Starting with the most basic notions, this text introduces all the key elements needed to read and understand current research in the field. The first part of the book focuses on core components, including subalgebras, congruences, lattices, direct and subdirect products, isomorphism theorems, clones, and free algebras. The second part covers topics that demonstrate the power and breadth of the subject, such as Jónsson’s lemma, finitely and nonfinitely based algebras, primal and quasiprimal algebras, Murskiĭ’s theorem, and directly representable varieties. Examples and exercises are included throughout the text.
Author(s): Clifford Bergman
Publisher: Chapman and Hall/CRC
Year: 2011
Language: English
Pages: 320
Tags: Математика;Общая алгебра;Универсальная алгебра;
Preface ......Page 11
I Fundamentals of Universal Algebra ......Page 15
1.1 Operations ......Page 17
1.2 Examples ......Page 18
1.3 More about subs, horns and prods ......Page 21
1.4 Generating subalgebras ......Page 24
1.5 Congruences and quotient algebras ......Page 27
2.1 Ordered sets ......Page 35
2.2 Distributive and modular lattices ......Page 38
2.3 Complete lattices ......Page 44
2.4 Closure operators and algebraic lattices ......Page 48
2.5 Galois connections ......Page 52
2.6 Ideals in lattices ......Page 54
3.1 The isomorphism theorems ......Page 61
3.2 Direct products ......Page 66
3.3 Subdirect products ......Page 69
3.4 Case studies ......Page 74
3.5 Varieties and other classes of algebras ......Page 85
4.1 Clones ......Page 93
4.2 Invariant relations ......Page 102
4.3 Terms and free algebras ......Page 108
4.4 Identities and Birkhoff’s theorem ......Page 118
4.5 The lattice of subvarieties ......Page 125
4.6 Equational theories and fully invariant congruences ......Page 131
4.7 Maltsev conditions ......Page 135
4.8 Interpretations ......Page 144
II Selected Topics ......Page 149
5.1 Ultrafilters and ultraproducts ......Page 153
5.2 Jonsson’s lemma ......Page 159
5.3 Model theory ......Page 163
5.4 Finitely based and nonfinitely based algebras ......Page 170
5.5 Definable principal (sub)congruences ......Page 174
6.1 Large clones ......Page 183
6.2 How rare are primal algebras? ......Page 192
7.1 Directly representable varieties ......Page 203
7.2 The centralizer congruence ......Page 211
7.3 Abelian varieties ......Page 219
7.4 Commutators ......Page 230
7.5 Directly representable varieties revisited ......Page 238
7.6 Minimal varieties ......Page 247
7.7 Functionally complete algebras ......Page 253
8.1 Minimal algebras ......Page 259
8.2 Localization and induced algebras ......Page 266
8.3 Centralizers again! ......Page 277
8.4 Applications ......Page 288
Bibliography ......Page 305
