; TeX output 1994.06.30:1218By%j cmti9[PageN<1]1 A3"VG cmbx10ConrvexityU>Theories0conrt.}QFXoundations$2':ff cmti10BoCdo&Pareigis,DieterPumpl/tun,andHelmutRohrlGt : cmbx9Abstract.o cmr9TheR8mainissueofthispapAerisanaxiomatizationofthenotionofabso- lutelyecon9vergentseriesinvolvingasetofsummandsof xed(butunrestricted)in nitecardinalit9y5" cmmi9N.Thisnotionisusedtode nethecategoryN 0ercmmi7RbpnSmoAd2-=ٓRcmr71:ofR>-prenormedR>-semimoAdulesk?withN-summationwhosehomomorphismsarecon9tractive.k?Basedonthisw9eintroAduceleftN-convexitytheoriesandthecategoryCofleft-convexmoAdules.W:esho9wthattheclosedunitballfunctorNRbpnSmoAd3-=1;l cmsy9A!bSet#,theforgetfulfunctorC5!ƬSetk,5andtheassoAciated-con9vex5modulefunctorNRbpnSmod2.՟-=19=>(!ƬCha9ve5leftadjoin9ts.1991TMathematicsSubjectClassi cation:16Y60,52A01.9J.;"Vff cmbx10In=trouductionK`y cmr10Thebasicde nitionsofthispapGerarecontainedin !", cmsy10x\L1.Theyconcernanaxiomatic approachDtothenotionof\absolutelyconvergentseries"inprenormedsemiringsandhprenormedsemimoGdules.Thetypeofseriesweareinterestedinhaveasetofsummands ofa xed,butarbitrary*,in nitecardinality b> cmmi10N.SemimoGdulesequippedwith=afamilyofsuch\summable"seriesaresaidtohaveN-summation.Thesectionconcludeswithfewstatementsdirectlyrelatedtothebasicde nitions.x2consistsof}severalresultsinvolvingsemimoGduleswithN-summation.Inadditionweintro-duce&mapsfroma xed,butarbitrary*,settoasemimoGdulewithN-summationthatare\summable".Thesemapsareusedinx3tode ne,foranarbitrarysetA,aGthefreesemimoGduleL^N(A)withN-summationonthesetA.L^N(A)isagen-eralizationVofthewellknownfunctional-analyticconceptof`1|s-spaceonthesetA.MThefunctorSet3 A7!L^N(A)2NRbpnSmoGd6o1>kisMtheleftadjointofthefor-getfulw&functorNRbpnSmoGd6ȟ1@ XF'!Set;here,NRbpnSmoGd6ȟ1?astandsforthecategoryof1Rǫ-prenormedR-semimoGduleswithN-summationandtheircontractive1homo-morphisms.Insection4wedealwithN-convexitytheoriesoverprenormedsemiringslkwithN-summation(generalizingthecorrespGondingconceptin[5],x43andA|xE7),A|derivefewofthepropGertiesof-convexmoGdules,andgiveanexplicitconstruction!ofthefree-convex!moGdules.Thisconstructiondi ersfromandismore[IpGerspicuousthantheonegivenin[3],x5.Inx5weintroGducethefunctorOܫ:tNRbpnSmoGd5ᶟ1==C!tC, whereforagivenN-convexitytheorythecategory*2F ެSectionT3: UPB.P9areigis,D.PumplA;un,andH.R`ohrl1 Cis#thecategoryof-convex#moGdulesandtheirhomomorphisms.Oqȫ(M)isthe closed"unitballofM9ЫequippGedwiththeobviousoperationofontheclosedunitballocofM.InadditionweexhibittheleftadjointS^ 'ofOqȫ.ThepapGerendswithaUUsectionpresentingseveralexamplesforthepreviouslyintroGducedconcepts.;Q`1.U>Thebasicde nitionsLetR_bGeasemiring(inthesenseof[5],xa1).LetfurthermoreNbea xedsetofcardinalityr@0|s.MapsfromNtoRǫ,thatiselementsofRǟ^Nx,willbGedenotedbylowerdcasegreekletterswithalowerdplaceholdersymbGol,e.g. O!cmsy7or ÐYu7msam7Y;occasionallywe!willwrite:f n:'ln2Ng!orf z(n)'l:n2Ng!insteadof .Ifwede ne +G ` as\themapN8?3!$n7v! nq~+Gq~ 8 n2R#then\Rǟ^N 4ԫbGecomesasemiring.IfrhA2!$RandifwedenotetheconstantmapN) F !|RXwithvqaluerɮbyruthenR&C3|rY7!r`2Rǟ^N isUUahomomorphismofsemirings.9LetMOMdjbGeasemimodule(inthesenseof[5],x1)overMOthesemiringRǫ.ThenwecanagainformM^N̫.Let _2Rǟ^N 'and^0;2M^N̫,andde ne and^0*y+pGointwise|asinthecaseofR|thenM^N TbecomesaRǟ^Nx-semimodule.Asbeforeweletm,m2M,bGetheconstantmapwithvqaluem.ThenM33m7!m_2M^NisUUahomomorphismofRǫ-semimoGdules.9IfRisapartiallyorderedsemiringandM8isapartiallyorderedRǫ-semimoGdule(inthesenseof[5],x1)thenwede ne^0.,where^0;2.M^N̫,as^0፴nnforalln p2N.ThismakesM^N 7apartiallyorderedsemimoGduleoverthepartiallyorderedcZsemiringRǟ^Nx.Obviously*,R<3ur%7!rwY2Rǟ^N ҫandcZM3m7!m2M^N &areorderרpreserving.902LM^N tissaidtobGeboundediftherearem^09;m^002LMëwithm^0jѫm^00r,[andafamilyf^i፺j:i2IgiscalleduniformlybGoundediftherearem^09;m^00㊷2MlpwithUUm^0_^i፺m^00qǫforUUalli2I.9IftMisanRǫ-prenormedR-semimoGduleoverttheprenormedsemiringRVandifkxkdenotestheprenormwithvqalueconeC/(see[5],xM2)then,with2M^N̫,wedenotenbykkthemapN3n7!knq~k2C;nhencekkisinC^NLͫ.ThismakesM^N īamRǟ^Nx-prenormedRǟ^N-semimoGduleovermtheprenormedsemiringRǟ^N pwithvqalueconeUUC^NLͫ.9FinallygaconstructionthatwillbGeusedlater.LetAbesomesetand:N3 !Aasetmap.LetfurthermoreM:bGeanRǫ-semimoduleand_2M^N̫.Givena2A}rweUUdenoteby:r0ncmsy5Zcmr51 QϮ(a)[themapN3 !MlpgivenbynoN33n7!^u cmex10G nI,UUif(n)=a; 0I,UUotherwise.#(InUUotherwords,:r1 QϮ(a)[isgivenbytheformulaeÍ)˱:r1 QϮ(a)[g^j1 t(a)=j1(a)UWand>#:r1 QϮ(a)[g^jNOL msbm10Lr817(a)=0jNLr81 t(a):^Ы3/J NotethatanysubsetofNcanbGeobtainedas^1 t(a),providedthatAcontains atleasttwoelements,andthateachpartitionofNcanbGewrittenasf^1 t(a):a2A^09gUUforsomesubsetA^0#ofA,providedthatAislargeenough.9In0thefollowingde nitionwerefertotheconceptofpGositivesemiring.Ac-cordingkto[5],xʫ1,apGositivesemiringisapartiallyorderedsemiringwith0asitssmallestUUelement."V cmbx10(1.1)De nition.QLetCӫbGeapositivesemiring.Bya$': cmti10leftN-summationforCis+meantapair(SCڱ;C)+consistingofaC-subsemimoGduleSC ٫ofC^N x̫andaC-homomorphismUUC :SCh!C qsuchUUthat (i)sC^(N,):=iAf  %2C^N :supp[  is nitegiscontainedinSC andforalls  5ݷ2C^(N,)ϫ,oDCګ( )=P+4޺0mf n w:n2suppñ g,oDwhereP޺0:standsforthesusualUUsumof nitelymanyelementsinC; (ii)sforTall  ʷ2pSC and 2pC^N with p ; isinSC andCګ( )psCګ( ); Wr(iii)sforCevery #2S?SC andeverymap'S?:NjZ !N; : z'r1 QϮ(n)[OisCinSC forallsnn2N,Vandthemap 1ɍ z'r17̍givenbyNꉷ3nn7!Cګ( : z'r1 QϮ(n)[OH)2CrisVinSCsandUUsatis esCګ( 1ɍ z'r17̍ȫ)=C( );(iv)sif{ ~isinC^N gandthereexistsamap':N :q!Nsuch{that : z'r1 QϮ(n)[isinsSC /forUUalln2NlpandUUthat 1ɍ z'r17̍sisinSCthen 9isinSCګ.G(1.2)Lemma.L}'etCLbeapositivesemiringwithleftN-summation(SCڱ;C). Thenthec}'onditions(1.1),(i)-(iii),implythatforevery Ʒ2,SC theinequalities nVFȫCګ( );n2N,Jhold;inp}'articular,ifsupf n:ȱn2NgJexists,supf n:ȱn2NgCګ( )issatis e}'d.ַut9The nextde nitionusesthenotionofprenormedsemiring.Dueto[5],(2.1),this)isasemiringR=togetherwithamapkk:R߷U!C,)whereCisapGositiveandcompleteUU(withrespGecttothepartialorder)semiring,suchthat (o)sk0k=0 UVandk1k=1; (i)skr1S+8r2|skkr1k8+kr2kԫ,UUforallr1;r2C2Rǫ; (ii)skr1|sr2kkr1|skkr2kw,UUforallr1|s;r2C2Rǫ.TheKnsemiringCiscalledthevqalueconeofR_5andkkissaidtobGetheprenormofRǫ.(1.3)mDe nition.TLetRbGeaprenormedsemiringwithprenormkk:R߷U!C(:TByaleftN-summationforRʫismeantaleftN-summation(SCڱ;C)forCntogetherwithДapair(SRb;R)ДconsistingofanRǫ-subsemimoGduleSR _ofR^N z andanR-homomorphismUUR Vz:SRr!RisuchUUthat (o)s _2Rǟ^N ͫisUUinSR ䷫ifandonlyifk kisinSCګ; (i)sRǟ^(N,)_:=uf  2Rǟ^N :suppgh  %isA nitegiscontainedinSR andforalls  =ŷ2Rǟ^(N,) z,tRb( )=P3޺0Uf n _:n2supp g,twhereP=޺0DxstandsforthesusualUUsumof nitelymanyelementsinRǫ; (ii)sforIall _2SR band 2Rǟ^N xwithk kk k,I isinSR bandkRb ksCڷk k;"4F ެSectionT3: UPB.P9areigis,D.PumplA;un,andH.R`ohrl1 Wr(iii)sforevery 2]SR ?andevery']:Ntڷ P!N; : z'r1 QϮ(n)[isinSR ?foralln]2N, sandthemap 1ɍ z'r17̍ϫgivenbyN33n7!Rb( : z'r1 QϮ(n)[OH)2RΫisinSR 3iandsatis essRb( 1ɍ z'r17̍ȫ)=R( ).:9De nition(1.4)requiresthenotionofRǫ-prenormedleftR-semimoGduleover thevprenormedsemiringR=withprenormkk:Rj !C.vBythisismeantaleftRǫ-semimoGduleUUMlptogetherwithamapkk:M3 !C qsuchUUthat (o)sk0k=0; (i)skm1S+8m2|skkm1k8+km2krJ9,UUforallm1;m2C2M; (ii)skrGmkkrkkmk%,UUforallr52R;m2M.(1.4)Elemen=taryresultsj(2.1)qLemma.L}'etqMbeanR-prenormedR-semimodulewithleftN-summation(SM\;M).L}'etfurthermore_2SM ,and2M^N b}'esuchthatthereisabijection~feg' O:/supp!nȈ !/suppfwith͝n"=/}feB6'B:(n),foralln2supp!n.ThenfisinSMandM\()=M().Pr}'oof.hExtend~feg'tosomemap'8:NS !N.Thenonechecksquicklythat=㍱1ɍ'r17̍Aholds.-Hence(1.1),(iii),showsthatƌisinSM andthatM\()=M()isUUvqalid.;Suu;SutI6F ެSectionT3: UPB.P9areigis,D.PumplA;un,andH.R`ohrl1 (2.2)8Corollary .L}'etMSbeanR-prenormedR-semimodulewithleftN-summation (SM\;M).L}'etfurthermoref˚: A ԁ!M0beamapwithcard(suppʱf) cardN.ThenMther}'eisamap`:N{ !A,Ma D2`M^N,andsetsA^0VandN^0 2qwithsuppʱfڧA^0QA,supp_N^0lN,andcard~A^0cardűNsuchthatWa)sA^0Q(N)andA^09jjN^0y;isabije}'ctionb)sn8=f((n))w,foralln2N^0T. Mor}'eover,if~feAşg F,~fegqƺ ],}feA >0^bGeanRǫ-prenormedR-semimoGdulewithleftN-summa-tionn~(SM\;M).LetfurthermoreAbGeanyset.Thenwede neSMQ;AD1asthesetofmapsfI:5A R!MsuchthatcardH(suppʱf)card 9Nandforsomedata;;A^09,andnN^0T8in(2.2),2SM O@holds.Moreover,nwede neMQ;Aճ(f):=M\(),forallfڧ2SMQ;Aճ,UUtoobtainamapMQ;A˫:SMQ:AA!M.9Note֙thatSMQ;N1=I1SM andMQ;N=I1M hold.W*ewillwriteoGccasionallyMQ;Aճff(a):a2AgUUinsteadofMQ;A(f):9F*orUUfڧ:A!MlpwedenotebykfkthemapA3a7!kf(a)k2C.!(2.5)Lemma.F;or3everysetAandeveryR-pr}'enormed3R-semimodule3MJwithleftN-summation,SMQ;ApisanR-subsemimo}'duleofM^A 5andMQ;AЫ:SMQ;AF!MisanR-homomorphismofR-pr}'enormedsemimodules;inparticular,kMQ;Aճ(f)kCI;A1(kfk):Pr}'oof.&ALetifandgPBbGeinSMQ;Aճ:Choosethedatain(2.2)toservebothfandg(seeE(2.3)).If)correspGondstofԫandcorrespGondstogIthen,obviously*,+h±correspGondsbtofU>+Ag[٫.Hencef+AgebGelongstoSMQ;Aճ.Similarlyoneshowsthatrf,r52Rǫ,UUfڧ2SMQ;Aճ,alsobGelongstoSMQ;A.Moreover,.MQ;Aճ(fLo+8g[٫)=M\(ī+)=M\()8+M\()=MQ;Aճ(f)8+MQ;A(g[٫)[^Ы7/J and|HewMQ;Aճ(rGf)=M\(r)=rM\()=rMQ;Aճ(f):Finally*,BIkMQ;Aճ(f)k=kM\()kCګ(kk)=CI;A1(kfk):[K msam10K Pu(2.6)qLemma.L}'etqMbeanR-prenormedR-semimodulewithleftN-summation (SM\;M).!F;urthermor}'eletAbeanyset.ThenfH2ǹM^A Then,foranysetA;M^(A)ū:=ffڧ2M^A aë:suppf{ish> nitegisc}'ontainedinuSMQ;AKQandforallfs2_M^(A) ڭ;MQ;Aճ(f)=P8޺0qff(a):a2supp efg,uwher}'ePٟ޺0standsfortheusualsumof nitelymanyelementsinM.Pr}'oof.(1.4),UU(i),and(2.2).uuut(2.8)qLemma.L}'etqMbeanR-prenormedR-semimodulewithleftN-summation(SM\;M).r{F;urthermor}'eletAbeanyset.Iff isinSMQ;AH.andg"2M^A &withkg[ٷkkfkthengisinSMQ;AiandkMQ;Aճ(g[٫)kCI;A1(kfk):Pr}'oof.(1.4),(ii),UUand(2.2).醭u醭t(2.9)qLemma.L}'etqMbeanR-prenormedR-semimodulewithleftN-summation(SM\;M).F;urthermor}'eletAbeanyset.IffoisinSMQ;Aand J:A [!Aisany Vmapthen,foranya2A,themapf^ @Lr1 (a)givenby7JlsA3b7!^G f(b)G,if [٫(b)=a; 0G,otherwise.@jisinSMQ;Aճ;mor}'eover,themapf^ @Lr11hgivenbyA3a7!MQ;A(f^ @Lr1 (a)ɫ)2M,isinSMQ;AiandMQ;Aճ(f^ @Lr1pW)=MQ;A(f). PuPr}'oof.(1.4),(iii),UUand(2.2).濐u濐t(2.10)rCorollary .L}'etDM[beanR-prenormedR-semimodulewithleftN-summa- tioni(SM\;M).F;urthermor}'eletAbeanysetand:$A Ӛ!Ab}'eanybijection.o8F ެSectionT3: UPB.P9areigis,D.PumplA;un,andH.R`ohrl1 ThenK,f&2M^A isinSMQ;A ifandonlyiff^ :=f<isinSMQ;Aճ,inwhichc}'ase MQ;Aճ(f^)=MQ;A(f):֍Pr}'oof.ImmediateUUconsequenceof(2.9)sust(2.11)Lemma.L}'etuMbeanR-prenormedR-semimodulewithleftN-summation(SM\;M).MF;urthermor}'eletAandBbeanysets.F;orfڧ2SMQ;AB! de nef(a;->):BG c!Mr}'esp.f(-;b):A!MasthemapsDG6׽BG3b7!f(a;b)2Mr}'esp.>A3a7!f(a;b)2M:Thenaf(a;->)ininSMQ;B4/;foralla2A,aandf(-;b)isinSMQ;Aճ;forallb2Bq,aandthemaps%BA3a7!MQ;B4/(f(a;->))2Mr}'esp.>BG3b7!MQ;Aճ(f(-;b))2Mar}'einSMQ;Airesp.SMQ;Bandsatisfy  MQ;AճfMQ;B4/ff(a;b):b2Bqg:a2Agګ=MQ;B4/fMQ;Aճff(a;b):a2Ag:b2Bqgګ=MQ;ABw(f):&Pr}'oof.׫Bye(2.2),(2.11)canbGereducedtothecaseA[=B=N.eInthissituationchoGoseUUabijectionN^2Zw!N,use(2.1),andapply(1.4),(iii),twice.4u4tDG9AsUUaspGecialcaseof(2.11)weobtain֍(2.12)rCorollary .L}'etDM[beanR-prenormedR-semimodulewithleftN-summa-tionB7(SM\;M).If ÷2߱SR љandm2MYRthenthemap mgivenbyN3n7! nq~m2MDis)inSM andM\( m)=(Rb )m.)Similarlyifrb2Randȷ2SMthen&vthemaprGZgivenbyN33n7!rn82M=is&vinSM andM\(r)=r(M\):ut(2.13) Corollary .L}'etYf++2M^A,':A 4!B@amap.F;orb2B@letkfk^'r1 QϮ(b)b}'ethemapfA3a7!^G kf(a)k,if'(a)=b; 0,otherwise.5Supp}'osethatkfk^'r1 QϮ(b)l{isinSCI;ACforeveryb2BXandthatthemapQBG3b7!^ CI;A1(kfk^'r1 QϮ(b)ؔ),ifb2'(A); 0,otherwise;isinSCI;B .ThenfvisinSMQ;Aճ.Pr}'oof.ThisUUisanimmediateconsequenceof(1.1),(iii'),(1.4),(o),and(2.2).?xu?xt {^Ы9/J 9IfshouldbGepointedoutthat,withallindexsetsassumedtohavecardinality zcard{'N,*thereisacorrespGondencebetween*certainaxiomsin[7],x?6,andsomeofUUtheresultsobtainedhere.ThiscorrespGondenceisasfollows:(@KXEquivqalentUUF*amiliesAxiomԋm(2.9),|/UnaryUUSumAxiomԋm(1.4),UU(i),PV(GeneralizedUUPartitionAxiomԋm(1.4),UU(iii),\(W*eakUUDoubleSumAxiomԋm(2.11).INH83.U>TheclosedunitballfunctorAsin[5]onede nestheclosedunitballfunctorBN ui:߸NRbpnSmoGd6MZ1>D!߸SetH.ItsvqalueUUontheob8jectMlpofNRbpnSmoGd51=isNlggBN(M):=fm2M3:kmk1g:댍(3.1)TTheorem.BN \ɫ:NRbpnSmoGd541hasaleftadjointL^N.Pr}'oof.W*eUUputL^N(;):=f0g.UUIfA6=;UUisanynonemptyset,weput{asaset{4ѷLN(A):=SR ,;A RǟAW:DueUUto(2.5),L^N(A)isanRǫ-subsemimoGduleofR^AW. 9NextUUwede neaprenormjjjjjj:L^N(A)!C qbyUUputting(3:2)Etjjjfjjj:=CI;A1(kfk)=CI;A1fkf(a)k:a2Ag"J;fڧ2LN(A):Since/kf+cg[ٷk2kfk+kg[ٷk,/fC*andgtinL^N(A),weobtainfrom(2.5)and(2.8)jjjf4f+ ױg[ٷjjj jjjfjjj+jjjg[ٷjjj. bjjjrGfjjj krk jjjfjjj,HrR2 Rǫ,f2L^N(A),followssimilarly*.UUThismeansthatL^N(A)isanRǫ-prenormedR-semimoGdule.9Itremainstode ne(SL O \cmmi5Nlu(A)Ʊ;LNlu(A)).LetF_2(Rǟ^AW)^N.Givena2AwedenotethemapN3ٱn7!Fnq~(a)2R琫byɱF(a).Dueto(1.3),(o),F(a)isinSR c+ifandonly~ifkF(a)k,thatisthemapN#l3 Qn7!kFnq~(a)k2C,~isinSCګ.InthiscasewecanUUformthemapCڷkFk:A!C qthatUUisgivenbyq(CڷkFk)(a):=C(kF(a)k)Cm;a2A:WithUUthesenotationswehaveɍSZ:=SLNlu(A)n=fF_2LN(A)N \ɫ:kF(a)k2SCڱ;forUUall"1ȱa2A;andqCkFk2SCI;A1g:Inordertode ne:=LNlu(A)ƫ,letFWbGeinS.SincekF(a)kisinSC xwehavethatF(a),2thatisthemapN33n7!Fnq~(a)2Rǫ,2isinSRb.Dueto(1.3),(ii),weobtainkRbF(a)k9<CڷkF(a)k.IfRbF2denotesthemapA9<3a7!RF(a)2RthenweFhavekRbFkCڷkFk.FSincethelatterfunctionisinSCI;A1,(2.8)showsthat 10CݬSectionT3: UPB.P9areigis,D.PumplA;un,andH.R`ohrl1 kRbFk"isinSCI;A1,whenceRFisinSR ,;Adueto(2.6).Inotherwords,RFis inUUL^N(A),andweputF_:=RbF.9AtthispGointwehavetoshowthat(ST;)isaleftN-summationforL^N(A).So,let2^FBandGbGeinS.ThenF֫+GisthemapN33n7!Fndp+Gn82L^N(A)2^andhenceF1!+=G_2L^N(A)^N.Moreover,(Fn +=Gnq~)(a)=Fn(a)=+Gn(a)foralla2A.Thus,g-k(Fݩ+DűG)(a)kkF(a)kDū+kG(a)k.g-SincebGothkF(a)kandkG(a)kareinSCګ,(1.1)showsthatkF(a)k]+kG(a)kisinSCګ.Therefore(1.1),(ii),impliesthatUUk(Fī+8G)(a)kisalsoinSC /andthatFCګ(k(Fī+8G)(a)k)C(kF(a)k8+kG(a)k)=Cګ(kF(a)k)8+Cګ(kG(a)k)holds[foralla 2A.[ThismeansthatCڷkF%+=AGk CkFk=A+CkGk[isvqalid. ByVassumptionbGothCڷkFkandCkGkareinSC,whenceCkFk9`+CkGkis&inSCګ.Thus(1.1),(ii),showsthatCڷkFi+GkisinSC.ThereforeFi+GZ isinS.VSimilarly*,butmoresimply,oneshowsthatFbJ2fSDandr2Rj~impliesrGFbJ2S.ThusUUwehaveshownthatSisanRǫ-subsemimoGduleofL^N(A)^N.9NextCweneedtoprovethatisahomomorphismofRǫ-semimoGdules.AgainletRFӫandGbGeinS.ThenRb(F(a)4+G(a))=RbF(a)4+RbG(a)RandthusRb(FHW+sG)=RFHW+sRG.Moreover,(Rb(FHW+sG))_=(RF)HW+s(RG),whenceF&:(Fī+8G)\;=Rb((R(Fī+8G)))=Rb((RF)ī+8(RG))\;=Rb((RF))8+Rb((RG))=Fī+8G:8jSimilarlysoneobtainsrGF׫=wrF,sandisrecognizedasahomomorphismofRǫ-semimoGdules.9Nowwewishtoverify(1.4),(o).F*orthis,letF QcbGeinL^N(A)^N.W*ehavetoQGshowthatF 2jSԫisequivqalentwithjjjFjjjj2SCګ,QGwherejjjFjjjisthemapN`3In7!jjjFnq~jjj2C.Let(F)denotethemapAmN3I(a;n)7!kFnq~(a)k2C.SuppGosewthatFl2S Yholds.Let':AOڷN 4!Aw̫bGethepro8jectionontothe rstfactor.UUThen,foreverya2A,q[(F)'r1 QϮ(a)(b;n)=^G kFnq~(a)k.5,UUifb=a, 0.5,UUotherwise.SincekF(a)kisinSCګ,(F)^'r1 QϮ(a)gثbGelongstoSCI;AN dueto(2.1).Moreover, VCڷkF(a)kw=CI;AN(F)^'r1 QϮ(a).XSinceCI;A1kFkisinSCI;A1,(2.12)showsthat(F)~isinSCI;AN.Thus(2.11)impliesthat,with ":A2N3 !N1the~pro8jectiononto*thesecondfactor,(F)^ @Lr1 (n)IisinSCI;AN.foreveryn2N.*Butdueto(2.1)*CI;AN(F) @Lr1 (n)U`=CI;A1fkFnq~(a)k:a2Ag=jjjFnjjjA;n2N:FHence(2.11)showsthatjjjFjjjisinSCګ.ThesametypGeofargumentshowsthatjjjFjjj2SC /impliesUUF_2S.9On:to(1.4),(i).HerewearedealingwithFD#2?L^N(A)^N swith nitesuppGort.HenceqfkF(a)khas nitesuppGortandisthereforeinSCګ,foralla2A.qfInparticular,TUHCڷkF(a)k=X8޺0fkFnq~(a)k:n2suppFg YЫ11/J andUUhenceaߍ`CڷkFk=X8޺0fkFnq~k:n2suppFg:ӪSince!SCI;APisanRǫ-semimoGdule,therighthandsideofthelastequation,andthus CڷkFk,isinSCI;A1.ThereforeFxisinS.Moreover,withFhaving nitesuppGort,soUUdoGesF(a),foralla2A,UUand̐_^RbF(a)=X8޺0fFnq~(a):n2suppFgCm;a2A:Consequently*,>W[@(F_=RbF=X8޺0fFn8:n2suppFg;a㍫andUU(1.4),(i),issatis ed.9Nextmcomes(1.4),(ii).LetF8ͷ2SjandG2L^N(A)^N mwithjjjGjjjjjjFjjj.Byw(1.4),(o),jjjFjjj2SC 1fholds.wDueto(1.1),(ii),jjjGjjjisinSCګ.By(1.4),(o),Gūis|inS.Since(RbG)(a) =R(G(a))=RfGnq~(a):n2Ng,|itfollowsfrom(1.3),UU(ii),thataߍ=nk(RbG)(a)kCڷfkGnq~(a)k:n2Ng=CkG(a)k:ThereforeUUweobtain,asintheproGofof(1.4),(o),ߍIjjjGjjj+=CI;A1fk(RbG)(a)k:a2AgCI;AfCڷfkGnq~(a)k:n2Ng:a2Ag+=CI;AN((G))=CڷjjjGjjjand,UUthrough(1.1),(iii),aߍfjjjGjjjCڷjjjGjjjCڷjjjFjjj:Thereӵremains(1.4),(iii).LetF22SgBandlet':Nѷ G!NЫbGeamap.By(1.7), F:c'r1 QϮ(n)[?isinS,foralln2b2N,andF1ɍc'r17̍ isinS.Moreover,foreverya2b2A,due toUU(1.3),(iii),&jȍe(F)(a)9`=Rb(F(a))=R(F(a)'r1N)=Rf(R(F(a)'r1 QϮ(n)EΫ)):n2Ng 9`=Rbf(R(F:c'r1 QϮ(n)[])(a)):n2Ng=Rf(F:c'r1 QϮ(n)[])(a):n2Ng9`=(fF:c'r1 QϮ(n)[pu:n2Ng)(a)=(Fc'r1፺wݫ)(a);*andUUthus(F1ɍc'r17̍wݫ)=(F),UUashadtobGeshown.9At+thispGointweknowthatL^N(A)isanRǫ-prenormedR-semimoGdulewithleftN-summationUU(SLNlu(A)Ʊ;LNlu(A)).9F*orNa`2Alet`^a -bGetheDiracfunctionataonA,thatisthemapA !Rwith%`^a7S(a)"N=1and`^a7S(b)=0,foralla6=b2A.By(1.3),(i),`^a \isinL^N(A)anddjjj`^a7Sjjj=1holds.ThemapA3a7!`^aٷ2BN(L^N(A))isdenotedbyEGandiscalledtheDiracmaponA.W*eclaimthat):A }!BN(L^N(A))isuniversalwithrespGecttoNRbpnSmod61; .F*orthisletM/beanRǫ-prenormedR-semimoGdulewith leftN-summation(SM\;M) andleth:A ;i!BN(M) bGeanysetmap.Letfurthermoremfڧ2L^N(A)andconsiderthemapfhgivenbyA3a7!f(a)h(a)2M. w12CݬSectionT3: UPB.P9areigis,D.PumplA;un,andH.R`ohrl1 Then̷kfhk(a)b=kf(a)h(a)kkf(a)kkh(a)kkf(a)k,foreveryab2A.By (1.3),av(o),f޷2OSR ,;AThimplieskfk2SCI;A1.Thelastinequalitiestogetherwith(1.1),(ii),showkfhk02SCI;A1.Hence(1.4),(o),impliesfh0ַ2SMQ;Aճ.Hencewehave fe$h(f):=MQ;Aճ(fh)2M.-Thusweobtainthemap fe$h :L^N(A)!M.-By(1.4),(i), fe$h(`^a7S)=h(a),foralla2A.SinceMQ;AHisahomomorphismofRǫ-semimoGdules,thesameistruefor fe$h}x.W*ewanttoshowthat fe$h 7isahomomorphismofRǫ-prenormedR-semimoGdules= withleftN-summation.LetF_2S.ThenjjjFjjj2SC dueto(1.4),g(o)._Since,foranyf20L^N(A),jjj fe$h(f)jjjjjjfjjj_˫holds,wehavejjj fe$h"N X(F)jjj0jjjFjjjcandthusjjj fe$h"N X(F)jjj2SC byc(1.1),(ii),andhence fe$h %"N(F)2SC byc(1.4),(o).O Thatis, fe$h "N(S)SM\.O NowconsiderFasamapA,KN3 !Rǫ.O SincejjjFjjj isbinSCګ,itfollowsfrom(2.12)thatFisinSR ,;ANH.Letpc:AA˷N~ !cAbbGethe - rstdpro8jectionandput\q~h 5:=cڱhwp.dThenFx\q~wh isinM^AN],anditfollowsfrom(2.8)UUthatFķ\qS>~8h Q,isinSMQ;AN+e.Hence(2.11)and(2.12)leadto"˦ÍMQ;AN+e(Fķ\qS>~8h׫)I=MQ;AճfM\fFnq~(a)8h(a):n2Ng:a2AgI=MQ;Aճf(RbfFnq~(a):n2Ng)8h(a):a2AgqI=MQ;Aճf(RbF(a))8h(a):a2Ag=MQ;Aճ((F)h)= fe$h(F):!OnUUtheotherhand 0ܠMMQ;AN+e(Fķ\qS>~8h׫){L=M\fMQ;AճfFnq~(a)h(a):a2Ag:n2Ngg{L=M\f fe$h(Fnq~):n2Ng=M( fe$h"N X(F)):pThismeansthatM\( fe$h"N X(F))= fe$h(F),whichistheformulain(1.5),(i).Finally*,gasUUjjj fe$h"N X(F)jjjjjjFjjjUUhasbGeenshownbefore,(1.4),(ii),impliesGkM\ fe$hS"N9(F)kCڷjjj fe$h"N X(F)jjj(CڷjjjFjjj)81;MshowingUU(1.5),(ii),aswellasprovingthat fe$h misacontraction.9Thevery nalstepisnowtoshowthat fe$h eisunique.So,leth^0H:L^N(A) 1!MbGeacontractivehomomorphismofRǫ-prenormedR-semimoGduleswithleftN-summationsuchthathp=h^0bַ`.Letf2pL^N(A).Asin(2.2)wehavesuppлfA^0߷禱A,hݱN^0N,withcardSA^0߷cardSN,and:N 7!Asatisfying(2.2),a).Letf9bGeUUthemapVбN33n7!^ f((n))`^(n)gf,UUifn2N^0 0gf,UUotherwise.7DObviously*,8f*isin(Rǟ^AW)^N.SincefǫisinSR ,;A,fisinS$ūandf_=f.Sinceh^0N (f)isUUthemapBP(N33n7!^ h^09(f((n))`^(n))=f((n))h^09(`^(n))ö,UUifn2N^0T; 0ö,UUotherwise;weUUhave;W卍-h09(f)F=h09(f)=M\(h0N (f))=MQ;Aճff(a)h09(`a7S):a2AgqFܫ=MQ;Aճff(a)h(a):a2Ag= fe$h(f); YЫ13/J provingUUtherequireduniqueness.ΓjuΓjt2iE`*4.U>,UUforall _2, ^Y፺ 52^N,x^_2X^N^.(4.3)WDe nition.猫LetbGealeftN-convexitytheory*.Byahomomorphismofleft-convexUUmoGdulesX p!X^0pismeantamapfڧ:X p!X^0psuchthat?f(h ;xi)=h ;fN@(x)iS;forUUall"1ȱ _2;x^2X^N^:9LetbGeanN-convexitytheory*.Thenthetotalityofleft-convexmoGdulesand theirhomomorphisms,withcompGositiontheset-theoreticalone,formacategoryC,Ȫthecategoryofleft-convexȪmoGdules.Clearly*,Cƫisanalgebraiccategory.Sinceithasarank([2],p.56),ithasfreeob8jectsonanyset.However,wewanttoaconstructsuchfreeob8jectsexplicitly*.First,threetechnicalstatementsabGout-convexmoGdules.Theycorrespondto[4],(2.4),(iii),(iv),and(viii),andtheproGofsUUtherecarryoverUUtothecurrentsituationwithnominalchangesonly*.xl(4.4)eLemma.L}'et5Xbealeft-convexmodule,let  ;2 Wwithsupp N0 ·mON, Handlety[ٟ^;zp^ vʷ2X^N hb}'esuchthaty[ٟ^n :=zp^n,foralln2N0|s.Thenh ;y[ٟ^i=h ;zp^ {i. u t(4.5)xLemma.L}'etIԱXbealeft-convexmodule,let _2andx^2X.F;oranybije}'ction":N3 !Nde ne^} ,r}'esp.^x^,asthemaps8GuN33n7! :@L(n)[.2Rnr}'esp.DN3n7!x@L(n)[.2X::14CݬSectionT3: UPB.P9areigis,D.PumplA;un,andH.R`ohrl1 Thenh ;x^i=h^b ;^ >x^i.·ut!z(4.6)ͯLemma.L}'etʱXVbealeft-convexmodule, &and in,andx^andy[ٟ^ inX.L}'etfurthermore':N3 !Nbeaninjectionandassumeō*ֱ n8= 㐴'1 QϮ(n)$~;n2'(N);OandA n=0 8;n=2 8ܱ'(N);v*ֱxn8=y[ٟ'r1 QϮ(n)$ڇ;n2'(N):;Thenh ;y[ٟ^i=h ;x^i.𩵷ut9NextUUwehave(4.7)NTheorem. (see[3],5.4)The;for}'getfulfunctorV i/:gC !Sethas;aleftadjointF9^p.Pr}'oof.LetXRkɫbGetheprenormedsemiringwithleftN-summation(SRb;R)XthatappGears5inthede nition(4.1)of.GivenanysetAwede neF9^p(A)asthesetofЀmapsf:_A !RGsuchЀthatcard-(suppʱf)_card N盫andЀforsomedata, ,A^09,andN^0\in(2.2), _2holds.InordertomakeF9^p(A)aleft-convexmoGdulewehavetode neh ;f^si,forall >2andf^ R$2F9^p(A)^N.Accordingto(2.3)wecanchoGose,A^0MandN^0esuchthatforeveryf^n ·2 F9^p(A)SR ,;A,n2N,theconditionsd&in(2.2)aresatis ed.SuppGosethat ^n፺ \isassociatedtof^n 3viathedata,#A^0andN^0T.Then ^n፺ 2,foralln2N.Dueto(4.1),(ii),h ÐY; ^Y፺Giisinand,e asisseeneasily*,suppV׷h ÐY; ^Y፺GiKS 6fsuppʱ ^n፺:n2NgN^0T.e Hencethereisaڏuniquef2"F9^p(A)withsuppYf"A^0ȫandf((n))=h ÐY; ^Y፺Gi,foralln2N^0T.Denoteithisfbyh ;f^si.By(2.2),fdoGesnotdependonthedatachosen.Itisclearfromtheconstructionthat(4.2),(i),issatis ed.Inordertoverify(4.2),(ii),itUUsucestoshowtheexistenceandequalityofthetwotermsS#h ÐY;h Y፺G; 8፮ɷiiUWand>#hh ÐY; Y፺Gi; 8፮ɷifor[_all ÐY *2), ^Y፺ F2^N, ^8b 2^N.However,[_theexistenceoftheseexpressionsisanimmediateconsequenceof(4.1),(ii).Asforequality*,considerthemap ÐY ^Y፺G ^8፴tgivenUUbylԞNO8N33(n;p)7! nq~ np 1ɍ8pD>t2R:Sinceby(1.2),k nq~ ^np 1ɍ8pD>t-kk nk,itfollowsfrom(1.1),(o)and(ii),and(1.4),(o),that0themapN㞷3̃n7! nq~ ^np 1ɍ8pD>t 2Ris0inSRb,whenceRf nq~ ^np 1ɍ8pD>t :̃n2Ng0isde ned.UUSince,dueto(1.2)and(1.3),(ii),UQ9зkRbf nq~ np 1ɍ8pD>t:n2NgkۜCڷfk nq~kk npkk 1ɍ8pD>t-k:n2NgUQۜCڷfk nq~kk npk:n2Ng;andm&sinceby(1.1),(iv),therighthandsideofthisinequality(asafunctionofp)isUUinSCګ,itfollowsfrom(1.3),(o),thatdEձN33p7!Rbf nq~ np 1ɍ8pD>t:n2Ng2RisinSRb.Hence(1.1),(iii'),andtheuseofabijectionN^2Zw!Nɫshowthat ÐY ^Y፺G ^8፴tisinSR ,;N,NZū.Therefore(2.11)leadstothedesiredequality*.Whichmeans,that YЫ15/J F9^p(A)%isaleft-convex%moGdule.TheDiracmap':A!F9^(A)%assignstoeach a{2AtheDiracfunction`^aata.W*eclaimthatv:{A !F9^p(A)isauniversalarrow.۬LetXbGealeft-convexmoGduleandleth:A s!Xbe۬asetmap.W*ewantItode neanappropriatemaph^0Q:F9^p(A)!X.ILetfڧ2F9^(A).ChoGoseforfhtheUUdata, ,A^09,andN^0:anddenotebyh^9themapg@N33n7!^Id h((n))eL,UUifn2N^0T; x0eL,UUifn=2 8ܱN^0T;}Awherex0issomeelementofX.Itfollowsfrom(4.4){(4.6)thath ;h^iisin-depGendentk#ofx0疫andofthedatachosen,andweputh^09(f)o:=h ;h^i.k#Obviously*,h^09(`^a7S)0=h(a),a2A,whenceh^01Jcv=hholds.Nextwecheckthath^0bثisahomo-morphismٷofleft-convexٷmoGdules.Let <2andf^ P-2F9^p(A)^N.By(2.3)wecan MchoGosedatasothattheyserveforallf^n ,n 2N. MIf ^n፺ istheelementofassoGciatedUUwithf^n ,n2N,UUviasuchdatathen}9rDAh09(h ;fsi)=hh ÐY; Y፺Gi;hi;rwDhil2`enhh ;h0N (fs)i=h ÐY;h Y፺G;hii;whenceatheequalityofthelefthandsidesofthelasttwoequationsfollowsfrom (4.2),r(ii),showingthath^0ѫisahomomorphismofleft-convexmoGdules.AsforuniquenessEoftherequiredfactorization,let\q~h hT:F9^p(A)!X'bGeEahomomorphismofcleft-convexcmoGduleswith\q~h 5=>ױh.F*orfRf2F9^p(A)choGosethedataasabGovetoobtain 72hSassociatedwithf.Denotefurthermoreby`^()themapN33n7!`^(n)ؘ2F9^p(A).UUAneasycomputationshowsthatfڧ=h ;`^()8i.Hence}9\q4R~3h9(f)=\qv~h(h ;`()8i)=h ;\q~hNP(`()8)i=h ;hi=h09(f):[K5}9g<5.U>Theassouciatedۨ-con=vexU>modulefunctor r(5.1)fPropQosition.L}'et4bealeftN-convexitytheoryoverR.ThenthereisafunctorO8:NRbpnSmoGd541(XJ)ͫ:=q[ٟ^N(SLNlu(XJ)Z).SuppGosenowthatf,f^0areinSLNlu(XJ)I and "nsatisfybq[ٟ^N(f){-=q[ٟ^N(f^0),bthatisq[٫(fnq~){-=q(f^0፴nq~)bforalln{-2N.bThen(fn;f^0፴n){-2andoVhence(fnq~;f^0፴n)2ShݫforoVallcontractinghomomorphismsh:L^N(X)!MqthatareUUS-compatible,foralln2N.UUThisshowsthath^N(f)=h^N(f^0)UUandthereforevH-h(LNlu(XJ)Zf)=M\(hN(f))=M\(hN(f0))=h(LNlu(XJ)Zf0);thatisweobtaintherelation(LNlu(XJ)Zf;LNlu(XJ)f^0)T02Sh؅forallsuchh,whence (LNlu(XJ)Zf;LNlu(XJ)f^0)2UUholds.Thuswecande nevEQS(>(XJ)쵫(s):=q[٫(LNlu(XJ)Zf)(߳;s_2SQS(>(XJ)쵱;whereKif_2S^NbL)>(X)ischosensuchthatg[ٟ^N(f)=s.KiInparticular,q[٫(LNlu(XJ)Z(f))=QS(>(XJ)쵫(q[ٟ^N(f))*&forallf mU2qSLNlu(XJ)Z.Onechecks*&easilythatS^bSU(X)isanRǫ-subsemimoGduleLofS^1ī(X)^N aandthatQS(>(XJ)z:]SQS(>(XJ)!!S^1ī(X)Lisahomomor- phismUUofRǫ-semimoGdules.9Next`%weverify(1.4),(o),forS^1ī(X).Letsr2SQS(>(XJ)LګandchoGosefr2SLNlu(XJ)withq[ٟ^N(f)ޣ=s.Thenjjjsnq~jjjޣjjjfnjjj,n2N.SincejjjfjjjisinSCګ,soisjjjsjjj+dueto(1.1),(ii).Conversely*,+assumes`_2{S^1ī(X)^N ܫandjjjsjjj{2SCګ.+W*eapply(OP)wtoc :=3jjjsjjjandobtaind =withthepropGertiesstatedthere.Ifjjjsnq~jjjMisamaximalelementofC,choGosef B2^L^N(X)suchthatq[٫(fnq~)^=sn.Then|jjjfnq~jjj0X=jjjsnjjj.|IfjjjsnjjjisnotamaximalelementofC,(LD)limpliestheexistenceofanfn]27L^N(X)withq[٫(fnq~)=sn ~andjjjsnq~jjjjjjfnjjjjjjsnjjje+dnq~.Hence]ƱfisinL^N(X)andjjjfjjj)jjjsjjj>+dn 2)SCګ,]whencejjjfjjjisinSC dueto(1.1),(ii),andthusfSisinSLNlu(XJ)by(1.4),(o).Thereforesȫ=oq[ٟ^N(f)isinSQS(>(XJ)쵫.9(1.4),UU(i),istriviallysatis edinthecurrentsituation."18CݬSectionT3: UPB.P9areigis,D.PumplA;un,andH.R`ohrl1 9OnGAto(1.4),(ii).Itfollowsfrom(1.5)thats_2SQS(>(XJ)3andt2S^1ī(X)^N with ۍjjjtjjjjjjsjjj}impliest_2SQS(>(XJ)쵫.Hencethereareg2SLNlu(XJ)׫witht=q[ٟ^N(g) andQ+jjjQS(>(XJ)쵱tjjj=jjjq[٫(LNlu(XJ)Zg)jjjjjjLNlu(XJ)ZgjjjCڷjjjgjjj;sdueUUto(1.4),(ii),forL^N(X).Hencewehaveȍ׍3jjjQS(>(XJ)쵱tjjjminfƷfCڷjjjgjjj:q[ٟN(g)=tgۍmҫ=Cګ(inf *fjjjgjjj:q[ٟN(g)=tg)=Cڷjjjtjjj:Ǎ9Finallyo{(1.4),(iii).Let':N3 !NbGeo{amapandlets_2SQS(>(XJ)쵫.Thenthereis?anfбf}2;SLNlu(XJ)8*withs=;q[ٟ^N(f).Sinces:'r1 QϮ(n)[* =q[ٟ^N(f:'r1 QϮ(n)[Y])andsincef:'r1 QϮ(n)[DisinSLNlu(XJ)Z,dueto(1.4),(iii),appliedtoL^N(X),wehaves:'r1 QϮ(n)[2FݱSQS(>(XJ)forevery7n;2N.Moreover,byde nition,QS(>(XJ)쵱s:'r1 QϮ(n)[ =;q[٫(LNlu(XJ)Zf:'r1 QϮ(n)[Y]).Sincef1ɍ'r17̍'ݫ,{athatisthemapN3n7!LNlu(XJ)Zf:'r1 QϮ(n)[_޷2L^N(X),{aisinSLNlu(XJ)LitfollowsthatUUs1ɍ'r17̍iisinSQS(>(XJ)쵫.FinallyL1QS(>(XJ)쵱s'r1፺f=q[٫(LNlu(XJ)Zf'r1፺'ݫ)=q(LNlu(XJ)Zf)=QS(>(XJ)쵱s:ThusdwehaveshownthatS^1ī(X)isanRǫ-prenormedR-semimoGdulewithleftN- summationUU(SQS(>(XJ)쵱;QS(>(XJ)). ۍ9Additionallyweclaimthatq:L^N(X) ]!S^1ī(X)isacontractinghomomor-phismbofRǫ-prenormedR-semimoGduleswithleftN-summation.(1.5),(i),isobviousfromNtheconstructionofS^1ī(X),while(1.5),(ii),{withcg =1N{wasestablishedinUUtheabGoveUUveri cationof(1.4),(ii).9WhatIremainstobGedoneistoshowthatBN(q[٫):^ѱX' D)!Oqȫ(S^1ī(X))Iisaluniversalarrow.SinceqEisahomomorphismofRǫ-prenormedR-semimoGduleswithN-summation,BN(q[٫)ķ isahomomorphismofleft-convexmoGdules.Leth:X r!Oqȫ(M)%bGesuchahomomorphism.Dueto(3.1)thereisacontrac-tiveChomomorphismh^04ګ:fL^N(X) !M^ofCRǫ-prenormedR-semimoGduleswithN-summation,withBN(h^09)o`=}h.,Sincehisahomomorphismofleft-convexmoGdules,h^0isS-compatibleandhencegivesrisetoafactorizationh^0= h^00׋q[٫,whereh^00  :蘷S^1ī(X) !MisacontractivehomomorphismofRǫ-prenormedR-semimoGdulesGwithN-summation.Henceh=BN(h^00:Xq[٫)`,Gwhichistherequired \sfactorization.W*eclaimthathdetermines fe$huniquely.So,let\qRýeh Bq[= hbGeanothernfactorization.EachsD@2S^1ī(X)ncanbGewrittenasq[٫(f),withfWϷ2D@L^N(X).Due3to(2.4),fGequalsLNlu(XJ)Z( `^()8),where:N3 !A3isasuitablemapand _2SR ䷫isUUchosenappropriately*.Hencewehave;㎍\qL:deLg"hR*(s)ahO=\qZeh(q[٫(f))=\qZeh(q[٫(LNlu(XJ)Z( `()8)))WahO=M\( (\qBeh׷8q[٫)N(`()8))=M( \ql&eh()3ޫ);&mwhere\q`eHh^()isHthemapN33n7!h((n))2Oqȫ(M).HThushdetermines fe$h ݇uniquely*.ut;MYЫ19/J  6.U>ExamplesClearly*,? everypGositivesemiringC'has(C^(N,)ϱ;Pލ 80% 8C),withPލ F0% FC+theusualsuminC, asRPaleftN-summation.Similarly*,everyprenormedsemiringRfhas(Rǟ^(N,) z;Pލ 80% 8RE)asbaleftN-summation,justas(M^(N,)α;Pލ 80% 8NΔ)isaleftN-summationforeveryRǫ-prenormedRǫ-semimoGdule.Thismeansthatthepositive(resp.prenormed)semi-rings,?theRǫ-prenormedR-semimoGdules,andthe nitaryconvexity?theoriesdis-cussedNin[5],xE1-x3,arespGecialcasesofthenotionsinvestigatedNinthepresentpapGer.9OnechecksquiteeasilythattheBanachsemiringsRdiscussedin[5],xA6,areanother2instanceoftheconceptstreatedhere.TheconeofsuchaBanachsemiringRmhastosatisfysuitablepropGerties(see[5],4.14and4.15);SC _isthenthesubsetofC^Zqymsbm7ZN*consisting^ofallthose forwhich hasalimit(inthesenseof[5],xТ4),whileCڱ :=AsupP8f n{:An2LNg.Inaddition,SR .tisthesetofthose 2ARǟ^ZN Aforwhich oisַanabsoluteCauchyַsum(inthesenseof[5],xH4),whileRb isthelimit(inthesenseof[5],xy4)ofthein nitesumP .Analogouslyoneobtains(SM\;M)for$eachBanachRǫ-semimoGduleM;inparticular,eachBanachRǫ-semimoGduleisanRǫ-prenormedWR-semimoGdulewithLN-summationincurrentterminology*,whiletheconverseZingeneralfailstobGecorrect.However,ZifCv:=uLR+ c=fr2LR:r0g,R߫:=LR,kk:LR!LR+ 2:istheusualabsolutevqalue,andSC Histhesetofall _2LR^ZN፮+forowhichP convergesandCڱ _=P thentheBanachspacesoverLR(inthesenseoffunctionalanalysis)arepreciselytheLR-prenormedLR-semimoGduleswithLn-summationasfollowsfromawellknowncharacterizationofBanachspaces([6],3.1.2).9NowzwewanttocharacterizeexplicitlytheconceptsofthepresentpapGerinthecasewherethesemiringinvolvedisthesmallestsemiringthatisnotaring.De neonUUthetwo-elementUUsetf0;1gadditionUUbyXF08+0=0;08+1=18+0=18+1=1;multiplicationUUbyXF080=081=180=0;181=1;partialUUorderbyXF0<1:Thenf0;1gequippGedwiththisstructureisacomplete,commutative,andunitalsemiringbLD.De neanN-summationonLDbyputtingSZD :=LD^N 3andZD5  :=maxvf n:An2Ng.OnecheckseasilythatthesedatamakeLDapGositivesemiringwithUUN-summation.9Put ˱R-:=0LDandde nekk:R- Jm!LDbykrGk:=r, ˱raM2Rǫ.LetfurthermoreSR Vz:=LD^N LandAR:=ZD5.AThenR߫=LDisaprenormed(evennormed)semiringwithN-summation.AξLD-prenormedLD-semimoGduleMiscommutative,idempotent(i.e.m=+m=moforallm2M)omonoidtogetherwithasubmonoidM0S(correspGondingto1fm2M3:kmk=0g);onecheckeasilythatM0mcanbGeanarbitrarysubmonoidofM.ThemonoidMhastheadditionalpropGertythat0istheonlyelementofM&tthatYpGossessesan(additive)inverse.Nextwede ne\m1Cm2|s"as\thereisanm2Mt!with]m1s+>m=m2|s",forallm1;m2P\2M.OnecheckseasilythatthisisapartialuzorderrelationonM(inparticular,m1Cm2andm2m1implym1=m2|s)thatiscompatiblewiththeadditivemonoidstructureonM.WithrespGecttothisorderrelation,M߫has nitesupremaandsupfm1|s;:::;mnq~g=m1N4+::: N)+mnq~.HenceR20CݬSectionT3: UPB.P9areigis,D.PumplA;un,andH.R`ohrl1 M0+׫isdclosedunder nitesuprema.IftheLD-prenormedLD-semimoGduleMhasN- summation.\(SM\;M)then(1.4),(o),impliesSM t=M^N̫.If_2M^N (andT*Nwedenoteby^T፺ NIthemapgivenby^T፺LjT*=jTe"and^T፺jNxLr]T*=0jNLrTc.Then(1.4),l@(iii),showsthatM\^T፺ ηM$holdsl@forall_2M^N  andT*N.Itfollowsfromz(3.1)thatforeverym2M,zm=M\m^isvqalid.Animmediateconsequenceof*thisisM\ =8%supF`fn:8%n2Ng.*ThismeansthatMEhasN-suprema,thatissupremaUUofallsubsetsofMlpofcardinalitycardűN.9Conversely`onechecks`easilythatanycommutative,idempGotent,partiallyor-deredmmonoidMӈwithadistinguishedsubmonoidM08suchthatthepartialorderiscompatibleowiththemonoidstructureandhasN-supremaisinfactaLD-prenormedLD-semimoGduleUUwithN-summation(M^Ṉ;M\),whereM_=supSfn8:n2Ng.9Let*ZDɯ:=;zLD^N.ItiseasytoseethatZD )_isanN-convexity*theoryover*LD.LetX1bGe!NbGeaninjectiveמmap,andde ne presp.x^asthemaps(withy0C2Xchosenarbitrarily){N33n7!^ 1,UUifn2im ㌱'; 0,UUotherwise;niresp.bN3n7!^G yUT,UUifn='(y[٫); y0UT,UUotherwise.ThenUUsupc(Y8)=h ;x^i.9Conversely*,UUifX7isapartiallyorderedsetthathasN-suprema,de ne6ӭh ;xi:=supSfxn8:n2supp g; _2UUandx^2X^N^:AMsimpleMcomputationshowsthatthismakesXaZD5-convexmoGdule.Finallyone concludeskfrom(4.3)thatamapfЫ:AX`# |!X^0bGetweenkZD5-convexmoGdulesisa@homomorphismofZD5-convex@moGdulespreciselywhenforeachsubsetYKNgXofcardinality٫cardĆN,sup(f(Y8))=f(sup;(Y8))gilt.HencethecategoryZD5Cis6isomorphictothecategoryofpartiallyorderedsetswithN-supremaandN-supremaUUpreservingmaps.9InsteadofZD ֫onecouldtakethesetZD; {֫ofall з2$ZDwithcardxN(suppʱ )$ cmmi10Zcmr5ٓRcmr7o cmr9K`y cmr10.