Continuous Lattices and Domains

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

• ISBN13: 9780521803380
• ISBN10: 0521803381
• Format: Hardcover
• Copyright: 2003-04-21
• Publisher: Cambridge University Press

Summary

Information content and programming semantics are just two of the applications of the mathematical concepts of order, continuity and domains. This authoritative and comprehensive account of the subject will be an essential handbook for all those working in the area. An extensive index and bibliography make this an ideal sourcebook for all those working in domain theory.

Table of Contents

Preface	xi
Acknowledgments	xxi
Foreword to A Compendium of Continuous Lattices	xxiii
Introduction to A Compendium of Continuous Lattices	xxvii
A Primer on Ordered Sets and Lattices	1	(47)
Generalities and Notation	1	(7)
Exercises	7	(1)
Old notes	8	(1)
Completeness Conditions for Lattices and Posets	8	(14)
Exercises	17	(4)
Old notes	21	(1)
New notes	22	(1)
Galois Connections	22	(14)
Exercises	31	(4)
Old notes	35	(1)
Meet Continuous Lattices and Semilattices	36	(5)
Exercises	39	(2)
Old notes	41	(1)
T0 Spaces and Order	41	(7)
Exercises	45	(2)
New notes	47	(1)
Order Theory of Domains	48	(83)
The ``Way-below'' Relation	49	(30)
The way-below relation and continuous posets	49	(8)
Auxiliary relations	57	(5)
Important examples	62	(9)
Exercises	71	(4)
Old notes	75	(3)
New notes	78	(1)
Products, Substructures and Quotients	79	(16)
Products, projection, kernel and closure operators on domains	79	(4)
Equational theory of continuous lattices	83	(7)
Exercises	90	(3)
Old notes	93	(1)
New notes	94	(1)
Irreducible elements	95	(20)
Open filters and irreducible elements	95	(3)
Distributivity and prime elements	98	(8)
Pseudoprime elements	106	(2)
Exercises	108	(6)
Old notes	114	(1)
Algebraic Domains and Lattices	115	(16)
Compact elements, algebraic and arithmetic domains	115	(4)
Products, kernel and closure operators	119	(6)
Completely irreducible elements	125	(2)
Exercises	127	(2)
Old notes	129	(1)
New notes	129	(2)
The Scott Topology	131	(77)
The Scott Topology	132	(25)
Scott convergence	132	(6)
The Scott topology of domains	138	(6)
The Hofmann--Mislove Theorem	144	(7)
Exercises	151	(4)
Old notes	155	(1)
New notes	156	(1)
Scott-Continuous Functions	157	(19)
Scott-continuous functions	157	(4)
Function spaces and cartesian closed categories of dcpos	161	(4)
FS-domains and bifinite domains	165	(6)
Exercises	171	(5)
Old notes	176	(1)
New notes	176	(1)
Injective Spaces	176	(11)
Injective and densely injective spaces	177	(5)
Monotone convergence spaces	182	(3)
Exercises	185	(2)
Old notes	187	(1)
New notes	187	(1)
Function Spaces	187	(21)
The Isbell topology	187	(3)
Spaces with a continuous topology	190	(7)
On dcpos with a continuous Scott topology	197	(7)
Exercises	204	(2)
Old notes	206	(1)
New notes	207	(1)
The Lawson Topology	208	(56)
The Lawson Topology	209	(10)
Exercises	216	(2)
Old notes	218	(1)
Meet Continuity Revisited	219	(7)
Exercises	224	(1)
Old notes	225	(1)
New notes	226	(1)
Quasicontinuity and Liminf Convergence	226	(14)
Quasicontinuous domains	226	(5)
The Lawson topology and Liminf convergence	231	(5)
Exercises	236	(4)
Old notes	240	(1)
New notes	240	(1)
Bases and Weights	240	(13)
Exercises	249	(3)
Old notes	252	(1)
New notes	252	(1)
Compact Domains	253	(11)
Exercises	261	(2)
New notes	263	(1)
Morphisms and Functors	264	(130)
Duality Theory	266	(14)
Exercises	279	(1)
Old notes	279	(1)
Duality of Domains	280	(10)
Exercises	289	(1)
New notes	290	(1)
Morphisms into Chains	290	(15)
Exercises	301	(3)
Old notes	304	(1)
Projective Limits	305	(13)
Exercises	317	(1)
Old notes	317	(1)
Pro-continuous and Locally Continuous Functors	318	(12)
Exercises	329	(1)
Old notes	330	(1)
New notes	330	(1)
Fixed-Point Constructions for Functors	330	(13)
Exercises	340	(2)
New notes	342	(1)
Domain Equations and Recursive Data Types	343	(16)
Domain equations for covariant functors	344	(7)
Domain equations for mixed variance functors	351	(4)
Examples of domain equations	355	(2)
Exercises	357	(1)
New notes	358	(1)
Powerdomains	359	(15)
The Hoare powerdomain	361	(2)
The Smyth powerdomain	363	(1)
The Plotkin powerdomain	364	(8)
Exercises	372	(2)
New notes	374	(1)
The Extended Probabilistic Powerdomain	374	(20)
Exercises	391	(1)
New notes	392	(2)
Spectral Theory of Continuous Lattices	394	(45)
The Lemma	395	(5)
Exercises	399	(1)
Old notes	399	(1)
Order Generation and Topological Generation	400	(3)
Exercises	402	(1)
Old notes	403	(1)
Weak Irreducibles and Weakly Prime Elements	403	(5)
Exercises	406	(1)
Old notes	407	(1)
Sober Spaces and Complete Lattices	408	(7)
Exercises	414	(1)
Old notes	415	(1)
Duality for Distributive Continuous Lattices	415	(16)
Exercises	423	(6)
Old notes	429	(2)
Domain Environments	431	(8)
Exercises	437	(1)
New notes	437	(2)
Compact Posets and Semilattices	439	(53)
Pospaces and Topological Semilattices	440	(5)
Exercises	444	(1)
Old notes	445	(1)
Compact Topological Semilattices	445	(5)
Exercises	449	(1)
Old notes	450	(1)
The Fundamental Theorem of Compact Semilattices	450	(12)
Exercises	457	(5)
Old notes	462	(1)
Some Important Examples	462	(6)
Old notes	467	(1)
Chains in Compact Pospaces and Semilattices	468	(6)
Exercises	472	(1)
Old notes	473	(1)
Stably Compact Spaces	474	(12)
Exercises	484	(2)
New notes	486	(1)
Spectral Theory for Stably Compact Spaces	486	(6)
Exercises	489	(2)
Old notes	491	(1)
Topological Algebra and Lattice Theory: Applications	492	(31)
One-Sided Topological Semilattices	493	(6)
Exercises	498	(1)
Old notes	499	(1)
Topological Lattices	499	(9)
Exercises	504	(3)
Old notes	507	(1)
New notes	508	(1)
Hypercontinuity and Quasicontinuity	508	(7)
Exercises	515	(1)
New notes	515	(1)
Lattices with Continuous Scott Topology	515	(8)
Exercises	521	(1)
Old notes	522	(1)
Bibliography	523	(1)
Books, Monographs, and Collections	523	(3)
Conference Proceedings	526	(2)
Articles	528	(31)
Dissertations and Master's Theses	559	(5)
Memos Circulated in the Seminar on Continuity in Semilattices (SCS)	564	(4)
List of Symbols	568	(4)
List of Categories	572	(3)
Index	575