Universal Algebra

Universal Algebra

by Clifford Bergman
Released September 2011
Publisher(s): Chapman and Hall/CRC
ISBN: 9781439851302

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Book description

Starting with the most basic notions, Universal Algebra: Fundamentals and Selected Topics introduces all the key elements needed to read and understand current research in this field. Based on the author's two-semester course, the text prepares students for research work by providing a solid grounding in the fundamental constructions and concepts o

Table of contents

  1. Cover
  2. Title Page
  3. Series Page
  4. Copyright Page
  5. Dedication Page
  6. Table of Contents
  7. Preface
  8. I Fundamentals of Universal Algebra
    1. 1 Algebras
      1. 1.1 Operations
      2. 1.2 Examples
      3. 1.3 More about subs, homs and prods
      4. 1.4 Generating subalgebras
      5. 1.5 Congruences and quotient algebras
    2. 2 Lattices
      1. 2.1 Ordered sets
      2. 2.2 Distributive and modular lattices
      3. 2.3 Complete lattices
      4. 2.4 Closure operators and algebraic lattices
      5. 2.5 Galois connections
      6. 2.6 Ideals in lattices
    3. 3 The Nuts and Bolts of Universal Algebra
      1. 3.1 The isomorphism theorems
      2. 3.2 Direct products
      3. 3.3 Subdirect products
      4. 3.4 Case studies
      5. 3.5 Varieties and other classes of algebras
    4. 4 Clones, Terms, and Equational Classes
      1. 4.1 Clones
      2. 4.2 Invariant relations
      3. 4.3 Terms and free algebras
      4. 4.4 Identities and Birkhoff’s theorem
      5. 4.5 The lattice of subvarieties
      6. 4.6 Equational theories and fully invariant congruences
      7. 4.7 Maltsev conditions
      8. 4.8 Interpretations
  9. II Selected Topics
    1. 5 Congruence Distributive Varieties
      1. 5.1 Ultrafilters and ultraproducts
      2. 5.2 Jónsson’s lemma
      3. 5.3 Model theory
      4. 5.4 Finitely based and nonfinitely based algebras
      5. 5.5 Definable principal (sub)congruences
    2. 6 Arithmetical Varieties
      1. 6.1 Large clones
      2. 6.2 How rare are primal algebras?
    3. 7 Maltsev Varieties
      1. 7.1 Directly representable varieties
      2. 7.2 The centralizer congruence
      3. 7.3 Abelian varieties
      4. 7.4 Commutators
      5. 7.5 Directly representable varieties revisited
      6. 7.6 Minimal varieties
      7. 7.7 Functionally complete algebras
    4. 8 Finite Algebras and Locally Finite Varieties
      1. 8.1 Minimal algebras
      2. 8.2 Localization and induced algebras
      3. 8.3 Centralizers again!
      4. 8.4 Applications
  10. Bibliography
  11. Index of Notation
  12. Index

Product information

  • Title: Universal Algebra
  • Author(s): Clifford Bergman
  • Release date: September 2011
  • Publisher(s): Chapman and Hall/CRC
  • ISBN: 9781439851302