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1、W.B.VASANTHAKANDASAMYSMARANDACHELOOPSL(8)15BB15AA115{e}AMERICANRESEARCHPRESSREHOBOTH2002SmarandacheLoopsW.B.VasanthaKandasamyDepartmentofMathematicsIndianInstituteofTechnology,MadrasChennai–600036,IndiaLA7A8A10A11A6A1A2A3A4A5AmericanResearchPressRehoboth,NM20021Thepictureonthecoverrepresentsthelatt
2、iceofsubgroupsoftheSmarandacheloopL(8).15ThelatticeofsubgroupsofthecommutativeloopL(8)isanon-modularlatticewith2215elements.ThisisaSmarandacheloopwhichsatisfiestheSmarandacheLagrangecriteria.ButfortheSmarandacheconceptsonewouldn'thavestudiedthecollectionofsubgroupsofaloop.Thisbookcanbeorderedinapap
3、erboundreprintfrom:BooksonDemandProQuestInformation&Learning(UniversityofMicrofilmInternational)300N.ZeebRoadP.O.Box1346,AnnArborMI48106-1346,USATel.:1-800-521-0600(CustomerService)http://wwwlib.umi.com/bod/andonlinefrom:PublishingOnline,Co.(Seattle,WashingtonState)at:http://PublishingOnline.comThi
4、sbookhasbeenpeerreviewedandrecommendedforpublicationby:Dr.M.Khoshnevisan,SharifUniversityofTechnology,Tehran,Iran.Dr.J.Dezert,OfficeNationald’EtudesetdeRecherchesAeorspatiales(ONERA),29,AvenuedelaDivisionLeclerc,92320Chantillon,France.ProfessorC.Corduneanu,TexasStateUniversity,DepartmentofMathemati
5、cs,Arlington,Texas76019,USA.Copyright2002byAmericanResearchPressandW.B.VasanthaKandasamyRehoboth,Box141NM87322,USAManybookscanbedownloadedfrom:http://www.gallup.unm.edu/~smarandache/eBooks-otherformats.htmISBN:1-931233-63-2StandardAddressNumber:297-5092PRINTEDINTHEUNITEDSTATESOFAMERICA2Pref
6、ace51.GeneralFundamentals1.1BasicConcepts71.2Afewpropertiesofgroupsandgraphs81.3Latticesanditsproperties112.Loopsanditsproperties2.1Definitionofloopandexamples152.2Substructuresinloops172.3Specialidentitiesinloops222.4Specialtypesofloops232.5Representationandisotopesofloops292.6Onanewclassofloopsan
7、ditsproperties312.7ThenewclassofloopsanditsapplicationtoproperedgecolouringofthegraphK402n3.SmarandacheLoops3.1DefinitionofSmarandacheloopswithexamples473.2Smarandachesubstructuresinloops513.3Somenewclassic
