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1、w.b.vasanthakandasamysmarandachesemirings,semifields,andsemivectorspaces{a,b,c,d}{a,b,c}{a,b,d}{a,d,c}{b,d,c}{a,b}{a,c}{a,d}{b,c}{b,d}{d,c}{a}{d}{b}{c}{φ}americanresearchpressrehoboth2002SmarandacheSemirings,Semifields,andSemivectorspacesW.B.VasanthaKandasa
2、myDepartmentofMathematicsIndianInstituteofTechnology,MadrasChennai–600036,IndiaAmericanResearchPressRehoboth,NM20021ThepictureonthecoverisaBooleanalgebraconstructedusingthepowersetP(X)whereX={a,b,c}whichisafiniteSmarandachesemiringoforder16.Thisbookcanbeord
3、eredinapaperboundreprintfrom:BooksonDemandProQuestInformation&Learning(UniversityofMicrofilmInternational)300N.ZeebRoadP.O.Box1346,AnnArborMI48106-1346,USATel.:1-800-521-0600(CustomerService)http://wwwlib.umi.com/bod/andonlinefrom:PublishingOnline,Co.(Seatt
4、le,WashingtonState)at:http://PublishingOnline.comThisbookhasbeenpeerreviewedandrecommendedforpublicationby:Prof.MurtazaA.Quadri,Dept.ofMathematics,AligarhMuslimUniversity,India.Prof.B.S.Kiranagi,Dept.ofMathematics,MysoreUniversity,Karnataka,India.Prof.R.C.A
5、garwal,Dept.ofMathematics,LucknowUniversity,India.Copyright2002byAmericanResearchPressandW.B.VasanthaKandasamyRehoboth,Box141NM87322,USAManybookscanbedownloadedfrom:http://www.gallup.unm.edu/~smarandache/eBooks-otherformats.htmISBN:StandardAddressNumber:297
6、-5092PrintedintheUnitedStatesofAmerica2PrintedintheUnitedStatesofAmericaCONTENTSPreface51.Preliminarynotions1.1Semigroups,groupsandSmarandachesemigroups71.2Lattices111.3Ringsandfields191.4Vectorspaces221.5Groupringsandsemigrouprings252.Semiringsanditsproper
7、ties2.1Definitionandexamplesofsemirings292.2Semiringsanditsproperties332.3Semiringsusingdistributivelattices382.4Polynomialsemirings442.5Groupsemiringsandsemigroupsemirings462.6Somespecialsemirings503.Semifieldsandsemivectorspaces3.1Semifields533.2Semivecto
8、rspacesandexamples573.3Propertiesaboutsemivectorspaces584.Smarandachesemirings4.1DefinitionofS-semiringsandexamples654.2SubstructuresinS-semirings674.3SmarandachespecialelementsinS-semirings754.4Specia
