Contributions to Universal Algebra (eBook)

Front Cover 1
Contributions to Universal Algebra 2
Copyright Page 3
PREFACE 4
CONTENTS 6
LIST OF PARTICIPANTS 11
Chapter 1. ON THE CONGRUENCE LATTICE OF PSEUDO-SIMPLE ALGEBRAS 16
REFERENCES 21
Chapter 2. ELEMENTARY PROPERTIES OF LIMIT REDUCED POWERS WITH APPLICATIONS TO BOOLEAN POWERS 22
REFERENCES 25
Chapter 3. ON CONGRUENCE LATTICES OF 2-VALUED ALGEBRAS 28
REFERENCES 32
Chapter 4. THE c-IDEAL LATTICE AND SUBALGEBRA LATTICE ARE INDEPENDENT 34
1. PRELIMINARIES 34
2. THE CONCRETE CHARACTERIZATION 35
3. PROOF OF THE MAIN THEOREM 37
4. SOME REMARKS ON RELATED REPRESENTATIONPROBLEMS 39
REFERENCES 39
Chapter 5. /-GROUP-CONE AND BOOLEAN ALGEBRA. A COMMON ONE-IDENTITY-AXIOM 42
REFERENCES 57
Chapter 6. SPLITTING LATTICES AND CONGRUENCE MODULARITY* 58
§1. INTRODUCTION 58
§2. PRELIMINARIES 59
§3. THE CLASS 61
§5. CONCLUDING REMARKS 71
REFERENCES 72
Chapter 7. SOME REMARKS ON WEAK AUTOMORPHISMS 74
0. INTRODUCTION 74
1. HOLOMORPH AND WEAK AUTOMORPHISMS 76
2. REDUCIBLE ALGEBRAS 77
REFERENCES 81
Chapter 8. ON POLYNOMIAL ALGEBRAS 84
1. INTRODUCTION 84
2. SOME BASIC PROPERTIES OF K -POLYNOMIAL ALGEBRAS 85
3. ON THE CANONICAL HOMOMORPHISM FROM FREEALGEBRAS INTO POLYNOMIAL ALGEBRAS 88
4. SOME GENERALIZATIONS FROM POLYNOMIAL ALGEBRASTO UNIVERSAL PAIRS OVER PARTIAL ALGEBRAS 90
5. POLYNOMIAL ALGEBRAS AND POLYNOMIAL FUNCTION ALGEBRAS 92
REFERENCES 99
Chapter 9. DUALITY FOR ALGEBRAS 102
INTRODUCTION 102
TERMINOLOGY AND NOTATION 103
1. CHARACTERISTIC MAPS 103
2. THE REPRESENTATION THEOREM 106
3. SUFFICIENT CONDITIONS FOR BICENTRALITY 108
REFERENCES 112
Chapter 10. PROJECTIVE AND INJECTIVE VARIETIES OF ABELIAN O-ALGEBRAS 114
REFERENCES 132
Chapter 11. SOME VARIETIES OF MODULAR LATTICES NOT GENERATED BY THEIR FINITE DIMENSIONAL MEMBERS 134
REFERENCES 144
Chapter 12. ON WEAK HOMOMORPHISMS OF STONE ALGEBRAS 146
1. WEAK HOMOMORPHISMS OF GENERAL ALGEBRAS 146
2. REMARKS ON ALGEBRAIC OPERATIONS OF STONE ALGEBRAS 148
3. WEAK ISOMORPHISMS AND WEAK HOMOMORPHISMSOF STONE ALGEBRAS 155
REFERENCES 159
Chapter 13. ON THE SUMS OF DOUBLE SYSTEMS OF LATTICES AND DS-CONGRUENCES OF LATTICES 162
REFERENCES 166
Chapter 14. n-DISTRIBUTIVITY AND SOME QUESTIONS OF THE EQUATIONAL THEORY OF LATTICES 168
0. CONTENTS 168
1. PRELIMINARIES 169
2. APPLICATIONS OF n-DISTRIBUTIVITY TO THEEQUATIONAL THEORY OF LATTICES 170
3. ON THE LATTICE GENERATED BY THE VARIETIES 174
REFERENCES 178
Chapter 15. POLYNOMIAL NORMAL FORMS AND THE EMBEDDING OF POLYNOMIAL ALGEBRAS 180
1. INTRODUCTION 180
2. SUFFICIENT CONDITIONS FOR fAB TO BEAN EMBEDDING 181
3. EXAMPLES OF POLYNOMIAL NORMAL FORMS 182
REFERENCES 188
Chapter 16. COVERINGS IN THE LATTICE OF VARIETIES 190
REFERENCES 203
Chapter 17. EMBEDDING SEMIGROUPS IN SEMIGROUPS GENERATED BY IDEMPOTENTS 206
1. INTRODUCTION 206
2. EMBEDDING THEOREM 207
3. COUNTABLE SEMIGROUPS 208
REFERENCES 209
Chapter 18. ENDOMORPHISM SEMIGROUPS AND SUBGROUPOID LATTICES 210
Chapter 19. A NOTE ON IMPLICATIONAL SUBCATEGORIES 214
THE GALOIS-CORRESPONDENCE INDUCED BY INJECTIVITY WITH RESPECT TO A FIXED CLASS OF EPIMORPHISMS 218
SOME EXAMPLES IN CATEGORIES OF PARTIAL ALGEBRAS 220
REFERENCES 222
Chapter 20. A REPORT ON SUBLATTICES OF A FREE LATTICE 224
1. INTRODUCTION 224
2. A SUMMARY OF RESULTS 225
3. THE FINITELY GENERATED CASE OF THEOREM 2.1: (iii) IMPLIES (ii). 227
4. THE FINITELY GENERATED CASE:THE PROOF COMPLETED 230
5. THE INFINITELY GENERATED CASE 233
6. CYCLES IN FINITE SEMI-DISTRIBUTIVE LATTICES 236
7. CYCLES IN S-LATTICES: PRELIMINARIES 240
8. CYCLES IN S-LATTICES 245
9. A PROOF OF DAY'S THEOREM 247
10. SUMMARY, PROBLEMS, AND A COUNTEREXAMPLE 250
REFERENCES 257
Chapter 21. EXTENSIVE GROUPOID VARIETIES 260
1. INTRODUCTION 260
2. Mod (x = t), t BALANCED 262
10. MAIN THEOREMS 285
REFERENCES 286
Chapter 22. PRIMITIVE SUBSETS OF ALGEBRAS 288
INTRODUCTION 288
1. PRELIMINARIES AND STATEMENT OF RESULTS 289
2. PROOFS OF THE THEOREMS' 290
REFERENCES 294
Chapter 23. IDEALS, NORMAL SETS AND CONGRUENCES 296
0. SUMMARY AND INTRODUCTION 296
1. p-DETERMINED CONGRUENCES 298
2. POLYNOMIALS DETERMINING ALL CONGRUENCES 300
3. IDEALS 303
REFERENCES 310
Chapter 24. A CHARACTERIZATION OF COMPLETE MODULAR p-ALGEBRAS 312
1. PRELIMINARIES 313
2. TRIPLE CHARACTERIZATION OF COMPLETE MODULAR p-ALGEBRAS 316
3. COMPLETE HOMOMORPHISMS AND COMPLETE SUBALGEBRASOF COMPLETE MODULAR p-ALGEBRAS 320
4. FILL-IN THEOREMS 326
REFERENCES 329
Chapter 25. JACOBSON'S DENSITY THEOREM IN UNIVERSAL ALGEBRA 332
REFERENCES 341
Chapter 26. CERTAIN QUESTIONS OF THE THEORY OF HOMOTOPY OF UNIVERSAL ALGEBRAS 342
1. BASIC CONCEPTS 343
2. HOMOTOPIES AND CONGRUENT FAMILIES OF EQUIVALENCES 345
3. HOMOTOPIES OF SOME CLASSICAL ALGEBRAIC SYSTEMS 350
4. SPECIAL MORPIDSMS IN A CATEGORY OF QUASIGROUPS 354
REFERENCES 355
Chapter 27. A THEOREM ON FINITE SUBLATTICES OF FREE LATTICES 358
REFERENCES 362
Chapter 28. A NOTE ON A PROBLEM OF GORALCÍK 364
Chapter 29. QUASI-DECOMPOSITIONS, EXACT SEQUENCES, AND TRIPLE SUMS OF SEMIGROUPS. I. GENERAL THEORY 366
§1. TRIPLES 368
§2. OUASI-DECOMPOSITIONS 370
§3. GLIVENKO OPERATORS, EXACTNESS 372
§4. LOCAL MULTIPLICATIONS, LIMITS 381
§5. SEMIGROUPS OF SEMIGROUPS 385
§6. SEMIMODULES, NAGATA'S IDEALIZATION PRODUCT 388
§7. SEMILATTICES OF SEMIGROUPS 393
REFERENCES 395
Chapter 30. QUASI-DECOMPOSITIONS, EXACT SEQUENCES, AND TRIPLE SUMS OF SEMIGROUPS. II. APPLICATIONS 400
§8. CLOSURE RETRACTIONS OF SEMILATTICES, LEFT ANDRIGHT BROUWERIAN ELEMENTS 401
§9. QUASI-DECOMPOSITIONS OF PSEUDO-COMPLEMENTEDSEMILATTICES 415
§10. QUASI-DECOMPOSITIONS OF BROUWERIAN SEMILATTICES: 421
Chapter 31. ENDOMORPHICALLY COMPLETE GROUPS 430
REFERENCES 435
Chapter 32. CONCRETE CATEGORIES WITH NON-INJECTIVE MONOMORPHISM 436
REFERENCES 440
Chapter 33. ON ALGEBRAS AS TREE AUTOMATA 442
INTRODUCTION 442
1. PRELIMINARIES 443
2. CONTEXT-FREE GRAMMARS AND RECOGNIZERS 446
3. CLASSES OF ALGEBRAS AND FAMILIES OF LANGUAGES 448
4. GROUPOID-RECOGNIZERS 450
5. CANTOR ALGEBRAS 453
REFERENCES 455
Chapter 34. ON AFFINE MODULES 458
REFERENCES 464
Chapter 35. EQUATIONAL LOGIC 466
1. ALGEBRAS 467
2. FREE ALGEBRAS 467
3. EQUATIONALLY DEFINED CLASSES 469
4. EQUATIONAL THEORIES 472
5. SUBDIRECT REPRESENTATION 473
6. EXAMPLES OF !– 474
7. BASES AND GENERIC ALGEBRAS 475
8. FINITELY BASED THEORIES 476
9. EQUIVALENT VARIETIES 478
10. ONE-BASED THEORIES 479
11. MINIMAL BASES 480
12. THE LATTICE OF VARIETIES (BEGINNING) 481
13. EFFECTIVELY QUESTIONS 483
14. THE LATTICE OF VARIETIES (CONTINUED) 485
15. SOME FURTHER INVARIANTS OF THE EQUIVALENCE CLASS OF VARIETY 490
16. GENERATION OF VARIETIES 493
17. MALCEV CONDITIONS 497
18. CONNECTIONS WITH TOPOLOGY 498
REFERENCES 499
Chapter 36. ON REGULAR ALGEBRAS 504
INTRODUCTION 504
NOTATIONS 505
§1. U-REGULARITY 505
§2. ATOMREGULARITY 510
§3. DEVIATION FROM THE CLASSICAL CASE 512
REFERENCES 514
Chapter 37. REMARKS ON FULLY INVARIANT CONGRUENCES 516
§0. INTRODUCTION 516
§1. ALGEBRAS WITH FEW F.I. CONGRUENCES 519
§2. NUCLEAR CONGRUENCES 533
§3. FULLY PRESERVING ENDOMORPHISMS 544
§4. FULLY INVARIANT SUBSETS 549
REFERENCES 554
Chapter 38. VARIETIES GENERATED BY QUASI-PRIMAL ALGEBRAS HAVE DECIDABLE THEORIES 556
§ 1. SUBSHEAVES OF CONSTANT SHEAVES: 558
§2. EMBEDDING SHEAVES INTO CONSTANT SHEAVES 561
§3. THE MAIN EMBEDDING THEOREM 563
§4. VARIETIES GENERATED BY OUASI-PRIMAL ALGEBRAS 566
§5. THE INTUITIONISTIC VERSION OF A CYLINDRIC ALGEBRA 571
REFERENCES 575
Chapter 39. ON THE POLYNOMIAL COMPLETENESS DEFECT OF UNIVERSAL ALGEBRAS 578
REFERENCES 580
Chapter 40. ON LATTICES FREELY GENERATED BY FINITE PARTIALLY ORDERED SETS 582
REFERENCES 593
Chapter 41. PROPER AND IMPROPER FREE ALGEBRAS 596
REFERENCES 602
PROBLEMS 604