Ordered sets: an introduction
"Ordered Sets: An Introduction is an excellent text for undergraduate and graduate students and a good resource for the interested researcher. Readers will discover order theory's role in discrete mathematics as a supplier of ideas as well as an attractive source of applications."--BO...
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Boston [u.a.]
Birkhäuser
2003
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Summary: | "Ordered Sets: An Introduction is an excellent text for undergraduate and graduate students and a good resource for the interested researcher. Readers will discover order theory's role in discrete mathematics as a supplier of ideas as well as an attractive source of applications."--BOOK JACKET. |
| Physical Description: | XVII, 391 S. graph. Darst. |
| ISBN: | 0817641289 3764341289 |
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MARC
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| 100 | 1 | |a Schröder, Bernd S. W. |d 1966- |e Verfasser |0 (DE-588)135918871 |4 aut | |
| 245 | 1 | 0 | |a Ordered sets |b an introduction |c Bernd S. W. Schröder |
| 264 | 1 | |a Boston [u.a.] |b Birkhäuser |c 2003 | |
| 300 | |a XVII, 391 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 520 | 1 | |a "Ordered Sets: An Introduction is an excellent text for undergraduate and graduate students and a good resource for the interested researcher. Readers will discover order theory's role in discrete mathematics as a supplier of ideas as well as an attractive source of applications."--BOOK JACKET. | |
| 650 | 4 | |a Ensembles ordonnés | |
| 650 | 4 | |a Ordered sets | |
| 650 | 0 | 7 | |a Geordnete Menge |0 (DE-588)4156748-1 |2 gnd |9 rswk-swf |
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Record in the Search Index
| _version_ | 1804129250684436480 |
|---|---|
| adam_text | Contents
Table of
Contents..........................
v
Concept Map of the Contents
.................... ix
Preface
............................... xi
Acknowledgements
........................ . xvi
1
The Basics
1
1.1
Definition and Examples
...................... 1
1.2
The Diagram
............................ 4
1.3
Order-Preserving Mappings/Isomorphism
............. 9
1.4
Fixed Points
............................. 12
1.5
Ordered Subsets/The Reconstruction Problem
........... 15
Exercises
.............................. 19
Remarks and Open Problems
.................... 22
2
Chains, Antichains and Fences
25
2.1
Chains and Zorn s Lemma
..................... 25
2.2
Well-ordered Sets
.......................... 29
2.3
A Remark on Duality
....................... . 32
2.4
The Rank of an Element
...................... 34
2.5
Antichains and Dilworth s Chain Decomposition Theorem
.... 36
2.6
Dedekind Numbers
......................... 39
2.7
Fences and Crowns
......................... 42
2.8
Connectivity
............................. 44
Exercises
.............................. 47
Remarks and Open Problems
.................... 51
Upper and Lower Bounds
55
3.1
Extremal Elements
......................... 55
3.2
Covers
................................ 62
3.3
Lowest Upper and Greatest Lower Bounds
............. 64
3.4
Chain-Completeness and the Abian-Brown Theorem
....... 69
Exercises
.............................. 72
Remarks and Open Problems
.................... 76
Retractions
77
4.1
Definition and Examples
...................... 77
4.2
Fixed Point Theorems
........................ 81
4.2.1
Removing Points
...................... 82
4.2.2
The Comparable Point Property
.............. 84
4.3
Dismantlability
........................... 86
4.3.1
(Connectedly) Collapsible Ordered Sets
.......... 90
4.4
The Fixed Point Property for Ordered Sets of Width
2
or Height
1 91
4.4.1
Width
2........................... 91
4.4.2
Height
1........................... 92
4.4.3
Minimal Automorphic Ordered Sets
............ 93
4.4.4
A Fixed Point Property for Graphs?
............ 94
4.5
Li and Milner s Structure Theorem
................. 95
4.6
Isotone Relations
.......................... 101
Exercises
.............................. 103
Remarks and Open Problems
.................... 108
Lattices 111
5.1
Definition and Examples
......................
Ill
5.2
Fixed Point Results/The Tarski-Davis Theorem
..........114
5.2.1
Preorders/The
Bernstein-Cantor-Schröder
Theorem
... 114
5.2.2
Other Results on the Structure of Fixed Point Sets
.....116
5.3
Embeddingsfflie
Dedekind-MacNeille Completion
....... . 119
5.4
Irreducible Points in Lattices
....................122
5.5
Finite Ordered Sets vs. Distributive Lattices
............123
5.6
More on Distributive Lattices
....................126
Exercises
..............................130
Remarks and Open Problems
....................135
Truncated Lattices
137
6.1
Definition and Examples
......................137
6.2
Recognizability and More
......................138
6.3
The Fixed Clique Property
.....................141
6.3.1
Simplicial Complexes
...................143
6.3.2
The Fixed Point Property for Truncated Lattices
......147
6.4
Triangulations
of
5 .........................148
6.5
Cutsets
................................152
6.6
Truncated
Noncomplemented
Lattices
...............154
Exercises
.............................. 156
Remarks and Open Problems
....................158
7
The Dimension of Ordered Sets
161
7.1
(Linear) Extensions of Orders
................... 161
7.2
Balancing Pairs
........................... 164
7.3
Defining the Dimension
.......................168
7.3.1
A Characterization of Realizers
..............171
7.4
Bounds on the Dimension
......................173
7.5
Ordered Sets of Dimension
2....................176
Exercises
..............................178
Remarks and Open Problems
....................182
8
Interval Orders
185
8.1
Definition and Examples
......................185
8.2
The Fixed Point Property for Interval Orders
............189
8.3
Dedekind s Problem for Interval Orders and Reconstruction
. ... 190
8.4
Interval Dimension
.........................193
Exercises
..............................195
Remarks and Open Problems
....................198
9
Lexicographic Sums
201
9.1
Definition and Examples
......................201
9.2
The Canonical Decomposition
...................206
9.3
Comparability
Invariance
......................208
9.4
Lexicographic Sums and Reconstruction
..............213
9.5
An Almost Lexicographic Construction
..............217
Exercises
.............................. 219
Remarks and Open Problems
....................224
10
Sets PQ
= Hom(ß,
Ρ)
and Products
227
10.1
Sets PQ
- Hom(ß,
Ρ)
.......................227
10.2
Finite Products
...........................230
10.2.1
Finite Products and the Fixed Point Property
....... 233
10.3
Infinite Products
...........................237
10.4
Hashimoto s Theorem and Automorphisms of Products
...... 240
10.4.1
Automorphisms of Products
................ 248
10.5
Arithmetic of Ordered Sets
.....................251
Exercises
..............................255
Remarks and Open Problems
....................259
11
Enumeration
263
11.1
Graded Ordered Sets
........................264
11.2
The Number of Graded Ordered Sets
................265
11.3
The Asymptotic Number of Graded Ordered Sets
.........268
11.4
The Number of Nonisomorphic Ordered Sets
...........277
11.5
The Number of Automorphisms
..................282
Exercises
..............................285
Remarks and Open Problems
....................289
12
Algorithmic Aspects
295
12.1
Algorithms
.............................296
12.2
Polynomial Efficiency
........................299
12.3
NP problems
............................303
12.3.1
Constraint Satisfaction Problems (CSPs)
..........305
12.3.2
Expanded Constraint Networks/
Cliques in r-colored Graphs
................308
12.4
NP-completeness
..........................309
12.5
So It s NP-complete
...........................312
12.5.1
Search Algorithms
.....................313
12.5.2
Algorithms That Enforce Local Consistency
........318
12.6
A Polynomial Algorithm for the Fixed Point Property in Graded
Ordered Sets of Bounded Width
..................324
Exercises
..............................326
Remarks and Open Problems
....................332
A A Primer on Algebraic Topology
337
A.I Chain Complexes
..........................338
A.2 The Lefschetz Number
.......................341
A.3 (Integer) Homology
.........................342
A.4 A Homological Reduction Theorem
................344
Remarks and Open Problems
....................349
В
Order vs. Analysis
351
B.I The Spaces
Ζ^(Ω,Κ)
........................351
В.
2
Fixed Point Theorems
........................352
B.3 An Application
...........................354
Remarks and Open Problems
....................356
References
359
Index
377
|
| any_adam_object | 1 |
| author | Schröder, Bernd S. W. 1966- |
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| dewey-full | 511.3/2 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 511 - General principles of mathematics |
| dewey-raw | 511.3/2 |
| dewey-search | 511.3/2 |
| dewey-sort | 3511.3 12 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| institution | BVB |
| isbn | 0817641289 3764341289 |
| language | English |
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| spelling | Schröder, Bernd S. W. 1966- Verfasser (DE-588)135918871 aut Ordered sets an introduction Bernd S. W. Schröder Boston [u.a.] Birkhäuser 2003 XVII, 391 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier "Ordered Sets: An Introduction is an excellent text for undergraduate and graduate students and a good resource for the interested researcher. Readers will discover order theory's role in discrete mathematics as a supplier of ideas as well as an attractive source of applications."--BOOK JACKET. Ensembles ordonnés Ordered sets Geordnete Menge (DE-588)4156748-1 gnd rswk-swf Geordnete Menge (DE-588)4156748-1 s DE-604 Digitalisierung UB Passau application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009844717&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Schröder, Bernd S. W. 1966- Ordered sets an introduction Ensembles ordonnés Ordered sets Geordnete Menge (DE-588)4156748-1 gnd |
| subject_GND | (DE-588)4156748-1 |
| title | Ordered sets an introduction |
| title_auth | Ordered sets an introduction |
| title_exact_search | Ordered sets an introduction |
| title_full | Ordered sets an introduction Bernd S. W. Schröder |
| title_fullStr | Ordered sets an introduction Bernd S. W. Schröder |
| title_full_unstemmed | Ordered sets an introduction Bernd S. W. Schröder |
| title_short | Ordered sets |
| title_sort | ordered sets an introduction |
| title_sub | an introduction |
| topic | Ensembles ordonnés Ordered sets Geordnete Menge (DE-588)4156748-1 gnd |
| topic_facet | Ensembles ordonnés Ordered sets Geordnete Menge |
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