Universal algebra and coalgebra:
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New Jersey [u.a.]
World Scientific
2009
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XII, 278 S graph. Darst. |
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| 245 | 1 | 0 | |a Universal algebra and coalgebra |c Klaus Denecke ; Shelly L Wismath |
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Record in the Search Index
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| adam_text | Titel: Universal algebra and coalgebra
Autor: Denecke, Klaus
Jahr: 2009
Contents
Preface vii
Introduction 1
1. Algebras and Identities 7
1.1 Subalgebras and Homomorphic Images........... 8
1.2 Direct and Subdirect Products............... 17
1.3 Term Algebras, Identities and Free Algebras........ 21
1.4 The Galois Connection {Id, Mod).............. 26
1.5 Multi-based Algebras and Clones.............. 29
1.6 Exercises for Chapter 1 ................... 32
2. State-based Systems 35
2.1 Black Boxes.......................... 36
2.2 Data Streams......................... 37
2.3 Data Types.......................... 38
2.4 Automata........................... 40
2.5 Automata with Error Conditions.............. 46
2.6 Object-oriented Programming................ 47
2.7 Kripke Structures....................... 48
2.8 The Concept of a Coalgebra................. 50
2.9 Homomorphisms of State-based Systems.......... 52
2.10 Behavioural Equivalence................... 59
2.11 Exercises for Chapter 2 ................... 66
3. Basic Concepts from Category Theory 69
3.1 The Concept of a Category................. 69
x Universal Algebra and Coalgebra
3.2 Special Morphisms...................... 70
3.3 Terminal and Initial Objects ................ 72
3.4 Sums and Products...................... 72
3.5 Equalizers and Coequalizers................. 74
3.6 Pushouts and Pullbacks................... 76
3.7 The Category Sei....................... 79
3.8 Functors............................ 84
3.9 Natural Transformations................... 87
3.10 Exercises for Chapter 3 ................... 87
4. F-Coalgebras 89
4.1 Coalgebras and Homomorphisms.............. 89
4.2 Subcoalgebras......................... 93
4.3 Homomorphic Images and Factorizations.......... 94
4.4 Congruences and Factor Coalgebras ............ 96
4.5 Colimits in SeiF....................... 97
4.6 Bisimulations......................... 101
4.7 Epi s and Mono s in SetF.................. 107
4.8 Congruences.......................... 108
4.9 Covarieties .......................... 110
4.10 Exercises for Chapter 4 ................... Ill
5. F-Algebras 113
5.1 The Concept of an F-Algebra................ 113
5.2 Homomorphisms of F-Algebras............... 114
5.3 Subalgebras of F-Algebras.................. 117
5.4 Homomorphic Images and Factorizations.......... 117
5.5 Factor Algebras........................ 119
5.6 Limits in SeiF ........................ 120
5.7 Congruences.......................... 123
5.8 Mono s and Epi s in SeiF.................. 125
5.9 Varieties of F-Algebras ................... 125
5.10 Exercises for Chapter 5 ................... 126
6. (F1,F2)-Coalgebras 129
6.1 Tree Automata as (Fl,F2)-Coalgebras........... 129
6.2 (Fi,F2)-Coalgebras and Their Homomorphisms...... 135
6.3 Limits and Colimits in Sei(Flir2).............. 137
Contents xi
6.4 Exercises for Chapter 6 ................... 144
7. Terminal Coalgebras 145
7.1 Terminal Coalgebras and Bisimulations .......... 145
7.2 Terminal Automata ..................... 146
7.3 Existence of Terminal Coalgebras.............. 148
7.4 Weakly Terminal Coalgebras ................ 149
7.5 Restricted Functors...................... 150
7.6 Exercises for Chapter 7 ................... 153
8. Cofree F-coalgebras and Coequations 155
8.1 Cofree F-coalgebras..................... 155
8.2 Coequations.......................... 159
8.3 The Co-Birkhoff Theorem.................. 161
8.4 Exercises for Chapter 8 ................... 164
9. Coalgebras of Type r 167
9.1 Homomorphisms, Subcoalgebras and Sums ........ 168
9.2 Congruences and Quotient Coalgebras........... 171
9.3 Bisimulations......................... 176
9.4 Terminal Coalgebras of Type r............... 177
9.5 Coterms............................ 178
9.6 Superposition of Co-operations............... 179
9.7 Coidentities and Covarieties................. 180
9.8 Exercises for Chapter 9 ................... 185
10. Clones of Operations and Co-operations 187
10.1 Clones as Algebraic Structures............... 189
10.2 The Lattice of All Boolean Clones............. 193
10.3 The Functional Completeness Problem........... 197
10.4 Clones of Algebras and Varieties.............. 200
10.5 Superposition and Clones of Co-operations ........ 201
10.6 Clones of Boolean Co-operations.............. 205
10.7 Exercises for Chapter 10................... 211
11. Semigroups of Operations and Co-operations 213
11.1 Basic Definitions and Properties .............. 213
11.2 Unary Operations ...................... 215
xii Universal Algebra and Coalgebra
11.3 From Unary to n-ary..................... 221
11.4 Bands of n-ary Operations.................. 226
11.5 Semigroups of Boolean Operations............. 228
11.6 Semigroups of Boolean Co-operations ........... 237
11.7 Elements of Semigroups of n-ary Co-operations...... 240
11.8 Bands of n-ary Co-operations................ 244
11.9 Green s Relations £ and U................. 246
11.10 Exercises for Chapter 11................... 247
12. Cohyperidentities and M-solid Classes of Coalgebras 251
12.1 Clones of Coterms...................... 251
12.2 Cohypersubstitutions and Cohyperidentities........ 253
12.3 Separation of Clones of Boolean Co-operations by Cohy-
peridentities ...................... .... 257
12.4 Exercises for Chapter 12................... 263
Bibliography 265
Glossary 271
Index 273
|
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| language | English |
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| spelling | Denecke, Klaus Verfasser aut Universal algebra and coalgebra Klaus Denecke ; Shelly L Wismath New Jersey [u.a.] World Scientific 2009 XII, 278 S graph. Darst. txt rdacontent n rdamedia nc rdacarrier Algebra, Universal Koalgebra (DE-588)4164324-0 gnd rswk-swf Universelle Algebra (DE-588)4061777-4 gnd rswk-swf Universelle Algebra (DE-588)4061777-4 s Koalgebra (DE-588)4164324-0 s DE-604 Wismath, Shelly L. Verfasser aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017720795&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Denecke, Klaus Wismath, Shelly L. Universal algebra and coalgebra Algebra, Universal Koalgebra (DE-588)4164324-0 gnd Universelle Algebra (DE-588)4061777-4 gnd |
| subject_GND | (DE-588)4164324-0 (DE-588)4061777-4 |
| title | Universal algebra and coalgebra |
| title_auth | Universal algebra and coalgebra |
| title_exact_search | Universal algebra and coalgebra |
| title_full | Universal algebra and coalgebra Klaus Denecke ; Shelly L Wismath |
| title_fullStr | Universal algebra and coalgebra Klaus Denecke ; Shelly L Wismath |
| title_full_unstemmed | Universal algebra and coalgebra Klaus Denecke ; Shelly L Wismath |
| title_short | Universal algebra and coalgebra |
| title_sort | universal algebra and coalgebra |
| topic | Algebra, Universal Koalgebra (DE-588)4164324-0 gnd Universelle Algebra (DE-588)4061777-4 gnd |
| topic_facet | Algebra, Universal Koalgebra Universelle Algebra |
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