Ordered Sets

1. Front Matter

   Pages i-xii

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2. Fundamental notions of set theory

   Pages 1-9

3. Fundamental notions

   Pages 11-47

4. General relations between posets and their chains and antichains

   Pages 49-70

5. Linearly ordered sets

   Pages 71-83

6. Products of orders

   Pages 85-141

7. Universally ordered sets

   Pages 143-158

8. Applications of the splitting method

   Pages 159-201

9. The dimension of posets

   Pages 203-230

10. Well-founded posets, pwo-sets and trees

    Pages 231-284

11. On the order structure of power sets

    Pages 285-329

12. Comparison of order types

    Pages 331-352

13. Comparability graphs

    Pages 353-368

14. Back Matter

    Pages 369-386

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This book is written particularly for mathematics students and, of course, for mathematicians interested in set theory. Only some fun- mental parts of naive set theory are presupposed, - not more than is treated in a textbook on set theory, even if this restricts us only to the most basic facts of this field. We have summarized all of this in Chapter 0 without longer discusssions and explanations, because there are s- eral textbooks which can be consulted by the reader, e.g. HrbacekIJech [88], KneeboneIRotman [99], ShenIVereshchagin [159]. Besides this only elementary facts of analysis are used. The theory of ordered sets can be divided into two parts, depending on whether the sets under consideration are finite or infinite. The first part is grounded mainly in combinatorics and graph theory and does not make essential use of set-theorical concepts, whereas the second part presupposes a knowledge of the fundamental notions of set theory, in particular of the system of ordinal and cardinal numbers. In this book we mainly deal with general infinite ordered sets. In this field the textbook literature is still very small. Therefore this book supplements the existing literature.

• Finite
• combinatorics
• field
• graph
• mathematics
• order theory
• ordinal
• set theory

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)

• University of Düsseldorf, Germany

  Egbert Harzheim

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