资源描述:
《基于柯西理想意义下的连续$omega$-domains 》由会员上传分享,免费在线阅读,更多相关内容在工程资料-天天文库。
1、基于柯西理想意义下的连续Ω-domains赖洪亮四川大学数学学院,成都610064摘要:Ω-范畴为研究量化domain理论提供了一个很好的基本框架.本文在假设完备剩余格(Ω,∗)的承载格Ω本身还是一个连续格的条件下,选取柯西理想描述定向完备性,引入了liminf完备的Ω-范畴的连续性理论.在此基础之上,进一步描述了waybelow关系,代数对象以及收缩等基本概念以及它们与连续性的关系.这些结果说明,这种连续的Ω-范畴确实可以看做量化的连续domain.关键词:范畴论,liminf完备的Ω-范畴,连续性,柯西
2、网,柯西理想,伴随,waybelow关系中图分类号:O189.13ContinuousΩ-domainsbasedonCauchyidealsLAIHong-LiangDepartmentofMathematics,SichuanUniversity,Chengdu610064Abstract:Categoriesenrichedonacommutative,unitalquantale(Ω,∗)makeagoodframeworkofquantitativedomaintheory.Inthispaper
3、,foracompleteresiduatedlattice(Ω,∗)withΩbeingacontinuouslattice,anotionofcontinuityinliminfcompleteΩ-categoriesisintroducedbasedonthedirectnessbeingdescribedbyCauchyideals.Moreover,thewaybelowrelations,algebraicobjectsandretractionsarecharacterizedandthere
4、lationshipswithcontinuityisstudied.ItisshownthatcontinuousliminfcompleteΩ-categoriescanbeviewedascontinuousΩ-domains.Keywords:Categorytheory,liminfcompleteΩ-category,continuity,Cauchynet,Cauchyideal,adjunction,waybelowrelation.0IntroductionAcompleteresidua
5、tedlatticeisapair(Ω,∗)(alsodenotedΩforshort),whereΩisacompletelattice;∗:Ω×Ω−→Ω,calledatensor,isacommutative,associativebinaryoperationonΩsuchthat(1)∗ismonotoneoneachvariable,(2)Foreachp∈Ω,themonotonefunctionp∗(−):Ω−→Ωhasarightadjointp→(−):Ω−→Ωand(3)thetope
6、lement1inΩisaunitelementfor∗,i.e.p∗1=pforeveryp∈Ω.Thebinaryoperation→:Ω×Ω−→Ωgivenby→(p,q)=p→q,iscalledthecotensor(correspondingto∗).基金项目:NaturalScienceFoundationofChina(11101297),FundamentalResearchFoundationofSichuanUniversity(2010SCU21009)andDoctoralFund
7、ofMinistryofEducationofChina(20100181120046)作者简介:Correspondenceauthor:LaiHongliang(1978-),male,associateprofessor,majorresearchdirection:fuzzytopologyandfuzzyorder.-1-Acompleteresiduatedlatticeisaspecialkindofsymmetricmonoidalclosedcompletesmallcategory.Ac
8、ategoryenrichedoverΩ,oranΩ-category,isasetAtogetherwithanassignmentofanelementA(a,b)∈Ωtoeveryorderedpair(a,b)∈A×A,suchthat(1)1≤A(a,a)foreverya∈A(reflexivity)and(2)A(a,b)∗A(b,c)≤A(a,c)foralla,b,c∈A(transitivity