Ordered Sets

Cover......Page 1
Advances in Mathematics VOLUME 7......Page 3
Ordered Sets......Page 4
0387242198......Page 5
Contents......Page 6
Preface\r......Page 10
0.1 Sets and functions......Page 14
0.2 Cardinalities and operations with sets......Page 16
0.3 Well-ordered sets......Page 17
0.4 Ordinals......Page 19
0.5 The alephs......Page 21
1.1 Binary relations on a set......Page 24
1.2 Special properties of relations......Page 25
1.3 The order relation and variants of it......Page 26
1.4 Examples......Page 29
1.5 Special remarks......Page 31
1.6 Neighboring elements. Bounds......Page 32
1.7 Diagram representation of finite posets......Page 37
1.8 Special subsets of posets. Closure operators......Page 42
1.9 Order-isomorphic mappings. Order types......Page 47
1.10 Cuts. The Dedekind-MacNeille completion......Page 53
1.11 The duality principle of order theory......Page 60
2.1 Components of a poset......Page 62
2.2 Maximal principles of order theory......Page 63
2.3 Linear Extensions of posets......Page 65
2.4 The linear kernel of a poset......Page 67
2.5 Dilworth\'s theorems......Page 69
2.6 The lattice of antichains of a poset......Page 75
2.7 The ordered set of initial segments of a poset......Page 79
3.1 Cofinality......Page 84
3.2 Characters......Page 90
3.3 η_a - sets\r......Page 93
4.1 Construction of new orders from systems of given posets......Page 98
4.2 Order properties of lexicographic products......Page 104
4.3 Universally ordered sets and the sets H_a of\rnormal type η_a......Page 110
4.4 Generalizations to the case of a singular ω_a\r......Page 121
4.5 The method of succesively adjoining cuts......Page 123
4.6 Special properties of the sets T_λ for indecomposable λ......Page 127
4.7 Relations between the order types of lexicographic products\r......Page 135
4.8 Cantor\'s normal form. Indecomposable ordinals......Page 150
5.1 Adjoining IF-pairs to posets......Page 156
5.2 Construction of an \\aleph_α- universally ordered set......Page 158
5.3 Construction of an injective <-preserving mapping of U_α into H_α......Page 165
6.1 The general splitting method......Page 172
6.2 Embedding theorems based on the order types of the well- and inversely well-ordered subsets\r......Page 179
6.3 The change number of dyadic sequences......Page 185
6.4 An application in combinatorial set theory......Page 194
6.5 Cofinal subsets......Page 202
6.6 Scattered sets......Page 206
7.1 The topology of linearly ordered sets and their products......Page 216
7.2 The dimension of posets......Page 219
7.3 Relations between the dimension of a poset\rand certain subsets......Page 226
7.4 Interval orders......Page 241
8.1 Well-founded posets......Page 244
8.2 The notions well-quasi-ordered and partially well-ordered set\r......Page 257
8.3 Partial ordinals......Page 263
8.4 The theorem of De Jongh and Parikh......Page 266
8.5 On the structure of \\mathfrak{J}(P), where P is well-founded or pwo......Page 271
8.6 Sequences in wqo-sets......Page 275
8.7 Trees......Page 279
8.8 Aronszajn trees and Specker chains\r......Page 285
8.9 Suslin chains and Suslin trees......Page 291
9.1 Antichains in power sets......Page 298
9.2 Contractive mappings in power sets......Page 310
9.3 Combinatorial properties of choice functions......Page 322
9.4 Combinatorial theorems on infinite power\rsets......Page 332
10.1 Some general theorems on order types......Page 344
10.2 Countable order types......Page 349
10.3 Uncountable order types......Page 351
10.4 Homogeneous posets......Page 356
11.1 General remarks......Page 366
11.2 A characterization of comparability graphs......Page 370
11.3 A characterization of the comparability\rgraphs of trees......Page 377
References\r......Page 382
Index\r......Page 392
List of symbols......Page 398