Lattice Theory: Foundation
This book started with Lattice Theory, First Concepts, in 1971. Then came General Lattice Theory, First Edition, in 1978, and the Second Edition twenty years later. Since the publication of the first edition in 1978, General Lattice Theory has become the
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Lattice Theory: Foundation
George Grätzer Department of Mathematics University of Manitoba Winnipeg, Manitoba R3T 2N2 Canada [email protected]
2010 Mathematics Subject Classification 06-01, 06-02
ISBN 978-3-0348-0017-4 DOI 10.1007/978-3-0348-0018-1
e-ISBN 978-3-0348-0018-1
Library of Congress Control Number: 2011921250 © Springer Basel AG 2011 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the right of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use, permission of the copyright owner must be obtained. Cover design: deblik, Berlin Printed on acid-free paper Springer Basel AG is part of Springer Science+Business Media www.birkhauser-science.com
To Cheryl and David, for the support they gave me, when it was most needed . . .
Short Contents
Preface
xvii
Foreword
xix
Glossary of Notation
xxiii
I
First Concepts 1 Two Definitions of Lattices 2 How to Describe Lattices 3 Some Basic Concepts 4 Terms, Identities, and Inequalities 5 Free Lattices 6 Special Elements
II
Distributive Lattices 1 Characterization and Representation Theorems 2 Terms and Freeness 3 Congruence Relations 4 Boolean Algebras R-generated by Distributive Lattices 5 Topological Representation 6 Pseudocomplementation
109 109 126 138 149 166 191
III
Congruences 1 Congruence Spreading 2 Distributive, Standard, and Neutral Elements 3 Distributive, Standard, and Neutral Ideals 4 Structure Theorems
207 207 223 234 244
vii
1 1 21 28 66 75 97
viii
Contents
IV
Lattice Constructions 1 Adding an Element 2 Gluing 3 Chopped Lattices 4 Constructing Lattices with Given Congruence Lattices 5 Boolean Triples
255 255 262 269 276 294
V
Modular and Semimodular Lattices 1 Modular Lattices 2 Semimodular Lattices 3 Geometric Lattices 4 Partition Lattices 5 Complemented Modular Lattices
307 307 329 342 359 373
VI
Varieties of Lattices 1 Characterizations of Varieties 2 The Lattice of Varieties of Lattices 3 Finding Equational Bases 4 The Amalgamation Property
409 409 423 438 454
VII Free Products 1 Free Products of Lattices 2 The Structure of Free Lattices 3 Reduced Free Products 4 Hopfian Lattices
467 467 493 508 526
Afterword
533
Bibliography
539
Index
589
Contents
Preface
xvii
Foreword
xix
Glossary of Notation I
xxiii
First Concepts 1 Two Definitions of Lattices 1.1 Orders 1.2 Equivalence relations and preorderings 1.3 Basic order concepts 1.4 Ordering and covers 1.5 Order diagrams 1.6 Order constructions 1.7 Two more numeric invariants 1.8 Lattices as orders 1.9 Algebras 1.10 Lattices as algebras Exercises 2 How to Describe Lattices 2.1 Lattice diagrams 2.2 Join- and meet-tables 2.3 Combinations Exercises 3 Some Basic Concepts 3.1 The concept of isomorphism 3.2 Homomorphisms 3.3 Sublattices and extensions 3.4 Ideals 3.5 Intervals ix
1 1 1 2 4 5 6 7 8 9 11 12 15 21 21 21 22 24 28 28 30 31 31 35
x
II
Contents
3.6 Congruence
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