Continuous Lattices and Domains (Encyclopedia of Mathematics and its Applications)

629 Pages·2003·2.279 MB·English

by  G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. W. Mislove, D. S. Scott

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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
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23 J.R.CannonTheonedimensionalheatequation
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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
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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-MorrisPade´approximants
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
  
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First published in print format 2003
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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