Ordered Sets

✍ Scribed by Egbert Harzheim

• Publisher: Springer
• Year: 2005
• Tongue: English
• Leaves: 400
• Series: Advances in Mathematics (Book 7)
• Edition: 1
• Category: Library

✦ Synopsis

Editorial Reviews

Review
From the reviews: "The exposition of material in Ordered Sets is generally quite clear. … The list of symbols is useful. … the book contains an unusual mix of topics that reflects both the author’s varied interests and developments in the theory of infinite ordered sets, particularly concerning universal orders, the splitting method, and aspects of well-quasi ordering. It will be of greatest interest to readers who want a selective treatment of such topics." (Dwight Duffus, SIAM Review, Vol. 48 (1), 2006) "The textbook literature on ordered sets is rather limited. So this book fills a gap. It is intended for mathematics students and for mathematicians who are interests in ordered sets." (Martin Weese, Zentralblatt MATH, Vol. 1072, 2005) "This book is a comprehensive introduction to the theory of partially ordered sets. It is a fine reference for the practicing mathematician, and an excellent text for a graduate course. Chains, antichains, linearly ordered sets, well-ordered sets, well-founded sets, trees, embedding, cofinality, products, topology, order types, universal sets, dimension, ordered subsets of power sets, comparability graphs, a little partition calculus … it’s pretty much all here, clearly explained and well developed." (Judith Roitman, Mathematical Reviews, Issue 2006 e)

From the Back Cover
The textbook literature on ordered sets is still rather limited. A lot of material is presented in this book that appears now for the first time in a textbook. Order theory works with combinatorial and set-theoretical methods, depending on whether the sets under consideration are finite or infinite. In this book the set-theoretical parts prevail. The book treats in detail lexicographic products and their connections with universally ordered sets, and further it gives thorough investigations on the structure of power sets. Other topics dealt with include dimension theory of ordered sets, well-quasi-ordered sets, trees, combinatorial set theory for ordered sets, comparison of order types, and comparability graphs. Audience This book is intended for mathematics students and for mathematicians who are interested in set theory. Only some fundamental parts of naive set theory are presupposed. Since all proofs are worked out in great detail, the book should be suitable as a text for a course on order theory.

CONTENTS

Ch 0. Fundamental notions of set theory
0.1 Sets and functions
0.2 Cardinalities and operations with sets
0.3 Well-ordered sets
0.4 Ordinals
0.5 The alephs
Ch 1. Fundamental notions
1.1 Binary relations on a set
1.2 Special properties of relations
1.3 The order relation and variants of it
1.4 Examples
1.5 Special remarks
1.6 Neighboring elements Bounds
1.7 Diagram representation of finite posets
1.8 Special subsets of posets Closure operators
1.9 Order-isomorphic mappings. Order types
1.10 Cuts. The Dedekind-MacNeille completion
1.11 The duality principle of order theory
Ch 2. General relations between posets and their chains and antichains
2.1 Components of a poset
2.2 Maximal principles of order theory
2.3 Linear extensions of posets
2.4 The linear kernel of a poset
2.5 Dilworth's theorems
2.6 The lattice of antichains of a poset
2.7 The ordered set of initial segments of a poset
Ch 3. Linearly ordered sets
3.1 Cofinality
3.2 Characters
3.3 η_α - sets
Ch 4. Products of orders
4.1 Construction of new orders from systems of given posets
4.2 Order properties of lexicographic products
4.3 Universally ordered sets and the sets H_a of normal type n_α
4.4 Generalizations to the case of a singular ω_α
4.5 The method of successively adjoining cuts
4.6 Special properties of the sets T_λ for indecomposable λ
4.7 Relations between the order types of lexicographic products
4.8 Cantor's normal form Indecomposable ordinals
Ch 5. Universally ordered sets
5.1 Adjoining IF-pairs to posets
5.2 Construction of an א_α -universally ordered set
5.3 Construction of an injective <-preserving mapping of U_α into H_α
Ch 6. Applications of the splitting method
6.1 The general splitting method
6.2 Embedding theorems based on the order types of the well- and inversely well-ordered subsets
6.3 The change number of dyadic sequences
6.4 An application in combinatorial set theory
6.5 Cofinal subsets
6.6 Scattered sets
Ch 7. The dimension of posets
7.1 The topology of linearly ordered sets and their products
7.2 The dimension of posets
7.3 Relations between the dimension of a poset and certain subsets
7.4 Interval orders
Ch 8. Well-founded posets, pwo-sets and trees
8.1 Well-founded posets
8.2 The notions well-quasi-ordered and partially well-ordered set
8.3 Partial ordinals
8.4 The theorem of de Jongh and Parikh
8.5 On the the structure of ב(P), where P is well-founded or pwo
8.6 Sequences in wqo-sets
8.7 Trees
8.8 Aronszajn trees and Specker chains
8.9 Suslin chains and Suslin trees
Ch 9. On the order structure of power sets
9.1 Antichains in power sets
9.2 Contractive mappings in power sets
9.3 Combinatorial properties of choice functions
9.4 Combinatorial theorems on infinite power sets
Ch 10. Comparison of order types
10.1 Some general theorems on order types
10.2 Countable order types
10.3 Uncountable order types
10.4 Homogeneous posets
Ch 11. Comparability graphs
11.1 General remarks
11.2 A characterization of comparability graphs
11.3 A characterization of the comparability graphs of trees
References
Index
List of symbols

✦ Table of Contents

Front Cover......Page 1
Contents......Page 7
Preface......Page 11
0.1 Sets and functions......Page 15
0.2 Cardinalities and operations with sets......Page 17
0.3 Well-ordered sets......Page 18
0.4 Ordinals......Page 20
0.5 The alephs......Page 22
1.1 Binary relations on a set......Page 25
1.2 Special properties of relations......Page 26
1.3 The order relation and variants of it......Page 27
1.4 Examples......Page 30
1.5 Special remarks......Page 32
1.6 Neighboring elements. Bounds......Page 33
1.7 Diagram representation of finite posets......Page 38
1.8 Special subsets of posets. Closure operators......Page 43
1.9 Order-isomorphic mappings. Order types......Page 48
1.10 Cuts. The Dedekind-MacNeille completion......Page 54
1.11 The duality principle of order theory......Page 61
2.1 Components of a poset......Page 63
2.2 Maximal principles of order theory......Page 64
2.3 Linear extensions of posets......Page 66
2.4 The linear kernel of a poset......Page 68
2.5 Dilworth's theorems......Page 70
2.6 The lattice of antichains of a poset......Page 76
2.7 The ordered set of initial segments of a poset......Page 80
3.1 Cofinality......Page 85
3.2 Characters......Page 91
3.3 η_α - sets......Page 94
4.1 Construction of new orders from systems of given posets......Page 99
4.2 Order properties of lexicographic products......Page 105
4.3 Universally ordered sets and the sets H_a of normal type n_α......Page 111
4.4 Generalizations to the case of a singular ω_α......Page 122
4.5 The method of successively adjoining cuts......Page 124
4.6 Special properties of the sets T_λ for indecomposable λ......Page 128
4.7 Relations between the order types of lexicographic products......Page 136
4.8 Cantor's normal form. Indecomposable ordinals......Page 151
5.1 Adjoining IF-pairs to posets......Page 157
5.2 Construction of an א_α -universally ordered set......Page 159
5.3 Construction of an injective <-preserving mapping of U_α into H_α......Page 166
6.1 The general splitting method......Page 173
6.2 Embedding theorems based on the order types of the well- and inversely well-ordered subsets......Page 180
6.3 The change number of dyadic sequences......Page 186
6.4 An application in combinatorial set theory......Page 195
6.5 Cofinal subsets......Page 203
6.6 Scattered sets......Page 207
7.1 The topology of linearly ordered sets and their products......Page 217
7.2 The dimension of posets......Page 220
7.3 Relations between the dimension of a poset and certain subsets......Page 227
7.4 Interval orders......Page 242
8.1 Well-founded posets......Page 245
8.2 The notions well-quasi-ordered and partially well-ordered set......Page 258
8.3 Partial ordinals......Page 264
8.4 The theorem of de Jongh and Parikh......Page 267
8.5 On the the structure of ב(P), where P is well-founded or pwo......Page 272
8.6 Sequences in wqo-sets......Page 276
8.7 Trees......Page 280
8.8 Aronszajn trees and Specker chains......Page 286
8.9 Suslin chains and Suslin trees......Page 292
9.1 Antichains in power sets......Page 299
9.2 Contractive mappings in power sets......Page 311
9.3 Combinatorial properties of choice functions......Page 323
9.4 Combinatorial theorems on infinite power sets......Page 333
10.1 Some general theorems on order types......Page 345
10.2 Countable order types......Page 350
10.3 Uncountable order types......Page 352
10.4 Homogeneous posets......Page 357
11.1 General remarks......Page 367
11.2 A characterization of comparability graphs......Page 371
11.3 A characterization of the comparability graphs of trees......Page 378
References......Page 383
Index......Page 393
List of symbols......Page 399