Universal Algebra / Edition 1

by P.M. Cohn

Overview

The present book was conceived as an introduction for the user of universal algebra, rather than a handbook for the specialist, but when the first edition appeared in 1965, there were practically no other books entirely devoted to the subject, whether introductory or specialized. Today the specialist in the field is well provided for, but there is still a demand for an introduction to the subject to suit the user, and this seemed to justify a reissue of the book. Naturally some changes have had to be made; in particular, I have corrected all errors that have been brought to my notice. Besides errors, some obscurities in the text have been removed and the references brought up to date. I should like to express my thanks to a number of correspondents for their help, in particular C. G. d'Ambly, W. Felscher, P. Goralcik, P. J. Higgins, H.-J. Hoehnke, J. R. Isbell, A. H. Kruse, E. J. Peake, D. Suter, J. S. Wilson. But lowe a special debt to G. M. Bergman, who has provided me with extensivecomments. particularly on Chapter VII and the supplementary chapters. I have also con­ sulted reviews of the first edition, as well as the Italian and Russian translations.

Product Details

ISBN-13:	9789027712134
Publisher:	Springer Netherlands
Publication date:	04/30/1981
Series:	Mathematics and Its Applications , #6
Edition description:	1981
Pages:	412
Product dimensions:	6.10(w) x 9.25(h) x 0.04(d)

Table of Contents

I: Sets and Mappings.- 1. The Axioms of Set Theory.- 2. Correspondences.- 3. Mappings and Quotient Sets.- 4. Ordered Sets.- 5. Cardinals and Ordinals.- 6. Categories and Functors.- II: Algebraic Structures.- 1. Closure Systems.- 2.—-Algebras.- 3. The Isomorphism Theorems.- 4. Lattices.- 5. The Lattice of Subalgebras.- 6. The Lattice of Congruences.- 7. Local and Residual Properties.- 8. The Lattice of Categories of—-Algebras.- III. Free Algebras.- 1. Universal Functors.- 2.—-Word Algebras.- 3. Clones of Operations.- 4. Representations in Categories of—-Algebras.- 5. Free Algebras in Categories of—-Algebras.- 6. Free and Direct Composition of—-Algebras.- 7. Derived Operators.- 8. Presentations of—-Algebras.- 9. The Word Problem.- IV. Varieties.- 1. Definition and Basic Properties.- 2. Free Groups and Free Rings.- 3. The Generation of Varieties.- 4. Representations in Varieties of Algebras.- V. Relational Structures and Models.- 1. Relational Structures over a Predicate Domain.- 2. Boolean Algebras.- 3. Derived Predicates.- 4. Closed Sentence Classes and Axiomatic Model Classes.- 5. Ultraproducts and the Compactness Theorem.- 6. The Model Space.- VI. Axiomatic Model Classes.- 1. Reducts and Enlargements.- 2. The Local Determination of Classes.- 3. Elementary Extensions.- 4. p-Closed Classes and Quasivarieties.- 5. Classes Admitting Homomorphic Images.- 6. The Characterization of Axiomatic Model Classes.- VII. Applications.- 1. The Natural Numbers.- 2. Abstract Dependence Relations.- 3. The Division Problem for Semigroups and Rings.- 4. The Division Problem for Groupoids.- 5. Linear Algebras.- 6. Lie Algebras.- 7. Jordan Algebras.- Foreword to the Supplements.- VIII. Category Theory and Universal Algebra.- 1. The Principle of Duality.- 2. Adjoint Pairs of Functors.- 3. Monads.- 4. Algebraic Theories.- IX. Model Theory and Universal Algebra.- 1. Inductive Theories.- 2. Complete Theories and Model Complete Theories.- 3. Model Completions.- 4. The Forcing Companion.- 5. The Model Companion.- 6. Examples.- X. Miscellaneous Further Results.- 1. Subdirect Products and Pullbacks.- 2. The Reduction to Binary Operations.- 3. Invariance of the Rank of Free Algebras.- 4. The Diamond Lemma for Rings.- 5. The Embedding of Rings in Skew Fields.- XI. Algebra and Language Theory.- 1. Introduction.- 2. Grammars.- 3. Machines.- 4. Transductions.- 5. Monoids.- 6. Power Series.- 7. Transformational Grammars.- Bibliography and Name Index.- List of Special Symbols.