北京邮电大学计算机学院--离散数学--数学结构--群论-chap9-2.ppt

SemigroupsandGroups

〔半群与群〕

22024/8/31CollegeofComputerScienceTechnology,BUPTContentApplicationofsemigroupsandgroupsBinaryoperations〔二元运算〕Semigroups〔半群〕Productsandquotientsofsemigroups〔积半群和商半群〕Groups〔群〕Productsandquotientsofgroups〔积群和商群〕

32024/8/31CollegeofComputerScienceTechnology,BUPTContentBinaryoperationsBinaryoperationonasetAPropertiesofbinaryoperationsSemigroupsSemigroupFreesemigroupgeneratedbyAMonoid〔独异点〕Subsemigroup〔子半群〕andsubmonoid〔子独异点〕Isomorphism〔同构〕

42024/8/31CollegeofComputerScienceTechnology,BUPTSemigroup–半群AsemigroupisanonemptysetStogetherwithanassociativebinaryoperation*definedonS.Denotethesemigroupby(S,*)orS.a*bisreferredastheproductofaandb.Thesemigroup(S,*)issaidtobecommutativeif*isacommutativeoperation.

52024/8/31CollegeofComputerScienceTechnology,BUPTExamples(Z,+)ThesetP(S),whereSisaset,togetherwiththeoperationofunionThesetZwiththebinaryoperationofsubtraction

62024/8/31CollegeofComputerScienceTechnology,BUPTExample4LetSbeafixednonemptyset,andletSsbethesetofal1functionsf:S?S.IffandgareelementsofSs,wedefinef*gasf?g,thecompositefunction.*isabinaryoperationonSs*isassociative.(Ss,*)isasemigroup.ThesemigroupSsisnotcommutative.

72024/8/31CollegeofComputerScienceTechnology,BUPTExample5Let(L,?)bealattice.DefineabinaryoperationonLbya*b=a?b.ThenLisasemigroup.

82024/8/31CollegeofComputerScienceTechnology,BUPTExample6LetA={al,a2,...,an}beanonemptyset.A*isthesetofallfinitesequencesofelementsofA.?and?beelementsofA*.Thecatenationisabinaryoperation?onA*.if?=a1a2...anand?=b1b2…bk???=a1a2...anb1b2…bkif?,?,and?areanyelementsofA*,??(???)=(???)??.

92024/8/31CollegeofComputerScienceTechnology,BUPTExample6?isanassociativebinaryoperation,and(A*,?)isasemigroup.T