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Contents
Special Notation xv
Preliminaries 1
I Lattices 5
§1. Definitions of Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
§2. Isomorphic Lattices, and Sublattices . . . . . . . . . . . . . . . . . . . . . . 10
§3. Distributive and Modular Lattices . . . . . . . . . . . . . . . . . . . . . . . . 12
§4. Complete Lattices, Equivalence Relations, and Algebraic Lattices . . . . . . 17
§5. Closure Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
II The Elements of Universal Algebra 25
§1. Definition and Examples of Algebras . . . . . . . . . . . . . . . . . . . . . . 25
§2. Isomorphic Algebras, and Subalgebras . . . . . . . . . . . . . . . . . . . . . 31
§3. Algebraic Lattices and Subuniverses . . . . . . . . . . . . . . . . . . . . . . . 33
§4. The Irredundant Basis Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 35
§5. Congruences and Quotient Algebras . . . . . . . . . . . . . . . . . . . . . . . 38
§6. Homomorphisms and the Homomorphism and Isomorphism Theorems . . . . 47
§7. Direct Products, Factor Congruences, and Directly Indecomposable Algebras 55
§8. Subdirect Products, Subdirectly Irreducible Algebras, and Simple Algebras . 62
§9. Class Operators and Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . 66
§10. Terms, Term Algebras, and Free Algebras . . . . . . . . . . . . . . . . . . . 68
§11. Identities, Free Algebras, and Birkhoff’s Theorem . . . . . . . . . . . . . . . 77
§12. Mal’cev Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
§13. The Center of an Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
§14. Equational Logic and Fully Invariant Congruences . . . . . . . . . . . . . . . 99
III Selected Topics 111
§1. Steiner Triple Systems, Squags, and Sloops . . . . . . . . . . . . . . . . . . . 111
§2. Quasigroups, Loops, and Latin Squares . . . . . . . . . . . . . . . . . . . . . 114
§3. Orthogonal Latin Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
xiiixiv Contents
§4. Finite State Acceptors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
IV Starting from Boolean Algebras . . . 129
§1. Boolean Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
§2. Boolean Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
§3. Filters and Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
§4. Stone Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
§5. Boolean Powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
§6. Ultraproducts and Congruence-distributive Varieties . . . . . . . . . . . . . . 163
§7. Primal Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
§8. Boolean Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
§9. Discriminator Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
§10. Quasiprimal Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
§11. Functionally Complete Algebras and Skew-free Algebras . . . . . . . . . . . 199
§12. Semisimple Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
§13. Directly Representable Varieties . . . . . . . . . . . . . . . . . . . . . . . . . 212
V Connections with Model Theory 217
§1. First-order Languages, First-order Structures, and Satisfaction . . . . . . . . 217
§2. Reduced Products and Ultraproducts . . . . . . . . . . . . . . . . . . . . . . 234
§3. Principal Congruence Formulas . . . . . . . . . . . . . . . . . . . . . . . . . 252
§4. Three Finite Basis Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
§5. Semantic Embeddings and Undecidability . . . . . . . . . . . . . . . . . . . 271
Recent Developments and Open Problems 283
§1. The Commutator and the Center . . . . . . . . . . . . . . . . . . . . . . . . 283
§2. The Classification of Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . 284
§3. Decidability Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
§4. Boolean Constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
§5. Structure Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
§6. Applications to Computer Science . . . . . . . . . . . . . . . . . . . . . . . . 289
§7. Applications to Model Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 289
§8. Finite Basis Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290
§9. Subdirectly Irreducible Algebras . . . . . . . . . . . . . . . . . . . . . . . . . 290
Bibliography 291
§1. Books and Survey Articles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
§2. Research Papers and Monographs . . . . . . . . . . . . . . . . . . . . . . . . 293
Author Index 303
Subject Index 306

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