Ordered sets
type: Item
instance of: scholarly article
title: Ordered sets
zbMATH Open document ID: 1072.06001
author: Egbert Harzheim
published in: Advances in Mathematics (Springer)
publication date: 2005-04-05T00:00:00Z
review text: 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, …
reviewed by: Martin Weese
Mathematics Subject Classification ID: 06-01
Mathematics Subject Classification ID: 03-01
Mathematics Subject Classification ID: 06A06
Mathematics Subject Classification ID: 06A05
Mathematics Subject Classification ID: 03E05
Mathematics Subject Classification ID: 03E10
zbMATH DE Number: 2152358
zbMATH Keywords: ordered set
zbMATH Keywords: partial order
zbMATH Keywords: linear order
MaRDI profile type: Publication
full work available at URL: b104891
OpenAlex ID: W4247579482
DOI: 10.1007/B104891
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description: scientific article; zbMATH DE number 2152358