Continuous Lattices and Domains
Cambridge University Press, Mar 6, 2003 - Mathematics - 591 pages
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.
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A Primer on Ordered Sets and Lattices
The Scott Topology
The Lawson Topology
Morphisms and Functors
Spectral Theory of Continuous Lattices
Compact Posets and Semilattices
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Page 538 - PJ Freyd. Algebraically complete categories. In A. Carboni, MC Pedicchio, and G. Rosolini, editors, Category Theory, volume 1488 of Lecture Notes in Mathematics, pages 131-156.