Szabó / Szendrei Lectures in Universal Algebra

1. Auflage 2016
ISBN: 978-1-4832-9540-4
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band Volume 1, 655 Seiten, Web PDF

Reihe: Colloquia Mathematica Societatis Janos Bolyai

These 34 papers cover topics ranging from various problems on varieties and other classes of algebras including categorical aspects and duality theory to the structure of finite algebras and clones on finite (or infinite) sets.As well as survey articles by invited speakers, the papers contain full proofs of new results not published elsewhere. The volume ends with a list of problems.

Autoren/Hrsg.

Szabó, L.

Szendrei, A.

Inhaltsverzeichnis

1;Front Cover;1
2;Lectures in Universal Algebra;2
3;Copyright Page;3
4;Table of Contents;6
5;Preface;4
6;Scientific Program;9
7;List of Participants;14
8;Chapter 1. On the number of clones containing all constants (A problem of R. McKenzie);22
8.1;THEOREM;22
8.2;DEFINITION;23
8.3;LEMMA;23
8.4;ACKNOWLEDGEMENTS;26
9;Chapter 2. Transferable tolerances and weakly tolerance regular lattices;28
9.1;DEFINITION 1;29
9.2;DEFINITION 2;29
9.3;DEFINITION 3;29
9.4;THEOREM 1;30
9.5;THEOREM 2;31
9.6;COROLLARY 1;31
9.7;COROLLARY 2;33
9.8;LEMMA 1;33
9.9;LEMMA 2;34
9.10;THEOREM 3;35
9.11;THEOREM 4;36
9.12;THEOREM 5;36
9.13;REFERENCES;40
10;Chapter 3. Epimorphisms in discriminator varieties;42
10.1;THEOREM;42
10.2;LEMMA;43
10.3;PROOF OF THEOREM;44
10.4;COROLLARY 1;45
10.5;COROLLARY 2;46
10.6;COROLLARY 3;46
10.7;COROLLARY 4;47
10.8;PROPOSITION;47
10.9;REFERENCES;48
11;Chapter 4. On conservative minimal operations;50
11.1;1. INTRODUCTION;50
11.2;2. PREPARATORY REMARKS;51
11.3;3. RESULTS;56
11.4;THEOREM 1;56
11.5;THEOREM 2;57
11.6;REFERENCES;61
12;Chapter 5. Piggyback-dualities;62
12.1;NATURAL PROTO-DUALITIES;63
12.2;PIGGYBACK-DUALITY-THEOREM;68
12.3;EXAMPLES;69
12.4;1. OCKHAM-ALGEBRAS;70
12.5;2. DISTRIBUTIVE ñ-ALGEBRAS;71
12.6;3. DOUBLE STONE ALGEBRAS;75
12.7;4. HEYTING-ALGEBRAS;76
12.8;REFERENCES;82
13;Chapter 6. On the depth of infinitely generated subalgebras of Post's iterative algebra p3;86
13.1;THEOREM;87
13.2;LEMMA 1;89
13.3;LEMMA 2;89
13.4;LEMMA 3;91
13.5;LEMMA 4;92
13.6;LEMMA 5;94
13.7;LEMMA 6;95
13.8;PROOF OF THE THEOREM;95
13.9;REFERENCES;96
14;Chapter 7. Tolerance-free algebras having majority term functions and admitting no proper subalgebras;98
14.1;1. INTRODUCTION;98
14.2;2. PRELIMINARIES;99
14.3;3. LEMMAS;100
14.4;4. RESULTS;105
14.5;REFERENCES;108
15;Chapter 8. Polynomial pairs characterizing principality;110
15.1;1. PRELIMINARIES;110
15.2;LEMMA 1;110
15.3;2. VARIETIES OF ALGEBRAS WITH PRINCIPAL COMPACT BLOCKS;111
15.4;DEFINITION 1;111
15.5;LEMMA 2;111
15.6;THEOREM 1;112
15.7;3. A NOTE ON VARIETIES WITH PRINCIPAL COMPACT CONGRUENCES;114
15.8;DEFINITION 2;114
15.9;THEOREM 2;115
15.10;4. A GENERALIZATION OF PCB AND PCC;116
15.11;DEFINITION 3;117
15.12;THEOREM 3;117
15.13;5. THE CONCEPT OF PRINCIPALITY ON VARIETIES OF RINGS WITH UNIT;120
15.14;LEMMA 3;120
15.15;THEOREM 4;121
15.16;REFERENCES;122
16;Chapter 9. On the connection of cylindrical homomorphisms and point functions for Crs a's;124
16.1;2. THE GENERAL CASE;128
16.2;THEOREM 1;128
16.3;THEOREM 2;131
16.4;THEOREM 3;134
16.5;THEOREM 4;134
16.6;3. HOMOMORPHISM FROM FULL ALGEBRA;136
16.7;THEOREM 5;136
16.8;THEOREM 6;139
16.9;REFERENCES;142
17;Chapter 10. A universality condition of 0,1-lattices;144
17.1;INTRODUCTION;144
17.2;THE UNIVERSALITY CONDITION;145
17.3;THEOREM;146
17.4;APPLICATIONS OF THE UNIVERSALITY CONDITION;150
17.5;COROLLARY 1;150
17.6;COROLLARY 2;151
17.7;REFERENCES;154
18;Chapter 11. On the join of some varieties of algebras;156
18.1;1. The following properties are shown in;156
18.2;LEMMA 1;157
18.3;LEMMA 2;157
18.4;LEMMA 3;158
18.5;LEMMA 4;158
18.6;THEOREM 1;158
18.7;THEOREM 2;158
18.8;THEOREM 3;159
18.9;COROLLARY;160
18.10;REFERENCES;160
19;Chapter 12. The Stone-Cech compactification of a pospace;162
19.1;1. DEFINITIONS;162
19.2;2. LEMMA;164
19.3;3. PROPOSITION;165
19.4;4. THEOREM;165
19.5;5. RECALL;167
19.6;6. DEFINITION;167
19.7;7. DEFINITION;169
19.8;8. PROPOSITION;169
19.9;9. LEMMA;170
19.10;10. DEFINITION;171
19.11;11. LEMMA;172
19.12;12. THEOREM;172
19.13;13. EXAMPLE;173
19.14;14. REMARKS;173
19.15;15. DEFINITION;175
19.16;16. THEOREM;175
19.17;17. THEOREM;176
19.18;REFERENCES;176
20;Chapter 13. Constructions of non–commutative algebras;178
20.1;1. INTRODUCTION;178
20.2;2. RESULTS;178
20.3;DEFINITION;179
20.4;LEMMA 1;179
20.5;LEMMA 2;179
20.6;THEOREM 1;180
20.7;REMARK;183
20.8;THEOREM 2;184
20.9;COROLLARY;185
20.10;DEFINITION;185
20.11;LEMMA 3;185
20.12;LEMMA 4;185
20.13;LEMMA 5;186
20.14;THEOREM 3;186
20.15;REFERENCE;188
21;Chapter 14. Fully invariant algebraic closure systems of congruences and quasivarieties of algebras;190
21.1;1. INTRODUCTION;190
21.2;2. QUASIVARIETIES AND QUASIEQUATIONS;192
21.3;3. PROOF OF PROPOSITION;194
21.4;4. PROOF OF PROPOSITIONS 2.5 and 2.6;196
21.5;5. SPECIAL CASES;198
21.6;6. GENERATION OF QUASIVARIETIES FROM ALGEBRAS VIA FULLY INVARIANT ALGEBRAIC CLOSURE SYSTEMS;200
21.7;7. ALGEBRAIC DERIVATION OF THE CLOSURE Qeq Mod R FROM A SET OF QUASIEQUATIONS R;203
21.8;8. GENERALIZATION TO THE CASE OF PARTIAL ALGEBRAS;205
21.9;REFERENCES;206
22;Chapter 15. On lattices with restrictions on their interval lattices;210
22.1;THEOREM 1;211
22.2;THEOREM 2;214
22.3;THEOREM 3;215
22.4;REFERENCES;216
22.5;L–continuous partial functions;218
22.6;1. INTRODUCTION AND NOTATION;218
22.7;2. L–CONTINUITY;221
22.8;3. L-CONVERGENCE OF POINT AND FUNCTIONAL SEQUENCES;226
22.9;4. THE CASE L = Con(Z);230
22.10;5. THE CASE L = Con(Zn);235
22.11;REFERENCES;240
23;Chapter 16. Infinite image homomorphisms of distributive bounded lattices;242
23.1;I. INTRODUCTION;242
23.2;II. EMBEDDING IN T2;249
23.3;III. MINIMAL UNIVERSAL VARIETIES;265
23.4;IV. INFINITE IMAGE HOMOMORPHISMS OF DISTRIBUTIVE LATTICES;276
23.5;REFERENCES;281
24;Chapter 17. Description of partial algebras by segments;284
24.1;1. INTRODUCTION;284
24.2;2. BASIC NOTIONS;285
24.3;3. SEGMENTS IN PARTIAL ALGEBRAS;286
24.4;4. DESCRIPTION OF PARTIAL ALGEBRAS;288
24.5;REFERENCES;292
25;Chapter 18. Tame congruences;294
25.1;THEOREM 1;296
25.2;THEOREM 2;296
25.3;THEOREM 3;299
25.4;THEOREM 4;300
25.5;THEOREM 5;301
25.6;THEOREM 6;301
25.7;THEOREM 7;302
25.8;THEOREM 8;302
25.9;THEOREM 9;303
25.10;THEOREM 10;303
25.11;THEOREM 11;304
25.12;THEOREM 12;304
25.13;THEOREM 13;305
25.14;REFERENCES;306
26;Chapter 19. Fifteen possible previews in equational logic;308
26.1;§ 1. PROLOGUE;308
26.2;§ 2. REVIEWS OF FIFTEEN PAPERS;308
26.3;§ 3. APOGLOGUE;323
26.4;§ 4. A LIST OF "THEOREMS";324
26.5;REFERENCES;327
27;Chapter 20. On strongly non-regular and trivializing varieties of algebras;334
27.1;THEOREM 1;337
27.2;COROLLARY 1;338
27.3;THEOREM 2;340
27.4;COROLLARY 2;340
27.5;THEOREM 3;341
27.6;COROLLARY 3;341
27.7;THEOREM 4;342
27.8;COROLLARY 4;344
27.9;COROLLARY 5;344
27.10;REFERENCES;344
28;Chapter 21. On varieties of semigroups satisfying x3 = x;346
28.1;INTRODUCTION;346
28.2;RESULTS;347
28.3;LEMMA 1;347
28.4;LEMMA 2;348
28.5;LEMMA 3;348
28.6;LEMMA 4;350
28.7;LEMMA 5;353
28.8;LEMMA 6;354
28.9;LEMMA 7;354
28.10;LEMMA 8;358
28.11;LEMMA 9;359
28.12;LEMMA 10;360
28.13;LEMMA 11;360
28.14;LEMMA 12;363
28.15;THEOREM;363
28.16;REFERENCES;364
29;Chapter 22. Cryptomorphisms of non-indexed algebras and relational systems;366
29.1;1. INTRODUCTION;366
29.2;ACKNOWLEDGEMENTS;369
29.3;2. ^ -CRYPTOMORPHISMS FOR ALGEBRAS;370
29.4;3. ^-CRYPTOMORPHISMS OF RELATIONAL SYSTEMS;377
29.5;4. ^-CRYPTOISOMORPHISMS AND K/-CRYPTOISOMORPHISMS;381
29.6;5. EXAMPLES;390
29.7;6. CONGRUENCES, FACTORS AND THE HOMOMORPHISM THEOREM;396
29.8;REFERENCES;403
30;Chapter 23. Minimal clones I: the five types;406
30.1;1. INTRODUCTION;406
30.2;2. DEFINITIONS AND THE MAIN RESULT;408
30.3;3. CONCLUDING REMARKS AND PROBLEMS;420
30.4;REFERENCES;425
31;Chapter 24. Quasi-boolean lattices and associations;430
31.1;THEOREM 1;433
31.2;LEMMA 1;433
31.3;LEMMA 2;434
31.4;LEMMA 3;435
31.5;LEMMA 4;436
31.6;LEMMA 5;437
31.7;LEMMA 6;440
31.8;LEMMA 7;440
31.9;THEOREM 2;441
31.10;LEMMA 8;442
31.11;LEMMA 9;443
31.12;REFERENCES;455
32;Chapter 25. Monoids and their local closures;456
32.1;1. INTRODUCTION;456
32.2;2. REPRESENTATIONS AND LOCAL CLOSURE;456
32.3;LEMMA 1;458
32.4;3. CHARACTERISATION OF L(M);459
32.5;THEOREM 1;459
32.6;LEMMA 2;462
32.7;LEMMA 3;464
32.8;THEOREM 2·;467
32.9;BIBLIOGRAPHY;468
33;Chapter 26. The congruence lattice as an act over the endomorphism monoid;470
33.1;THEOREM 1;474
33.2;THEOREM 2;476
33.3;THEOREM 3;477
33.4;LEMMA 1;480
33.5;LEMMA 2;481
33.6;LEMMA 3;481
33.7;LEMMA 4;490
33.8;REFERENCES;496
34;Chapter 27. Interpolation in idempotent algebras;498
34.1;1. INTRODUCTION;498
34.2;2. PRELIMINARIES;499
34.3;3. RESULTS;502
34.4;THEOREM 1;502
34.5;THEOREM 2;503
34.6;THEOREM 4;503
34.7;THEOREM 5;505
34.8;COROLLARY 6;507
34.9;REFERENCES;507
35;Chapter 28. Demi-primal algebras with a single operation;510
35.1;INTRODUCTION;510
35.2;1. CHARACTERIZATION OF DEMI-PRIMAL ALGEBRAS;512
35.3;THEOREM 1;513
35.4;LEMMA 1;515
35.5;LEMMA 2;516
35.6;LEMMA 3;516
35.7;COROLLARY 1;520
35.8;COROLLARY 2;520
35.9;COROLLARY 3;521
35.10;2. GENERALIZATION OF ROUSSEAU'S THEOREM;521
35.11;THEOREM 2;525
35.12;LEMMA 4;525
35.13;REFERENCES;530
36;Chapter 29. Perfect chamber systems;534
36.1;1. BASIC DEFINITIONS;534
36.2;2. CONSTRUCTIONS;538
36.3;3. EXAMPLES;541
36.4;4. AUTOMORPHISMS;543
36.5;REFERENCES;548
37;Chapter 30. More ideals in universal algebra;550
37.1;1. INTRODUCTION;550
37.2;2. f–NORMAL SUBSETS;551
37.3;3. f–IDEALS;554
37.4;4. Bf(A)IN if(A,K);558
37.5;REFERENCES;560
38;Chapter 31. A duality for the lattice variety generated by M3;562
38.1;THEOREM 1;568
38.2;COROLLARY 2;568
38.3;THEOREM 3;569
38.4;REFERENCES;573
39;Chapter 32. Generation of finite partition lattices;574
39.1;INTRODUCTION;574
39.2;1. CL–GENERATION;575
39.3;2. GENERATION IN THE SENSE OF LATTICES;581
39.4;REFERENCES;586
40;Chapter 33. Unitary congruence adjunctions;588
40.1;1. INTRODUCTION;588
40.2;2. PRELIMINARIES;590
40.3;3. PARTIALLY ORDERED SETS WITH ADJUNCTION;593
40.4;4. NILPOTENCY AND SOLVABILITY;599
40.5;5. RESIDUATED LATTICES;602
40.6;6. ALGEBRAIC RESIDUATED LATTICES;606
40.7;7. CONGRUENCE ADJUNCTIONS;614
40.8;8. DECOMPOSITION THEOREMS;633
40.9;9. APPLICATIONS;639
40.10;REFERENCES;645
41;PROBLEMS;650