《代数系统群》ppt课件

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1、AlgebraicSystemsandGroupsDiscreteMathematicalAlgebraicOperationsFunctionƒ:AnBiscalledann-naryoperationfromAtoB.Binaryoperation:ƒ:AAB(ƒ:AAA)Anexample:anewoperation“*”definedonthesetofrealnumber,usingcommonarithmeticoperations:x*y=x+y-xyNote:2*3=-1；0.5*0.7=0.85ClosenessofOperationsForanyope

2、rationƒ:AnB,ifBA,thenitissaidthatAisclosedwithrespecttoƒ.Or,wesaythatƒisclosedonA.Example:SetA={1,2,3,…,10},gcdisclosed,butlcmisnot.OperationTableOperationtablecanbeusedtodefineunaryorbinaryoperationsonafiniteset(usuallyonlywithseveralelements)Howmanybinaryoperationscanbedefinedhere?Associati

3、onOperation“⃘”definedonthesetAisassociativeifandonlyif:Foranyx,y,zA,(x⃘y)⃘z=x⃘(y⃘z)If“⃘”isassociative,thenx1⃘x2⃘x3⃘…⃘xncanbecomputedbyanyorderofamongthe(n-1)operations,withtheconstraintthattheorderofalloperandsarenotchanged.CommutationOperation“⃘”definedonthesetAisassociativeifandonlyif:Forany

4、x,yA,x⃘y=y⃘xIf“⃘”iscommutativeandassociative,thenx1⃘x2⃘x3⃘…⃘xncanbecomputedbyanyorderoftheoperations,andinanypermutationofalloperands.DistributionTwodifferentoperationsmustbedefinedforanalgebraicsystemfordiscussionofdistribution.Operation“⃘”isdistributiveover“”(bothoperationsdefinedonthesetA)

5、ifandonlyif:Foranyx,y,zA,x⃘(yz)=(x⃘y)(x⃘z)(Exactlyspeaking,thisisthefirstdistributiveproperty)IdentityofanAlgebraicSystemForarithmeticmultiplicationonthesetofrealnumber,thereisaspecificrealnumber1，satisfyingthatforanyrealnumberx,1∙x=x∙1=xAnelementeiscalledtheidentityelementofanalgebraicsyste

6、m(S,⃘)ifandonlyif:ForanyxS,e⃘x=x⃘e=x。Denotation:1S,orsimply1,butrememberthatitisnotthat“1”.Itisnotthateveryalgebraicsystemhasitsidentityelement.LeftIdentityandRightIdentityeliscalledaleftidentityofanalgebraicsystemS,ifandonlyif:ForanyxS,el⃘x=xRightidentityer。canbedefinedsimilarly.MoreaboutIde

7、ntityForanyalgebraicsystemS:Theremayormaynotbeleftorrightidentity.Theremaybemorethanoneleftorrightidentities.IfShasaleftidentityandarightidentityaswell,thentheymustbeequal,andthiselementisalsoanidentityofthesystem:el=el⃘er=erIfexi