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1、AlgebraicSystems andGroupsDiscreteMathematicalAlgebraicOperationsFunction?:An?Biscalledann-naryoperationfromAtoB.Binaryoperation:?:A?A?B(?:A?A?A)Anexample:anewoperation“*”definedonthesetofrealnumber,usingcommonarithmeticoperations:x*y=x+y-xyNote:2*3=-1；0.5*0.7=0.85ClosenessofOperationsForanyope
2、ration?:An?B,ifB?A,thenitissaidthatAisclosedwithrespectto?.Or,wesaythat?isclosedonA.Example:SetA={1,2,3,…,10},gcdisclosed,butlcmisnot.OperationTableOperationtablecanbeusedtodefineunaryorbinaryoperationsonafiniteset(usuallyonlywithseveralelements)Howmanybinaryoperationscanbedefinedhere?Associati
3、onOperation“?”definedonthesetAisassociativeifandonlyif:Foranyx,y,z?A,(x?y)?z=x?(y?z)If“?”isassociative,thenx1?x2?x3?…?xncanbecomputedbyanyorderofamongthe(n-1)operations,withtheconstraintthattheorderofalloperandsarenotchanged.CommutationOperation“?”definedonthesetAisassociativeifandonlyif:Forany
4、x,y?A,x?y=y?xIf“?”iscommutativeandassociative,thenx1?x2?x3?…?xncanbecomputedbyanyorderoftheoperations,andinanypermutationofalloperands.DistributionTwodifferentoperationsmustbedefinedforanalgebraicsystemfordiscussionofdistribution.Operation“?”isdistributiveover“?”(bothoperationsdefinedonthesetA)
5、ifandonlyif:Foranyx,y,z?A,x?(y?z)=(x?y)?(x?z)(Exactlyspeaking,thisisthefirstdistributiveproperty)IdentityofanAlgebraicSystemForarithmeticmultiplicationonthesetofrealnumber,thereisaspecificrealnumber1，satisfyingthatforanyrealnumberx,1?x=x?1=xAnelementeiscalledtheidentityelementofanalgebraicsyste
6、m(S,?)ifandonlyif:Foranyx?S,e?x=x?e=x。Denotation:1S,orsimply1,butrememberthatitisnotthat“1”.Itisnotthateveryalgebraicsystemhasitsidentityelement.LeftIdentityandRightIdentityeliscalledaleftidentityofanalgebraicsystemS,ifandonlyif:Foranyx?S,el?x=xRightidentityer。canbedefinedsimilarly.MoreaboutIde
7、ntityForanyalgebraicsystemS:Theremayormaynotbeleftorrightidentity.Theremaybemorethanoneleftorrightidentities.IfShasaleftidentityandarightidentityaswell,thentheymustbeequal,andthiselementisalsoanidentityofthesystem:el=el?er=erIfexi
