DISSERTATIONES MATHEMATICAE - Journals IM PAN - Instytut ...

P O L S K A A K A D E M I A N A U K, I N S T Y T U T M A T E M A T Y C Z N Y
D I S S E R T A T I O N E S
M A T H E M A T I C A E
(ROZPRAWY MATEMATYCZNE)
428
RALPH KUMMETZ
Partially ordered sets with projections
and their topology
W A R S Z A W A 2004

Ralph Kummetz
Institut für Algebra
Technische Universität Dresden
D-01062 Dresden, Germany
Current affiliation:
3SOFT GmbH
Frauenweiherstr. 14
D-91058 Erlangen, Germany
E-mail: ralph.kummetz@3soft.de
Published by the Institute of Mathematics, Polish Academy of Sciences
Typeset using TEX at the Institute
Printed in Poland by
Warszawska Drukarnia Naukowa
Nakład 400 egz.
Available online at http://journals.impan.gov.pl
c○ Copyright by Instytut Matematyczny PAN, Warszawa 2004
ISSN 0012-3862

C O N T E N T S
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.1. Basic notation from order theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.2. Topological and uniform spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.3. Dependence graphs and traces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2. Continuous domains via approximating mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.1. Definition and basic properties of F-posets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2. Continuous domains and convergence of monotone nets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.3. Compact approximating F-posets and FS-domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3. From posets with projections to algebraic domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.1. Projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.2. F-posets with projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.3. Complete approximating pop’s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.4. Domains with compact-valued projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.4.1. Topological characterizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.4.2. Order-theoretic characterizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.5. Characterizations of bifinite domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4. Homomorphisms and function spaces of posets with projections . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.1. Homomorphisms and substructures of indexed pop’s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.2. ω-pop’s and length functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.3. Function spaces of indexed pop’s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
4.3.1. Function spaces from a topological viewpoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
4.3.2. Cartesian closed categories with weak homomorphisms. . . . . . . . . . . . . . . . . . . . . . . 99
4.3.3. Cartesian closed categories with homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
4.3.4. D∞-models for the untyped λ-calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5. Completion of posets with projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
5.1. Existence and uniqueness of the pop completion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
5.2. Domain completion and ideal completion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.3. Comparison of the completions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
6. The topology of traces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
2000 Mathematics Subject Classification: 54E15, 54F05, 06B35, 06F30, 68Q55.
Key words and phrases: poset with approximating mappings, poset with projections, continuous
poset, algebraic poset, dcpo, FS-domain, bifinite domain, P-domain, F-uniformity,
F-topology, pop uniformity, pop topology, convergence of monotone nets, pop homomorphism,
non-expansive map, weight function, cartesian closed category, model for the untyped
λ-calculus, pop completion, domain completion, Mazurkiewicz traces, real traces, α-traces,
δ-traces, topology of traces.
Received 15.7.2002; revised version 2.10.2003.
[3]

Acknowledgements
The present monograph is a revised and extended version of my PhD thesis submitted to and
defended at the Department of Mathematics, Dresden University of Technology, 2000.
First of all, I would like to thank my supervisor, Manfred Droste. His questions on the
topology of Mazurkiewicz traces marked the starting point of my research. I am very grateful
for his guidance and his perpetual interest.
Several people assisted me with my research. In particular, I owe many thanks to Dietrich
Kuske. I benefited a lot from the stimulating discussions with him. After all, they have led to the
characterization of the topology of real traces in Chapter 6. I am also indebted to Achim Jung.
His suggestion to extend the notion of a “pop” has culminated in a separate chapter (Chapter 2).
With their invaluable suggestions, the referees of my PhD thesis—Manfred Droste, Achim
Jung, and Ralph Kopperman—helped improve this monograph a lot.
I greatly acknowledge the support of the PhD programme “Specification of discrete processes
and systems of processes by operational models and logics” (Department of Computer Science)
as well as the pleasant atmosphere at the Institute of Algebra (Department of Mathematics) at
Dresden University of Technology.
Finally, I wish to thank Szisza Zvada for her emotional support and encouragement.
[4]