New trends in quantum structures:
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| Main Authors: | , |
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| Format: | Book |
| Language: | English |
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Netherlands
Springer
2010
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| Edition: | [1. ed, softcover version] |
| Series: | Mathematics and its applications
516 |
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| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XVI, 541 S. graph. Darst. |
| ISBN: | 9789048155255 9048155258 |
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| 245 | 1 | 0 | |a New trends in quantum structures |c by Anatolij Dvurečenskij and Sylvia Pulmannová |
| 250 | |a [1. ed, softcover version] | ||
| 264 | 1 | |a Netherlands |b Springer |c 2010 | |
| 300 | |a XVI, 541 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
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| adam_text | Contents
Preface
vii
Introduction
1
CHAPTER
1
D-posets and Effect Algebras
9
1.1
D-posets
...................................................................10
1.2
Effect Algebras
............................................................15
1.3
Morphisms and Categorical Equivalence of D-posets and Effect Algebras
.... 20
1.4
Group Representations of Effect Algebras. Interval Effect Algebras
..........22
1.4.1
Examples of Interval Algebras
.......................................32
1.5
Generalized Orthoalgebras and OMP as Orthomodular Structures
...........35
1.6
Examples and Applications
.................................................42
1.7
Effect Algebras with the Riesz Decomposition Property
.....................58
1.8
D-lattices and Lattice-ordered Effect Algebras
..............................69
1.9
Some Basic Constructions with D-posets/Effect Algebras
....................80
1.9.1
Subalgebras and Ideals
..............................................80
1.9.2
Center and Direct Products
.........................................82
1.9.3
Principal and Sharp Elements in Effect Rings
........................86
1.9.4
Finite Distributive Lattice Ordered Effect Algebras
..................87
1.9.5
Direct Products of Interval Algebras
................................91
1.9.6
Horizontal Sums
....................................................92
1.9.7
Direct Limits of D-posets/Effect Algebras
...........................94
1.10
Compatibility and
Observables
............................................97
1.10.1
Compatibility in Orthomodular Structures
.........................97
1.10.2
Compatibility in GDLs
...........................................102
1.10.3
Observables
......................................................109
1.11
Order Properties of £(H)
................................................117
1.12
Exercises
................................................................121
CHAPTER
2
MV-algebras and QMV-algebras
129
2.1
S-algebras
................................................................130
2.2
MV-algebras
..............................................................132
2.2.1
Basic Properties
...................................................132
2.2.2
Congruences and Ideals in MV-algebras
............................143
2.2.3
Subdirect
Representation Theorem
.................................147
2.2.4 Semisimple MV-algebras
and Bold Fuzzy Set Theory
...............148
2.3
QMV-algebras
............................................................152
2.3.1
Basic Properties of QMV-algebras
..................................153
2.3.2
Examples of QMV-algebras
........................................158
2.3.3
Commutativity in QMV-algebras
...................................162
2.3.4
Congruences and Ideals in Quantum MV-algebras
..................167
2.3.5
Lattices of Ideals in QMV-algebras
.................................177
2.3.6
Quotients of QMV-algebras
........................................181
2.3.7
Quasilinearity and Weak Linearity
.................................185
2.3.8
Open Problems
....................................................189
CHAPTER
3
Quotients of Partial Abelian Monoids
191
3.1
Congruences and Ideals on Partial Abelian Monoids
.......................192
3.1.1
Congruences and Quotients
........................................193
3.1.2
Riesz Ideals on CPAMs
............................................ 198
3.1.3
Congruences and Ideals in Effect Algebras
..........................202
3.2
Some Lattices of Ideals in Positive CPAMs
................................211
3.2.1
Лі
-ideals and Riesz Ideals
..........................................211
3.3
Applications to Dimension Theory
.........................................220
3.3.1
Congruences and Dimensional Equivalence
.........................221
3.3.2
Relations to Ko of AF C*-algebras
.................................223
3.4
Exercises
.................................................................227
CHAPTER
4
Tensor Product of D-Posets and Effect Algebras
231
LI D-test Spaces and Difference Posets
.......................................232
4.1.1
D-test Spaces
......................................................232
4.1.2
D-algebraic D-test Spaces
..........................................234
4.1.3
D-Test Spaces versus Difference Posets
.............................236
4.1.4
Examples of D-Test Spaces
........................................237
4.1.5
Weights on D-test spaces
..........................................240
4.1.6
Morphisms and Bimorphisms
...................................... 241
4.1.7
Tensor Products of D-test Spaces
..................................243
4.2
Tensor Product of Difference Posets
........................................244
4.2.1
Tensor Product
....................................................244
4.2.2
Bounded Boolean Powers and Tensor Products
......................247
4.2.3
Tensor Product of D-posets and D-test Spaces
......................251
4.2.4
State Tensor Product
..............................................253
4.2.5
Weight Tensor Product of D-test Spaces
............................259
4.2.6
Examples of State Tensor Products
.................................264
4.3
Partition Logics, Orthoalgebras and Automata
.............................267
4.3.1
Boolean Atlases
...................................................267
4.3.2
Quasi Orthoalgebras and Orthoalgebras
............................268
4.3.3
Examples of Orthoalgebras
........................................269
4.3.4
Relations among Boolean Atlases and Quasi Orthoalgebras
.........274
4.3.5
Partition Logics
...................................................275
4.3.6
Partition Logics and Automata Logics
..............................277
4.3.7
Partition Logics in Examples
.......................................279
4.3.8
Partition Test Spaces
..............................................283
4.3.9
Product of Partition Logics
........................................285
4.3.10
Exercises
.........................................................289
CHAPTERS BCK-algebras
293
5.1
Elements of BCK-algebras
................................................293
5.1.1
Definitions and Elementary Properties of BCK-algebras
............293
5.1.2
Examples of BCK-algebras
.........................................296
5.1.3
Commutative BCK-algebras
.......................................299
5.1.4
BCK-algebras with the Condition (S)
...............................304
5.1.5
Union BCK-algebras
............................................... 306
5.1.6
Direct Product of BCK-algebras
...................................307
5.1.7
Ideals of a BCK-algebra
...........................................308
5.1.8
Measures on BCK-algebras
........................................312
5.1.9
Exercises
..........................................................319
5.2
Commutative BCK-algebras with the Relative Cancellation Property
.......320
5.2.1
Basic Properties of BCK-algebras with the R. C. Property
..........321
5.2.2
BCK-algebras and the Riesz Decomposition Property
...............324
5.2.3
Ideals and Prime Ideals
............................................326
5.2.4
Subdirect
Product of Linear Commutative BCK-algebras
...........328
5.2.5
Embedding of Commutative BCK-algebras into ¿-groups
............330
5.2.6
Characterizations
of
Comm. BCK-algebras
with the
R.C.
Property
. 334
5.2.7
Universal Groups for Commutative BCK-algebras
..................336
5.2.8
BCK-hull of Commutative BCK-algebras
...........................339
5.2.9
Relations among BCK-algebras and Universal Groups
..............342
5.2.10
Ideals of BCK-algebras and f-ideals
...............................345
5.2.11
Quasi Strong Units of BCK-algebras
..............................350
5.2.12
Exercises
.........................................................351
5.3
Representation of Commutative BCK-algebras
.............................352
5.3.1
Categorical Equivalences of Commutative BCK-algebras
............353
5.3.2
Categorical Representation of Bounded Commutative BCK-algebras
357
5.3.3
Categorical Representation of Comm. BCK-algebras with Q. S. Unit
358
5.3.4
Product on MV-algebras
...........................................359
5.3.5
Product on Commutative BCK-algebras
............................370
5.3.6
Ideals in Product BCK-algebras and BCKf-algebras
................373
5.3.7
Exercises
..........................................................376
CHAPTER
6
BCK-algebras in Applications
379
6.1
Algebraic Properties of Commutative BCK-algebras
.......................379
6.1.1 Semisimple BCK-algebras ..........................................379
6.1.2
Topologies and Semisimplicity
.....................................382
6.1.3
Fuzzy set representation of BCK-algebras
..........................385
6.1.4
Atomic and
Semisimple BCK-algebras..............................386
6.1.5
Measure-Morphisms and Maximal Ideals
...........................390
6.1.6
State Space of a BCK-algebra
......................................393
6.1.7
Simple Commutative BCK-algebras
................................399
6.1.8
Exercises
..........................................................403
6.2
Dedekind Complete BCK-algebras
.........................................405
6.2.1
Dedekind Complete Commutative BCK-algebras
...................405
6.2.2
Decomposition of Dedekind Complete Atomic BCK-algebras
........413
6.2.3
Exercises
..........................................................417
6.3
Connections between BCK-algebras and Difference Posets
..................419
6.3.1
Exercises
..........................................................427
6.4
Pseudo
MV-algebras
......................................................428
6.4.1
On Partial Addition in
Pseudo
MV-algebras
........................428
6.4.2
Commutativity of Atomic
Pseudo
MV-algebras
.....................435
6.4.3
Commutativity of Linear
Pseudo
MV-algebras
......................438
6.4.4
Ideals of
Pseudo
MV-algebras
......................................439
6.4.5
Exercises
..........................................................444
CHAPTER
7
Loomis-Sikorski Theorems for MV-algebras and
BCK-algebras
447
7.1
Loomis-Sikorski s Theorem for a-complete MV-algebras
...................448
7.1.1
Bold Algebras of Fuzzy Sets
.......................................448
7.1.2
a-complete MV-algebras and Tribes
................................451
7.1.3
Loomis-Sikorski s Theorem
........................................458
7.1.4
Loomis-Sikorski Theorem for Dedekind
σ
-complete
¿-groups
........465
7.2
Loomis-Sikorski s Theorem for MV-algebras and BCK-algebras
............468
7.2.1
Loomis-Sikorski s Theorem for Perfect MV-algebras
................468
7.2.2
Loomis-Sikorski s Theorem for Product BCK-algebras
..............471
7.3
Weakly Divisible MV-algebras
.............................................474
7.4
MV-observables
...........................................................478
7.4.1
MV-observables
...................................................478
7.4.2
Calculus of MV-observables
........................................483
7.4.3
Exercises
..........................................................488
Bibliography
491
Index of Symbols
531
Index
535
List of Figures and Tables
Fig.
1.1 ....................................................................13
Fig.
1.2 ....................................................................43
Fig.
1.3 ....................................................................44
Fig.
1.4 ....................................................................47
Fig.
1.5 ....................................................................47
Fig.
1.6 ....................................................................48
Fig.
1.7,
Diamond
..........................................................88
Fig.
1.8,
Frazer cube
.......................................................99
Fig.
1.9,
Fano
plane
.......................................................101
Tab.
2.0.
S-algebra vith non-transitive order
...............................132
Fig.
2.1 ...................................................................177
Fig.
2.2
Me
..............................................................182
Fig.
4.1,
Firefly in a box
..................................................270
Fig.
4.2 ...................................................................270
Fig.
4.3 ...................................................................271
Tab. 4.4 ..................................................................271
Fig. 4.5,
Firefly in a three chamber box
....................................272
Fig.
4.6,
Wright triangle
...................................................273
Fig.
4.7...................................................................273
Tab.
4.8 ..................................................................274
Fig.
4.9,
Fano
plane
.......................................................279
Tab.
4.10 .................................................................280
Tab.
4.11 .................................................................280
Fig.
4.12 ..................................................................281
Tab.
4.13 .................................................................281
Tab.
4.14 .................................................................282
Fig.
4.15 ..................................................................282
Fig.
4.16 ..................................................................283
Fig.
4.17..................................................................287
Fig.
4.18 ..................................................................289
Tab.
5.1 ..................................................................297
Fig.
5.1 ...................................................................297
Tab.
5.2 ..................................................................298
Fig.
5.2 ...................................................................298
Tab.
5.3 ..................................................................298
Fig.
5.3 ...................................................................298
Tab.
5.4 ..................................................................306
Fig.
5.4 ...................................................................306
Tab.
5.5 ..................................................................308
Tab.
5.5 ..................................................................308
Tab.
5.6 ..................................................................321
Fig.
5.6 ...................................................................321
Tab.
5.7 ..................................................................331
|
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| id | DE-604.BV036851582 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T22:49:23Z |
| institution | BVB |
| isbn | 9789048155255 9048155258 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-020767536 |
| oclc_num | 706032854 |
| open_access_boolean | |
| owner | DE-739 |
| owner_facet | DE-739 |
| physical | XVI, 541 S. graph. Darst. |
| publishDate | 2010 |
| publishDateSearch | 2010 |
| publishDateSort | 2010 |
| publisher | Springer |
| record_format | marc |
| series | Mathematics and its applications |
| series2 | Mathematics and its applications |
| spelling | Dvurečenskij, Anatolij 1949- Verfasser (DE-588)172681979 aut New trends in quantum structures by Anatolij Dvurečenskij and Sylvia Pulmannová [1. ed, softcover version] Netherlands Springer 2010 XVI, 541 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Mathematics and its applications 516 Quantentheorie (DE-588)4047992-4 gnd rswk-swf Mathematisches Modell (DE-588)4114528-8 gnd rswk-swf Mathematik (DE-588)4037944-9 gnd rswk-swf Quantentheorie (DE-588)4047992-4 s Mathematisches Modell (DE-588)4114528-8 s DE-604 Mathematik (DE-588)4037944-9 s Pulmannová, Sylvia Verfasser aut Mathematics and its applications 516 (DE-604)BV008163334 516 Digitalisierung UB Passau application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020767536&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Dvurečenskij, Anatolij 1949- Pulmannová, Sylvia New trends in quantum structures Mathematics and its applications Quantentheorie (DE-588)4047992-4 gnd Mathematisches Modell (DE-588)4114528-8 gnd Mathematik (DE-588)4037944-9 gnd |
| subject_GND | (DE-588)4047992-4 (DE-588)4114528-8 (DE-588)4037944-9 |
| title | New trends in quantum structures |
| title_auth | New trends in quantum structures |
| title_exact_search | New trends in quantum structures |
| title_full | New trends in quantum structures by Anatolij Dvurečenskij and Sylvia Pulmannová |
| title_fullStr | New trends in quantum structures by Anatolij Dvurečenskij and Sylvia Pulmannová |
| title_full_unstemmed | New trends in quantum structures by Anatolij Dvurečenskij and Sylvia Pulmannová |
| title_short | New trends in quantum structures |
| title_sort | new trends in quantum structures |
| topic | Quantentheorie (DE-588)4047992-4 gnd Mathematisches Modell (DE-588)4114528-8 gnd Mathematik (DE-588)4037944-9 gnd |
| topic_facet | Quantentheorie Mathematisches Modell Mathematik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020767536&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV008163334 |
| work_keys_str_mv | AT dvurecenskijanatolij newtrendsinquantumstructures AT pulmannovasylvia newtrendsinquantumstructures |