FacultyofSciences,PalackyUniversity(UP),
77147Olomouc,CzechRepublic
3DepartmentofMathematics,HanyangUniversity,Seoul133-791,RepublicofKorea
4DepartmentofMathematicsEducation,ChinjuNationalUniversityofEducation,
Chinju660-756,RepublicofKorea
5DepartmentofMathematics,HubeiInstituteforNationalities,Enshi,Hubei445000,China
6DepartmentofComputerScience,TheBucharestAcademyofEconomicStudies,
010374Bucharest,Romania
CorrespondenceshouldbeaddressedtoYoungBaeJun,[email protected]
Received5November2011;Accepted10November2011
Copyrightq2011YoungBaeJunetal.ThisisanopenaccessarticledistributedundertheCreative
CommonsAttributionLicense,whichpermitsunrestricteduse,distribution,andreproductionin
anymedium,providedtheoriginalworkisproperlycited.
BCK/BCI-algebras are algebraic structures, introduced by K. Ise´ki in 1966, that describe
fragments of the propositional calculus involving implication known as BCK/BCI-logics. It
isknownthattheclassofBCK-algebrasisapropersubclassoftheclassofBCI-algebras.We
referthereaderforusefultextbooksforBCK/BCI-algebrato(cid:2)1–3(cid:3).
Theaimofthisspecialissuewastopromotetheexchangeofideasbetweenresearchers
and to spread new trends in this area. It is focused on all aspects of BARAS, from their
foundationstoapplicationsincomputersciencesandinformatics.
This special issue contains nine papers. In the paper entitled “Commutative pseudo
valuationsonBCK-algebras,”M.I.DohandM.S.Kangintroducedthenotionofacommu-tative
pseudovaluationonaBCK-algebraandinvestigateditscharacterizations.Theydiscussedthe
relationship between a pseudo valuation and a commutative pseudo valua-tion. They also
providedconditionsforapseudovaluationtobeacommutativepseudovaluation.
Neggers et al. (cid:2)4(cid:3) introduced the notion of Q-algebras which are a generalization of
BCK/BCI/BCH-algebras, obtained several properties, and discussed quadratic Q-algebras.
In the paper, entitled “A construction of mirror Q-algebras,” K. S. So introduced the notion
of mirror algebras to Q-algebras, and she investigated how to construct mirror Q-algebras
2 InternationalJournalofMathematicsandMathematicalSciences
from a Q-algebra. She also obtained the necessary conditions for a left mirror algebra
(cid:4)M(cid:4)X(cid:5),∗,(cid:4)0,0(cid:5)(cid:5) of (cid:4)X,∗,0(cid:5)tobeaQ-algebra.
Manyauthorsstudiedthegraphtheoryinconnectionwith(cid:4)commutative(cid:5)semigroups
and(cid:4)commutativeandnoncommutative(cid:5)ringsasshowninthereferencesthatwereferthe
readerto.Forexample,Beck(cid:2)5(cid:3)associatedtoanycommutativeringRitszero-divisorgraph
G(cid:4)R(cid:5)whoseverticesarethezero-divisorsofR(cid:4)including0(cid:5),withtwoverticesa,bjoinedbyan
edgeincaseab(cid:6)0.Also,DeMeyeretal.(cid:2)6(cid:3)definedthezero-divisorgraphofacommutative
semigroup S with zero (cid:4)0x (cid:6) 0 ∀x ∈ S(cid:5). Motivated by these works, in the paper, entitled
“Graphs based on BCK/BCI-algebras,”, Y. B. Jun and K. S. Lee studied graph theory based on
BCK-algebras. They tried to discuss the associated graphs of BCK/BCI-algebras. To do so,
theyfirstintroducedthenotionsof(cid:4)l-prime(cid:5)quasi-idealsandzero-divisorsandinvestigated
related properties. They introduced the concept of associative graph of a BCK/BCI-algebra
and provided several examples. They provided conditions for a proper (cid:4)quasi(cid:5)ideal of a
BCK/BCI-algebra to be l-prime. Finally they showed that the associative graph of a BCK-
algebraisaconnectedgraphinwhicheverynonzerovertexisadjacentto0,buttheassociative
graphofaBCI-algebraisnotconnectedbyprovidinganexample.
Inthepaper,entitled,“FuzzyfilterspectrumofaBCKalgebra,”X.L.Xinetal.investigated
both the topological structure and fuzzy structure on BCK-algebras. They introduced the
concept of fuzzy s-prime filters and discussed some related properties. Using the fuzzy s-
primefilters,theyestablishedafuzzytopologicalstructureonboundedcommutativeBCK-
algebrasandboundedimplicativeBCK-algebras,respectively.
In the paper, entitled “Generalized derivations and bilocal jordan derivations of nest
algebras,”D.YanandC.ZhangdiscussedgeneralizedandbilocalJordanderivationsofnest
algebras.Theyshowedthatinnestalgebra,
(cid:4)1(cid:5)(cid:4)bi(cid:5)localJordanderivationsareinnerderivationsand,
(cid:4)2(cid:5)generalizedderivationsaregeneralizedinnerderivations.
In the paper, entitled “On algebraic approach in quadratic systems,” M. Mencinger
consideredhomogeneousquadraticsystemsviatheso-calledMarkusapproach.Heusedthe
one-to-onecorrespondencebetweenhomogeneousquadraticdynamicalsystemsandalgebra
whichwasoriginallyintroducedbyMarkusin(cid:2)7(cid:3)andconsideredsomegeneralconnections
andtheinfluenceofpowerassociativityinthecorrespondingquadraticsystem.
In (cid:2)8(cid:3), H. S. Kim and Y. H. Kim introduced the notion of a BE-algebra as a
generalizationofaBCK-algebraandgaveanequivalentconditionofthefilterinBE-algebras
by using the notion of upper sets. In (cid:2)9, 10(cid:3), Ahn and So introduced the notion of ideals in
BE-algebras and proved several characterizations of such ideals. In the paper, entitled “On
BE-semigroups,”S.S.AhnandY.H.KimcombinedBE-algebrasandsemigroupstointroduce
thenotionofBE-semigroups.Theydefinedleft(cid:4)resp.,right(cid:5)deductivesystems(cid:4)(cid:4)LDS,resp.,
RDS(cid:5)forshort(cid:5)ofaBE-semigroup,andthentheydescribedLDSgeneratedbyanonempty
subsetinaBE-semigroupasasimpleform.
In the paper entitled “Two new types of rings constructed from quasiprime ideals,” M.
Ghanem and H. Al-Ezeh generalized the well-known concepts of regular and PF-rings to
ordinarydifferentialringsandconsideredwell-knownpropertiesofregularandPF-ringsin
theirsituations.
In the paper entitled “Rough filters in BL-algebras,” L. Torkzadeh and S. Ghorbani
applied the rough set theory to BL-algebras, and introduced the notion of rough filters
(cid:4)subalgebras(cid:5)ofBL-algebrasasageneralizationoffilters(cid:4)subalgebras(cid:5)ofBL-algebras.
InternationalJournalofMathematicsandMathematicalSciences 3
Although the selected topics and papers are not an exhaustive representation of the
areaofBARASs,theypresenttherichandmany-facetedknowledgethatwehavethepleasure
ofsharingwiththereaders.
Acknowledgments
It was our honor to receive submissions from many authors. We would like to thank
the authors for their excellent contributions and patience in assisting us. Moreover, the
fundamentalworkofallreviewersonthesepapersisalsoverywarmlyacknowledged.Last,
but not least, the publishing staff have worked diligently with us on this special issue. We
alsowouldliketoexpressoursinceregratitudetothepublishingstaff.
YoungBaeJun
IvanChajda
HeeSikKim
EunHwanRoh
JianmingZhan
AfroditaIorgulescu
References
(cid:2)1(cid:3) Y.S.Huang,BCI-Algebra,SciencePress,Beijing,China,2006.
(cid:2)2(cid:3) A.Iorgulescu,AlgebrasofLogicasBCKAlgebras,EdituraASE,Bucharest,Romania,2008.
(cid:2)3(cid:3) J.MengandY.B.Jun,BCK-Algebras,KyungMoonSaCo.,Seoul,Korea,1994.
(cid:2)4(cid:3) J. Neggers, S. S. Ahn, and H. S. Kim, “On Q-algebras,” International Journal of Mathematics and
MathematicalSciences,vol.27,no.12,pp.749–757,2001.
(cid:2)5(cid:3) I.Beck,“Coloringofcommutativerings,”JournalofAlgebra,vol.116,no.1,pp.208–226,1988.
(cid:2)6(cid:3) F.R.DeMeyer,T.McKenzie,andK.Schneider,“Thezero-divisorgraphofacommutativesemigroup,”
SemigroupForum,vol.65,no.2,pp.206–214,2002.
(cid:2)7(cid:3) L. Markus, “Quadratic differential equations and non-associative algebras,” vol. 45 of Annals of
MathematicsStudies,PrincetonUniversityPress,Princeton,NJ,USA,1960,pp.185–213.
(cid:2)8(cid:3) H.S.KimandY.H.Kim,“OnBE-algebras,”ScientiaeMathematicaeJaponicae,vol.66,no.1,pp.113–116,
2007.
(cid:2)9(cid:3) S.S.AhnandK.S.So,“OnidealsanduppersetsinBE-algebras,”ScientiaeMathematicaeJaponicae,vol.
68,no.2,pp.279–285,2008.
(cid:2)10(cid:3) S.S.AhnandK.S.So,“OngeneralizeduppersetsinBE-algebras,”BulletinoftheKoreanMathematical
Society,vol.46,no.2,pp.281–287,2009.
HindawiPublishingCorporation
InternationalJournalofMathematicsandMathematicalSciences
Volume2011,ArticleID474375,13pages
doi:10.1155/2011/474375
Research Article
BL
Rough Filters in -Algebras
Lida Torkzadeh1 and Shokoofeh Ghorbani2
1DepartmentofMathematics,IslamicAzadUniversity,KermanBranch,Kerman,Iran
2DepartmentofMathematicsofBam,ShahidBahonarUniversity,Kerman,Iran
CorrespondenceshouldbeaddressedtoShokoofehGhorbani,sh [email protected]
Received22December2010;Revised10March2011;Accepted4April2011
AcademicEditor:YoungBaeJun
Copyrightq2011L.TorkzadehandS.Ghorbani.Thisisanopenaccessarticledistributedunder
the Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproductioninanymedium,providedtheoriginalworkisproperlycited.
WeapplytheroughsettheorytoBL-algebras.Asageneralizationoffilters(cid:2)subalgebras(cid:3)ofBL-
algebras, we introduce the notion of rough filters (cid:2)subalgebras(cid:3) of BL-algebras and investigate
someoftheirproperties.
1. Introduction
The rough sets theory introduced by Pawlak (cid:4)13(cid:5) has often proved to be an excellent
mathematicaltoolfortheanalysisofavaguedescriptionofobjects(cid:2)calledactionsindecision
problems(cid:3). Manydifferentproblemscanbeaddressedbyroughsetstheory.Duringthe last
few years this formalism has been approached as a tool used in connection with many
different areas of research. There have been investigations of the relations between rough
sets theory and the Dempster-Shafertheory and betweenrough sets and fuzzysets. Rough
sets theory has also provided the necessary formalism and ideas for the development of
somepropositionalmachinelearningsystems.Ithasalsobeenusedfor,amongmanyothers,
knowledge representation; data mining; dealing with imperfect data; reducing knowledge
representationandforanalyzingattributedependencies.Thenotionsofroughrelationsand
rough functions are based on rough sets theory and can be applied as a theoretical basis
forroughcontrollers,amongothers.Analgebraicapproachtoroughsetshasbeengivenby
Iwinski(cid:4)1(cid:5).Roughsettheoryisappliedtosemigroupsandgroups(cid:2)see(cid:4)2,3(cid:5)(cid:3).In1994,Biswas
andNanda(cid:4)4(cid:5)introducedanddiscussedtheconceptofroughgroupsandroughsubgroups.
Jun (cid:4)5(cid:5) applied rough set theory to BCK-algebras. Recently, Rasouli (cid:4)6(cid:5) introduced and
studiedthenotionofroughnessinMV-algebras.
BL-algebras are the algebraic structures for Ha`jek Basic Logic (cid:2)BL-logic(cid:3) (cid:4)7(cid:5), arising
from the continuous triangular norms (cid:2)t-norms(cid:3), familiar in the frameworks of fuzzy set