Table Of Content

This page intentionally left blank ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS FOUNDING EDITOR G.-C. ROTA EditorialBoard R.S.Doran,P.Flajolet,M.Ismail,T.-Y.Lam,E.Lutwak Volume93 ContinuousLatticesandDomains ENCYCLOPEDIAOFMATHEMATICSANDITSAPPLICATIONS http://publishing.cambridge.org/stm/mathematics/com 4 W.Miller,Jr.Symmetryandseparationofvariables 6 H.MincPermanents 11 W.B.JonesandW.J.ThronContinuedfractions 12 N.F.G.MartinandJ.W.EnglandMathematicaltheoryofentropy 18 H.O.FattoriniTheCauchyproblem 19 G.G.Lorentz,K.JetterandS.D.RiemenschneiderBirkhoffinterpolation 21 W.T.TutteGraphtheory 22 J.R.BastidaFieldextensionsandGaloistheory 23 J.R.CannonTheonedimensionalheatequation 25 A.SalomaaComputationandautomata 26 N.White(ed.)Theoryofmatroids 27 N.H.Bingham,C.M.GoldieandJ.L.TeugelsRegularvariation 28 P.P.PetrushevandV.A.PopovRationalapproximationofrealfunctions 29 N.White(ed.)Combinatorialgeometrics 30 M.PohstandH.ZassenhausAlgorithmicalgebraicnumbertheory 31 J.AczelandJ.DhombresFunctionalequationscontainingseveralvariables 32 M.Kuczma,B.ChozewskiandR.GerIterativefunctionalequations 33 R.V.AmbartzumianFactorizationcalculusandgeometricprobability 34 G.Gripenberg,S.-O.LondenandO.StaffansVolterraintegralandfunctionalequations 35 G.GasperandM.RahmanBasichypergeometricseries 36 E.TorgersenComparisonofstatisticalexperiments 37 ANeumaierIntervalsmethodsforsystemsofequations 38 N.KorneichukExactconstantsinapproximationtheory 39 R.A.BrualdiandH.J.RyserCombinatorialmatrixtheory 40 N.White(ed.)Matroidapplications 41 S.SakaiOperatoralgebrasindynamicalsystems 42 W.HodgesModeltheory 43 H.StahlandV.TotikGeneralorthogonalpolynomials 44 R.SchneiderConvexbodies 45 G.DaPratoandJ.ZabczykStochasticequationsininfinitedimensions 46 ABjorner,M.LasVergnas,B.Sturmfels,N.WhiteandG.ZieglerOrientedmatroids 47 E.A.EdgarandL.SuchestonStoppingtimesanddirectedprocesses 48 C.SimsComputationwithfinitelypresentedgroups 49 T.PalmerBanachalgebrasandthegeneraltheoryof*-algebras 50 F.BorceuxHandbookofcategoricalalgebraI 51 F.BorceuxHandbookofcategoricalalgebraII 52 F.BorceuxHandbookofcategoricalalgebraIII 54 A.KatokandB.HassleblattIntroductiontothemoderntheoryofdynamicalsystems 55 V.N.SachkovCombinatorialmethodsindiscretemathematics 56 V.N.SachkovProbabilisticmethodsindiscretemathematics 57 P.M.CohnSkewFields 58 RichardJ.GardnerGeometrictomography 59 GeorgeA.Baker,Jr.andPeterGraves-MorrisPadeápproximants 60 JanKrajicekBoundedarithmetic,propositionallogic,andcomplextheory 61 H.GromerGeometricapplicationsofFourierseriesandsphericalharmonics 62 H.O.FattoriniInfinitedimensionaloptimizationandcontroltheory 63 A.C.ThompsonMinkowskigeometry 64 R.B.BapatandT.E.S.RaghavanNonnegativematricesandapplications 65 K.EngelSpernertheory 66 D.Cvetkovic,P.RowlinsonandS.SimicEigenspacesofgraphs 67 F.Bergeron,G.LabelleandP.LerouxCombinatorialspeciesandtree-likestructures 68 R.GoodmanandN.WallachRepresentationsoftheclassicalgroups 69 T.Beth,D.JungnickelandH.LenzDesignTheoryvolumeI2ed. 70 APietschandJ.WenzelOrthonormalsystemsandBanachspacegeometry 71 GeorgeE.Andrews,RichardAskeyandRanjanRoySpecialFunctions 72 R.TicciatiQuantumfieldtheoryformathematicians 76 A.A.IvanovGeometryofsporadicgroupsI 78 T.Beth,D.JungnickelandH.LenzDesignTheoryvolumeII2ed. 80 O.StormarkLie’sStructuralApproachtoPDESystems 81 C.F.DunklandY.XuOrthogonalpolynomialsofseveralvariables 82 J.MayberryThefoundationsofmathematicsinthetheoryofsets 83 C.Foias,R.Temam,O.ManleyandR.MartinsdaSilvaRosaNavier-Stokesequationsandturbulence 84 B.PolsterandG.SteinkeGeometriesonSurfaces 85 D.KaminskiandR.B.ParisAsymptoticsandMellin–Barnesintegrals 86 RobertJ.McElieceThetheoryofinformationandcoding,2ed. 87 BruceA.MagurnAnalgebraicintroductiontoK-theory Continuous Lattices and Domains G. GIERZ K. H. HOFMANN K. KEIMEL J. D. LAWSON M. MISLOVE D. S. SCOTT    Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge  , United Kingdom Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521803380 © Cambridge University Press 2003 This book is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2003 -  isbn-13 978-0-511-06356-5 eBook (NetLibrary) -  isbn-10 0-511-06356-3 eBook (NetLibrary) -  isbn-13 978-0-521-80338-0 hardback -  isbn-10 0-521-80338-1 hardback Cambridge University Press has no responsibility for the persistence or accuracy of s for external or third-party internet websites referred to in this book, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. Contents Preface pagexi Acknowledgments xxi ForewordtoACompendiumofContinuousLattices xxiii IntroductiontoACompendiumofContinuousLattices xxvii O APrimeronOrderedSetsandLattices 1 O-1 GeneralitiesandNotation 1 Exercises 7 Oldnotes 8 O-2 CompletenessConditionsforLatticesandPosets 8 Exercises 17 Oldnotes 21 Newnotes 22 O-3 GaloisConnections 22 Exercises 31 Oldnotes 35 O-4 MeetContinuousLatticesandSemilattices 36 Exercises 39 Oldnotes 41 O-5 T SpacesandOrder 41 0 Exercises 45 Newnotes 47 I OrderTheoryofDomains 48 I-1 The“Way-below”Relation 49 Theway-belowrelationandcontinuousposets 49 Auxiliaryrelations 57 Importantexamples 62 v vi Contents Exercises 71 Oldnotes 75 Newnotes 78 I-2 Products,SubstructuresandQuotients 79 Products,projection,kernelandclosureoperatorson domains 79 Equationaltheoryofcontinuouslattices 83 Exercises 90 Oldnotes 93 Newnotes 94 I-3 Irreducibleelements 95 Openfiltersandirreducibleelements 95 Distributivityandprimeelements 98 Pseudoprimeelements 106 Exercises 108 Oldnotes 114 I-4 AlgebraicDomainsandLattices 115 Compactelements,algebraicandarithmeticdomains 115 Products,kernelandclosureoperators 119 Completelyirreducibleelements 125 Exercises 127 Oldnotes 129 Newnotes 129 II TheScottTopology 131 II-1 TheScottTopology 132 Scottconvergence 132 TheScotttopologyofdomains 138 TheHofmann–MisloveTheorem 144 Exercises 151 Oldnotes 155 Newnotes 156 II-2 Scott-ContinuousFunctions 157 Scott-continuousfunctions 157 Functionspacesandcartesianclosedcategoriesof dcpos 161 FS-domainsandbifinitedomains 165 Exercises 171 Oldnotes 176 Newnotes 176 Contents vii II-3 InjectiveSpaces 176 Injectiveanddenselyinjectivespaces 177 Monotoneconvergencespaces 182 Exercises 185 Oldnotes 187 Newnotes 187 II-4 FunctionSpaces 187 TheIsbelltopology 187 Spaceswithacontinuoustopology 190 OndcposwithacontinuousScotttopology 197 Exercises 204 Oldnotes 206 Newnotes 207 III TheLawsonTopology 208 III-1 TheLawsonTopology 209 Exercises 216 Oldnotes 218 III-2 MeetContinuityRevisited 219 Exercises 224 Oldnotes 225 Newnotes 226 III-3 QuasicontinuityandLiminfConvergence 226 Quasicontinuousdomains 226 TheLawsontopologyandLiminfconvergence 231 Exercises 236 Oldnotes 240 Newnotes 240 III-4 BasesandWeights 240 Exercises 249 Oldnotes 252 Newnotes 252 III-5 CompactDomains 253 Exercises 261 Newnotes 263 IV MorphismsandFunctors 264 IV-1 DualityTheory 266 Exercises 279 Oldnotes 279 viii Contents IV-2 DualityofDomains 280 Exercises 289 Newnotes 290 IV-3 MorphismsintoChains 290 Exercises 301 Oldnotes 304 IV-4 ProjectiveLimits 305 Exercises 317 Oldnotes 317 IV-5 Pro-continuousandLocallyContinuous Functors 318 Exercises 329 Oldnotes 330 Newnotes 330 IV-6 Fixed-PointConstructionsforFunctors 330 Exercises 340 Newnotes 342 IV-7 DomainEquationsandRecursiveDataTypes 343 Domainequationsforcovariantfunctors 344 Domainequationsformixedvariancefunctors 351 Examplesofdomainequations 355 Exercises 357 Newnotes 358 IV-8 Powerdomains 359 TheHoarepowerdomain 361 TheSmythpowerdomain 363 ThePlotkinpowerdomain 364 Exercises 372 Newnotes 374 IV-9 TheExtendedProbabilisticPowerdomain 374 Exercises 391 Newnotes 392 V SpectralTheoryofContinuousLattices 394 V-1 TheLemma 395 Exercises 399 Oldnotes 399 V-2 OrderGenerationandTopologicalGeneration 400 Exercises 402 Oldnotes 403