Lattice theorynvolume 1, Special topics and applications

472 Pages·2014·3.012 MB·English

by  George Grätzer, Friedrich Wehrung (eds.)

George Grätzer
Friedrich Wehrung
Editors
Lattice Theory:
Special Topics
and Applications
Volume 1

George Grätzer • Friedrich Wehrung
Editors
Lattice Theory: Special
Topics and Applications
Volume 1
Editors
George Grätzer Friedrich Wehrung
Department of Mathematics Department of Mathematics
University of Manitoba University of Caen
Winnipeg, MB, Canada Caen, France

ISBN 978-3-319-06412-3 ISBN 978-3-319-06413-0 (eBook)
DOI 10.1007/978-3-319-06413-0
Springer Cham Heidelberg New York Dordrecht London

Library of Congress Control Number: 2014947337

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Introduction
George Gr¨atzer started writing his General Lattice Theory in 1968. It was
published in 1978. It set out “to discuss in depth the basics of general lattice
theory.” Almost 900 exercises, 193 research problems, and a detailed Further
Topics and References for each chapter completed the picture.
AsT.S.BlythwroteintheMathematicalReviews: “GeneralLatticeTheory
hasbecomethelatticetheorist’sbible. Now,twodecadeson,wehavethesecond
edition, in which the old testament is augmented by a new testament that is
epistolic. The new testament gospel is provided by leading and acknowledged
experts in their fields.”
Another decade later, Gr¨atzer considered updating the second edition to
reflect some exciting and deep developments. “When I started on this project,
it did not take me very long to realize that what I attempted to accomplish in
1968–1978, I cannot even try in 2009. To lay the foundation, to survey the
contemporary field, to pose research problems, would require more than one
volume or more than one person. So I decided to cut back and concentrate in
this volume on the foundation.”
SoLattice Theory: Foundation (referencedinthisvolumeasLTF)provides
the foundation. Now we complete this project with Lattice Theory: Special
Topics and Applications, written by a distinguished group of experts, to cover
some of the vast areas not in LTF.
As in LTF, Theorems (lemmas) presented without proofs are often marked
by the diamond symbol ♦.
v
vi
This Volume 1 is divided into three parts.
Part I. Topology and Lattices
Chapter 1. Continuous and Completely Distributive Lattices
by Klaus Keimel and Jimmie Lawson ...............................5
Chapter 2. Frames: Topology Without Points
by Aleˇs Pultr and Jiˇr´ı Sichler ......................................55
Part II. Special Classes of Finite Lattices
Chapter 3. Planar Semimodular Lattices: Structure and Diagram
by G´abor Cz´edli and George Gra¨tzer ...............................91
Chapter 4. Planar Semimodular Lattices: Congruences
by George Gr¨atzer ................................................131
Chapter 5. Sectionally Complemented Lattices
by George Gr¨atzer ................................................167
Chapter 6. Combinatorics in Finite Lattices
by Joseph P.S. Kung .............................................195
Part III. Congruence Lattices of Infinite Lattices, and Beyond
Chapter 7. Schmidt and Pudl´ak’s Approaches to CLP
by Friedrich Wehrung ............................................235
Chapter 8. Congruences of Lattices and Ideals of Rings
by Friedrich Wehrung ............................................297
Chapter 9. Liftable and Unliftable Diagrams
by Friedrich Wehrung ............................................337
Chapter 10. Two More Topics on Congruence Lattices of Lattices
by George Gr¨atzer ................................................393
Volume 2 will follow with contributions by K. Adaricheva, N. Caspard,
R. Freese, P. Jipsen, J.B. Nation, H. Priestley, N. Reading, L. Santocanale,
and F. Wehrung.
George Gr¨atzer and Friedrich Wehrung, editors
Contents
I Topology and Lattices 1
Introduction 3
1 Continuous Lattices
K. Keimel and J. Lawson 5
1-1 Introduction 5
1-1.1 Some terminology 7
1-2 Basics 7
1-3 The equational theory 12
1-4 The Scott topology 19
1-5 Function spaces and Cartesian closed categories 23
1-6 The Lawson topology 28
1-7 Generation by irreducibles and primes 34
1-8 Fixed point theorems and domain equations 38
1-9 Appendix: Galois adjunctions 45
1-10 Exercises 49
2 Frames: Topology Without Points
A. Pultr and J. Sichler 55
2-1 Introduction 55
2-2 Topological spaces and lattices of open sets: frames 57
2-2.1 Spaces and frames 57
2-2.2 Sobriety 57
2-2.3 Spectrum 58
2-2.4 The spectrum adjunction 59
2-3 Sublocales (generalized subspaces) 60
2-3.1 On the definition 60
2-3.2 Frame congruences 60
2-3.3 Sublocales 61
2-3.4 Open and closed sublocales 62
vii
viii CONTENTS
2-3.5 How to construct the congruence generated by a relation 64
2-4 Free frames. Coproduct 65
2-4.1 Free frames: the down-set functor 65
2-4.2 The construction 66
2-4.3 Coproducts of frames 66
2-4.4 The basic elements ⊕ a 68
i i
2-4.5 Products on Loc compared with topological products 68
2-5 Separation axioms 69
2-5.1 Normal, regular, and completely regular frames 69
2-5.2 Subfitness 70
2-5.3 Hausdorff axiom 71
2-5.4 More about regular frames 71
2-6 Compactness and compactification 72
2-6.1 A few concepts 72
2-6.2 Properties 72
2-6.3 Two more counterparts of classical Hausdorff facts 73
2-6.4 A simple but not very satisfactory compactification 73
2-7 Continuous frames; locally compact spaces.
Hofmann–Lawson duality 76
2-7.1 Continuous lattices 76
2-7.2 Locally compact spaces and continuous frames 77
2-7.3 Adjustment of the spectrum construction 77
2-7.4 Scott topology 77
2-8 Notes on uniform frames 79
2-8.1 Covers and systems of covers 79
2-8.2 Uniformities 80
2-8.3 Completeness and completion 81
2-8.4 An application. Behaviour of paracompact frames 82
2-9 Exercises 83
2-10 Problems 86
II Special Classes of Finite Lattices 89
3 Planar Semimodular Lattices
G. Cz´edli and G. Gr¨atzer 91
3-1 Introduction 91
3-2 ♦♦♦ Some related results 92
3-3 Planarity and diagrams 93
3-4 Slim lattices, the basics 97
3-5 Construction with forks 103
3-6 Construction with resections 106
3-7 ♦♦♦ Rectangular lattices 108
3-8 ♦♦♦ A description by matrices 111
CONTENTS ix
3-9 Description by permutations 114
3-10 Variants of the Jordan–Ho¨lder Theorem 117
3-11 Exercises 121
4 Planar Semimodular Lattices: Congruences
G. Gr¨atzer 131
4-1 Introduction 131
4-2 Congruence structure and N sublattices 133
7
4-2.1 Congruences in finite lattices 134
4-2.2 Finite semimodular lattices 136
4-2.3 Proof of the Tight N Theorem 137
7
4-3 Congruence lattices of rectangular lattices 141
4-3.1 Preliminaries 141
4-3.2 The construction and proof 142
4-3.3 The size of L 145
n
4-4 More on tight N -s 147
7
4-5 The Lower Bound Theorem 150
4-6 Proof of Theorem 4-1.2(ii) 156
4-7 A brief survey of recent results 157
4-8 Exercises 158
4-9 Problems 164
5 Sectionally Complemented Lattices
G. Gr¨atzer 167
5-1 Introduction 167
5-2 Chopped lattices 168
5-2.1 Basic definitions 168
5-2.2 Compatible vectors 169
5-2.3 From the chopped lattice to the ideal lattice 170
5-2.4 Sectional complementation 170
5-3 The representation theorem 171
5-3.1 Constructing M, congruences 171
5-3.2 L is sectionally complemented 174
5-4 An algorithmic construction of sectional complements 175
5-4.1 A crude algorithm 175
5-4.2 Incompatibilities and failures 176
5-4.3 Failures, cuts, and the algorithm 177
5-4.4 The result 179
5-4.5 Proving the failure lemmas 179
5-4.6 Proving the main result 180
V-compatibility 181
C-compatibility 182
H-compatibility 182
5-4.7 Sectional complement 183