Preface ........................................................ ix
I Fundamentals of Universal Algebra ........................... 1
1 Algebras ..................................................... 3
1.1 Operations .............................................. 3
1.2 Examples ................................................ 4
1.3 More about subs, homs and prods ......................... 7
1.4 Generating subalgebras ................................. 10
1.5 Congruences and quotient algebras ...................... 13
2 Lattices .................................................... 21
2.1 Ordered sets ........................................... 21
2.2 Distributive and modular lattices ...................... 24
2.3 Complete lattices ...................................... 30
2.4 Closure operators and algebraic lattices ............... 34
2.5 Galois connections ..................................... 38
2.6 Ideals in lattices ..................................... 40
3 The Nuts and Bolts of Universal Algebra ..................... 47
3.1 The isomorphism theorems ............................... 47
3.2 Direct products ........................................ 52
3.3 Subdirect products ..................................... 55
3.4 Case studies ........................................... 60
3.5 Varieties and other classes of algebras ................ 71
4 Clones, Terms, and Equational Classes ....................... 79
4.1 Clones ................................................. 79
4.2 Invariant relations .................................... 88
4.3 Terms and free algebras ................................ 94
4.4 Identities and Birkhoff's theorem ..................... 104
4.5 The lattice of subvarieties ........................... 111
4.6 Equational theories and fully invariant congruences ... 117
4.7 Maltsev conditions .................................... 121
4.8 Interpretations ....................................... 130
II Selected Topics ........................................... 135
5 Congruence Distributive Varieties .......................... 139
5.1 Ultrafilters and ultraproducts ........................ 139
5.2 Jonsson's lemma ....................................... 145
5.3 Model theory .......................................... 149
5.4 Finitely based and nonfinitely based algebras ......... 156
5.5 Definable principal (sub)congruences .................. 160
6 Arithmetical Varieties ..................................... 169
6.1 Large clones .......................................... 169
6.2 How rare are primal algebras? ......................... 178
7 Maltsev Varieties .......................................... 189
7.1 Directly representable varieties ...................... 189
7.2 The centralizer congruence ............................ 197
7.3 Abelian varieties ..................................... 205
7.4 Commutators ........................................... 216
7.5 Directly representable varieties revisited ............ 224
7.6 Minimal varieties ..................................... 233
7.7 Functionally complete algebras ........................ 239
8 Finite Algebras and Locally Finite Varieties ............... 245
8.1 Minimal algebras ...................................... 245
8.2 Localization and induced algebras ..................... 252
8.3 Centralizers again! ................................... 263
8.4 Applications .......................................... 274
Bibliography .................................................. 291
Index of Notation ............................................. 299
Index ......................................................... 303
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