Metamathematics of Algebraic Systems (eBook)

Front Cover 1
The Metamathematics of Algebraic Systems 4
Copyright Page 5
Contents 14
Translator’s foreword 8
Biographical note 12
Chapter 1. Investigations in the realm of mathematical logic 20
Chapter 2. A general method for obtaining local theorems in group theory 34
Chapter 3. Representations of models 41
Chapter 4. Quasiprimitive classes of abstract algebras 46
Chapter 5. Subdirect products of models 51
Chapter 6. Derived operations and predicates 56
Chapter 7. Classes of models with an operation of generation 63
Chapter 8. Defining relations in categories 70
Chapter 9. The structural characterization of certain classes of algebras 75
Chapter 10. Certain classes of models 80
Chapter 11. Model correspondences 85
Introduction 85
§1. Multibase models 86
§2. Fundamental properties of projective correspondences 93
§3. Quasiuniversal subclasses 99
Chapter 12. Regular products of models 114
Introduction 114
§1. Splitting correspondences 115
§2. Regular products 120
Chapter 13. Small models 133
Chapter 14. Free solvable groups 138
Chapter 15. A correspondence between rings and groups 143
§1. The direct mapping 144
§2. Groups with distinguished elements 145
§3. The inverse mapping 146
§4. The reciprocity of the correspondences s and t 148
§5. Some special cases 150
§6. Reductions and interpretations of classes of models 152
§7. The undecidability of sundry classes of metabelian groups 154
§8. Nilpotent groups 155
Chapter 16. The undecidability of the elementary theories of certain fields 157
§1. The field of rational functions 158
§2. Fields of formal power series 161
Chapter 17. A remark concerning “The undecidability of the elementary theories of certain fields” [XVI] 166
Chapter 18. Constructive algebras. I 167
Introduction 167
§1. Algebraic systems 170
§2. Numbered sets 184
§3. Numbered algebraic systems 206
§4. Finitely generated algebras 220
Chapter 19. The undecidability of the elementary theory of finite groups 234
Chapter 20. Elementary properties of linear groups 240
Introduction 240
§1. The elementary nature of the Segre characteristic 242
§2. Elementary (arithmetic) types of linear and projective groups 257
Chapter 21. The effective inseparability of the set of valid sentences from the set of finitely refutable sentences in several elementary theories 267
Chapter 22. Closely related models and recursively perfect algebras 274
§1. Closely related models 274
§2. Recursively perfect algebras 275
§3. Linear groups 277
Chapter 23. Axiomatizable classes of locally free algebras of various types 281
§1. Locally absolutely free algebras 281
§2. Ordered groupoids 284
§3. G -algebras 285
§4. Special formulas 286
§5 . Standard formulas 290
§6. The reduction of negations of standard formulas 292
§7. The reduction of closed formulas 295
Chapter 24. Recursive abelian groups 301
Chapter 25. Sets with complete numberings 306
§1. Complete numberings 307
§2. Isomorphism. Factor numberings 310
§3. Enumerable families of elements 312
§4. Completely numbered sets whose every family of nonspecial elements is totally enumerable 314
§5. Universal series of sets 317
§6. Totally enumerable families of partial recursive functions 319
§7. Projective families of functions. Computable numberings 324
§8. Quasiordered families 327
§9. Intrinsically productive families 328
Chapter 26. Problems in the theory of classes of models 332
Introduction 332
§1. Fundamental concepts 332
§2. Axiomatizable classes of models 338
§3. Some special axiomatizable classes 346
§4. Ultraproducts 354
§5. A few second-order classes of models 361
Chapter 27. Toward a theory of computable families of objects 372
§1. principal and complete numberings 372
§2. The a-order and a-topology 375
§3. Normal and subnormal numberings 378
§4. Effectively principal numberings 382
§5. Standard families and precomplete numberings 384
§6. Special and subspecial numberings 388
§7. Totally enumerable families 393
Chapter 28. Positive and negative numberings 398
Chapter 29. Identical relations in varieties of quasigroups 403
§1. The problem of identical relations 403
§2. Algebras with unary operations 405
§3. Partial quasigroups 409
§4. Varieties of quasigroups 411
Chapter 30. Iterative algebras and Post varieties 415
§1. Iterative algebras 417
§2. Iterative algebras of partial functions 418
§3. Congruences on pi and Eliw 419
§4. Automorphisms 423
§5. Representations of iterative algebras 426
§6. Post variables 427
§7. Selector representations of pre-iterative algebras 430
§8. Subalgebras 432
Chapter 31. A few remarks on quasivarieties of algebraic systems 435
§1. Identities and quasidentities 435
§2. Quasivarieties of algebraic systems 437
Chapter 32. Multiplication of classes of algebraic systems 441
§1. The basic definition 441
§2. Products and axiomatizable classes 448
§3. Multiplication in special classes of systems 454
§4. Additional observations 459
Chapter 33. Universally axiomatizable subclasses of locally finite classes of models 466
§1. Conditions for universal axiomatizability 466
§2. Independent axiomatizability 468
§3. Graphs of finite degree 473
§4. Uniformly locally finite classes 474
Cxapter 34. Problems on the border between algebra and logic 479
§1. The algorithmic nature of theories 483
§2. Varieties and quasivarieties 489
Bibliography 493
Part I. The articles translated in this collection 494
Part II. Previous English translations 496
Part III. Other works of A.I. Mal’cev 497
Part IV. Reviews cited in editor’s notes 498
Part V. Obituaries with bibliographies 499
Part VI. General references 500
Topic table 508
Index 509