Continuous Lattices: Proceedings of the Conference on Topological and Categorical Aspects of Continuous Lattices (Workshop IV) Held at the University of Bremen, Germany, November 9–11, 1979

423 Pages·1981·7.093 MB·English-German

by  Bernhard Banaschewski (auth.), Bernhard Banaschewski, Rudolf-Eberhard Hoffmann (eds.)

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
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© by Springer-Verlag Berlin Heidelberg 1891
Printed in Germany
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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.