Algebras, Lattices, Varieties

496 Pages·2022·5.211 MB·English

by  Ralph S. Freese, Ralph N. McKenzie, George F. McNulty, Walter F. Taylor

Mathematical
Surveys
and
Monographs
Volume 268
Algebras, Lattices,
Varieties
Volume II
Ralph S. Freese
Ralph N. McKenzie
George F. McNulty
Walter F. Taylor
Algebras, Lattices,
Varieties
Volume II

Mathematical
Surveys
and
Monographs
Volume 268
Algebras, Lattices,
Varieties
Volume II
Ralph S. Freese
Ralph N. McKenzie
George F. McNulty
Walter F. Taylor
EDITORIAL COMMITTEE
Ana Caraiani Natasa Sesum
Michael A. Hill Constantin Teleman
Bryna Kra (chair) Anna-Karin Tornberg
2020 Mathematics Subject Classification. Primary 08-02, 08Bxx, 03C05, 06Bxx.
For additional informationand updates on this book, visit
www.ams.org/bookpages/surv-268
Library of Congress Cataloging-in-Publication Data
Names: McKenzie, Ralph, author. | McNulty, George F., 1945- author. | Taylor, W. (Walter),
1940-author.
Title: Algebras,lattices,varieties/RalphN.McKenzie,GeorgeF.McNulty,WalterF.Taylor.
Description: Providence,RhodeIsland: AmericanMathematicalSociety: AMSChelseaPublish-
ing, 2018-| Series: | Originally published: Monterey, California : Wadsworth & Brooks/Cole
AdvancedBooks&Software,1987. |Includesbibliographicalreferencesandindexes.
Identifiers: LCCN2017046893|ISBN9781470467975(paperback)|9781470471293(ebook)
Subjects: LCSH: Algebra, Universal. | Lattice theory. | Varieties (Universal algebra) | AMS:
General algebraic systems – Algebraic structures – Algebraic structures. | General algebraic
systems–Varieties–Varieties. |Mathematicallogicandfoundations–Modeltheory–Equa-
tional classes, universal algebra. | Order, lattices, ordered algebraic structures – Lattices –
Lattices. | Order, lattices, ordered algebraic structures – Modular lattices, complemented
lattices–Modularlattices,complementedlattices.
Classification: LCCQA251.M432018|DDC512–dc23
LCrecordavailableathttps://lccn.loc.gov/2017046893
Volume2ISBN:978-1-4704-6797-5
Volume2ISBN(Electronic): 978-4704-7129-3
Volume3ISBN:978-1-4704-6798-2
Volume3ISBN(Electronic): 978-1-4704-7130-9
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Contents
ListofFigures ix
PrefaceforVolumesIIandIII xi
Acknowledgments xvii
Chapter6. TheClassificationofVarieties 1
6.1. Introduction 1
6.2. PermutabilityofCongruences 13
6.3. GeneratingCongruenceRelations 22
6.4. CongruenceSemidistributiveVarieties 26
6.5. CongruenceModularity 40
6.6. CongruenceRegularityandUniformity 50
6.7. LinearMal’tsevConditions,Derivations,StrongMal’tsevConditions 62
6.8. TaylorClassesofVarieties 81
6.9. CongruenceIdentities 95
6.10. Relationships 119
6.11. Hamiltonian,Semidegenerate,andAbelianVarieties 134
Chapter7. EquationalLogic 145
7.1. TheSetUp 145
7.2. TheDescriptionof ΘModΣ: TheCompletenessTheorem 151
7.3. EquationalTheoriesthatarenotFinitelyAxiomatizable 169
7.4. EquationalTheoriesthatareFinitelyAxiomatizable 196
7.5. FirstInterlude: AlgebraicLatticesasCongruenceLattices 211
7.6. TheLatticeofEquationalTheories 225
7.7. SecondInterlude: theRudimentsofComputability 251
7.8. UndecidabilityinEquationalLogic 254
7.9. ThirdInterlude: ResidualBounds 277
7.10. AFiniteAlgebraofResidualCharacterℵ 281
1
7.11. UndecidablePropertiesofFiniteAlgebras 294
Chapter8. RudimentsofModelTheory 313
8.1. TheFormalismofElementaryLogic 313
8.2. UltraproductsandtheCompactnessTheorem 327
8.3. ApplicationsoftheCompactnessTheorem 336
8.4. Jónsson’sLemmaforCongruenceDistributiveVarieties 355
8.5. DefinableCongruencesandBaker’sFiniteBasisTheorem 373
8.6. BooleanPowers 377
vii
viii CONTENTS
8.7. UniversalClassesandQuasivarieties 396
8.8. SentencesTrueinReducedProducts 410
8.9. ElementaryChains,Amalgamation,andInterpolation 421
8.10. SentencesPreservedunderHomomorphicImages 435
8.11. SentencesPreservedunderSubdirectProducts 438
Bibliography 443
Index 467
List of Figures
6.1 ThePermutabilityParallelogram 5
6.2 Thetermtreefor 𝐻(𝐵(𝑥,𝐵(𝑥,𝑦)),𝑧,𝐵(𝑦,𝐵(𝑥,𝑦))) 6
6.3 Adigraphrepresentationfor 𝐻(𝐵(𝑥,𝐵(𝑥,𝑦)),𝑧,𝐵(𝑦,𝐵(𝑥,𝑦))) 7
6.4 Adigraphrepresentationfor𝑡 7
3
6.5 TheCongruenceLatticeofPolin’sAlgebra 16
6.6 Twoslendertermsforrepresentingtranslations 23
6.7 TheIntervalabovetheMeetoftwoCoatomsof𝐂𝐨𝐧𝐒. 35
6.8 Day’sPentagoninCon 𝐅 (𝑥,𝑦,𝑧,𝑢) 42
𝒱
6.9 Gumm(Modular)DifferenceTerm 46
6.10 Kiss’s4-VariableDifferenceTerm 47
6.11 TheShiftingLemma 49
6.12 TheLattice𝐋 97
14
6.13 ThePentagon,𝐍 ,Labeled. 98
5
6.14 TheLattice𝐋 ,Labeled 98
14
6.15 Con(𝐏× 𝐏) 99
𝛽
6.16 Desargues’Configurationin3Dimensions 107
6.17 TheCongruenceLatticeof𝐏× 𝐏 115
𝛾
6.18 𝐂𝐨𝐧(𝐅 (1)) 116
𝒫
6.19 RelationshipsbetweenClassesofVarieties 120
6.20 ASubalgebraof𝐀 ×𝐀 127
0 1
7.1 TheGaloisConnectionUnderlyingEquationalLogic 148
7.2 Lyndon’sAlgebra𝐋 172
7.3 Lee’sAlgebra𝐋∗ 173
7.4 The𝜏-DecomposableSubalgebra 183
7.5 TheEvaluationTreeofthePolynomial𝑔(𝑥) 184
7.6 McKenzie’sautomaticalgebra𝐑 191
7.7 Murskiı̌’salgebra𝐌 192
7.8 Anillustrationof𝜋 (𝑥 ,𝑦 ,𝑥 ,𝑦 ) 205
𝑚 0 0 1 1
7.9 ProofoftheDefiniteAtomsProperty 208
7.10 Analgebraiclatticewithapinchpoint 224
ix