A Course in Universal Algebra

Front Cover......Page 1
Preface......Page 5
Table of Contents......Page 13
Special Notation......Page 15
Preliminaries......Page 17
1. Definitions of Lattices......Page 21
2. Isomorphic Lattices, and Sublattices......Page 26
3. Distributive and Modular Lattices......Page 28
4. Complete Lattices, Equivalence Relations, and Algebraic Lattices......Page 33
5. Closure Operators......Page 36
1. Definition and Examples of Algebras......Page 41
2. Isomorphic Algebras, and Subalgebras......Page 47
3. Algebraic Lattices and Subuniverses......Page 49
4. The Irredundant Basis Theorem......Page 51
5. Congruences and Quotient Algebras......Page 54
6. Homomorphisms and the Homomorphism and Isomorphism Theorems......Page 63
7. Direct Products, Factor Congruences, and Directly Indecomposable Algebras......Page 71
8. Subdirect Products, Subdirectly Irreducible Algebras, and Simple Algebras......Page 78
9. Class Operators and Varieties......Page 82
10. Terms, Term Algebras, and Free Algebras......Page 84
11. Identities, Free Algebras, and Birkhoff's Theorem......Page 93
12. Mal'cev Conditions......Page 101
13. The Center of an Algebra......Page 107
14. Equational Logic and Fully Invariant Congruences......Page 115
1. Steiner Triple Systems, Squags, and Sloops......Page 127
2. Quasigroups, Loops, and Latin Squares......Page 130
3. Orthogonal Latin Squares......Page 131
4. Finite State Acceptors......Page 135
1. Boolean Algebras......Page 145
2. Boolean Rings......Page 152
3. Filters and Ideals......Page 158
4. Stone Duality......Page 168
5. Boolean Powers......Page 175
6. Ultraproducts and Congruence-distributive Varieties......Page 179
7. Primal Algebras......Page 185
8. Boolean Products......Page 190
9. Discriminator Varieties......Page 202
10. Quasiprimal Algebras......Page 207
11. Functionally Complete Algebras and Skew-free Algebras......Page 215
12. Semisimple Varieties......Page 223
13. Directly Representable Varieties......Page 228
1. First-order Languages, First-order Structures, and Satisfaction......Page 233
2. Reduced Products and Ultraproducts......Page 250
3. Principal Congruence Formulas......Page 268
4. Three Finite Basis Theorems......Page 275
5. Semantic Embeddings and Undecidability......Page 287
1. The Commutator and the Center......Page 299
2. The Classification of Varieties......Page 300
3. Decidability Questions......Page 301
4. Boolean Constructions......Page 303
5. Structure Theory......Page 304
7. Applications to Model Theory......Page 305
9. Subdirectly Irreducible Algebras......Page 306
1. Books and Survey Articles......Page 307
2. Research Papers and Monographs......Page 309
Author Index......Page 319
Subject Index......Page 322