离散数学课件(英文版)----Semigroup.ppt

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1、AlgebraicSystemsandGroupsLecture13DiscreteMathematicalStructuresAlgebraicOperationsFunctionƒ: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

2、*0.7=0.85ClosenessofOperationsForanyoperationƒ:AnB,ifBA,thenitissaidthatAisclosedwithrespecttoƒ.Or,wesaythatƒisclosedonA.Example:SetA={1,2,3,…,10},gcdisclosed,butlcmisnot.OperationTableOperationtablecanbeusedtodefineunaryorbinaryoperationsonafiniteset(usuallyonlywithsevera

3、lelements)Howmanybinaryoperationscanbedefinedhere?AssociationOperation“⃘”definedonthesetAisassociativeifandonlyif:Foranyx,y,zA,(x⃘y)⃘z=x⃘(y⃘z)If“⃘”isassociative,thenx1⃘x2⃘x3⃘…⃘xncanbecomputedbyanyorderofamongthe(n-1)operations,withtheconstraintthattheorderofalloperandsareno

4、tchanged.CommutationOperation“⃘”definedonthesetAisassociativeifandonlyif:Foranyx,yA,x⃘y=y⃘xIf“⃘”iscommutativeandassociative,thenx1⃘x2⃘x3⃘…⃘xncanbecomputedbyanyorderoftheoperations,andinanypermutationofalloperands.DistributionTwodifferentoperationsmustbedefinedforanalgebraic

5、systemfordiscussionofdistribution.Operation“⃘”isdistributiveover“”(bothoperationsdefinedonthesetA)ifandonlyif:Foranyx,y,zA,x⃘(yz)=(x⃘y)(x⃘z)(Exactlyspeaking,thisisthefirstdistributiveproperty)IdentityofanAlgebraicSystemForarithmeticmultiplicationonthesetofrealnumber,ther

6、eisaspecificrealnumber1，satisfyingthatforanyrealnumberx,1∙x=x∙1=xAnelementeiscalledtheidentityelementofanalgebraicsystem(S,⃘)ifandonlyif:ForanyxS,e⃘x=x⃘e=x。Denotation:1S,orsimply1,butrememberthatitisnotthat“1”.Itisnotthateveryalgebraicsystemhasitsidentityelement.LeftIdentit

7、yandRightIdentityeliscalledaleftidentityofanalgebraicsystemS,ifandonlyif:ForanyxS,el⃘x=xRightidentityer。canbedefinedsimilarly.MoreaboutIdentityForanyalgebraicsystemS:Theremayormaynotbeleftorrightidentity.Theremaybemorethanoneleftorrightidentities.IfShasaleftidentityandarigh

8、tidentityaswell,thentheymustbeequal,andthiselementisalsoanidentityofthesyst