资源描述：
《Set representations of abstract lattices组抽象格表示.ppt》由会员上传分享，免费在线阅读，更多相关内容在教育资源-天天文库。
1、SetrepresentationsofabstractlatticesZhaoDongsheng2009.6OutlineA.SetrepresentationB.TopologicalrepresentationC.RepresentationassetoflowersemicontinuousfunctionsD.RepresentationsasScottclosedsetsE.SomeproblemsA.SetrepresentationAlatticeisasetlatticeifitselementsaresets,itsorderrelationisgiven
2、bysetinclusion,anditisclosedundertakingfiniteunionsandintersections.AsetrepresentationofalatticeLisapair((C,⊆),f)where(C,⊆)isasetlatticeandfisanisomorphismfromLto(C,⊆).Whichlatticeshaveapropersetrepresentation?Someclassicalresults:BirkhoffAfinitelatticehasasetlatticerepresentationiffitisdis
3、tributive.2.StoneEveryBooleanalgebrahasasetlatticerepresentation3.PriestlyEveryboundeddistributivelatticehasasetlatticerepresentation.RepresentationasfamiliesofclosedsetsGivenatopologicalspaceX,letC(X)bethesetofallclosedsetsofX.(C(X),⊆)isasetlattice.IfalatticeLisisomorphicto(C(X),⊆)foratopo
4、logicalspaceX,thenXiscalledatopologicalrepresentationofL.AlatticethathasatopologicalrepresentationiscalledaC-lattices.QuestionsB-1)Whichlatticeshaveatopologicalrepresentation?B-2)Whichspaces(X,C)canbereconstructedfromthelattice(C,⊆)?B-3)Whichspace(X,C)havetheproperty:foranyspace(Y,E),if(E,⊆
5、)isisomorphicto(C,⊆),thenXishomeomorphictoY?B-4)HowtoconstructalltopologicalrepresentationsofagivenlatticeL?AnelementrofalatticeLisanirreducibleelementifr=x∨yimpliesr=xorr=y.ThesetofallreducibleelementsofLisdenotedby∆(L).Theorem1(W.J.Thron)Alatticehasatopologicalrepresentationiffitiscomplet
6、eanddistributive,andallirreducibleelementsforma(join)base.AtopologicalspaceXissoberifforanyirreducibleelementAof(C(X),⊆),thereisauniquepointxofX,suchthatA=cl({x}).Theorem2ForanysoberspaceX,Xishomeomorphictothespace∆(C(X)),where{↓A∩∆(C(X)):AisfromC(X)}isthesetofclosedsetsof∆(C(X)).*Everysobe
7、rspacespacesXcanbereconstructedfromthelattice(C(X),⊆).Theorem3IfXisHausdorffspace,thenforanyspaceY,C(X)≈C(Y)impliesthatXishomeomorphictoY.(E,⊆)(E,⊆)(X,C)(Y,E)Theorem4LetLbeaC-lattice.1)ForanybaseB⊆∆(L),(B,C(B))atopologicalrepresentationofL,whereC(B)isthe
