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1、w.b.vasanthakandasamySMARANDACHEFUZZYALGEBRAgroupoidsemigroupoidgroupsemigroupsemigroupsemigroupgroupgrouplooploopgroupgroupAMERICANRESEARCHPRESSREHOBOTH2003SMARANDACHEFUZZYALGEBRAW.B.VasanthaKandasamyDepartmentofMathematicsIndianInstituteofTechnologyMadrasChennai–600036,Indiae-mail:va
2、santha@iitm.ac.inweb:http://mat.iitm.ac.in/~wbvAMERICANRESEARCHPRESSREHOBOTH20031ThepictureonthecoverisasimplegraphicillustrationdepictingtheclassicalalgebraicstructureswithsinglebinaryoperationsandtheirSmarandacheanalogues.Thepicturesontheleft,composedofconcentriccircles,depictsthetra
3、ditionalconceptionofalgebraicstructures,andthepicturesoftheright,withtheirliberalintersections,describeSmarandachealgebraicstructures.Infact,SmarandacheAlgebra,likeitspredecessor,FuzzyAlgebra,arosefromtheneedtodefinestructureswhichweremorecompatiblewiththerealworldwherethegreyareasmatt
4、ered.LoftiAZadeh,thefatheroffuzzysets,remarkedthat:"So,thiswholethingstartedbecauseofmyperceptionatthattime,thattheworldofclassicalmathematics–wasalittletoomuchofablackandwhiteworld,thattheprincipleofthe'excludedmiddle'meantthateverypropositionmustbeeithertrueorfalse.Therewasnoallowanc
5、eforthefactthatclassesdonothavesharplydefinedboundaries."So,hereisthisbook,whichisanamalgamationofalternatives.Thisbookcanbeorderedinapaperboundreprintfrom:BooksonDemandProQuestInformation&Learning(UniversityofMicrofilmInternational)300N.ZeebRoadP.O.Box1346,AnnArborMI48106-1346,USATel.
6、:1-800-521-0600(CustomerService)http://wwwlib.umi.com/bod/andonlinefrom:PublishingOnline,Co.(Seattle,WashingtonState)at:http://PublishingOnline.comThisbookhasbeenpeerreviewedandrecommendedforpublicationby:Copyright2003byAmericanResearchPressandW.B.VasanthaKandasamyRehoboth,Box141NM8732
7、2,USAManybookscanbedownloadedfrom:http://www.gallup.unm.edu/~smarandache/eBooks-otherformats.htmISBN:1-931233-74-8StandardAddressNumber:297-5092PrintedintheUnitedStatesofAmerica2CONTENTSPreface5PARTONE1.SOMERESULTSONFUZZYALGEBRA1.1Fuzzysubsets91.2Groupsandfuzzysubgroups111.3Fuzzysub-
