基于柯西理想思义下的连续Q-domains赖洪亮四川大学数学学院，成都610064摘要：Q-范畴为研究量化domain理论提供了一个很好的棊本框架.本文在假设完备剩余格(Q,*)的承载格Q本身还是一个连续格的条件下，选収柯西理想描述定向完备性，引入了liminf完备的d范畴的连续性理论.在此基础Z上，进一步描述了waybelow关系，代数对象以及收缩等基木概念以及它们与连续性的关系.这些结果说明，这种连续的(2■范畴确实可以看做量化的连续domain.关键词：范畴论,liminf完备的Q-范畴，连续性，柯西网，柯西理想，伴随,waybelow关系中图分类号：0189.13ContinuousdomainsbasedonCauchyidealsLAIHong-LiangDepartmentofMathematics,SichuanUniversity,Chengdu610064Abstract:Categoriesenrichedonacommutative,unitalquantalc(Q,*)makeagoodframeworkofquantitativedomaintheory.Inthispaper,foracompleteresiduatedlattice(Q,*)with(2beingacontinuouslattice,anotionofcontinuityinliminfcompleteQ-categoriesisintroducedbasedonthedirectnessbeingdescribedbyCauchyideals・Moreover,thewaybelowrelations,algebraicobjectsandretractionsarccharacterizedandtherelationshipswithcontinuityisstudied.ItisshownthatcontinuousliminfcompleteQ-categoriescanbeviewedascontinuousQ-domains・Keywords:Categorytheory,liminfcompleteQ-category,continuity,Cauchynet,Cauchyideal,adjunction,waybelowrelation.0IntroductionAcompleteresiduatedlatticeisapair(Q,*)(alsodenotedQforshort),whereQisacompletelattice;*:QxQ—>Q,calledatensor,isacommutative,associativebinaryoperationonQsuchthat(1)*ismonotoneoneachvariable,(2)ForeachpW(2,themonotonefunctionp*(一):(2>(2hasarightadjointp—>(一):Q>(2and(3)thetopelement1in(2isaunitelementfor*,i.e./?*1=〃foreveryp丘Q.Thebinaryoperation—►:QxQ—>Qgivenbyf(p,q)=pfq,iscalledthecotensor(correspondingto*)・垒金项口:NaturalScienceFoundationofChina(11101297),FundamentalResearchFoundationofSichuanUniversity(2010SCU21009)andDoctoralFundofMinistryofEducationofChina(20100I81120046)作者简介：Correspondenceauthor：LaiHongliang(I978-)・male,associateprofessor,majorresearchdirection：fuzzytopologyandfuzzyorder.Acompleteresiduatedlatticeisaspecialkindofsymmetricmonoidalclosedcompletesmallcategory.AcategoryenrichedoverQ,o『anQ-category,isasetAtogetherwithanassignmentofanelementA(a,b)WQtoeveryorderedpair(a,b)&彳xA.suchthat(1)1<A(afa)foreveryawA(reflexivity)and(2)A(a,b)*A(b,c)<A(a,c)foralla,b,cA(transitivity).Q-categoriesareinterestingobjectsformathematiciansandtheoreticalcomputerscientists.In1973,Lawvere[I]observedthatthetheoryofQ-categoriesunifiespreorderedsets(((2,*)=(2,/I),thetwopointlattice),generalizedmetricspaces((Q,*)=([0,+)),andmanyothermathematicalstructuresintooneframework.Andalsointhispioneeringpaper,Lawveredemonstratedthatbasicconceptssuchaslogic,distance,andcategoriesarecloselyrelatedtoeachother.Thiscanberoughlyexplainedasfollows:theadjunctiona*b<c<==>h<acbetweenthetensor*andthecotensor—wecaninterpretthecompletelattice(2asasetoftruthvalues,thetensor兴andthecotensor—>canbeinterpretedasthelogicconnectivesconjunctionandimplicationrespectively.So,thetheoryofQ-catcgoricshasastrongmanyvaluedlogicflavor.ThisfeatureofQ-categoriesleadstothepointthatQ-categoriescanbeinvestigatedasmany-valuedpreorderedsetsorquantitativedomains[2,3,4,5,6,7,&9].DirectedcompletenessofQ-categorieshasreceivedmuchattentionintheliterature[2,3,4,6,7,8]sinceitisthefirststeptostudyquantitativedomains・However,asdemonstratedin[4],thenotionofdirectedcompletenessforQ-categoriesisveryflexible.Itdependsonwha...