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Lattice Theory:
George Grätzer
Foundation
www.birkhauser-science.com
George Grätzer
University of ManitobaWinnipeg, Manitoba R3T 2N2Canada
Department of Mathematics
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© Springer Basel AG 2011
Cover design: deblik, Berlin
2010 Mathematics Subject Classification 06-01, 06-02
ISBN 978-3-0348-0017-4 e-ISBN 978-3-0348-0018-1DOI 10.1007/978-3-0348-0018-1
Library of Congress Control Number: 2011921250
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
1 Two Definitions of Lattices 1
2 How to Describe Lattices 21
3 Some Basic Concepts 28
4 Terms, Identities, and Inequalities 66
5 Free Lattices 75
6 Special Elements 97
II Distributive Lattices 109
1 Characterization and Representation Theorems 109
2 Terms and Freeness 126
3 Congruence Relations 138
4 Boolean Algebras R-generated by Distributive Lattices 149
5 Topological Representation 166
6 Pseudocomplementation 191
III Congruences 207
1 Congruence Spreading 207
2 Distributive, Standard, and Neutral Elements 223
3 Distributive, Standard, and Neutral Ideals 234
4 Structure Theorems 244
vii
viii Contents
IV Lattice Constructions 2551 Adding an Element 2552 Gluing 2623 Chopped Lattices 2694 Constructing Lattices with Given Congruence Lattices 2765 Boolean Triples 294
V Modular and Semimodular Lattices 3071 Modular Lattices 3072 Semimodular Lattices 3293 Geometric Lattices 3424 Partition Lattices 3595 Complemented Modular Lattices 373
VI Varieties of Lattices 4091 Characterizations of Varieties 4092 The Lattice of Varieties of Lattices 4233 Finding Equational Bases 4384 The Amalgamation Property 454
VII Free Products 4671 Free Products of Lattices 4672 The Structure of Free Lattices 4933 Reduced Free Products 5084 Hopfian Lattices 526
Afterword 533
Bibliography 539
Index 589
Contents
Preface xvii
Foreword xix
Glossary of Notation xxiii
I First Concepts 11 Two Definitions of Lattices 1
1.1 Orders 11.2 Equivalence relations and preorderings 21.3 Basic order concepts 41.4 Ordering and covers 51.5 Order diagrams 61.6 Order constructions 71.7 Two more numeric invariants 81.8 Lattices as orders 91.9 Algebras 111.10 Lattices as algebras 12Exercises 15
2 How to Describe Lattices 212.1 Lattice diagrams 212.2 Join- and meet-tables 212.3 Combinations 22Exercises 24
3 Some Basic Concepts 283.1 The concept of isomorphism 283.2 Homomorphisms 303.3 Sublattices and extensions 313.4 Ideals 313.5 Intervals 35
ix
x Contents
3.6 Congruences 363.7 Congruences and homomorphisms 403.8 Congruences and extensions 413.9 Congruences and quotients 423.10 ♦Tolerances 433.11 Direct products 453.12 Closure systems 473.13 Galois connections 493.14 Complete lattices 503.15 Algebraic lattices 523.16 ♦Continuous lattices by Jimmie D. Lawson 543.17 ♦Algebraic lattices in universal algebra 57Exercises 59
4 Terms, Identities, and Inequalities 664.1 Terms and polynomials 664.2 Identities and inequalities 684.3 Distributivity and modularity 71Exercises 73
5 Free Lattices 755.1 The formal definition 755.2 Existence 775.3 Examples 825.4 Partial lattices 835.5 Free lattices over partial lattices 895.6 ♦Finitely presented lattices 91Exercises 92
6 Special Elements 976.1 Complements 976.2 Pseudocomplements 996.3 Other types of special elements 1016.4 ♦Axiomatic games 102Exercises 104
II Distributive Lattices 1091 Characterization and Representation Theorems 109
1.1 Characterization theorems 1091.2 Structure theorems, finite case 1121.3 ♦Structure theorems, finite case, categorical variant 1151.4 Structure theorems, infinite case 1161.5 Some applications 1181.6 Automorphism groups 1201.7 ♦Distributive lattices and general algebra 122Exercises 123
Contents xi
1262 Terms and Freeness 126
2.1 Terms for distributive lattices 1262.2 Boolean terms 1282.3 Free constructs 1302.4 Boolean homomorphisms 1312.5 ♦Polynomial completeness of lattices by Kalle Kaarli 133Exercises 136
3 Congruence Relations 1383.1 Principal congruences 1383.2 Prime ideals 1413.3 Boolean lattices 1423.4 Congruence lattices 145Exercises 146
4 Boolean Algebras R-generated by Distributive Lattices 1494.1 Embedding results 1494.2 The complete case 1544.3 Boolean lattices generated by chains 156Exercises 164
5 Topological Representation 1665.1 Distributive join-semilattices 1675.2 Stone spaces 1685.3 The characterization of Stone spaces 1705.4 Applications 1755.5 Free distributive products 1775.6 ♦Priestley spaces by Hilary A. Priestley 1805.7 ♦Frames by Ales Pultr 184Exercises 185
6 Pseudocomplementation 1916.1 Definitions and examples 1916.2 Stone algebras 1936.3 Triple construction 1946.4 A characterization theorem for Stone algebras 1966.5 Two representation theorems for Stone algebras 1976.6 ♦Generalizing Stone algebras 2026.7 ♦Background 202Exercises 202
III Congruences 2071 Congruence Spreading 207
1.1 Congruence-perspectivity 2071.2 Principal congruences 2091.3 The join formula 2121.4 Finite lattices 213
xii Contents
1.5 Congruences and extensions 2141.6 Congruence-preserving extensions 2171.7 Weakly modular lattices 2181.8 Representable congruences 219Exercises 220
2 Distributive, Standard, and Neutral Elements 2232.1 The three element types 2232.2 Distributive elements 2232.3 Standard elements 2242.4 Neutral elements 2262.5 Connections 2282.6 The set of distributive, standard, and neutral elements 230Exercises 232
3 Distributive, Standard, and Neutral Ideals 2343.1 Defining the three ideal types 2343.2 Characterization theorems 2353.3 The associated congruences 238Exercises 241
4 Structure Theorems 2444.1 Direct decompositions 2444.2 Indecomposable and simple factors 2464.3 Boolean congruence lattices 2484.4 ♦ Infinite direct decompositions of complete lattices
by Friedrich Wehrung 251Exercises 252
IV Lattice Constructions 2551 Adding an Element 255
1.1 One-Point Extension 2551.2 Doubling elements and intervals 259Exercises 260
2 Gluing 2622.1 Definition 2632.2 Congruences 2652.3 ♦Generalizations 266Exercises 267
3 Chopped Lattices 2693.1 Basic definitions 2693.2 Compatible vectors of elements 2703.3 Compatible congruence vectors 2723.4 From the chopped lattice to the ideal lattice 273Exercises 275
4 Constructing Lattices with Given Congruence Lattices 2764.1 The finite case 276
Contents xiii
4.2 Construction and proof 2794.3 Sectional complementation 2804.4 ♦Finite lattices by J. B. Nation 2824.5 ♦Finite lattices in special classes 2854.6 ♦Two finite lattices 2864.7 ♦More than two finite lattices 2874.8 ♦ Independence theorem for finite lattices 2884.9 ♦General lattices 2894.10 ♦Complete lattices 291Exercises 292
5 Boolean Triples 2945.1 The general construction 2955.2 Congruence-preserving extension 2975.3 The distributive case 2995.4 ♦Tensor products 3005.5 ♦Congruence-permutable, congruence-preserving
extensions by Friedrich Wehrung 301Exercises 303
V Modular and Semimodular Lattices 3071 Modular Lattices 307
1.1 Equivalent forms 3071.2 The Isomorphism Theorem for Modular Lattices 3081.3 Two applications 3091.4 Congruence spreading 3111.5 Congruences in the finite case 3161.6 Von Neumann independence 3161.7 Sublattice theorems 3191.8 ♦Pseudocomplemented modular lattices
by Tibor Katrinak 3211.9 ♦ Identities and quasi-identities in submodule lattices
by Gabor Czedli 323Exercises 325
2 Semimodular Lattices 3292.1 The basic definition 3292.2 Equivalent formulations 3312.3 The Jordan-Holder Theorem 3332.4 Independence of atoms 3342.5 M-symmetry 3352.6 ♦Consistency by Manfred Stern 338Exercises 340
3 Geometric Lattices 3423.1 Definition and basic properties 3423.2 Structure theorems 344
xiv Contents
3.3 Geometries 3493.4 Graphs 3523.5 Whitney numbers 353Exercises 355
4 Partition Lattices 3594.1 Basic properties 3594.2 Type 3 representations 3624.3 Type 2 representations 3654.4 Type 1 representations 3674.5 ♦Type 2 and 3 congruence lattices in algebras 3694.6 ♦Sublattices of finite partition lattices 3704.7 ♦Generating partition lattices 371Exercises 371
5 Complemented Modular Lattices 3735.1 Congruences 3735.2 Modular geometric lattices 3735.3 Projective spaces 3755.4 The lattice PG(D,m) 3785.5 Desargues’ Theorem 3795.6 Arguesian lattices 3835.7 The Coordinatization Theorem 3845.8 Frink’s Embedding Theorem 3875.9 A weaker version of the arguesian identity 3905.10 Projective planes 3925.11 ♦Coordinatizing sectionally complemented modular
lattices by Friedrich Wehrung 3945.12 ♦The dimension monoid of a lattice
by Friedrich Wehrung 3975.13 ♦Dilworth’s Covering Theorem by Joseph P. S. Kung 401Exercises 403
VI Varieties of Lattices 4091 Characterizations of Varieties 409
1.1 Basic definitions and results 4091.2 Fully invariant congruences 4111.3 Formulas for Var(K) 4121.4 Jonsson’s Lemma 415Exercises 419
2 The Lattice of Varieties of Lattices 4232.1 Basic properties 4232.2 ♦Varieties of finite height 4252.3 Join-irreducible varieties 4262.4 2ℵ0 lattice varieties 4282.5 ♦The covers of the pentagon 429
Contents xv
2.6 ♦Products of varieties 4302.7 ♦Lattices of equational theories and quasi-equational
theories by Kira Adaricheva 4312.8 ♦Modified Priestley dualities as natural dualities
by Brian A. Davey and Miroslav Haviar 434Exercises 437
3 Finding Equational Bases 4383.1 UDE-s and identities 4383.2 Bounded sequences of intervals 4433.3 The modular varieties covering M3 445Exercises 450
4 The Amalgamation Property 4544.1 Basic definitions and elementary results 4544.2 Lattice varieties with the Amalgamation Property 4584.3 The class Amal(K) 461Exercises 464
VII Free Products 4671 Free Products of Lattices 467
1.1 Introduction 4671.2 The basic definitions 4701.3 Covers 4711.4 The algorithm 4721.5 Computing the algorithm 4721.6 Representing the free product 4741.7 The Structure Theorem for Free Products 4761.8 Sublattices of a free product satisfying (W) 4801.9 Minimal representations 4811.10 Sublattices of a free product satisfying (SD∨) 4841.11 The Common Refinement Property 4851.12 ♦Bounded and amalgamated free products 4871.13 ♦Distributive free products 488Exercises 488
2 The Structure of Free Lattices 4932.1 The structure theorem 4932.2 ♦The word problem for modular lattices 4942.3 Applications 4942.4 Sublattices 4982.5 ♦More covers 5012.6 ♦Finite sublattices and transferable lattices 5022.7 ♦Semidistributive lattices by Kira Adaricheva 503Exercises 506
3 Reduced Free Products 5083.1 Basic definitions 508
xvi Contents
3.2 The structure theorem 5083.3 Getting ready for applications 5113.4 Embedding into uniquely complemented lattices 5143.5 ♦Dean’s Lemma 5173.6 Some applications of Dean’s Lemma 518Exercises 521
4 Hopfian Lattices 5264.1 Basic definitions 5264.2 Free product of hopfian lattices 528Exercises 531
Afterword 533
Bibliography 539
Index 589
Preface
My book, General Lattice Theory, was published in 1978. Its goal: “to discussin depth the basics of general lattice theory”. Each chapter concluded witha section, Further Topics and References, providing brief outlines of, andreferences to, related topics. Each chapter contained a long list of openproblems.
The second edition appeared twenty years later, in 1998. It included thematerial of the first edition, and a series of appendices. The first, Retrospective,reviewed developments of the 20 years between the two editions, especially,solutions of the open problems proposed in the first edition. The otherseven appendices surveyed new fields. They were written by the best expertsavailable. Obviously, I could no longer command an overview of all of latticetheory. The book provided foundation, the appendices surveyed contemporaryresearch.
The explosive growth of the field continued. While the nineteen sixtiesprovided under 1,500 papers and books, the seventies 2,700, the eighties over3,200, the nineties almost 3,600, and the first decade of this century about4,000. As a result, it became almost inevitable that we split the book into twovolumes.
This book, Lattice Theory: Foundation, lays the foundations of the field.There are no Retrospectives and no lists of open problems. Its companionvolume, Lattice Theory: Special Topics and Applications, completes the picture;it is written by experts in the various topics covered.
To help the readers of this book to acquire a wider view, almost a thousandexercises are provided. And there are over forty diamond sections, brief sectionsmarked by the symbol ♦, that provide brief glimpses into research fields beyondthe horizon of this book.
xvii
xviii Preface
Contributors
The following mathematicians contributed diamond sections:
• Kira Adaricheva (Sections VI.2.7 and VII.2.7);• Gabor Czedli (Section V.1.9);• Brian A. Davey and Miroslav Haviar (Section VI.2.8);• Kalle Kaarli (Section II.2.5);• Jimmie D. Lawson (Section I.3.16);• Joseph P. S. Kung (Section V.5.13);• Tibor Katrinak (Section V.1.8);• J. B. Nation (Section IV.4.4);• Hilary A. Priestley (Section II.5.6);• Ales Pultr (Section II.5.7);• Manfred Stern (Section V.2.6);• Friedrich Wehrung (Sections III.4.4, IV.5.5, V.5.11, and V.5.12).
I am deeply appreciative to all of them.
Acknowledgements
To keep this Preface short, I put the history of this book and the very extensiveacknowledgements into the Afterword. But let me repeat one point made there.I started writing this book in 1968. In the forty plus years of this endeavor,I received help from hundreds of mathematicians. I am forever grateful.
Foreword
In the first half of the nineteenth century, George Boole’s attempt to formalizepropositional logic led to the concept of boolean algebras. While investigat-ing the axiomatics of boolean algebras at the end of the nineteenth century,Charles S. Pierce and Ernst Schroder found it useful to introduce the latticeconcept. Independently, Richard Dedekind’s research on ideals of algebraicnumbers led to the same discovery. In fact, Dedekind also introduced modu-larity, a weakened form of distributivity. Although some of the early results ofthese mathematicians and of Edward V. Huntington are very elegant and farfrom trivial, they did not attract the attention of the mathematical community.
It was Garrett Birkhoff’s work in the mid-1930s that started the general de-velopment of lattice theory. In a brilliant series of papers, he demonstrated theimportance of lattice theory and showed that it provides a unifying frameworkfor hitherto unrelated developments in many mathematical disciplines. Birkhoffhimself, Valere Glivenko, Karl Menger, John von Neumann, Oystein Ore, andothers had developed enough of this new field for Birkhoff to attempt to “sell”it to the general mathematical community, which he did with astonishingsuccess in the first edition of his Lattice Theory. The further early developmentof the subject matter can best be followed by comparing the first, second, andthird editions of his book: G. Birkhoff [65] (1940), [70] (1948), and [71] (1967).
The goal of the present volume can be stated very simply: to discuss indepth the foundation of lattice theory. I tried to include the most importantresults and research methods that form the basis of all the work in this field.
Special topics and applications of lattice theory are presented in the com-panion volume. As I mentioned in the Preface, over forty diamond sectionswhet the appetite of the reader by providing brief glimpses into areas notcovered in this volume.
In my view, distributive lattices have played a many-faceted role in thedevelopment of lattice theory. Historically, lattice theory started with (boolean)distributive lattices; as a result, the theory of distributive lattices is one ofthe most extensive and most satisfying chapters of lattice theory. Distributive
xix
xx Foreword
lattices have provided the motivation for many results in general latticetheory. Several conditions on lattices and on elements and ideals of latticesare weakened forms of distributivity. Therefore, a thorough understanding ofdistributive lattices is indispensable for work in lattice theory.
This viewpoint moved me to break with the traditional approach to latticetheory, which proceeds from orders to general lattices, semimodular lattices,modular lattices, and, finally, to distributive lattices. My approach has theadded advantage that the reader reaches interesting and deep results early inthe book.
Chapter I develops the basic concepts of orders and lattices. Diagramsare emphasized because I believe that an important part of learning latticetheory is the acquisition of skill in drawing diagrams. This point of view isstressed throughout the book by about 130 diagrams (heeding Alice’s advice:“and what is the use of a book without pictures”, L. Carroll [1865]); the readerwould be well advised to draw lots more while reading the book.
A special feature of this chapter is a detailed development of free latticesgenerated by a partial lattice over an arbitrary variety; this is one of the mostimportant research tools of lattice theory.
Diamond section topics include tolerances, continuous lattices, the charac-terization theorem of congruence lattices of universal algebras, finitely presentedlattices, and various axiom systems for lattices.
Chapter II develops distributive lattices including representation theo-rems, congruences, boolean algebras, and topological representations. The lastsection is a brief introduction to the theory of distributive lattices with pseu-docomplementation. While the theory of distributive lattices is developedin detail, the reader should keep in mind that the purpose of this chapter is,basically, to serve as a model for the rest of lattice theory.
Diamond section topics include polynomial completeness, Priestley spaces,frames (a lattice theoretic approach to topology), and generalizations of Stonealgebras.
In Chapter III, we discuss congruences and ideals of general lattices.The various types of ideals discussed all imitate to some extent the behaviorof ideals in distributive lattices.
There is only one diamond section, discussing infinite direct decompositionsof complete lattices.
Lattice constructions play a central role in lattice theory. Chapter IVdiscusses a construction of old: gluing (1941) and the newer One-Point Ex-tension (1992), the crucial chopped lattices (from the 1970s), and the newestconstruction (1999): boolean triples.
Diamond section topics include generalized gluing constructions, congru-ence lattices of (i) finite lattices, (ii) finite lattices in special classes, (iii) morethan one finite lattice, (iv) general lattices, (v) complete lattices; further-more, independence results, tensor products, and congruence-permutable,congruence-preserving extensions.
Foreword xxi
After presenting the basic facts concerning modular and semimodularlattices, Chapter V investigates in detail the connection between latticetheory and geometry. We develop the theory of geometric lattices, in particular,direct decompositions and geometric lattices arising out of geometries andgraphs. As an important example, we investigate partition lattices. The lastsection deals with complemented modular lattices and projective geometries,including the Coordinatization Theorem and Frink’s Embedding Theorem.
Diamond section topics include pseudocomplementation in modular lattices,identities of submodule lattices, consistency (a generalization of modularitydifferent from semimodularity), type 2 and 3 congruence lattices for universalalgebras, special topics on partition lattices, coordinatization results of sec-tionally complemented modular lattices, the dimension monoid of a lattice(a precursor of the congruence lattice), and Dilworth’s covering theorem.
Chapter VI deals with varieties of lattices. It covers the basic properties,including Jonsson’s Lemma, the lattice of varieties of lattices, equational bases,and the Amalgamation Property.
Diamond section topics include products of varieties, lattices of (quasi-)equational theories, and modified Priestley dualities.
Chapter VII presents free products of lattices, including the StructureTheorem, the Common Refinement Property, sublattices of a free lattice,reduced free products, and hopfian lattices.
Diamond section topics include amalgamated free products, the wordproblem for modular lattices, transferable lattices and finite sublattices of afree lattice, semidistributive lattices, and Dean’s Lemma.
The exercises form an integral part of the book; do not leave a sectionwithout doing a good number of them.
The Bibliography contains about 700 entries; it is not a comprehensivebibliography of this field. With a few exceptions, it contains only items referredto in the text. To find the references for a topic, use the AMS online database,MathSciNet, or turn to Zentralblatt.
A very detailed Index and the Glossary of Notation should help the readerin finding where a concept or notation is first introduced. For names andconcepts, such as “Jonsson, B.” and “Priestley space”, use the Index ; symbols,such as ConL, rank(p), should be looked up in the Glossary.
I assume a rudimentary knowledge of basic set theory and algebra.
Notation
More difficult exercises are marked by *. Theorems (lemmas) presented withoutproofs are marked by the diamond symbol ♦.
Section 5 refers to a section in the chapter you are reading, whereasSection II.5 refers to a section in Chapter II. Exercise 5.2 refers to the secondexercise in Section 5 of the chapter you are in, while Exercise V.5.2 refers tothe second exercise in Section 5 of Chapter V. Finally, Lemma 403(ii) refers to
xxii Foreword
the second statement of Lemma 403 and Definition 41(i) to the first conditionof Definition 41.
If you are curious how the mathematical notational system used in thisbook developed, consult the Afterword.
Winnipeg, ManitobaNovember 2010
Glossary of Notation
Symbol Explanation Page
A, (A;F ) universal algebra 12(A,− ) geometry 349(A,L) projective plane 393Amal(K) amalgamation class of K 461(Assoc) associativity condition for a binary operation 10(ASym) antisymmetry condition for binary relations 1Atom(L) set of atoms of the lattice L 101AutL automorphism group of L 29breadth(P ) breadth of an order 8(B;∨,∧,′ , 0, 1) boolean algebra 99B1 boolean lattice with 2 elements 99Bn boolean lattice with n atoms 99BRL generalized boolean lattice R-generated by L 152con(a, b) smallest congruence under which a ≡ b 39con(H) smallest congruence collapsing H 39con(p) principal congruence for the prime interval p 213(C) covering condition for free product 472Cn n-element chain 4CenL center of the lattice L 250CEP Congruence Extension Property 42(CID) Complete Infinite Distributive Identity 164(Com) commutativity condition for a binary operation 10ConA congruence lattice of an algebra A 57ConL congruence lattice of a lattice L 38Con(ϕ) Con applied to a homomorphism ϕ 42ConJi L order of join-irreducible congruences of L 213Comp(A) set of complementary pairs in the lattice A 513
xxiii
xxiv Glossary of Notation
Symbol Explanation Page
CovP covering graph of order P 7D class (variety) of distributive lattices 15, 75dim(P ) order-dimension of an order P 9DistrL set of all distributive elements of L 223
Distrδ L set of all dually distributive elements of L 224DnsL dense set of L 101, 194DownP order of down-sets of the order P 7ext for K ≤ L, extension map: α 7→ conL(α) 216EndL endomorphism monoid of L 122, 515End{0,1} L {0, 1}-endomorphism monoid of L 122, 515EquA lattice of all equivalences on A 3(Ex) condition for the existence of a free lattice 77fil(a) filter generated by the element a 34fil(H) filter generated by the set H 34(Fil) a condition for partial lattices 88FilL filter lattice of a lattice L 34Fil0 L augmented filter lattice of a lattice L 34, 88Free(m) free lattice on m generators 76FreeP free lattice over the order P 76FreeA free lattice over the partial lattice A 90FreeD(3) free distributive lattice on three generators 82FreeK A free lattice over A in a variety K 89FreeK P free lattice over the order P in a variety K 76FreeM(3) free modular lattice on three generators 83Free(P ;J ,M) free lattice in Dean’s Lemma 517(GC) compactness condition on open sets 170(G;E) graph on set G with edges E 7height(a) height of an element 4H(K) class of homomorphic images of members of K 413id(a) principal ideal generated by the element a 32, 88id(H) ideal generated by the set H 32I(K) class of isomorphic copies of members of K 413IdL ideal lattice of L 33, 52, 270Id0 L augmented ideal lattice of L 33, 88(Id) for ideals in chopped lattices 270(Idem) idempotency condition for a binary operation 10Iden(K) set of identities holding in the class K 409(Idl) a condition for partial lattices 88inf H,
∧H greatest lower bound of H 5
JiL order of join-irreducible elements of L 102(JID) Join Infinite Distributive Identity 154
Glossary of Notation xxv
Symbol Explanation Page
Ker(ϕ) congruence kernel of ϕ 41L class (variety) of all lattices 75Λ lattice of all varieties of lattices 423LatG lattice representation of the graph G 515Lalg the (order) lattice L as an algebra 13Lord the (algebra) lattice L as an order 13len(P ) length of a finite order P 4(Lin) linearity condition for binary relations 1M class (variety) of modular lattices 15, 75M3 five-element modular nondistributive lattice 23M3 variety generated by M3 425M4 a modular lattice with four atoms 425M4 variety generated by M4 425M3,3 two copies of M3 glued together 425M3,3 variety generated by M3,3 425Max(M) maximal elements of a chopped lattice 270Merge(C,D) merging of lattices C and D 269MiL order of meet-irreducible elements of a lattice L 102(MID) Meet Infinite Distributive Identity 154Mod(Σ) class of all lattice models of Σ 409M3[L] order of boolean triples of the lattice L 295M3[α] congruence on M3[L] 297N5 five-element nonmodular lattice 23N5 the variety generated by N5 423N6 = N(p, q) six-element nonmodular lattice 277NeutrL set of all neutral elements of L 226(OP) condition for One-Point Extension 256p, q prime intervals 35P δ dual of P 5Pmax a partial lattice formed from the order P 90Pmin a partial lattice formed from the order P 90PartA partition lattice of A 3, 359PartfinA set of finite partitions 361PG(D,m) m-dimensional projective geometry over D 378PowX power set lattice of X 4P(K) class of direct products of members of K 413Ps(K) class of subdirect products of members of K 414Pu(K) class of ultraproducts of members of K 416PrInt(L) set of prime intervals of L 213Q chain of rational numbers 158
xxvi Glossary of Notation
Symbol Explanation Page
rank(p) rank of a term p 68re for K ≤ L, restriction map: α 7→ αeK 214rep(p) terms equivalent to p in a free product 474(Refl) reflexivity condition for binary relations 1sub(H) sublattice generated by H 31supH,
∨H least upper bound of H 5
spec(a) spectrum of an element a 112, 118(SD∨) join-semidistributive law 479(SD∧) meet-semidistributive law 479S(K) class of subalgebras of members of K 413Si(K) class of subdirectly irreducible members of K 418S1 three-element Stone algebra 201SB booleanization of the Stone space S 176SpecL spectrum of a distributive lattice L 116SkelL skeleton of L 99(SP∨) join-substitution property 36, 43, 269(SP∧) meet-substitution property 36, 43, 269SubA subalgebra lattice of an algebra A 57StandL set of all standard elements of L 224(Stone1)–(Stone3) Stone conditions on a topological space 171, 174SubL sublattice lattice (including ∅) of a lattice L 51T class (variety) of trivial lattices 92, 423Tn Tamari lattice 27Term(n) n-ary lattice terms 66TermB(n) n-ary lattice terms in B 129TermD(n) n-ary lattice terms in D 126tran(%) transitive closure for binary relation % 3(Trans) transitivity condition for binary relations 1(W) Whitman condition for free product 479Var(K) smallest variety containing the class K 414Wi Whitney number 353
Glossary of Notation xxvii
Symbol Explanation Page
Relations and
Congruences
A2 set of ordered pairs of A 2ε, %, τ , π, . . . binary relations 2equ(π) binary relation from a partition π 3α, β, . . . , θ congruences 360,1 zero and unit of PartA and ConL 36(a, b) ∈ ε a and b are in relation ε 3a ε b a and b are in relation ε 3a ≡ b (mod ε) a and b are in relation ε 3p ∈ α prime interval p collapsed by α 213a/π block containing a 3, 36A/π set of all block of π 3, 40L/α quotient lattice 40β/α quotient congruence, tolerance 42, 43, 198αeK restriction of α to the sublattice K 41πi projection map 46α× β direct product of congruences 46
Orders
≤, < ordering 2≥, > ordering, inverse notation 2K ≤ L K is a sublattice of L 31L ≥ K L is an extension of K 31≤Q ordering of P restricted to a subset Q 4a ‖ b a incomparable with b 4a ≺ b a is covered by b 6a � b a ≺ b or a = b 6b � a b covers a 6b � a b � a or b = a 60, 1 zero and unit of order 5a ∨ b join operation 10∨H least upper bound of H 5
a ∧ b meet operation 10∧H greatest lower bound of H 5
[a, b] interval 35down(H) down-set generated by H 7down(a), ↓ a down-set generated by a 7P ∼= Q order (lattice) P isomorphic to Q 4, 12, 28
xxviii Glossary of Notation
Symbol Explanation Page
Constructions
P ×Q direct product of P and Q 1, 7, 45P 2 P × P 7∏
(Li | i ∈ I ) direct product of {Li | i ∈ I } 46∏D(Li | i ∈ I) ultraproduct of {Li | i ∈ I } 416
P δ dual of an order (lattice) P 5, 14P +Q sum of P and Q 8
P.+Q glued sum of P and Q 8
↓H down-set generated by H 7↑H up-set generated by H 7A⊗B tensor product of A and B 300LI One-Point Extension of the lattice L 255L[I] interval doubling 259M3[L] order of boolean triples of the lattice L 295A ∗B free product of the lattices A and B 467
Perspectivities
[a, b] ∼ [c, d] perspective intervals 35
[a, b]up∼ [c, d] up-perspective intervals 35
[a, b]dn∼ [c, d] down-perspective intervals 35
[a, b] ≈ [c, d] projective intervals 35
[a, b]n≈ [c, d] projective intervals in n steps 35
[a, b]� [c, d] congruence-perspective intervals 208
[a, b]up� [c, d] congruence-perspective up intervals 208
[a, b]dn� [c, d] congruence-perspective down intervals 208
[a, b]⇒ [c, d] congruence-projective intervals 208[a, b]⇔ [c, d] [a, b]⇒ [c, d] and [c, d]⇒ [a, b] 208x ∼ y x is perspective to y 239x . y x is subperspective to y 239x ≤⊕ y x has a relative complement in [0, y] 239ProjRep(p) projective representative of p 344ProjCl(p) projective closure of p 344unit(p) unit of projective closure of p 344
Glossary of Notation xxix
Symbol Explanation Page
Miscellaneous
a∗ pseudocomplement of a 99a∗ unique lower cover of a 102a′ complement of a 99x,X closure of x and X 47, 49∅ empty set 5≡B equivalence of boolean terms 128p/B block of boolean terms 129≡D equivalence of distributive lattice terms 126p/D block of distributive lattice terms 126x+ y symmetric difference 132α ◦ β product of binary relations 2V ◦W product of the varieties V and W 430aM b modular pair 335p(i), p(i) upper and lower i-cover of p in a free product 471p, p upper and lower cover of p in P 517
Ab lattice A with two new bounds 471
