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1、W.B.VasanthaKandasamyGroupoidsandSmarandacheGroupoids01423A1B1Z1A1×A2B1×B2Z2A2B2AmericanResearchPressRehoboth2002W.B.VasanthaKandasamyDepartmentofMathematicsIndianInstituteofTechnologyMadras,Chennai–600036,IndiaGroupoidsandSmarandacheGroupoids∗e012345ee01
2、234500e30303115e52522241e14133303e30442525e55514141eAmericanResearchPressRehoboth20021Thisbookcanbeorderedinapaperboundreprintfrom:BooksonDemandProQuestInformation&Learning(UniversityofMicrofilmInternational)300N.ZeebRoadP.O.Box1346,AnnArborMI48106-1346,U
3、SATel.:1-800-521-0600(CustomerService)http://wwwlib.umi.com/bod/andonlinefrom:PublishingOnline,Co.(Seattle,WashingtonState)at:http://PublishingOnline.comThisbookhasbeenpeerreviewedandrecommendedforpublicationby:D.Constantinescu,CollegeofArts,Rm.Valcea,Rom
4、ania.Dr.MKhoshnevisan,GriffithUniversity,GoldCoast,Queensland,Australia.SebastianMartinRuiz,AvdadeRegla43,Chipiona11550,Spain.Copyright2002byAmericanResearchPressandW.B.VasanthaKandasamyRehoboth,Box141NM87322,USAManybookscanbedownloadedfrom:http://www.gal
5、lup.unm.edu/~smarandache/eBooks-otherformats.htmISBN:1-931233-61-6StandardAddressNumber:297-5092PrintedintheUnitedStatesofAmerica2Preface51.PreliminaryNotions1.1Integers71.2Groupoids81.3DefinitionofSemigroupwithExamples111.4SmarandacheGroupoids1
6、41.5LoopsanditsProperties162.GroupoidsanditsProperties2.1SpecialPropertiesinGroupoids192.2SubstructuresinGroupoids212.3SomeSpecialPropertiesofaGroupoid272.4InfiniteGroupoidsanditsProperties293.NewClassesofGroupoidsUsingZn3.1DefinitionoftheClassofGroupoids
7、Z(n)31∗3.2NewClassofGroupoidsZ(n)35∗∗3.3OnNewClassofgroupoidsZ(n)39∗∗∗3.4OnGroupoidsZ(n)413.5GroupoidswithIdentityUsingZn.434.SmarandacheGroupoids4.1SmarandacheGroupoids454.2SubstructuresinSmarandacheGroupoids484.3IdentitiesinSmarandacheGroupoids564.4More
8、PropertiesonSmarandacheGroupoids654.5SmarandacheGroupoidswithIdentity6735.SmarandacheGroupoidsusingZn5.1SmarandacheGroupoidsinZ(n)69∗5.2SmarandacheGroupoidsinZ(n)75∗∗5.3SmarandacheGroupoidsinZ(n)78∗∗∗5.4SmarandacheG
