; TeX output 2002.06.04:10077 VN cmbx12MathematischesInstitutSS2002derUniversitXatM`unchenSet7Prof.Dr.B.Pareigis;KrNff cmbx12Problemffsetfor&QuantumffGroupsandNoncommutativeGeometry, XQ cmr12(25)%Letthesetg cmmi12ZutogetherwiththemrultiplicationmtF:Z!", cmsy10 ZPjo!޷ZbSeamonoid.%ShorwthattheunitelementeUR2ZFisuniquelydetermined.%Let(Z5;m)bSeagroup.Shorwthatalsotheinversei7:ZV-!eZ⯹isuniquely%determined.%Shorwfthatunitelementandinversesofgroupsarepreservedbymapsthatare%compatiblewiththemrultiplication. (26)%FindanexampleofmonoidsY andZ7andamapf۹:sYL)!ZwithfG(y|{Ycmr81y2)s=%fG(y1)f(y2)forally1;y2V2URYp,butfG(e2cmmi8YP)6=eZ8. (27)%Let(G;m)bSeagroupinCandiX &: XG(X)"!,=G(X)bSetheinrverseforall%XF2URC5.8Shorwthatiisanaturaltransformation.%ShorwОthattheYVonedaLemmaprovidesamorphismS:ܽGH!GОsuchthat%iX r۹=URMorO K cmsy8C(XJg;S׹)UR=S(X)forallXF2URC5.%FVormrulateandprovepropSertiesofSofthetypSeS]idʞ=UR:::. (28)%LetCbSeacategorywith niteproducts.DmShorwthatamorphismf߹:[Guk!MG20%〹inCݹisahomomorphismofgroupsifandonlyifHYGGYIGI|{fd@O line10-:mK̓G20xG20KG2032fdNά-.&m-:!q% cmsy60HǠ*Ffe4̟Ǡ?`ffHڟǠ*Ffe Ǡ?`fe%〹commrutes.=}Duedate:8TVuesdary,11.06.2002,16:15inLectureHallE41*;7 !q% cmsy6 K cmsy8!", cmsy102cmmi8g cmmi12|{Ycmr8Nff cmbx12N cmbx12XQ cmr12O line10 o