Ordered sets. (English) Zbl 1072.06001

Advances in Mathematics (Springer) 7. New York, NY: Springer (ISBN 0-387-24219-8/hbk). xii, 386 p. (2005).
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 interestes in ordered sets. Only some fundamental parts of naive set theory are presupposed.
The theory of ordered sets can be divided into two parts, depending on whether finite or infinite orders are under consideration. In this book the author is mainly interested in the infinite ones. So the reader should have some knowledge of the fundamental notions of set theory such as well-ordering, ordinal and cardinal number.
The book starts with the fundamental notions of set theory and recalls some facts which are used in the following. It continues with the basics of ordered sets. The author introduces cuts and uses them to construct the Dedekind-MacNeille completion. In Chapter 2 he deals with chains and antichains. He proves the Dilworth theorem on the decomposition of posets with finite widths. He introduces the lattice of antichains of a given poset. In Chapter 3 the author considers linear orders and introduces the \(\eta_\alpha\)-sets. In Chapter 4 he investigates products of orders. So he introduces the lexicographic product and shows the Cantor normal-form theorem for ordinals. In the next chapter the author gives a construction of an \(\aleph_\alpha\)-universal ordered set.
In Chapter 6 he explains the so-called splitting method. This method is applied in combinatorial set theory. He introduces scattered sets and shows the Hausdorff theorem, which states that every linear order is an orderd sum of scattered sets over a dense linear order. In Chapter 7 the author investigates the dimension of posets and proves some facts on the relations between dimensions. In Chapter 8 he investigates well-founded posets, well-quasi-ordered and partially well-ordered sets. He shows the theorem of De Jongh and Parikh. He introduces trees and considers Aronszajn trees and Suslin trees and their counterparts in linear orders, the Specker chains and Suslin chains. In Chapter 9 the author investigates the order structure of a power set. He considers contractive mappings and properties of choice functions. Chapter 10 is devoted to the comparison of order types. The author investigates homogeneous and uniquely homogeneous linear orders. In the last chapters he considers the comparability of graphs.


06-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to ordered structures
03-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to mathematical logic and foundations
06A06 Partial orders, general
06A05 Total orders
03E05 Other combinatorial set theory
03E10 Ordinal and cardinal numbers
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