A Compendium of Continuous LatticesA mathematics book with six authors is perhaps a rare enough occurrence to make a reader ask how such a collaboration came about. We begin, therefore, with a few words on how we were brought to the subject over a ten-year period, during part of which time we did not all know each other. We do not intend to write here the history of continuous lattices but rather to explain our own personal involvement. History in a more proper sense is provided by the bibliography and the notes following the sections of the book, as well as by many remarks in the text. A coherent discussion of the content and motivation of the whole study is reserved for the introduction. In October of 1969 Dana Scott was lead by problems of semantics for computer languages to consider more closely partially ordered structures of function spaces. The idea of using partial orderings to correspond to spaces of partially defined functions and functionals had appeared several times earlier in recursive function theory; however, there had not been very sustained interest in structures of continuous functionals. These were the ones Scott saw that he needed. His first insight was to see that - in more modern terminology - the category of algebraic lattices and the (so-called) Scott-continuous functions is cartesian closed. |
Contents
1 | |
Complete lattices | 10 |
Galois connections | 18 |
Meetcontinuous lattices | 34 |
Lattice Theory of Continuous Lattices | 36 |
The waybelow relation | 42 |
The equational characterization | 66 |
Irreducible elements | 68 |
Fixed point construction for functors | 236 |
ΧΙ 1 1 | 238 |
8 | 240 |
Sober spaces and complete lattices | 251 |
18 | 263 |
Compact Posets and Semilattices | 271 |
Some important examples | 293 |
Topological lattices | 316 |
Algebraic lattices | 72 |
The Scott Topology | 97 |
The Scott topology | 115 |
Scottcontinuous functions | 120 |
Injective spaces | 124 |
Function spaces | 128 |
The Lawson Topology | 141 |
The Lawson topology 2 Meetcontinuous lattices revisited | 157 |
Liminf convergence | 164 |
Bases and weights | 169 |
Morphisms and Functors | 177 |
Morphisms into chains | 194 |
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Common terms and phrases
algebraic lattices closure operator compact Hausdorff compact pospace compact semilattices compact topological semilattice complete Heyting algebras complete lattice containing continuous lattice continuous poset continuous semilattice convergence convex COROLLARY define definition denote directed set distributive continuous lattices dual duality EXERCISE exists F-algebra finite infs full subcategory functor Gierz Hence HINT Hofmann homomorphisms implies interpolation property intersection isomorphism Lawson topology LEMMA lower adjoint lower sets maps Math meet-continuous lattice Mislove monotone morphisms nonempty open filter open sets open upper sets partial order patch topology preserves arbitrary infs preserves directed sups preserving arbitrary sups prime elements prime ideal Proof PROPOSITION Remark satisfies Scott continuous Scott topology Scott-open sets Section semicontinuous semilattice with small small semilattices sober spaces Spec statements are equivalent Stralka subsemilattice sup-closed Suppose surjective Theorem theory topological lattice topological space ultrafilter upper adjoint upper set way-below relation