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
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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 ?_? - 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_?
4.4 Generalizations to the case of a singular ?_?
4.5 The method of successively adjoining cuts
4.6 Special properties of the sets T_? for indecomposable ?
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 ?_? -universally ordered set
5.3 Construction of an injective <-preserving mapping of U_? into H_?
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 ?(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