**Ordered Sets**

**ISBN/ASIN**: *0387242198,9780387242194* | *2005* | *English* | *pdf* | *400/400* pages | *5.82* Mb**Publisher**: *Springer* | **Author**: *Egbert Harzheim* | **Edition**: *1*

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 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