Universal Algebra

ISBN-13: 9789027712547 / Angielski / Miękka / 1981 / 412 str.

Paul M. Cohn; P. M. Cohn; P. M. Cohn

Opis

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 entir~ly 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 extensive comments. 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.

Spis treści

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.