## Continuous Lattices and DomainsInformation 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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### Contents

O4 Meet Continuous Lattices and Semilattices | 36 |

The Scott Topology | 131 |

I13 Inject ve Spaces | 176 |

IH The Lawson Topology | 208 |

Morphisms and Functors | 264 |

Spectral Theory of Continuous Lattices | 394 |

Compact Posets and Semilattices | 439 |

Applications | 492 |

Bibliography | 523 |

559 | |

List of Symbols | 568 |

575 | |

### Common terms and phrases

algebraic domain algebraic lattices auxiliary relation bounded complete bounded complete domain closed sets closure operator compact elements compact pospace compact semilattices compact space complete lattice completely distributive Computer Science continuous lattice continuous semilattice convergence Corollary dcpo DCPOc defined Definition denote directed set distributive lattices duality Exercise exists F-algebra finite infs full subcategory function space functor Gierz hence Hint Hofmann homomorphism ideal implies intersection irreducible isomorphism Lawson topology Lemma liminf locally compact lower adjoint lower sets lower topology meet continuous monotone morphisms nonempty numbers open filter open sets open subsets open upper sets order preserving partial order patch topology poset powerdomain preserves directed sups prime Proof Proposition quasicontinuous domain resp satisfies Scott closed Scott open set Scott topology Scott-continuous Scott-continuous maps Section semilattice sober space Spec sup semilattice Suppose supremum surjective Theorem topological space ultrafilter upper bound way-below relation