
At a Glance
608 Pages
Revised
23.39 x 15.6 x 3.12
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Universal Algebra heralded as ". . . the standard reference in a field notorious for the lack of standardization . . .", has become the most authoritative, consistently relied on text in a field with applications in other branches of algebra and other fields such as combinatorics, geometry, and computer science.
Each chapter is followed by an extensive list of exercises and problems. The "state of the art" account also includes new appendices (with contributions from B. J nsson, R. Quackenbush, W. Taylor, and G. Wenzel) and a well selected additional bibliography of over 1250 papers and books which makes this an indispensable new edition for students, faculty, and workers in the field.
"This book will certainly be, in the years to come, the basic reference to the subject." The American Mathematical Monthly (First Edition)
"In this reviewer's opinion the author] has more than succeeded in his aim. The problems at the end of each chapter are well-chosen; there are more than 650 of them. The book is especially suitable for self-study, as the author frequently provides ample explanation not only of what he is proving, but also of how and why he is proving it. As a reference work for the specialist or a text for the student, the book is highly recommended." Mathematical Reviews (First Edition)
"Since the first day of its appearance in 1968, this book has been the standard reference in universal algebra, and no book since has reached its quality." Journal of Symbolic Logic (Second Edition)
| Table of Notation | p. xvii |
| Basic Concepts | |
| Sets and Relations | p. 1 |
| Mappings and Operations | p. 3 |
| Algebras and Relational Systems | p. 7 |
| Partially Ordered Sets | p. 10 |
| Structure of Mappings and Equivalence Relations | p. 17 |
| Ideals and Semilattices | p. 19 |
| Exercises | p. 27 |
| Subalgebras and Homomorphisms | |
| Basic Concepts | p. 33 |
| Polynomial Symbols and Polynomial Algebras | p. 37 |
| Structure of Subalgebras | p. 45 |
| Structure of Congruence Relations | p. 50 |
| The Homomorphism Theorem and Some Isomorphism Theorems | p. 57 |
| Homomorphisms | p. 62 |
| Exercises | p. 70 |
| Problems | p. 76 |
| Partial Algebras | |
| Basic Notions | p. 79 |
| Polynomial Symbols over a Partial Algebra | p. 84 |
| Extension of Congruence Relations | p. 91 |
| Subalgebras and Homomorphisms of Partial Algebras | p. 96 |
| The Characterization Theorem of Congruence Lattices: Preliminary Considerations | p. 100 |
| The Characterization Theorem of Congruence Lattices | p. 109 |
| Exercises | p. 113 |
| Problems | p. 116 |
| Constructions of Algebras | |
| Direct Products | p. 118 |
| Subdirect Products of Algebras | p. 122 |
| Direct and Inverse Limits of Algebras | p. 128 |
| Products Associated with the Direct Product | p. 139 |
| Operators on Classes of Algebras | p. 152 |
| Exercises | p. 153 |
| Problems | p. 160 |
| Free Algebras | |
| Definition and Basic Properties | p. 162 |
| Construction of Free Algebras | p. 166 |
| Identities and Free Algebras | p. 169 |
| Equational Completeness and Identities of Finite Algebras | p. 173 |
| Free Algebras Generated by Partial Algebras | p. 180 |
| Free Products of Algebras | p. 183 |
| Word Problem | p. 186 |
| Exercises | p. 189 |
| Problems | p. 194 |
| Independence | |
| Independence and Bases | p. 196 |
| Independence in Special Classes of Algebras | p. 201 |
| Some Invariants of Finite Algebras | p. 205 |
| The System of Independent Sets of an Algebra | p. 209 |
| Generalizations of the Notion of Independence | p. 212 |
| Exercises | p. 216 |
| Problems | p. 222 |
| Elements of Model Theory | |
| Structures and the First Order Logic | p. 223 |
| Satisfiability and the Case of Boolean Set Algebras | p. 227 |
| Elementary Equivalence and Elementary Extensions | p. 234 |
| Prime Products | p. 239 |
| Prime Powers | p. 246 |
| Two Algebraic Characterizations of Elementary Equivalence | p. 248 |
| Elementary and Axiomatic Classes | p. 255 |
| Exercises | p. 262 |
| Problems | p. 269 |
| Elementary Properties of Algebraic Constructions | |
| Extensions and Substructures | p. 270 |
| Generalized Atomic Sets of Formulas | p. 275 |
| Chain Unions and Homomorphisms | p. 278 |
| Direct Products | p. 283 |
| Direct Products (Continued) | p. 287 |
| Subdirect Products | p. 293 |
| Exercises | p. 295 |
| Problems | p. 299 |
| Free [Sigma]-Structures | |
| [Sigma]-Inverses and [Sigma]-Substructures | p. 300 |
| [Sigma]-Homomorphisms and Slender [Sigma]-Substructures | p. 305 |
| Free [Sigma]-Structures and the Uniqueness Theorem | p. 307 |
| On the Family of Free [Sigma]-Structures | p. 311 |
| On the Existence of Free [Sigma]-Structures | p. 315 |
| Strong Free [Sigma]-Structures and the Inverse Preserving Property | p. 322 |
| Exercises | p. 326 |
| Problems | p. 328 |
| General Survey | |
| A Survey by Sections | p. 331 |
| Related Structures | p. 335 |
| Miscellany | p. 338 |
| The Problems | |
| Solutions and Partial Solutions of 46 Problems | p. 342 |
| Congruence Varieties by Bjarni Jonsson | |
| Algebras and Their Congruence Lattices | p. 348 |
| Mal'cev Classes | p. 351 |
| Congruence Varieties | p. 360 |
| Congruence Distributivity and Finite Bases | p. 373 |
| Equational Logic by Walter Taylor | |
| Equationally Defined Classes | p. 378 |
| Equational Theories | p. 380 |
| Equivalent Varieties | p. 381 |
| Bases and Generic Algebras | p. 383 |
| Finitely Based Theories | p. 385 |
| One-Based Theories | p. 387 |
| Irredundant Bases | p. 389 |
| The Lattice of Equational Theories | p. 389 |
| Some Further Invariants of the Equivalence Class of a Variety | p. 392 |
| Mal'cev Conditions and Congruence Identities | p. 398 |
| Primality: The Influence of Boolean Algebras in Universal Algebra | |
| Introduction | p. 401 |
| Primal Algebras | p. 402 |
| Quasi-Primal Algebras | p. 403 |
| Arithmetical Algebras | p. 405 |
| Para-Primal Algebras | p. 406 |
| Dual-Discriminator Algebras | p. 408 |
| Functional Completeness | p. 409 |
| Representation Theory | p. 410 |
| Congruences | p. 411 |
| Injectivity and Projectivity | p. 412 |
| Further References and Comments | p. 415 |
| Equational Compactness | |
| Equational and Atomic Compactness-First Examples | p. 417 |
| Related Compactness Concepts and Characterizations | p. 420 |
| Connections with [vertical bar]A[vertical bar] | p. 425 |
| The Mycielski Question: Chromatic Numbers and Topology | p. 428 |
| Minimum Compactness | p. 432 |
| Compactification of Algebras | p. 435 |
| Application to Equational Classes of Algebras | p. 441 |
| Concluding Remarks | p. 444 |
| Some Problems | p. 446 |
| The Independence Proof | |
| Statement of the Main Results | p. 448 |
| Preliminaries | p. 449 |
| Cls-Expansions and Free Extensions | p. 451 |
| Cls-Expansions with Two Orbits | p. 457 |
| Three More Constructions | p. 469 |
| Proof of the Main Theorem | p. 471 |
| Bibliography | p. 475 |
| Additional Bibliography | p. 505 |
| Index | p. 565 |
| Epilogue | p. 583 |
| Table of Contents provided by Ingram. All Rights Reserved. |
ISBN: 9780387774862
ISBN-10: 0387774866
Published: 1st July 2008
Format: Paperback
Language: English
Number of Pages: 608
Audience: College, Tertiary and University
Publisher: Springer Nature B.V.
Country of Publication: GB
Edition Number: 2
Edition Type: Revised
Dimensions (cm): 23.39 x 15.6 x 3.12
Weight (kg): 0.86
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