Csákány / Schmidt Contributions to Universal Algebra

1. Auflage 2014
ISBN: 978-1-4831-0302-0
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 608 Seiten, Web PDF

Contributions to Universal Algebra focuses on the study of algebra. The compilation first discusses the congruence lattice of pseudo-simple algebras; elementary properties of limit reduced powers with applications to Boolean powers; and congruent lattices of 2-valued algebras. The book further looks at duality for algebras; weak homomorphisms of stone algebras; varieties of modular lattices not generated by their finite dimensional members; and remarks on algebraic operations of stone algebras. The text describes polynomial normal forms and the embedding of polynomial algebras; coverings in the lattice of varieties; embedding semigroups in semigroups generated by idempotents; and endomorphism semigroups and subgroupoid lattices. The book also discusses a report on sublattices of a free lattice, and then presents the cycles in finite semi-distributive lattices; cycles in S-lattices; and summary of results. The text also describes primitive subsets of algebras, ideals, normal sets, and congruences, as well as Jacobson's density theorem. The book is a good source for readers wanting to study algebra.

Autoren/Hrsg.

Csákány, B.

Schmidt, J.

Inhaltsverzeichnis

1;Front Cover;1
2;Contributions to Universal Algebra;2
3;Copyright Page;3
4;PREFACE;4
5;CONTENTS;6
6;LIST OF PARTICIPANTS;11
7;Chapter 1. ON THE CONGRUENCE LATTICE OF PSEUDO-SIMPLE ALGEBRAS;16
7.1;REFERENCES;21
8;Chapter 2. ELEMENTARY PROPERTIES OF LIMIT REDUCED POWERS WITH APPLICATIONS TO BOOLEAN POWERS;22
8.1;REFERENCES;25
9;Chapter 3. ON CONGRUENCE LATTICES OF 2-VALUED ALGEBRAS;28
9.1;REFERENCES;32
10;Chapter 4. THE c-IDEAL LATTICE AND SUBALGEBRA LATTICE ARE INDEPENDENT;34
10.1;1. PRELIMINARIES;34
10.2;2. THE CONCRETE CHARACTERIZATION;35
10.3;3. PROOF OF THE MAIN THEOREM;37
10.4;4. SOME REMARKS ON RELATED REPRESENTATIONPROBLEMS;39
10.5;REFERENCES;39
11;Chapter 5. /-GROUP-CONE AND BOOLEAN ALGEBRA. A COMMON ONE-IDENTITY-AXIOM;42
11.1;REFERENCES;57
12;Chapter 6. SPLITTING LATTICES AND CONGRUENCE MODULARITY*;58
12.1;§1. INTRODUCTION;58
12.2;§2. PRELIMINARIES;59
12.3;§3. THE CLASS;61
12.4;§5. CONCLUDING REMARKS;71
12.5;REFERENCES;72
13;Chapter 7. SOME REMARKS ON WEAK AUTOMORPHISMS;74
13.1;0. INTRODUCTION;74
13.2;1. HOLOMORPH AND WEAK AUTOMORPHISMS;76
13.3;2. REDUCIBLE ALGEBRAS;77
13.4;REFERENCES;81
14;Chapter 8. ON POLYNOMIAL ALGEBRAS;84
14.1;1. INTRODUCTION;84
14.2;2. SOME BASIC PROPERTIES OF K -POLYNOMIAL ALGEBRAS;85
14.3;3. ON THE CANONICAL HOMOMORPHISM FROM FREEALGEBRAS INTO POLYNOMIAL ALGEBRAS;88
14.4;4. SOME GENERALIZATIONS FROM POLYNOMIAL ALGEBRASTO UNIVERSAL PAIRS OVER PARTIAL ALGEBRAS;90
14.5;5. POLYNOMIAL ALGEBRAS AND POLYNOMIAL FUNCTION ALGEBRAS;92
14.6;REFERENCES;99
15;Chapter 9. DUALITY FOR ALGEBRAS;102
15.1;INTRODUCTION;102
15.2;TERMINOLOGY AND NOTATION;103
15.3;1. CHARACTERISTIC MAPS;103
15.4;2. THE REPRESENTATION THEOREM;106
15.5;3. SUFFICIENT CONDITIONS FOR BICENTRALITY;108
15.6;REFERENCES;112
16;Chapter 10. PROJECTIVE AND INJECTIVE VARIETIES OF ABELIAN O-ALGEBRAS;114
16.1;REFERENCES;132
17;Chapter 11. SOME VARIETIES OF MODULAR LATTICES NOT GENERATED BY THEIR FINITE DIMENSIONAL MEMBERS;134
17.1;REFERENCES;144
18;Chapter 12. ON WEAK HOMOMORPHISMS OF STONE ALGEBRAS;146
18.1;1. WEAK HOMOMORPHISMS OF GENERAL ALGEBRAS;146
18.2;2. REMARKS ON ALGEBRAIC OPERATIONS OF STONE ALGEBRAS;148
18.3;3. WEAK ISOMORPHISMS AND WEAK HOMOMORPHISMSOF STONE ALGEBRAS;155
18.4;REFERENCES;159
19;Chapter 13. ON THE SUMS OF DOUBLE SYSTEMS OF LATTICES AND DS-CONGRUENCES OF LATTICES;162
19.1;REFERENCES;166
20;Chapter 14. n-DISTRIBUTIVITY AND SOME QUESTIONS OF THE EQUATIONAL THEORY OF LATTICES;168
20.1;0. CONTENTS;168
20.2;1. PRELIMINARIES;169
20.3;2. APPLICATIONS OF n-DISTRIBUTIVITY TO THEEQUATIONAL THEORY OF LATTICES;170
20.4;3. ON THE LATTICE GENERATED BY THE VARIETIES;174
20.5;REFERENCES;178
21;Chapter 15. POLYNOMIAL NORMAL FORMS AND THE EMBEDDING OF POLYNOMIAL ALGEBRAS;180
21.1;1. INTRODUCTION;180
21.2;2. SUFFICIENT CONDITIONS FOR fAB TO BEAN EMBEDDING;181
21.3;3. EXAMPLES OF POLYNOMIAL NORMAL FORMS;182
21.4;REFERENCES;188
22;Chapter 16. COVERINGS IN THE LATTICE OF VARIETIES;190
22.1;REFERENCES;203
23;Chapter 17. EMBEDDING SEMIGROUPS IN SEMIGROUPS GENERATED BY IDEMPOTENTS;206
23.1;1. INTRODUCTION;206
23.2;2. EMBEDDING THEOREM;207
23.3;3. COUNTABLE SEMIGROUPS;208
23.4;REFERENCES;209
24;Chapter 18. ENDOMORPHISM SEMIGROUPS AND SUBGROUPOID LATTICES;210
25;Chapter 19. A NOTE ON IMPLICATIONAL SUBCATEGORIES;214
25.1;THE GALOIS-CORRESPONDENCE INDUCED BY INJECTIVITY WITH RESPECT TO A FIXED CLASS OF EPIMORPHISMS;218
25.2;SOME EXAMPLES IN CATEGORIES OF PARTIAL ALGEBRAS;220
25.3;REFERENCES;222
26;Chapter 20. A REPORT ON SUBLATTICES OF A FREE LATTICE;224
26.1;1. INTRODUCTION;224
26.2;2. A SUMMARY OF RESULTS;225
26.3;3. THE FINITELY GENERATED CASE OF THEOREM 2.1: (iii) IMPLIES (ii).;227
26.4;4. THE FINITELY GENERATED CASE:THE PROOF COMPLETED;230
26.5;5. THE INFINITELY GENERATED CASE;233
26.6;6. CYCLES IN FINITE SEMI-DISTRIBUTIVE LATTICES;236
26.7;7. CYCLES IN S-LATTICES: PRELIMINARIES;240
26.8;8. CYCLES IN S-LATTICES;245
26.9;9. A PROOF OF DAY'S THEOREM;247
26.10;10. SUMMARY, PROBLEMS, AND A COUNTEREXAMPLE;250
26.11;REFERENCES;257
27;Chapter 21. EXTENSIVE GROUPOID VARIETIES;260
27.1;1. INTRODUCTION;260
27.2;2. Mod (x = t), t BALANCED;262
27.3;10. MAIN THEOREMS;285
27.4;REFERENCES;286
28;Chapter 22. PRIMITIVE SUBSETS OF ALGEBRAS;288
28.1;INTRODUCTION;288
28.2;1. PRELIMINARIES AND STATEMENT OF RESULTS;289
28.3;2. PROOFS OF THE THEOREMS';290
28.4;REFERENCES;294
29;Chapter 23. IDEALS, NORMAL SETS AND CONGRUENCES;296
29.1;0. SUMMARY AND INTRODUCTION;296
29.2;1. p-DETERMINED CONGRUENCES;298
29.3;2. POLYNOMIALS DETERMINING ALL CONGRUENCES;300
29.4;3. IDEALS;303
29.5;REFERENCES;310
30;Chapter 24. A CHARACTERIZATION OF COMPLETE MODULAR p-ALGEBRAS;312
30.1;1. PRELIMINARIES;313
30.2;2. TRIPLE CHARACTERIZATION OF COMPLETE MODULAR p-ALGEBRAS;316
30.3;3. COMPLETE HOMOMORPHISMS AND COMPLETE SUBALGEBRASOF COMPLETE MODULAR p-ALGEBRAS;320
30.4;4. FILL-IN THEOREMS;326
30.5;REFERENCES;329
31;Chapter 25. JACOBSON'S DENSITY THEOREM IN UNIVERSAL ALGEBRA;332
31.1;REFERENCES;341
32;Chapter 26. CERTAIN QUESTIONS OF THE THEORY OF HOMOTOPY OF UNIVERSAL ALGEBRAS;342
32.1;1. BASIC CONCEPTS;343
32.2;2. HOMOTOPIES AND CONGRUENT FAMILIES OF EQUIVALENCES;345
32.3;3. HOMOTOPIES OF SOME CLASSICAL ALGEBRAIC SYSTEMS;350
32.4;4. SPECIAL MORPIDSMS IN A CATEGORY OF QUASIGROUPS;354
32.5;REFERENCES;355
33;Chapter 27. A THEOREM ON FINITE SUBLATTICES OF FREE LATTICES;358
33.1;REFERENCES;362
34;Chapter 28. A NOTE ON A PROBLEM OF GORALCÍK;364
35;Chapter 29. QUASI-DECOMPOSITIONS, EXACT SEQUENCES, AND TRIPLE SUMS OF SEMIGROUPS. I. GENERAL THEORY;366
35.1;§1. TRIPLES;368
35.2;§2. OUASI-DECOMPOSITIONS;370
35.3;§3. GLIVENKO OPERATORS, EXACTNESS;372
35.4;§4. LOCAL MULTIPLICATIONS, LIMITS;381
35.5;§5. SEMIGROUPS OF SEMIGROUPS;385
35.6;§6. SEMIMODULES, NAGATA'S IDEALIZATION PRODUCT;388
35.7;§7. SEMILATTICES OF SEMIGROUPS;393
35.8;REFERENCES;395
36;Chapter 30. QUASI-DECOMPOSITIONS, EXACT SEQUENCES, AND TRIPLE SUMS OF SEMIGROUPS. II. APPLICATIONS;400
36.1;§8. CLOSURE RETRACTIONS OF SEMILATTICES, LEFT ANDRIGHT BROUWERIAN ELEMENTS;401
36.2;§9. QUASI-DECOMPOSITIONS OF PSEUDO-COMPLEMENTEDSEMILATTICES;415
36.3;§10. QUASI-DECOMPOSITIONS OF BROUWERIAN SEMILATTICES:;421
37;Chapter 31. ENDOMORPHICALLY COMPLETE GROUPS;430
37.1;REFERENCES;435
38;Chapter 32. CONCRETE CATEGORIES WITH NON-INJECTIVE MONOMORPHISM;436
38.1;REFERENCES;440
39;Chapter 33. ON ALGEBRAS AS TREE AUTOMATA;442
39.1;INTRODUCTION;442
39.2;1. PRELIMINARIES;443
39.3;2. CONTEXT-FREE GRAMMARS AND RECOGNIZERS;446
39.4;3. CLASSES OF ALGEBRAS AND FAMILIES OF LANGUAGES;448
39.5;4. GROUPOID-RECOGNIZERS;450
39.6;5. CANTOR ALGEBRAS;453
39.7;REFERENCES;455
40;Chapter 34. ON AFFINE MODULES;458
40.1;REFERENCES;464
41;Chapter 35. EQUATIONAL LOGIC;466
41.1;1. ALGEBRAS;467
41.2;2. FREE ALGEBRAS;467
41.3;3. EQUATIONALLY DEFINED CLASSES;469
41.4;4. EQUATIONAL THEORIES;472
41.5;5. SUBDIRECT REPRESENTATION;473
41.6;6. EXAMPLES OF !–;474
41.7;7. BASES AND GENERIC ALGEBRAS;475
41.8;8. FINITELY BASED THEORIES;476
41.9;9. EQUIVALENT VARIETIES;478
41.10;10. ONE-BASED THEORIES;479
41.11;11. MINIMAL BASES;480
41.12;12. THE LATTICE OF VARIETIES (BEGINNING);481
41.13;13. EFFECTIVELY QUESTIONS;483
41.14;14. THE LATTICE OF VARIETIES (CONTINUED);485
41.15;15. SOME FURTHER INVARIANTS OF THE EQUIVALENCE CLASS OF VARIETY;490
41.16;16. GENERATION OF VARIETIES;493
41.17;17. MALCEV CONDITIONS;497
41.18;18. CONNECTIONS WITH TOPOLOGY;498
41.19;REFERENCES;499
42;Chapter 36. ON REGULAR ALGEBRAS;504
42.1;INTRODUCTION;504
42.2;NOTATIONS;505
42.3;§1. U-REGULARITY;505
42.4;§2. ATOMREGULARITY;510
42.5;§3. DEVIATION FROM THE CLASSICAL CASE;512
42.6;REFERENCES;514
43;Chapter 37. REMARKS ON FULLY INVARIANT CONGRUENCES;516
43.1;§0. INTRODUCTION;516
43.2;§1. ALGEBRAS WITH FEW F.I. CONGRUENCES;519
43.3;§2. NUCLEAR CONGRUENCES;533
43.4;§3. FULLY PRESERVING ENDOMORPHISMS;544
43.5;§4. FULLY INVARIANT SUBSETS;549
43.6;REFERENCES;554
44;Chapter 38. VARIETIES GENERATED BY QUASI-PRIMAL ALGEBRAS HAVE DECIDABLE THEORIES;556
44.1;§ 1. SUBSHEAVES OF CONSTANT SHEAVES:;558
44.2;§2. EMBEDDING SHEAVES INTO CONSTANT SHEAVES;561
44.3;§3. THE MAIN EMBEDDING THEOREM;563
44.4;§4. VARIETIES GENERATED BY OUASI-PRIMAL ALGEBRAS;566
44.5;§5. THE INTUITIONISTIC VERSION OF A CYLINDRIC ALGEBRA;571
44.6;REFERENCES;575
45;Chapter 39. ON THE POLYNOMIAL COMPLETENESS DEFECT OF UNIVERSAL ALGEBRAS;578
45.1;REFERENCES;580
46;Chapter 40. ON LATTICES FREELY GENERATED BY FINITE PARTIALLY ORDERED SETS;582
46.1;REFERENCES;593
47;Chapter 41. PROPER AND IMPROPER FREE ALGEBRAS;596
47.1;REFERENCES;602
48;PROBLEMS;604