Ordered Sets
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 set
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Advances in Mathematics VOLUME 7
Series Editor: J. Szep, Budapest University of Economics, Hungary
Advisory Board: S-N. Chow, Georgia Institute of Technology, U.S.A. G. Erjaee, Shiraz University, Iran W . Fouche, University of South Africa, South Africa
P. Grillet, Tulane University, U.S.A. H.J. Hoehnke, Institute of Pure Mathematics of the Academy of Sciences, Germany F. Szidarovszky, University of Airzona, U.S.A. P.G. Trotter, University of Tasmania, Australia P. Zecca, Universitci di Firenze, Italy
ORDERED SETS
EGBERT HARZHEIM University of Diisseldorf, Germany
- Springer
Library of Congress Cataloging-in-Publication Data A C.I.P. record for this book is available from the Library of Congress.
AMS Subject Classifications: 06-01, 06A05, 06A06, 06A07 ISBN 0-387-24219-8
e-ISBN 0-387-24222-8
Printed on acid-free paper.
O 2005 Springer Science+Business Media, Inc.
All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, Inc., 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now know or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if the are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed in the United States of America. 9 8 7 6 5 4 3 2 1
SPIN 11367116
Contents Preface Chapter 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 0.5Thealephs
ix 1 1 3 4 6 8
Chapter 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
11 11 12 13 16 18 19 24 29 34 40 47
Chapter 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
49 49 50 52 54 56 62 66
Chapter 3. Linearly ordered sets 3.1 Cofinality 3.2 Characters 3.3 r| a -sets
71 71 77 80
Chapter 4. Products of orders 4.1 Construction of new orders from systems of given posets 4.2 Order properties of lexicographic products
85 85 91
vi
4.3 Universally ordered sets and the sets Ha of normal type r|a 4.4 Generalizations to the case of a singular coa 4.5 The me
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