Lecture Notes ni Mathematics Edited yb .A Dold and .B Eckmann 178 IIIIIIIIIII II Continuous Lattices Proceedings of the Conference on Topological and Categorical Aspects of Continuous Lattices (Workshop )VI Held at the University of Bremen, Germany, November 9-11, 1979 Edited yb .B Banaschewski and .E-.R Hoffmann I1[ II galreV-regnirpS Berlin Heidelberg New York 1981 Editors Bernhard Banaschewski Mathematical Sciences, McMaster University Hamilton, Ontario L8S 4K1, Canada Rudolf-Eberhard Hoffmann Fachbereich Mathematik, Universit~t Bremen 2800 Bremen, Federal Republic of Germany AMS Subject Classifications (1980): 06 A 23, 06 B 30, 06 F 30, 18 B 99, 54 F 05, 54 D 45 ISBN 3-540-10848-3 Springer-Verlag Berlin Heidelberg NewYork ISBN 0-38?-10848-3 Springer-Verlag NewYork Heidelberg Berlin This work is subject to copyright. All rights are reserved, whether thew hole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar and means, storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich, © by Springer-Verlag Berlin Heidelberg 1891 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210 PREFACE The theory of continuous lattices has its grass roots in the work of Dana .S Scott on a mathematical theory of "Computation", and was com- municated to the public in mature form for the first time in his article "Continuous Lattices" which appeared in 1972 (Springer Lecture Notes in Mathematics 274). It took a few years before the intimate relationship of the concept of a continuous lattice with several other mathematical fields was recognized: Indeed, continuous lattices appear as certain compact Hausdorff spaces with an additional semi- lattice structure (K.H.Hofmann,A.R.Stralka) and as lattices of open subsets of locally quasi-compact (not necessarily HausdorffJ spaces(K.H.Hofmann,J.D.LawsonJ which have attracted some interest in functional analysis since the early sixties (J.M.G.Fell,J.Oixmierl). Also, during the further development of the new theory it became more and more apparent how useful and, in some oases, even indispens- able the concepts of category theory are in order to clarify the subject: Continuous lattices with a suitable (non-Hausdorff)topology are the "injective" objects with regard to embeddings in the category of topological To-spaces and continuous maps (O.S.Scott], a concept which first arose in the study of (divisible) abelian groups and was later formulated in categorical terms; the full subcategory of in- jeotive Te-spaces(~ continuous lattices) turns out to be ~ B O closed; the topological spaces X for which the lattice OX of open subsets is a continuous lattice are recognized as precisely those spaces for which the functor Xx- (defined on all topological spaces] has a !~f~ ~e~, i'e" for which X,f:X,Y ~ XxZ is a quotient map, whenever f:Y ~ Z is so (8.J.Oay - G.M.Kelly, J.R.Isbell). Further- more, the category of continuous lattices (with a stronger sort of morphisms than in the above results of Scott's) turns out to be the category of Eilenberg-Moore algebras for several m2oads(~ ~ s ) , a concept which originated in homological algebra. These "algebras" for a monad had attracted much interest in several sub- stantial papers in the second half of the sixties when - among other things - it became clear that practically all categories of algebras (with operations defined everywhere) can be represented in this fashion, once the induced monad is determined. In marked contrast to the monads associated with the familiar algebras(groups, rings, VI etc.) which seem to be somewhat artificial constructions, the monads arising in the context of continuous lattices have a gr'eat topologi- cal significance: The funetor part )s of the first of these assigns to a To-space its "open filter space" a( construction which has been employed by B.BanaschBwski in the study of extension problems about twenty years agog, the other functor part assigns to a compact Hausdorff space its space of closed subsets, introduced by .L Vie- toris as early as in a paper of 1922 which seems to have been much ahead of its time (A.Oay; O.Wyler, these proceedings). Indeed, there are some other interesting monads in the vicinity of these: A space of closed subsets in its lower (or weak) topology(R.-E, Hoff- mann), a space of open prime filters (H.Simmons) which turns an extension space studied by .J Ylachsmeyer in 1981 into the functor part of a monad, a poser of(extended) ideals of a given poser ["algebraic posets", R.-E, Hoffmann), and several other monads which have not yet been thoroughly studied. Besides this @legant inter- action between concepts of general topology and category theory, these observations make a convincing plea for non-Hausdorff spaces as a legitimate object of study in general topology. The present volume is the published proceedings of a conference Topol,,ogical and Categorical Aspects of Continuous Lattices held at the Department of Mathematics of the University of Bremen, November 9-11, 1978. This conference was the fourth in a series of workshops which were held at Tulane University (1977), Technische Hochschule Darmstadt {1978), and University of California at Riverside{spring 1979), respectively. Proceedings of these earlier workshops have not been published, but much of t~e research reported there has been incor- porated in the book A Compendium of Continuous Lattices (Springer Verlag 1980) which provides a clearly written, self- contained introduction into this field and which is strongly recom- *) A monad T = (T,n,p> consists of three data: an endofunctor T and two natural transformations q and p subject to certain axioms. V mended to everybody interested in the subject. While the com- pendium was still under preparation, several new concepts appeared as mathematically desirable generalizations or variations of the concept of a continuous lattice, the most remarkable of which is known as the concept of a continuous reset .G( Markowsky, R.-E. Hoffmann, J.O. Lawson, R.L. Wilson). During the Bremen workshop the emphasis was on such generalizations. Thus it seemed reasonable to include also a few related papers not read at the conference, which have had some impact on the development of the concept of a continuous poser, but are not yet published elsewhere nor superposed by the compendium. Whereas most of the articles of this volume develop categorical and topological aspects of continuous lattices, a few also exhibit the connections with iozic, lattice theory, theoretical computer science, and functional analysis. All the papers have been refereed, and our grateful thanks go to all those who helped the editors of this volume in doing this job. Also, we would like to thank the participants for a lively and successful conference, and the Department of Mathematics of the University of Bremen for some financial support. {R.-E.H.) Bernhard Banaschewski Rudolf-E. Hoffmann PARTICIPANTS x) Bernhard Banasohewski, McMaster University, Hamilton,Ont. L8S4K1, Cansda Hans-J.Bandelt, Universit~t Oldenburg, 0-29oo Oldenburg Heiko Bauer, Teohnische Hochschule Darmstadt, D-61oo Darmstadt Reinhard BOrger, Fern-Universit~t, D-SBoo Hagen Marcel Ern~, Universit~t Hannover, D-3ooo Hannover Gerhard U.Gierz, Technisohe Hochschule Oarmstadt, 0-61oo Oarmstadt Georg Greve, Fern-Universit~t, 0-58oo Hagen Horst Herrlich, Universit~t Bremen, D-28oo Bremen Rudolf-E.Hoffmann, Universit~t Bremen, 0-28oo Bremen Karl H.Hofmann, Tulane University, New Orleans, La.7o118, U.S.A. Klaus Keimel, Technische Hochschule Darmstadt, 0-61oo Oarmstadt Herald Lindner, Universit~t DOsseldorf, 0-4ooo OOsseldorf Jean-Marie McOill, presently Universit~t Bremen, D-28oo Bremen Axel MBbus, Univers~t~t DOsseldorf, D-4ooo DOsseldor? Evelyn Nelson, McMaster University, Hamilton,Ont,L8S 4KI, Canada Mario Petrich, presently Universit~t Oldenburg, 0-290o Oldenburg Hans-E.Porst, Universit~t Bremen, 0-2800 Bremen GOnther R~chter, Unlversit~t Bielefeld, 0-48oo BieIefeid Friedhelm Schwarz, Universit~t Hannover, D-3ooo Hennover x) Oena S.Scott, Merton College, Oxford, Great Britain x) O.Spoerel, FB Elektrotechnik, Hochschuie der Bundeswehr, 2ooo Hamburg Albert Stralka, University of California, Riverside, Cei.92521, U.S.A. Momme Jobs Thomsen, presently Universit~t Bremen, D-28oo Bremen Sibylle Weck, Universit~t Hannover, O-3ooo Hannover x) Elke Wilkeit, 0-29oo Oldenburg Manfred Wischnewsky, UniversitQt Bremen, D-28oo Bremen x) Unless explicitly indicated by x}, the address refers to the department of Mathematics. Address list of authors: Bernhard Banaschewskl Department of Mathematics, McMaster University, Hamilton, Ontario SBL 4KI, adanaC Heiko Bauer Fachbereich Mathematik, Technischs Hochschule Darmstadt, Arbeitsgruppe 1, SchloBgartenstr,7, O-61oo Darmstadt Marcel Ern@ Mathematisches Institut, Universit~t Hannover, Welfengarten I, 0-3o0o Hannover 1 Gerhard U.Gierz Fachbersich Mathematik, Technisohe Hochschule Darmstadt, Arbeitsgruppe ,I SchloBgartenstr.7, D-61oo Darmstadt Rudolf-E.Hoffmann Fachbereich Mathematik, UniversitQt Bremen, Kufsteiner Str., 0-28oo Bremen Karl H.Hofmann Department of Mathematics, Tulane University, New Orleans, Louisiana 7o118, U.S.A. J.Martin E.Hyland Department of Pure Mathematics, University of Cambridge, Cambridge CB2 ISB, Great Britain Peter T.Johnstone Oepartment of Pure Mathematics, University of Cambridge, Cambridge CB2 ISB, Great Britain Klaus Keimel Fachbereich Mathematik, Technische Hochschule Oarmstadt, Arbeitsgruppe ,I SchloBgartenstr.7, O-61oo Oarmstadt Roland KOhler Fachbereich Mathematik, Technische Hochschule Oarmstadt, Arbeitsgruppe ,1 SchloBgartenstr. ,7 0-61oo Darmstadt George Markowsky IBM, Thomas J.Watson Research Center, Yorktown Heights, New York Io598, USA. Michael W.Mislove Department of Mathematics, Tulane University, New Orleans, Louisiana 7o118, U.S.A. Evelyn Nelson Department of Mathematics, McMaster University, Hamilton, Ontario, LBS 4K1, Canada VIII Friedhelm Schwarz Mathematisches Institut, Universit~t Hannover, Welgengarten ,I 0-3ooo Hannover I Albert Stralka Department of Mathematics, University of California, Riverside, California 92521, U.S.A. Adrian Tang Department of Computer Science, University of Kansas, Lawrence, Kansas 66o45, U.S.A. Fred Watkins Department of Mathematics, Tulane University, New Orleans, Louisiana 7o118, U.S.A. Sibytle Weck Institut fSr Mathematik, Universit~t Hannover, Welfengarten ,I D-3ooo Hannover I Oswald Wylsr Oepartment of Mathematics, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, U.S.A. STNETNOC Coherent {rames by Bernhard Banaschewski The duality o$ distributive ~-continuous lattices by Bernhard Banaschswski 12 Ver{einerungs- und KOrzungss~tze {Qr Produkte geordneter topologischer R~ume und {Or Funktionen (-halb-)verb~nde by Heiko Bauer, Klaus Keimel und Roland K6hler 2o Completion-invariant extension o{ the concept o{ continuous lattices by Marcel Ern§ 45 Scott convergence and Scott topology in partially ordered sets, II. by Marcel Ern@ 16 Continuous ideal completions and compacti{ioations by Gerhard Gierz and Klaus Keimel 97 Projective sober spaces by Rudol{-E.Ho~{mann 125 Continuous posets, prime spectra o{ completely distributive complete lattices, and Hausdor{{ compactifications by Rudol{-E.Ho~{mann 159 Local compactness and continuous lattices by Karl H.Ho{mann and Michael W.Mislove 209 The spectrum as a {unctor by Karl H.Ho{mann and Fred Watkins 249 Function spaces in the category o{ locales by J.M.E.Hyland 264 Scott is not always sober by Peter T.Johnstone 282 Injective toposes by Peter T.Johnstone 284 A motivation and generalization o£ Scott's notion o{ a continuous lattice by George Markowsky 298 Propaedeutic to chain-complete posets with basis by George Markowsky 3o8 Z-continuous algebras by Evelyn Nelson 315 "Continuity" properties in lattices o~ topological structures by Friedhelm Sohwarz 335 Fundamental congruences on Lawson semilattices by Albert Stralka 348 Wadge reducibility and Hausdorgg diggerence hierarchy in P~ by A. Tang 36o Scott convergence and Scott topology in partially ordered sets, .I by Sibyile Weck 372 Dedekind complete posets end Scott topologies by Oswald Wyler 384 Algebraic theories og continuous lattices by Oswald Wyler 39o nA expanded version of O.S.Scott's paper "A space of retracts" s read at the conference, will appear elsewhere.