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Universal algebra:

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Bibliographic Details
Main Author:	Grätzer, George 1936- (Author)
Format:	Book
Language:	English
Published:	New York, NY Springer 2008
Edition:	2. ed. with updates
Subjects:	Algebra, Universal Universelle Algebra Algebra
Online Access:	Inhaltsverzeichnis
Physical Description:	XVIII, 586 S.
ISBN:	9780387774862 9780387774879

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MARC


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adam_text	CONTENTS PAGE Table of Notation xvii CHAPTER 0. BASIC CONCEPTS §1. Sets and Relations 1 §2. Mappings and Operations 3 §3. Algebras and Relational Systems 7 §4. Partially Ordered Sets 10 §5. Structure of Mappings and Equivalence Relations 17 §6. Ideals and Semilattices 19 Exercises 27 CHAPTER 1. SUBALGEBBAS AND HOMOMOBPHISMS §7. Basic Concepts 33 §8. Polynomial Symbols and Polynomial Algebras 37 §9. Structure of Subalgebras 45 §10. Structure of Congruence Relations 50 §11. The Homomorphism Theorem and Some Isomorphism Theorems 57 §12. Homomorphisms 62 Exercises 70 Probhms 76 CHAPTEB 2. FABTIAL ALGEBRAS §13. Basic Notions 79 §14. Polynomial Symbols over a Partial Algebra 84 §15. Extension of Congruence Relations 91 §16. Subalgebras and Homomorphisms of Partial Algebras 96 §17. The Characterization Theorem of Congruence Lattices: Preliminary Considerations 100 §18. The Characterization Theorem of Congruence Lattices 109 Exercises 113 Probhms 116 CONTENTS CHAPTER 3. CONSTRUCTIONS OF ALGEBRAS §19. Direct Products 118 §20. Subdireet Products of Algebras 122 §21. Direct and Inverse Limits of Algebras 128 §22. Products Associated with the Direct Product 139 §23. Operators on Classes of Algebras 152 Exercises 153 Problems 160 CHAPTER 4. FREE ALGEBRAS §24. Definition and Basic Properties 162 §25. Construction of Free Algebras 166 §26. Identities and Free Algebras 169 §27. Equational Completeness and Identities of Finite Algebras 173 §28. Free Algebras Generated by Partial Algebras 180 §29. Free Products of Algebras 183 §30. Word Problem 186 Exercises 189 Problems 194 CHAPTER 5. INDEPENDENCE §31. Independence and Bases 196 §32. Independence in Special Classes of Algebras 201 §33. Some Invariants of Finite Algebras 205 §34. The System of Independent Sets of an Algebra 209 §35. Generalizations of the Notion of Independence 212 Exercises 216 Problems 222 CHAPTER 6. ELEMENTS OB MODEL THEORY §36. Structures and the First Order Logic 223 §37. Satisfiability and the Case of Boolean Set Algebras 227 P8. Elementary Equivalence and Elementary Extensions 234 §39. Prime Products 239 §40. Prime Powers 246 §41. Two Algebraic Characterizations of Elementary Equivalence 248 §42. Elementary and Axiomatic Classes 255 Exercises 262 Problems 269 CHAPTER 7. ELEMENTARY PROPERTIES OF ALGEBRAIC CONSTRUCTIONS §43. Extensions and Substructures 270 §44. Generalized Atomic Sets of Formulas 275 CONTENTS то §45. Chain Unions and Homomorphisms 278 §46. Direct Products 283 §47. Direct Products {Continued) 287 §48. Subdirect Products 293 Exercises 295 Problems 299 CHAPTER 8. FREE Σ -STRUCTURES §49. Σ -Inverses and Σ -Substruetures 300 §50. Σ -Homomorphisms and Slender Σ -Substructures 305 §51. Free Σ -Structures and the Uniqueness Theorem 307 §52. On the Family of Free Σ -Structures 311 §53. On the Existence of Free Σ -Structures 315 §54. Strong Free Σ -Structures and the Inverse Preserving Property 322 Exercises 326 Problems 328 APPENDIX 1. GENERAL SURVEY §55. A Survey by Sections 331 §56. Related Structures 335 §57. Miscellany 338 APPENDIX 2. THE PROBLEMS §58. Solutions and Partial Solutions of 46 Problems 342 APPENDIX 3. CONGRUENCE VARIETIES by Bjarni Jonsson §59. Algebras and Their Congruence Lattices 348 §60. Mal eev Classes 351 §61. Congruence Varieties 360 §62. Congruence Distributivity and Finite Bases 373 APPENDIX 4. EQUATIONAL LOGIC by Walter Taylor §63. Equationally Defined Classes 378 §64. Equational Theories 380 §65. Equivalent Varieties 381 §66. Bases and Generic Algebras 383 §67. Finitely Based Theories 385 §68. One-Based Theories 387 §69. Irredundant Bases 389 §70. The Lattice of Equational Theories 389 §71. Some Further Invariants of the Equivalence Class of a Variety 392 §72. Mal eev Conditions and Congruence Identities 398 mi CONTENTS APPENDIX б. PRIMALITY: THE INFLUENCE OF BOOLEAN ALGEBBAS IN UNIVERSAL ALGEBBA by Robert W. Quackenbush §73. Introduction 401 §74. Primal Algebras 402 §75. Quasi-Primal Algebras 403 §76. Arithmetical Algebras 405 §77. Para-Primal Algebras 406 §78. Dual-Discriminator Algebras 408 §79. Functional Completeness 409 §80. Representation Theory 410 §81. Congruences 411 §82. Injeetivity and Projectivity 412 §83. Further References and Comments 416 APPENDIX 6. EQUATIONAL COMPACTNESS by Günter H. Wenzel §84. Equational and Atomic Compactness—First Examples 417 §85. Related Compactness Concepts and Characterizations 420 §86. Connections with A 425 §87. The Mycielski Question: Chromatic Numbers and Topology 428 §88. Minimum Compactness 432 §89. Compactification of Algebras 435 §90. Application to Equational Classes of Algebras 441 §91. Concluding Remarks 444 §92. Some Problems 446 APPENDIX 7. THE INDEPENDENCE PROOF by G·. Grätzer and W. A. Lampe §93. Statement of the Main Results 448 §94. Preliminaries 449 §95. Cls-Expansions and Free Extensions 451 §96. Cls-Expansions with Two Orbits 457 §97. Three More Constructions 469 §98. Proof of the Main Theorem 47I Bibliography 475 Adăitional Bibliography 595 Index 665 Epilogue 583
adam_txt	CONTENTS PAGE Table of Notation xvii CHAPTER 0. BASIC CONCEPTS §1. Sets and Relations 1 §2. Mappings and Operations 3 §3. Algebras and Relational Systems 7 §4. Partially Ordered Sets 10 §5. Structure of Mappings and Equivalence Relations 17 §6. Ideals and Semilattices 19 Exercises 27 CHAPTER 1. SUBALGEBBAS AND HOMOMOBPHISMS §7. Basic Concepts 33 §8. Polynomial Symbols and Polynomial Algebras 37 §9. Structure of Subalgebras 45 §10. Structure of Congruence Relations 50 §11. The Homomorphism Theorem and Some Isomorphism Theorems 57 §12. Homomorphisms 62 Exercises 70 Probhms 76 CHAPTEB 2. FABTIAL ALGEBRAS §13. Basic Notions 79 §14. Polynomial Symbols over a Partial Algebra 84 §15. Extension of Congruence Relations 91 §16. Subalgebras and Homomorphisms of Partial Algebras 96 §17. The Characterization Theorem of Congruence Lattices: Preliminary Considerations 100 §18. The Characterization Theorem of Congruence Lattices 109 Exercises 113 Probhms 116 CONTENTS CHAPTER 3. CONSTRUCTIONS OF ALGEBRAS §19. Direct Products 118 §20. Subdireet Products of Algebras 122 §21. Direct and Inverse Limits of Algebras 128 §22. Products Associated with the Direct Product 139 §23. Operators on Classes of Algebras 152 Exercises 153 Problems 160 CHAPTER 4. FREE ALGEBRAS §24. Definition and Basic Properties 162 §25. Construction of Free Algebras 166 §26. Identities and Free Algebras 169 §27. Equational Completeness and Identities of Finite Algebras 173 §28. Free Algebras Generated by Partial Algebras 180 §29. Free Products of Algebras 183 §30. Word Problem 186 Exercises 189 Problems 194 CHAPTER 5. INDEPENDENCE §31. Independence and Bases 196 §32. Independence in Special Classes of Algebras 201 §33. Some Invariants of Finite Algebras 205 §34. The System of Independent Sets of an Algebra 209 §35. Generalizations of the Notion of Independence 212 Exercises 216 Problems 222 CHAPTER 6. ELEMENTS OB' MODEL THEORY §36. Structures and the First Order Logic 223 §37. Satisfiability and the Case of Boolean Set Algebras 227 P8. Elementary Equivalence and Elementary Extensions 234 §39. Prime Products 239 §40. Prime Powers 246 §41. Two Algebraic Characterizations of Elementary Equivalence 248 §42. Elementary and Axiomatic Classes 255 Exercises 262 Problems 269 CHAPTER 7. ELEMENTARY PROPERTIES OF ALGEBRAIC CONSTRUCTIONS §43. Extensions and Substructures 270 §44. Generalized Atomic Sets of Formulas 275 CONTENTS то §45. Chain Unions and Homomorphisms 278 §46. Direct Products 283 §47. Direct Products {Continued) 287 §48. Subdirect Products 293 Exercises 295 Problems 299 CHAPTER 8. FREE Σ -STRUCTURES §49. Σ -Inverses and Σ -Substruetures 300 §50. Σ -Homomorphisms and Slender Σ -Substructures 305 §51. Free Σ -Structures and the Uniqueness Theorem 307 §52. On the Family of Free Σ -Structures 311 §53. On the Existence of Free Σ -Structures 315 §54. Strong Free Σ -Structures and the Inverse Preserving Property 322 Exercises 326 Problems 328 APPENDIX 1. GENERAL SURVEY §55. A Survey by Sections 331 §56. Related Structures 335 §57. Miscellany 338 APPENDIX 2. THE PROBLEMS §58. Solutions and Partial Solutions of 46 Problems 342 APPENDIX 3. CONGRUENCE VARIETIES by Bjarni Jonsson §59. Algebras and Their Congruence Lattices 348 §60. Mal'eev Classes 351 §61. Congruence Varieties 360 §62. Congruence Distributivity and Finite Bases 373 APPENDIX 4. EQUATIONAL LOGIC by Walter Taylor §63. Equationally Defined Classes 378 §64. Equational Theories 380 §65. Equivalent Varieties 381 §66. Bases and Generic Algebras 383 §67. Finitely Based Theories 385 §68. One-Based Theories 387 §69. Irredundant Bases 389 §70. The Lattice of Equational Theories 389 §71. Some Further Invariants of the Equivalence Class of a Variety 392 §72. Mal'eev Conditions and Congruence Identities 398 mi CONTENTS APPENDIX б. PRIMALITY: THE INFLUENCE OF BOOLEAN ALGEBBAS IN UNIVERSAL ALGEBBA by Robert W. Quackenbush §73. Introduction 401 §74. Primal Algebras 402 §75. Quasi-Primal Algebras 403 §76. Arithmetical Algebras 405 §77. Para-Primal Algebras 406 §78. Dual-Discriminator Algebras 408 §79. Functional Completeness 409 §80. Representation Theory 410 §81. Congruences 411 §82. Injeetivity and Projectivity 412 §83. Further References and Comments 416 APPENDIX 6. EQUATIONAL COMPACTNESS by Günter H. Wenzel §84. Equational and Atomic Compactness—First Examples 417 §85. Related Compactness Concepts and Characterizations 420 §86. Connections with \A\ 425 §87. The Mycielski Question: Chromatic Numbers and Topology 428 §88. Minimum Compactness 432 §89. Compactification of Algebras 435 §90. Application to Equational Classes of Algebras 441 §91. Concluding Remarks 444 §92. Some Problems 446 APPENDIX 7. THE INDEPENDENCE PROOF by G·. Grätzer and W. A. Lampe §93. Statement of the Main Results 448 §94. Preliminaries 449 §95. Cls-Expansions and Free Extensions 451 §96. Cls-Expansions with Two Orbits 457 §97. Three More Constructions 469 §98. Proof of the Main Theorem 47I Bibliography 475 Adăitional Bibliography 595 Index 665 Epilogue 583
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spelling	Grätzer, George 1936- Verfasser (DE-588)10907873X aut Universal algebra George Grätzer 2. ed. with updates New York, NY Springer 2008 XVIII, 586 S. txt rdacontent n rdamedia nc rdacarrier Algebra, Universal Universelle Algebra (DE-588)4061777-4 gnd rswk-swf Algebra (DE-588)4001156-2 gnd rswk-swf Universelle Algebra (DE-588)4061777-4 s DE-604 Algebra (DE-588)4001156-2 s 1\p DE-604 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016753993&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk
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subject_GND	(DE-588)4061777-4 (DE-588)4001156-2
title	Universal algebra
title_auth	Universal algebra
title_exact_search	Universal algebra
title_exact_search_txtP	Universal algebra
title_full	Universal algebra George Grätzer
title_fullStr	Universal algebra George Grätzer
title_full_unstemmed	Universal algebra George Grätzer
title_short	Universal algebra
title_sort	universal algebra
topic	Algebra, Universal Universelle Algebra (DE-588)4061777-4 gnd Algebra (DE-588)4001156-2 gnd
topic_facet	Algebra, Universal Universelle Algebra Algebra
url	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016753993&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
work_keys_str_mv	AT gratzergeorge universalalgebra

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