; TeX output 1995.05.31:1950`&M`HƷiK`yff cmdunh10LeftLinearTheories[2N cmbx12-AGeneralizationofMo`duleTheory-5 P cmu10HerrnPrqofessorDr.D.Pumpl#N9unzum60.Geburtstaggewidmet.gR- cmcsc10BodoP3areigisandHelmutR'ohrl&1""V 3 cmbx10AbstractMƍ-̻K`y 3 cmr10InfthispapMerw!eintroMduceleftlineartheoriesofexponen!t b> 3 cmmi10Nf(aset)on _theysetLasmapsL!", 3 cmsy10L0ercmmi7N u3j(l7);1) !l22Lysuc!hthatforalll2jL_and\;1:{2LN dtherelation(lt)=l7)()\holds,Fwhere:{2LN dis_giv!enby()B)(i)~=(i);1i2N1.$dWeeassumethatLhasaunit,that_isx/anelemen!tb2hKLN withl1=hKl7),foralllt2L,andd=,forall_92LN.2Next,left(resp.righ!t)L-moMdulesandL-M1-bimodulesandtheir_homomorphismsarede nedandleadtocategoriesL-MoMd,[Mod-L,and_L-M1-MoMd.pThese`categoriesarealgebraiccategoriesandtheirfreeobjects_are5describMedexplicitlye.IFinally,(Hom(X@<;1Yn)5andX &Y)$arein!troduced_andftheirpropMertiesarein!vestigated.-̻Keywtords: leftlineartheorye,?ba!ycentrictheory,?convexitytheory,_moMduleftheorye,tensorproduct,innerhom.-̻AMSMgclassi cationM91:GPrimary08C99,(Secondary16D10,18C05,_18D15,f52A01.&-ff cmcsc100. SIntroductionXQ cmr12TheopSerationsde ningang cmmi12RJ-moduleXaorveraringRJ, amonoidmoduleorveramonoid,9anespaces,certaintrypSesofbarycentriccaculi,9vXariousconvexitytheoriesPetc. harveincommon,Mythattheyformcertain"linearcombinations"subjecttosucrhlawsasdistributivityV, assoSciativity, oractionofaunit. TThegeneralxde nitionforsucrhopSerationsandtheiraxiomscanbederivredinthecaseofꨳRJ-moSdulesasfollorws.LetxXbSeanryunitalmoduleorverxagivrenringRJ.fOLetfurthermorer K cmsy8 d|denoteasequence.(r|{Ycmr81;ṟ2;:::ʞ).ofelemenrtsofRxwith nitesuppSort./ThenwecanassoSciatewithꨳr̷themapn9(r̷)UR:X2 2cmmi8INH !", cmsy10]!X+thatisgivrenby+ɩn9(r̷)(m )UR=(r̷)(m 1;m 2;:::ʞ)UR:=u cmex10X "%i2IN&Mr̴idڳm i;0m V2URX IN ;*`&M`HķwhereaGthisin nitesumisde nedbSecausethesupportofr̷!Kis nite. WVeinrterpretn9(r̷)4as"opSerationonX".Inparticular,Go(r̷)opSeratesonRJ2(IN As)w5,GothesetofallsequencesofelemenrtsofRwith nitesuppSort;thatisATQn9(r̷)(s 1ڍ;s 2ڍ;:::ʞ)UR=X "%i2IN&Mr̴idڳs iڍ: Withthesenotationsthefollorwinglawsaresatis ed:% r5(A̱0)0^ifꨳr̷,s21RA,s22RA,:::aresequencesasspSeci edaborveandm2V2URX2IN ,thenOn9((r̷)(s 1ڍ;s 2ڍ;:::ʞ))(m )UR=n9(r̷)((s 1ڍ)(m );(s 2ڍ)(m );:::ʞ);> r5(A̱1)0^ifꨳ2s2jRA.=UR(Os2jO1<;Os2jO2;:::ʞ),withOs2jqitheusualKronecrkersymbSol,then*n9( s2jڍ<)(m )UR=m jf ;0m V2X IN :IntermsoftheinitialmoSdulestructure,(A̱0)isrX "%pWi2IN}T(2X "%jv2IN+r̴jf sOjqi)m i,=X "%URjv2IN'}r̴j(/X "%i2INsOjqim idڰ);!aQcomrbinationoftheassoSciativelaw,thecommutativelawfortheaddition,andthedistributivrelaw.7ConrverselyV,if%asetXisgivrenandif,forevreryr̷)asabSove,ٳn9(r̷)UR:X2INH]!Xisa9mapsucrhthatA̱0andA̱1aresatis ed,]thenX,equippSedwiththecompositionsdb!m21j+m22:=ϳn9((1;1;0;:::ʞ))(m21;m22;:::ʞ);zrSm21:=ϳn9((rr;0;0;:::ʞ))(m21;m22;:::ʞ);isaunitalRJ-moSdule.*A~simplecomputationshorwsthatthestandardaxiomsforunitalꨳRJ-moSdulesareequivXalenrttotheaxioms(A̱0)and(A̱1)(see4.13).(A̱0)zand(A̱1)canbSecastinadi erenrtwayV.Theinitiallyde nedopSerationisexpressedasamapسË:URRJ (IN As)!ݶX INH]!XJg: HenceitgivresrisetoamapAT(RJ (IN As)w5) IN lX INH]!URX IN :Inparticularwrehaverܡ(RJ (IN As)w5) IN l(RJ (IN As)) INŶl!UR(RJ (IN As)) IN s: G`&M`HķLetŒustakrethismapasa"multiplication"on(RJ2(IN As)w5)2IN s.ɟThenassoSciativityofthismrultiplicationisexpressedby(A̱0),b8forXF<=TRJ2(IN As)w5,whileunitalitryofthemrultiplicationisgivenby(A̱1),*forX=dRJ2(IN As)w5. Thus(RJ2(IN As)w5)2INBisamonoid.Moreorver,(A̱0)and(A̱1),forX,statethatX2INisan(RJ2(IN As)w5)2IN s-moSdule.a6ThisviewofouraxiomssuggeststhatRJ2(IN As)6&bSereplacedbryanarbitrarysetLandGthatamonoidstructureonL2INybSederivredfromamapLL2INg!IL.Then,cofKcourse,themonoidmoSdulestructureonX2INhastocomefromamapLX2IN'@|!tJX.It)shouldbSenotedthatthisrequiremenrtputsanon-trivialrestrictiononthemonoidstructureonL2IN s,:andsimilarlyonthemoSdulestructureonꨳX2IN ,namelythefollorwing:Íthei2th"compSonenrt"ofaproductoftrwoelementsinL2IN TunderthemultiplicationvL2IN {'L2IN=!|L2INݰdepSendsjonlyonthei2thF"componenrt"ofthe rstfactorandingeneralonthefullsecondfactor.SevreralinstancesofthisverygeneralschemehaveappSearedintheliterature.TheBarycenrtricCalculusin[4]and[14]isavXariantofcasewhereRZisasuitableringandRtheelemenrtsofLarethoser̷a2]RJ2(IN As)JforwhichPPi2IN$1Kr̴ij7=]1.o(ThistheoryofR"baryzenrtrischerKalkSvul"waslaterelabSoratedin[5].Thetheoryof"AneR aume"2devrelopSedin[1],[2]iscloselyrelatedtothebarycentriccalculus. AInaddition,^vXariousGconrvexitytheories(see[8],^[9],[10],[12],[13],etc.)OarespSecialcases./Inϱ[3],applicationstovXariousphrysicalproblemsarediscussed.Additionalexamplesarelistedin1.4.WVecallleftlineartheoryLthespSecialtrypeofmonoidstructurewredescribedabSorveonthesetL2ND,nwherenorwthesetI0NݮofnaturalnumbSersisreplacedbyanarbitraryYsetN2=ofcardinalitrygreaterthantwo.LFVurthermorealeftL-moSduleis,bry de nition,asetXltogetherwithanopSerationLX2N Y!9Xsubject tothelarws(Ḻ0;X)and(Ḻ1;X)thatareobtainedfrom(A̱0)and(A̱1)bryreplacingI0NvbyN@.9So@2wreobtainacategoryofleftL-moSdules.WVestudythiscategoryinordertoIseewhicrhfurtherstructuresonX̰canbSederivedfromit.RThecategoryofL-L-bimoSdulesturnsouttobe(almost)monoidalwithaspeci ctensorproduct.One_thingwrecouldnot ndoutis,whethertheassoSciativityhomomorphism :$(X ̷L wYp) ̷LZs¶ !X ̷L(Yal ̷LZܞ)Tisanisomorphismingeneral(cf.6.17);horwever,@itfsatis estheusualcoherenceconditionsformonoidalcategories.0FVur-thermoreŖthereisarighrtadjoint,3i.e./aninnerhom-functorforthistensorproSduct.InwithentriesinthecommutatorofasubsetꨳS)URRisaunitalleftlineartheoryV.h);IfI0NꚰisthesetofnaturalnrumbSers;thenthesingletonsetf 2ӓginRJ2(IN As)KwithP ;i2IN6 2iͻ=UR1isaunitalsequenrtialleftlineartheoryV.i)7IfI0N㜰isthesetofnaturalnrumbSers7thenthesetofsequencesinRJ2(IN As)̰withatmostonenon-zerocoSecienrtisaunitalsequentialleftlineartheoryV.FVorcthefollorwingexamplesletRbSetheringofrealorcomplexnumbSersandI0NthesetofnaturalnrumbSers.j)Theset ̴R !ofallelemenrtsx2\ofRJ2INwKwithP*ɟi2IN#{Ķkx2idڶkUR1isaunitalsequentialleftlineartheoryV,calledtotalconrvexitytheory[9].k)Theset ̴sc ofallelemenrtsx2ذofRJ2INwithPni2IN# x2i,=UR1andnon-negativerealenrtries{isaunitalsequentialleftlineartheoryV,calledsupSerconvexitytheory[11].F`&M`Hķl)5Theset ̴R ;fin䅰ofallelemenrtsx2 P9ofRJ2(IN As)jwithPpi2IN%okkx2idڶk"15isaunitalsequenrtialleftlineartheoryV,called nitetotalconvexitytheory[9].m)m)Theset ̴cofallelemenrtsx2--ofRJ2(IN As)^withP di2IN#L_x2i,=UR1andnon-negativerealenrtriesisaunitalsequentialleftlineartheoryV,calledconvexitytheory[14].n)E$ThesetPAofallelemenrtsx2 (ofRJ2INwithP_i2IN$$Zx2iT0V1andnon-negativerealenrtriesұisaunitalsequentialleftlineartheoryV, calledpSositiveconvexitytheory[8],[15].o)ThesetP2+ Ӱofallelemenrtsx2ɰofRJ2INЂwith0h<

2L;2L2N:lC(0)2N=lC(02ND)=(lC02ND)UR=0ꨰhencebryuniqueness0UR=0.(b)[,ByassoSciativitrywehave8lXe2L;2L2N .:(lC02ND)=l(02ND)=l02N tphence[,bryuniquenessꨳlC02N n=UR0.0ffǟffffffffǎ3.3DZCorollary: ELet)XbSeanL-bimodule.IfXhasarighrtzerooraleftzero,thenꨳX+hasazero.3.4Lemma: 8ThefollorwingareequivXalentfortheL-bimoSduleL: v`&M`Hķ(a)Lhasarighrtzero.ud(b)Lhasaleftzero.Ü(c)Lhasazero.Proof:(c)F)(a):WVeharveFtoshorwuniquenessofthezero.zLet020kf2-Lsuchthatꨶ8l2URL:lC020N ϰ=0209,then020#=0020N ϰ=0bry(ii).(c))(b):WVeharvetoshorwuniquenessofthezero. Let020 02bLsuchthat8UR2L2N n:0209=020,then020#=UR02002N n=0bry(i).TheconrverseisthepreviouscorollaryV.1̟ffǟffffffffǎt3.5V)Lemma: Let=Lharve=azero0andletXbSealeftL-module.,Thenwrehave E׶8s;Ë2URX N `:0Ű=0n9:Proof:8Letꨳs;Ë2URX2N oandj%2N@.8De ne=S2X2N obryuh(i)UR:=f\(dcs(jӰ);=forOi=j;cn9(i);=forOi6=j:ZThenwithko6=RjO_wreget0"Ű=(0s2(jӰ)2ND)=0(s2(jӰ)2NDs)=0(jӰ)2N =0(s2(j)2ND)=(0s2(jӰ)2ND)=S=UR(0(kg)2N)=S=UR0((kg)2N)UR=0n9(kg)2N n=0(s2(k)2NDn9)=(0s2(k)2ND)Ë=0n9. "Dffǟffffffffǎ3.6Corollary: 8LetꨶLharveazero0.(a)IfXisanon-emptryleftL-moSdulethenXhasarighrtzero020 y26@Xand8Ŷ2URX2N `:0=0209.ud(b)If'nXisanon-emptryL-bimoSduleandhasaleftzero020then020isalsothe(unique)righrtzeroofX+and ˶8Ŷ2URX N `:0=0 09;ð8xUR2XFհ:x0 N n=0 09:ėÜ(c)IfꨳfQ:URXFն .!YisahomomorphismofleftL-moSdules,thenfG(0)=0.Proof:v(a) FVorsome^2Xde ne020X$:=0^2X.ThenlC020N h=l(0s)2N /=l02ND^=0=k3020żanduniquenessisobtainedasfollorws._pLet02002XbSearighrtzero._pThen0200qİ=UR00200N =0020N ϰ=0209.(b)Xhasarighrtzero0200bypart(a)whichisazeroby3.3hencebyuniquenessofttheleftzerowreget020ɰ=@0200 ]=0s.~Thustݳx02N Y԰=x(02ND)=(x02N)tݰimpliesbryuniquenessꨳx02N n=UR0209.(c)ꨳfG(0)UR=f(0s)=0f2N aC(s)=0.BffǟffffffffǎAnon-emptryrightL-moSdule,+however,willingeneralnotharvealeftzero.հTheelemenrtȳx02N 2XKsatis es(ii),)PbutitwillnotbSeunique,e.g.@X'=f0;0209gwherebSothelemenrtssatisfy(ii). `&M`Hķ84. S(Semi-)AdditivepTheoriesandScalars]The]elemenrtsofLcanbSeconsideredasoperatorswhicrhproduceallorwable]linearcomrbinationsofelementsinaleftL-moSdule.Thisistheessenceofalltheex-amplesZofconrvexityZtheories.)InsomesensethemrultiplicationofelementsinthemoSdulebrycertainelements(whichwewillcallscalarsandwhichwillbSediscussedlateron)andtheadditioninthemoSdulearehiddenamongtheseoperators.&Ob-servre,_however,thattheadditionpropSerisvreryoftennotallowableinconvexsets.InNsomecases,fhorwever,thereNwillbSeastructureofanadditiononthemodule.In,section3wresawthatazeroiscarriedoverfromLtomoSdulesoverL.wmWVewillsstudynorwhowmuchofanadditivestructurewillbSetransferredinasimilarwrayV. ThroughoutcthissectionwreshallassumethatLisaunitalleftlineartheorywhicrhhasazero:80UR2L.FVor"ųi;j%2URNcwithi6=jϘandforelemenrtsx;yinaleftL-moSduleXHlet"(x;yn9ji;jӰ)UR2X2N obSegivrenbySpVS"(x;yn9ji;jӰ)(kg)UR:=򍓫8 >< >:fd80;0鷰if;ko6=i;j;8x;0鷰if;ko=i;8y;0鷰if;ko=j:+tv4.1De nition:+L>>< >>>:&e80;0鷰if;t6=i;j;kg;8x;0鷰if;t=i;8yn9;0鷰if;t=j;8z;0鷰if;t=kg:Ҡ`&M`HķThenv(x+yn9)+zD`s=UR(s2(i)+(jӰ))"(x+yn9;zji;j)D`s=UR(s2(i)+(jӰ))"(((i)+(jӰ))"(x;yn9;zji;j;kg);(kg)"(x;yn9;zji;j;k)ji;jӰ)D`s=UR(s2(i)+(jӰ))"((i)+(jӰ);(kg)ji;jӰ)"(x;yn9;zji;j;k)D`s=UR((s2(i)+(jӰ))+(kg))"(x;yn9;zji;j;k)and8similarlyx?s+(y+z)UR=(s2(i)?s+((jӰ)+(kg)))"(x;yn9;zji;j;k).Thrus8assoSciativityisinheritedbrythemoSdules.FVurthermore~@x<+yË=UR(s2(i)+(jӰ))"(x;yn9ji;j)~@andy;u+X2N 쐶3ɰ(lC;s)7!lv<2Z3bSeӕgivren.WVeobservethatX2N"-:bP϶԰η=(X2N ǰ)2N"-:bAacmr61>andde ne؍niL N"-:aPX N"-:b)X3UR(;)7!2Zܞ N"-:a+b1bryG=(1;1;0;0;:::ʞ)sandtheadditionisassociativre,commrutative,Gwithzeroelemenrtandwithinverses,GͳMBݰisanAbSeliangroupbyProp. 4.5.Clearlyh(rr;0;:::ʞ)isa1-scalarinL. FVormh2M_Landr2R7wrede ner2mXV:=(rr;0;:::ʞ)mXV=(rr;0;:::ʞ)"(mj1). By4.12wegetr2(m̱1+m̱2)XV=(rr;0;:::ʞ)3(m̱1+m̱2)+=(r;0;:::ʞ)"(m̱1+3m̱2j1)+=(r;0;:::ʞ)a"(m̱1;m̱2j1;2)+=(rr;0;:::ʞ)a"(s2(1);(2)j1;2)"(m̱1;m̱2j1;2)@=a"(rSs2(1);r(2)j1;2)"(m̱1;m̱2j1;2)@=r6m̱1j+rm̱2.8FVurtherbry4.11wehave Sd (ṟ1j+ṟ2)mYɰ=l4Q(ṟ1j+ṟ2;0;:::ʞ)mUR=a"((ṟ1;0;:::ʞ);(ṟ2;0;:::ʞ)j1;2)"(mj1)Y=l4Qṟ1jm+ṟ2m:TheassoSciativitryfollowsfromd6_(rSs)mjzG=}(rSs;0;:::ʞ)mUR=(rr;0;:::)"((s;0;:::)j1)"(mj1)jzG=}(rr;0;:::ʞ)"(smj1)UR=r6(sm):!FinallyŁ1?m԰=(1;0;:::ʞ)"(mj1)=s2(1)"(mj1)=m:WVelearvetothereadertocrheckthatthisdescribSesaninrversetothefunctorofexample2.8.b).ڟffǟffffffffǎ25. SFreepModulesInthissectionwrewillexplicitlyconstructfreemoSdules.Theirexistenceisactuallyclearfromthefactthatwreareconsideringalgebraiccategoriesandthattheunderlyingofunctorsarealgebraicfunctorsinthesenseof[6]sotheyharveoleftadjoinrt)"free"functors. cButwearealsointerestedintheactualsizeoffreemoSdulesandintheircomputationalrules.LetsSetE̴N!RbSesthecategoryofnon-emptrysetsofcardinalitylessthanorequaltothecardinalitryofN@.xGWVe rstwanttoconstructfreeleftL-moSdules̷L XEFư(Yp)oversetsꨳYinSet=f1~g (x)l=fG(x)l=W~*f(x;)henceWyR~*fR=*~URg c./ffǟffffffffǎ`&M`Hķ5.6Prop`osition: 8LetꨳX+bSeaset.8ThenthefreeL-M-bimoduleorverꨳX+isA̷LF̷M (X)UR:=̷L ͳFư(X+M ND):Proof:8Inthefollorwinguniversalproblemdiagram=z􍍍֜jYX֜XX+M2Ny%f҄fd9ά-`M9, f7џX7џlX7џX7џlX7џX7џlX7џX7џlX7џX7џlX7џX7џlXXџXXџz՜Ž՜R̷LFͳFư(X+M2ND)f҄fd@ά-`3,)ôfǟ-:0޹lHlHlHlHlHlHɑHɑjM9F9%ܳM,@@fe,:Q@?0Ѵfǟ-:00themapfG200 wisahomomorphismofrighrtM-moSdulessincefG20)SiscompatiblewithmrultiplicationfromtherightwithelementsofM2ND.$xffǟffffffffǎ5.7De nition:|LetóX}FbSealeft,arighrt,orabi-moSdule.>AsubsetRelURXWԳXiscalleda(left,hrighrt,orbi-)congruencerelationifRelisanequivXalencerelationandaleft,righrt,orbi-sub-moSdule.5.8Prop`osition: 8LetꨳX+bSea(left,righrt,orbi-)module.(a)FVorxeacrhsubsetU62URXLɳXthereisasmallest(left,right,orxbi-)congruencerelationRelgwithꨳU6URRelwi.ud(b)FVorWyanry(left,right,orWybi-)congruencerelationRel thesetofequivXalenceclassesꨳX=Rel isa(left,righrt,orbi-)moSdule.Proof:(a)TVakreReldastheintersectionofallcongruencerelationscontainingU@.(b)TheproSofofthemodulepropertiesisstraighrtforward.l\ffǟffffffffǎ2J6.TensorpProductsFVoramapfQ:URX+Yp2N `_!ZFwrede nefG2N"S :URX2N oYp2N `_!Zܞ2N brypyfG N"(s;n9)(i)UR:=fG((i);n9);i2N:&6.1 ̩De nition:RLet fA̷LX̷M , '̷MbY̷K, 'and fA̷LZ̷KiװbSe fAbimodules.A dmapf7v:wXYp2N+!ZNisrcalledbilinearifforallxw2XJg;2X2N dz;]2Yp2N ;l32L;UR2M2ND;2K,`2NE:SfdȳfG(lCs;n9)ܑ=lCfG2N"(s;n9);iwfG(x;n9)ܑ=fG(x;n9);fG(x;n9)ܑ=fG(x;n9): `&M`HķAnꨶL-K,`-bimoSduleX+ ̷M HeYtogetherwithabilinearmapT_| UR:X+Yp N 3(x;n9)7!x ̷M HeË2X+ ̷MYis[calledatensorproSductifforevreryL-K,`-bimoduleZ8andforevrerybilinearmapVfа:fѳX Yp2Nq!ZgthereisexactlyonehomomorphismofL-K,`-bimoSdulesgË:URX+ ̷M HeY¶ G!ZFsucrhthatthediagram*ӼX+Yp2NӼ³X+ ̷M HeY$!2fd$ά-@E ,~Ѝjf^P P _Pר Pר q3F9Z@ feQ@?pAgcommrutes.AshiscustomarywrespSeakofXJK ̷M YذasthetensorproductofXandYذandomit#referencetothemapXqVӳYp2NuJ!j=X ̷MYp.&FVor#2X2N ǰ,rv2Y2Nslet ̷M XË2UR(Xr ̷MYp2N )2N VbSede nedbry( ̷M Xn9)(i)UR:=s(i) ̷MË2XrYp2N .2Thefollorwingrulesofcalculationfollowimmediately:;kۦf(lCs) ̷M Heǰ=8OlC(? ̷M Hen9);z(x) ̷M Heǰ=8Ox ̷M He(n9);x ̷M He(n9)=8O(x ̷M Hen9);ݲ(s) ̷M Heǰ=8O(? ̷M Hen9);s ̷M Heǰ=8O? ̷M Hen9;? ̷M He(n9)=8O(? ̷M Hen9):and̷MeY̷Kde ne#HNHom(XJg;Yp)UR:=ffQ:X N `r!Yj82K,` NE;Ŷ2X N `:fG(s)=f(s)g:&6.5Lemma: 8Hom(XJg;Yp)isaM-L-bimoSdulebrytheoperations!dS(( fG)(s)"n:=މ (fG(s))7(fG)(s)"n:=މfG(s) SandabifunctorconrtravXariantinX+andcorvariantinYinM-MoSd-L.Proof:8Straighrtforward.effǟffffffffǎ6.6tBProp`osition: -Letp̷L 8X̷M ,{̷M8Y̷K,{̷L :Z̷K bSepbimodules.5xThenthereisanaturalisomorphismGL-MoSd-T/K,`(X+ ̷M HeY;Zܞ)PUR԰n:=L-MoSd-M(XJg;Hom(Y;Zܞ)):)c`&M`HķProof:m!WVede ne:˶L-MoSd-T/K,`(X ̷M Z/Y;Zܞ) $!L-MoSd-M(XJg;Hom(Y;Zܞ))bryn9(fG)(x)()UR:=f(x ̷M Hen9).8Then/&e4pn9(fG)(x)(){ż=DfG(x ̷M He(n9))UR=f(x ̷M Hen9)=(fG)(x)();+(n9(fG)(x))(){ż=Dn9(fG)(x)()UR=f(x ̷M Hen9)=f(x ̷M Hen9){ż=Dn9(fG)(x)();';(lCn9(fG)2ND(s))(){ż=DlC(fG2N"(? ̷M Hen9))UR=fG(l? ̷M Hen9)=(fG)(lCs)():/URThrusꨳXiswell-de ned.8Theinversemapisde nedby5`n9 1 :URL-MoSd-T/M(XJg;Hom(Y;Zܞ)) !L-MoSd-T/K,`(X+ ̷M HeY;Zܞ)bry n921 ʵ(gn9)(xŶ ̷M Q)l:=gn9(x)().a Sinceg(x)()l=gn9(x)(),dg(lCs)()l=lgn92N}(s)(n9),andXwgn9(x)():=gn9(x)(),sthemap21 ʵ(g)isawrell-de nedbimoSdulehomomor-phismPonX ̷M -$Yp.4mObrviouslyn921istheinversemapton9.4mItiseasilyseenthatthesemapsarenaturaltransformations.Qffǟffffffffǎ6.7Prop`osition: 8Letꨟ̷L C#X̷M ,ꨟ̷MeY̷K,ꨟ̷K >Z̷I !bSebimodules.8Thenthemap ^ h:UR(X+ ̷M HeYp) ̷K >Z13(x ̷M Hen9) ̷K >=S7!x ̷M He( ̷K >)2X+ ̷M(YG ̷K >Zܞ)isabimoSdulehomomorphism,whicrhiscoherentinthesenseofmonoidalcate-gories.Proof:jWVe; rstshorwthatthemapinthepropSositioniswellde ned.FVorthatpurpSosewreconsiderthemap^' :URX+ ̷M HeY¶ G!Hom(Z5;X ̷M He(YG ̷K >Zܞ))withꨳ O(x ̷M Hen9)()UR:=x ̷M He( ̷K >).8Themap iswrell-de nedsince/&ee (lCs) ̷M He( ̷K >)=($lC(? ̷M He( ̷K >));iMx ̷M He( ̷K >)=($x ̷M He( ̷K >);j5x ̷M He(n9 ̷K >)=($x ̷M He( ̷K >);j Ix ̷M He( ̷K >#)=($(x ̷M He( ̷K >))#:The*adjoinrtmapis h:UR(X ̷M ܳYp)  ̷K#Z1 I!X ̷M ܰ(Y ̷K#Zܞ)*with ((x  ̷Mn9) ̷K#)UR=x ̷M He( ̷K >):Thecoherencediagramis1d((U댶 X) Yp) Zd(U댶 (X+ Yp)) Zw%%fd!Yά-MǴ 1(dU댶 ((X+ Yp) Zܞ)%fd!Yά-۬ ,A*@feA]Q@?H#75: ,[@fe[3Q@?}_Ѵ 1,d(U댶 X) (YG Zܞ),(dU댶 (X+ (YG Zܞ)):w%_҄fdά-c襴 6W`&M`HķByelemenrtwisecomputationsitiseasytoseethatthisdiagramcommutes.)şffǟffffffffǎWVewillseeinCor.86.17that 7issurjectivre.6.8XRemark:WVecannotshorwthat Yisanisomorphism.QGInfactweconjecturethatAitisnotingeneral.hHorwever,3#howAcloseitistobSeingbijectivre,canbSeseenbrypropSositions' 6.16and6.17.NevrerthelessL-Mod-T/Lbeharves' likeaclosedmonoidalcategoryV.TheYabSorvepSentagondiagramsucestogeneratecoherence.AndtheinnerQ+hom-functorsharveQ+theusualpropSerties.Beforewreprovethis,oweshowthatthereisatrwo-sidedunitforthetensorproSduct.6.9Prop`osition: 8TherearenaturalisomorphismsS:sUR:L ̷L #XPFն԰_=~XqsandF7d:X+ ̷M HeMP԰n:=Xwhicrhsatisfythecoherencediagramsformonoidalcategories(n񿛍D1(X+ ̷L #L) ̷LY g zt!&X+ ̷L #(L ̷LYp)PL ̷L #LM=`L ̷L #L?d X.LYYi&&.URXҷ X.Ls/ d7&`.URsokX+ ̷L #YNvLqD:/zProof:LWVe#de nes20#:URLX2N `r!Xbry#s209(lC;s):=ls.~Obrviously#s20isbilinear,KVsoitQfactorsuniquelythroughabimoSdulehomomorphismsonthetensorproduct.WVede nes21pް:bX[>!L ̷L \Xy8brys21 \|(x):=s2(1) ̷L \x2ND. Thenss21 \|(x)=s(s2(1)Ag ̷L x2ND)0=(1)x2N t=xhencess21*=0idԶj̴X 嚰ands21 \|s(l ̷L s)0=s21(lCs)=s2(1) ̷L q(lCs)2N {ڰ=b(1) ̷L qlC2N\ӳ =b(1)lC2N ɶ ̷L =bl ̷Ls.PAObrviouslysitissucienttoshorwQthats21 \|sandidagreeonelementsoftheforml ̷Ls,p^ifweknowthatd21Gisan,wL-bimoSdulehomomorphism.zWVeharve,ws21 \|(lCs)UR=s2(1)&/ ̷L~(ls)2N n=s2(1)&/ ̷L~l2N\ӳŰ=l7 ̷L #Ű=URlCڶ ̷L=URlC(s2(1) ̷Lx2NRAiD)=lC(s21 \|)2N(s),wherewreused9߳s2(i) ̷L #Ű=UR(jӰ)(i) N ̷LŰ=UR(jӰ) ̷L(i) NDŰ=UR(jӰ) ̷Lx Nڍiforallcrhoicesofi;jn2cN@.vFVurthermorewehaves21 \|(x)c=s2(1)}, ̷L է(x)2N |߰=s2(1)k ̷L+x2NDUR=s21 \|(x)..ConsequenrtlyԳsands21(Pareinversetoeachother..Theyareobrviouslynaturaltransformations.WVede nenorwd20#:URX_M2N n!X_byܳd209(x;):=x.d20isobrviouslybilinear,4ksoitfactorsWhuniquelythroughanL-bimoSdulehomomorphismdonthetensorproduct.WVe}de ned21:X a!X ̷M dM}bryd21 \|(x):=x# ̷Ms2.^Then}dd21 \|(x)=x=xand`d21 \|d(x ̷M j)^_=x ̷M jё=x ̷M j. Again`wremustshowthatd21ܰisanL-bimoSdulehomomorphism:Jd21 \|( s)= k ̷M еK= (d21 \|)2ND(s)andd21(x)=x ̷M HeȄ=URx ̷MUR=x ̷Ms2=d21 \|(x).rbffǟffffffffǎFw`&M`Hķ6.10BDe nition: ,E 2lL-MoSd-T/LtogetherwithL-bimoSdulehomomorphismsrUR:E^ ̷L E i ^!EandꋳË:L !Eiscalledamonoid,ifthefollorwingdiagramscommruteFu`Nq(E^ ̷L #E) ̷LEu`rE^ ̷L #ENaƨfdm`ά-\]r X.L1u`NqE^ ̷L #(E ̷LE)v?ۘ fev5qۘ?l Z u`,&'@*Ffe&[1@?挻+ r,v?@ fev5q@?tY1 X.Lr_ӍbM$E^ ̷L #E_Ӎ!E_҄fdά-۫Arand/g`%ӳE%8$E^ ̷L #E%fdJpά-٘(1 X.LI{)d-:1,<@fep1@?瑍k6ɱ(I{ X.L1)s-:1,#2%K`y cmr10idQPQPQ*?PQPQPwQPwQq,?@fe q@?Jr_ӍE^ ̷L #E_ӍmE:Ρ_҄fdFά-۫кӷr$6.11AProp`osition: |An95L-L-bimoSduleELisamonoidinL-Mod-T/L,LifandonlyifEisaleftlineartheoryand:5۶L 4!EisahomomorphismofleftlineartheoriesinducingtheL-bimoSdulestructureonE.Proof:Let$ḚbSealeftlineartheoryand:kL 9!EbSe$ahomomorphismofBleftlineartheories. @Thenthemrultiplicationm:EHγE2Nl f!Ede nesBahomomorphismꨶrUR:E^ ̷L #E i ^!EinL-MoSd-T/Lsince'궳m(e; 09)UR=(en9 N}()) 0#=URe(n9 N() 09)UR=m(e; 0);'m(lC; 09)UR=(n9(l)) 09)UR=n9(l)( 09)UR=lm ND(; 09);'m(e; 09)UR=e( 0xn9 N}())UR=(e 0)n9 N}()UR=m(e; 0):ThexinducedmaprisassoSciativre,~sincethemultiplicationisassoSciative.WVeusetheߔsamesymrbSolfortheunitsinLandinE,hencen9(s2)UR=.5/TheߔunitpropSertryfor amonoidfollorwsfroms2(1)jx2N q=X(1)x2N=Xx andxj=x=x. Hence(E;r;n9)isamonoid.ConrverselyletEbSeamonoid.8Thenthereisamrultiplication3ԍ|SmUR:E^E N g "x!"ZE ̷L #E )irg  i ^!Ewhicrh3obviouslyisassoSciative.FVromtheunitpropSertyofamonoidwegetn9(s2(i))SAUR=r(o ̷Lo1)(s2(i)6 ̷L)UR=r( ̷Lo1)(s2(1)6 ̷Le2NRAiD)UR=e̴i,=(i),usingxanequalitryfrom%theproSofofthepreviouspropositionandepkn92N}(s2)UR=r(E$ ̷Ln9)(epk ̷L)UR=eY=es2,eowherewreabbreviatedn92N}()Y=:.ThenistheunitofEŰandEisaleftlineartheoryV.ThemappKpreservrestheunitandinducesthebimoSdulesstructureofE,)3sincen9(lC)|=(lCs2)=ln9(s2)=ls2,)3hencen9(l)̹|=(ls2)̹|=landX`&M`Hķen92N}()s`=e(s2)s`=e.1 Finallyisahomomorphismofleftlineartheoriessinceꨳn9(lC)UR=(lCs2)=ls2=(ls2)Ȅ=UR(ls2)()UR=n9(lC)2N}().C.ffǟffffffffǎ6.12Remark:LetꨳX+andYbSeL-bimodules.8TheevXaluation?evË:URHom(XJg;Yp) ̷L #XFն .!Ywith=evn9(fL[ ̷L\׳s)UR:=fG()=isthecounitforthepairofadjoinrtfunctors\ ̷L\׳XandHom(XJg;).6.13GCorollary:p3Let ̷L ,X̷M ,Nc̷M Y̷K,Nc̷L ޳Z̷K נbSe bimodules. FVoreacrhbimodulehomomorphismufa:BbX ̷M 8YҶ4+!ZRthereisauniquebimoSdulehomomorphismgË:URXFն .!Hom(Y;Zܞ)sucrhthatthediagramcommutes:2g`%qҳX+ ̷M HeY,YζHom(Y;Zܞ) ̷M HeY@feџ@?ߍhMgI{ X.M :Y,@fQ@XQ*@XQ@XƴQ*@XдQ@XڴQ*@XQ@XQXQz,,Z5:%_҄fdPά-c[RevProof:°This,isastandardconsequenceofthefactthat׍ ̷M uJY isleft-adjoinrttoꨶHom(Y;).K ffǟffffffffǎ6.145Corollary:~TheBevXaluationevӋ:eRHom(XJg;Yp)B ̷MXVն.!YSde nesBacompSositionofmapsHom(Y;Zܞ) ̷M HeHom(XJg;Yp)UR3f ̷M He Ë7!(x7!fG( n9(s))2Hom(XJg;Zܞ);whicrhisassoSciativeinthesensethatthediagramI94u`O(Hom(Z5;U@) Hom(Y;Zܞ)) Hom(XJg;Yp)u`Hom(Y;U@) Hom(XJg;Yp)YAƨfdؐά- u`OHom(Z5;U@) (Hom(Y;Zܞ) Hom(XJg;Yp)){_ۘ fe|ۘ?u`,C@*FfeCџ@?,{_@ fe|@?,9۶Hom(Z5;U@) Hom(XJg;Zܞ),%kHom(XJg;U@)_҄fd`pά-`commrutes.Proof:аThe mapHom(Y;Zܞ)Q ̷M Hom(XJg;Yp)n P!nHom(XJg;Zܞ) inducedbrytheevXaluationisuniquelydeterminedbrythefollowingdiagramu`(Hom(Y;Zܞ) Hom(XJg;Yp)) Xu`Hom(Y;Zܞ) (Hom(XJg;Yp) X)eƨfd=ά- u`,X@*FfeXџ@?͍D״gI{ 1u`0Hom(Y;Zܞ) YCۘ feCџۘ?LmHiQ1 ev,C@ feCџ@?7HiQev,.hKHom(XJg;Zܞ) X,?Z5:A_҄fd$Ѝά-`iM`&M`HķItiseasytoseethatthecompSositionisdescribedasgivrenintheCorollaryV.ffǟffffffffǎ6.15؀Corollary:FVorabimoSdule̷L ,X̷MqthesetofinnerendomorphismsEónd(X)UR:=Hom(XJg;X)isaleftlineartheoryV.Proof:ByJBCorollary6.14Eónd(X)isamonoidinthequasi-monoidalcategoryL-MoSd-T/L.#^The#unitarylarwisgivenbythemapLUR3l7!(Ŷ7!lCs)2Eónd(X).#^ByPropSosition6.11itisaleftlineartheoryV. ffǟffffffffǎ6.16'Lemma: LeteNbSeanin niteset. Let(x̴iȶ ̷M 8̴idڰ)̴i2N 2UR(Xq ̷MYp)2ND.Thenthereareelemenrtss202URX2N ,n920Ķ2Yp2N\sucrhthatxܰ(x̴i ̷M He̴idڰ)̴i2N =UR(s 0b(i) ̷Mn9 0̴idڰ)̴i2Nvareintheimageof . ffǟffffffffǎ&Bibliographry1[1]0^Bos,TW.,and?WVol ,G.:AneR aumeI.Mitt.Math.Sem.Gieen,129,0^1978,1-115.U[2]0^|,|:8AneR aumeISI.Mitt.Math.Sem.Gieen,130,1978,0-83.[3]0^Gudder,ɤS.PV.,andcScrhroSeck,ɤF.:$=GeneralizedConvexityV.+SIAMXJ.Math.0^Anal.811,1980,984-1001.[4]0^Kneser,H.:8KonrvexeRaSvume.8Arch.d.Math.3,1952,198-206.[5]0^Ostermann,rNF.,andT7Scrhmidt,J.:DerbaryzenrtrischeKalkSvulalsaxioma-0^tiscrheGrundlagederanenGeometrie.J.Reineu.Angew.Math.224,0^1966,44-57.[6]0^Prareigis,!B.:;CategoriesandFVunctors.@iAcademicPress,NewYVork|0^London,1970.[7]0^|:(Non-additivre8ringandmoSduletheoryI.Generaltheoryofmonoids.0^Publ.Math.8(Debrecen)24,1977,189-204.wŠ`&M`Hķ[8]0^PumplSvun,sfD.:WRegularlyUOrderedBanacrhSpacesandPositivelyConvex0^Spaces.8ResultsinMath.7,1984,85-112.U[9]0^PumplSvun,2D.,and{R ohrl,H.:BanacrhSpacesandTVotallyConvexSpaces0^I.Comm.inAlg.812,1984,953-1019.ݚ[10]0^|,A|:-BanacrhASpacesandTVotallyConvexSpacesISI.Comm.inAlg.h13,0^1985,1047-1113.ݚ[11]0^RoSdrse,G.:8SuperkonvexeAnalysis.8Arch.Math.34,1980,452-462.ݚ[12]0^R ohrl,H.:?ConrvexityTheoriesI.-ConrvexSpaces. oIn:Constantin0^CarathrseoSdoryV.; An@InternationalTVribute.; WorldScienrti cPubl.; 1175-0^1209,1991.ݚ[13]0^Semadeni,zZ.:8MonadsjandtheirEilenrbSerg-MoorejAlgebrasinFVunc-0^tionalAnalysis.8Queen'sPrapSersPureAppl.Math.33,1973.ݚ[14]0^Stone,M.H.: Prostulates=fortheBarycentricCalculus.ȟAnn.Mat.Pura0^Appl.829,1949,25-30.ݚ[15]0^Wicrkenh auser,:JA.:JPositively*]ConvexSpaces.DiplomarbSeitFU*LHagen0^1987.dUԀBoSdoPrareigisMath.8Inst.,Univ.MSvuncrhen8000MSvuncrhen2GermanryԀԱhHelmrutR ohrlԱh9322LaJollaFVarmsRd.ԱhLaJolla,CA92037ԱhUSA;`&M %K`y cmr10$@ cmti12#}h! cmsl12""V 3 cmbx10!", 3 cmsy100ercmmi7 b> 3 cmmi10K`y 3 cmr10N cmbx12 P cmu10- cmcsc10-ff cmcsc10K`yff cmdunh10 q% cmsy6 K cmsy8 !", cmsy10 ;cmmi6 2cmmi8g cmmi12Aacmr6|{Ycmr8XQ cmr12O line10u cmex106