Lectures in Universal Algebra (eBook)

Front Cover 1
Lectures in Universal Algebra 2
Copyright Page 3
Table of Contents 6
Preface 4
Scientific Program 9
List of Participants 14
Chapter 1. On the number of clones containing all constants (A problem of R. McKenzie) 22
THEOREM 22
DEFINITION 23
LEMMA 23
ACKNOWLEDGEMENTS 26
Chapter 2. Transferable tolerances and weakly tolerance regular lattices 28
DEFINITION 1 29
DEFINITION 2 29
DEFINITION 3 29
THEOREM 1 30
THEOREM 2 31
COROLLARY 1 31
COROLLARY 2 33
LEMMA 1 33
LEMMA 2 34
THEOREM 3 35
THEOREM 4 36
THEOREM 5 36
REFERENCES 40
Chapter 3. Epimorphisms in discriminator varieties 42
THEOREM 42
LEMMA 43
PROOF OF THEOREM 44
COROLLARY 1 45
COROLLARY 2 46
COROLLARY 3 46
COROLLARY 4 47
PROPOSITION 47
REFERENCES 48
Chapter 4. On conservative minimal operations 50
1. INTRODUCTION 50
2. PREPARATORY REMARKS 51
3. RESULTS 56
THEOREM 1 56
THEOREM 2 57
REFERENCES 61
Chapter 5. Piggyback-dualities 62
NATURAL PROTO-DUALITIES 63
PIGGYBACK-DUALITY-THEOREM 68
EXAMPLES 69
1. OCKHAM-ALGEBRAS 70
2. DISTRIBUTIVE ñ-ALGEBRAS 71
3. DOUBLE STONE ALGEBRAS 75
4. HEYTING-ALGEBRAS 76
REFERENCES 82
Chapter 6. On the depth of infinitely generated subalgebras of Post's iterative algebra p3 86
THEOREM 87
LEMMA 1 89
LEMMA 2 89
LEMMA 3 91
LEMMA 4 92
LEMMA 5 94
LEMMA 6 95
PROOF OF THE THEOREM 95
REFERENCES 96
Chapter 7. Tolerance-free algebras having majority term functions and admitting no proper subalgebras 98
1. INTRODUCTION 98
2. PRELIMINARIES 99
3. LEMMAS 100
4. RESULTS 105
REFERENCES 108
Chapter 8. Polynomial pairs characterizing principality 110
1. PRELIMINARIES 110
LEMMA 1 110
2. VARIETIES OF ALGEBRAS WITH PRINCIPAL COMPACT BLOCKS 111
DEFINITION 1 111
LEMMA 2 111
THEOREM 1 112
3. A NOTE ON VARIETIES WITH PRINCIPAL COMPACT CONGRUENCES 114
DEFINITION 2 114
THEOREM 2 115
4. A GENERALIZATION OF PCB AND PCC 116
DEFINITION 3 117
THEOREM 3 117
5. THE CONCEPT OF PRINCIPALITY ON VARIETIES OF RINGS WITH UNIT 120
LEMMA 3 120
THEOREM 4 121
REFERENCES 122
Chapter 9. On the connection of cylindrical homomorphisms and point functions for Crs a's 124
2. THE GENERAL CASE 128
THEOREM 1 128
THEOREM 2 131
THEOREM 3 134
THEOREM 4 134
3. HOMOMORPHISM FROM FULL ALGEBRA 136
THEOREM 5 136
THEOREM 6 139
REFERENCES 142
Chapter 10. A universality condition of 0,1-lattices 144
INTRODUCTION 144
THE UNIVERSALITY CONDITION 145
THEOREM 146
APPLICATIONS OF THE UNIVERSALITY CONDITION 150
COROLLARY 1 150
COROLLARY 2 151
REFERENCES 154
Chapter 11. On the join of some varieties of algebras 156
1. The following properties are shown in 156
LEMMA 1 157
LEMMA 2 157
LEMMA 3 158
LEMMA 4 158
THEOREM 1 158
THEOREM 2 158
THEOREM 3 159
COROLLARY 160
REFERENCES 160
Chapter 12. The Stone-Cech compactification of a pospace 162
1. DEFINITIONS 162
2. LEMMA 164
3. PROPOSITION 165
4. THEOREM 165
5. RECALL 167
6. DEFINITION 167
7. DEFINITION 169
8. PROPOSITION 169
9. LEMMA 170
10. DEFINITION 171
11. LEMMA 172
12. THEOREM 172
13. EXAMPLE 173
14. REMARKS 173
15. DEFINITION 175
16. THEOREM 175
17. THEOREM 176
REFERENCES 176
Chapter 13. Constructions of non–commutative algebras 178
1. INTRODUCTION 178
2. RESULTS 178
DEFINITION 179
LEMMA 1 179
LEMMA 2 179
THEOREM 1 180
REMARK 183
THEOREM 2 184
COROLLARY 185
DEFINITION 185
LEMMA 3 185
LEMMA 4 185
LEMMA 5 186
THEOREM 3 186
REFERENCE 188
Chapter 14. Fully invariant algebraic closure systems of congruences and quasivarieties of algebras 190
1. INTRODUCTION 190
2. QUASIVARIETIES AND QUASIEQUATIONS 192
3. PROOF OF PROPOSITION 194
4. PROOF OF PROPOSITIONS 2.5 and 2.6 196
5. SPECIAL CASES 198
6. GENERATION OF QUASIVARIETIES FROM ALGEBRAS VIA FULLY INVARIANT ALGEBRAIC CLOSURE SYSTEMS 200
7. ALGEBRAIC DERIVATION OF THE CLOSURE Qeq Mod R FROM A SET OF QUASIEQUATIONS R 203
8. GENERALIZATION TO THE CASE OF PARTIAL ALGEBRAS 205
REFERENCES 206
Chapter 15. On lattices with restrictions on their interval lattices 210
THEOREM 1 211
THEOREM 2 214
THEOREM 3 215
REFERENCES 216
L–continuous partial functions 218
1. INTRODUCTION AND NOTATION 218
2. L–CONTINUITY 221
3. L-CONVERGENCE OF POINT AND FUNCTIONAL SEQUENCES 226
4. THE CASE L = Con(Z) 230
5. THE CASE L = Con(Zn) 235
REFERENCES 240
Chapter 16. Infinite image homomorphisms of distributive bounded lattices 242
I. INTRODUCTION 242
II. EMBEDDING IN T2 249
III. MINIMAL UNIVERSAL VARIETIES 265
IV. INFINITE IMAGE HOMOMORPHISMS OF DISTRIBUTIVE LATTICES 276
REFERENCES 281
Chapter 17. Description of partial algebras by segments 284
1. INTRODUCTION 284
2. BASIC NOTIONS 285
3. SEGMENTS IN PARTIAL ALGEBRAS 286
4. DESCRIPTION OF PARTIAL ALGEBRAS 288
REFERENCES 292
Chapter 18. Tame congruences 294
THEOREM 1 296
THEOREM 2 296
THEOREM 3 299
THEOREM 4 300
THEOREM 5 301
THEOREM 6 301
THEOREM 7 302
THEOREM 8 302
THEOREM 9 303
THEOREM 10 303
THEOREM 11 304
THEOREM 12 304
THEOREM 13 305
REFERENCES 306
Chapter 19. Fifteen possible previews in equational logic 308
§ 1. PROLOGUE 308
§ 2. REVIEWS OF FIFTEEN PAPERS 308
§ 3. APOGLOGUE 323
§ 4. A LIST OF "THEOREMS" 324
REFERENCES 327
Chapter 20. On strongly non-regular and trivializing varieties of algebras 334
THEOREM 1 337
COROLLARY 1 338
THEOREM 2 340
COROLLARY 2 340
THEOREM 3 341
COROLLARY 3 341
THEOREM 4 342
COROLLARY 4 344
COROLLARY 5 344
REFERENCES 344
Chapter 21. On varieties of semigroups satisfying x3 = x 346
INTRODUCTION 346
RESULTS 347
LEMMA 1 347
LEMMA 2 348
LEMMA 3 348
LEMMA 4 350
LEMMA 5 353
LEMMA 6 354
LEMMA 7 354
LEMMA 8 358
LEMMA 9 359
LEMMA 10 360
LEMMA 11 360
LEMMA 12 363
THEOREM 363
REFERENCES 364
Chapter 22. Cryptomorphisms of non-indexed algebras and relational systems 366
1. INTRODUCTION 366
ACKNOWLEDGEMENTS 369
2. ^ -CRYPTOMORPHISMS FOR ALGEBRAS 370
3. ^-CRYPTOMORPHISMS OF RELATIONAL SYSTEMS 377
4. ^-CRYPTOISOMORPHISMS AND K/-CRYPTOISOMORPHISMS 381
5. EXAMPLES 390
6. CONGRUENCES, FACTORS AND THE HOMOMORPHISM THEOREM 396
REFERENCES 403
Chapter 23. Minimal clones I: the five types 406
1. INTRODUCTION 406
2. DEFINITIONS AND THE MAIN RESULT 408
3. CONCLUDING REMARKS AND PROBLEMS 420
REFERENCES 425
Chapter 24. Quasi-boolean lattices and associations 430
THEOREM 1 433
LEMMA 1 433
LEMMA 2 434
LEMMA 3 435
LEMMA 4 436
LEMMA 5 437
LEMMA 6 440
LEMMA 7 440
THEOREM 2 441
LEMMA 8 442
LEMMA 9 443
REFERENCES 455
Chapter 25. Monoids and their local closures 456
1. INTRODUCTION 456
2. REPRESENTATIONS AND LOCAL CLOSURE 456
LEMMA 1 458
3. CHARACTERISATION OF L(M) 459
THEOREM 1 459
LEMMA 2 462
LEMMA 3 464
THEOREM 2· 467
BIBLIOGRAPHY 468
Chapter 26. The congruence lattice as an act over the endomorphism monoid 470
THEOREM 1 474
THEOREM 2 476
THEOREM 3 477
LEMMA 1 480
LEMMA 2 481
LEMMA 3 481
LEMMA 4 490
REFERENCES 496
Chapter 27. Interpolation in idempotent algebras 498
1. INTRODUCTION 498
2. PRELIMINARIES 499
3. RESULTS 502
THEOREM 1 502
THEOREM 2 503
THEOREM 4 503
THEOREM 5 505
COROLLARY 6 507
REFERENCES 507
Chapter 28. Demi-primal algebras with a single operation 510
INTRODUCTION 510
1. CHARACTERIZATION OF DEMI-PRIMAL ALGEBRAS 512
THEOREM 1 513
LEMMA 1 515
LEMMA 2 516
LEMMA 3 516
COROLLARY 1 520
COROLLARY 2 520
COROLLARY 3 521
2. GENERALIZATION OF ROUSSEAU'S THEOREM 521
THEOREM 2 525
LEMMA 4 525
REFERENCES 530
Chapter 29. Perfect chamber systems 534
1. BASIC DEFINITIONS 534
2. CONSTRUCTIONS 538
3. EXAMPLES 541
4. AUTOMORPHISMS 543
REFERENCES 548
Chapter 30. More ideals in universal algebra 550
1. INTRODUCTION 550
2. f–NORMAL SUBSETS 551
3. f–IDEALS 554
4. Bf(A)IN if(A,K) 558
REFERENCES 560
Chapter 31. A duality for the lattice variety generated by M3 562
THEOREM 1 568
COROLLARY 2 568
THEOREM 3 569
REFERENCES 573
Chapter 32. Generation of finite partition lattices 574
INTRODUCTION 574
1. CL–GENERATION 575
2. GENERATION IN THE SENSE OF LATTICES 581
REFERENCES 586
Chapter 33. Unitary congruence adjunctions 588
1. INTRODUCTION 588
2. PRELIMINARIES 590
3. PARTIALLY ORDERED SETS WITH ADJUNCTION 593
4. NILPOTENCY AND SOLVABILITY 599
5. RESIDUATED LATTICES 602
6. ALGEBRAIC RESIDUATED LATTICES 606
7. CONGRUENCE ADJUNCTIONS 614
8. DECOMPOSITION THEOREMS 633
9. APPLICATIONS 639
REFERENCES 645
PROBLEMS 650