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E-raamat: Algebras, Lattices, Varieties

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This book is the second of a three-volume set of books on the theory of algebras, a study that provides a consistent framework for understanding algebraic systems, including groups, rings, modules, semigroups and lattices. Volume I, first published in the 1980s, built the foundations of the theory and is considered to be a classic in this field.

The long-awaited volumes II and III are now available. Taken together, the three volumes provide a comprehensive picture of the state of art in general algebra today, and serve as a valuable resource for anyone working in the general theory of algebraic systems or in related fields.

The two new volumes are arranged around six themes first introduced in Volume I. Volume II covers the Classification of Varieties, Equational Logic, and Rudiments of Model Theory, and Volume III covers Finite Algebras and their Clones, Abstract Clone Theory, and the Commutator. These topics are presented in six chapters with independent expositions, but are linked by themes and motifs that run through all three volumes.
List of Figures
ix
Preface for Volumes II and III xi
Acknowledgments xvii
Chapter 6 The Classification of Varieties
1(144)
6.1 Introduction
1(12)
6.2 Permutability of Congruences
13(9)
6.3 Generating Congruence Relations
22(4)
6.4 Congruence Semidistributive Varieties
26(14)
6.5 Congruence Modularity
40(10)
6.6 Congruence Regularity and Uniformity
50(12)
6.7 Linear Mal'tsev Conditions, Derivations, Strong Mal'tsev Conditions
62(19)
6.8 Taylor Classes of Varieties
81(14)
6.9 Congruence Identities
95(24)
6.10 Relationships
119(15)
6.11 Hamiltonian, Semidegenerate, and Abelian Varieties
134(11)
Chapter 7 Equational Logic
145(168)
7.1 The Set Up
145(6)
7.2 The Description of 6 Mod 2: The Completeness Theorem
151(18)
7.3 Equational Theories that are not Finitely Axiomatizable
169(27)
7.4 Equational Theories that are Finitely Axiomatizable
196(15)
7.5 First Interlude: Algebraic Lattices as Congruence Lattices
211(14)
7.6 The Lattice of Equational Theories
225(26)
7.7 Second Interlude: the Rudiments of Computability
251(3)
7.8 Undecidability in Equational Logic
254(23)
7.9 Third Interlude: Residual Bounds
277(4)
7.10 A Finite Algebra of Residual Character
281(13)
7.11 Undecidable Properties of Finite Algebras
294(19)
Chapter 8 Rudiments of Model Theory
313(130)
8.1 The Formalism of Elementary Logic
313(14)
8.2 Ultraproducls and the Compactness Theorem
327(9)
8.3 Applications of the Compactness Theorem
336(19)
8.4 Jonsson's Lemma for Congruence Distributive Varieties
355(18)
8.5 Definable Congruences and Baker's Finite Basis Theorem
373(4)
8.6 Boolean Powers
377(19)
8.7 Universal Classes and Quasivarieties
396(14)
8.8 Sentences True in Reduced Products
410(11)
8.9 Elementary Chains, Amalgamation, and Interpolation
421(14)
8.10 Sentences Preserved under Homomorphic Images
435(3)
8.11 Sentences Preserved under Subdirect Products
438(5)
Bibliography 443(24)
Index 467
Ralph S. Freese, University of Hawaii, Honolulu, HI.

Ralph N. McKenzie, Vanderbilt Univesity, Nashville, TN.

George F. McNulty, University of South Carolina, Columbia, SC.

Walter F. Taylor, University of Colorado, Boulder, CO.