Information content and programming semantics are just two of the applications of the mathematical concepts of order, continuity and domains. The authors develop the mathematical foundations of partially ordered sets with completeness properties of various degrees, in particular directed complete ordered sets and complete lattices. Uniquely, they focus on partially ordered sets that have an extra order relation, modelling the notion that one element 'finitely approximates' another, something closely related to intrinsic topologies linking order and topology. Extensive use is made of topological ideas, both by defining useful topologies on the structures themselves and by developing close connections with numerous aspects of topology. The theory so developed not only has applications to computer science but also within mathematics to such areas as analysis, the spectral theory of algebras and the theory of computability. This authoritative, comprehensive account of the subject will be essential for all those working in the area.

• The only self-contained introduction to the mathematical theory of continuous lattices and domains • An authoritative resource for experts and students • Based on the authors' classic 'Compendium of Continuous Lattices', bringing together the major developments of the last 21 years

### Contents

Preface; Acknowledgements; Foreword; Introduction; 1. A primer on ordered sets and lattices; 2. Order theory of domains; 3. The Scott topology; 4. The Lawson Topology; 5. Morphisms and functors; 6. Spectral theory of continuous lattices; 7. Compact posets and semilattices; 8. Topological algebra and lattice theory: Applications; Bibliography; List of symbols; List of categories; Index.

### Reviews

'A highly recommendable handbook on a subject which links algebra and topology in a convincing and beautiful way.' H. Mitsch, Wien

'Being written in an expository and almost self-contained way, the book can serve as a handbook for those working in the area as well as a student textbook.' EMS Newsletter

'… this is the book for all those interested in the topic … I am pleased to recommend … this most authentic source of information on continuous lattices and domains.' Acta Scientiarum Mathematicarum