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Bernd Schröder Ordered Sets An Introduction with Connections from Combinatorics to Topology Second Edition Bernd Schröder Ordered Sets An Introduction with Connections from Combinatorics to Topology Second Edition Bernd Schröder Department of Mathematics University of Southern Mississippi Hattiesburg, MS, USA ISBN 978-3-319-29786-6 ISBN 978-3-319-29788-0 DOI 10.1007/978-3-319-29788-0 Library of Congress Control Number: 2016935408 Mathematics Subject Classification (2010): 06-01, 06-02, 05-01 © Springer International Publishing 2003, 2016 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper This book is published under the trade name Birkhäuser. The registered company is Springer International Publishing AG, CH Preface to the Second Edition Writing the second edition of this text was a wonderful opportunity for a more mature presentation than I was able to give 13 years ago. The intent still is to give a self-contained introduction to the theory of ordered sets and to its connections to other areas. I tried to shine a light on as many branches of the theory of ordered sets as possible, with the main obstacle being my limited cranial capacity. Indeed, although in this text I can present the most I have ever known about ordered sets, writing the text has also shown me how much I don’t know. However, that should not be a problem for anyone, as that which we don’t know will always exceed that which we do. Writing the second edition also was quite humbling in another way. Thorough re-reading revealed more typos than were reported on the posted errata, and some of them were rather embarrassing. The presentation remains modular. Specifically, we have the following: • Chapters 1, 2, 3, and Sections 4.1 and 4.2 form the core of the text. If you are new to ordered sets, you should read this part in the order in which it is presented here. (Skip Section 3.5 if you are not focusing on analysis.) • The remaining chapters can be read in just about any order. There have been some content rearrangements and additions. The automorphism problem (see Open Question 2.14), of which nothing appears to be known beyond the references given here, has been moved into Chapter 2 to feature it more prominently. Similarly, the chapter on algorithms has been “promoted” to Chapter 5 and expanded to focus even more strongly on constraint satisfaction problems. I believe that more results similar to the “(” direction of Theorem 5.57 can be proved for the fixed point property and for other constraint satisfaction problems. Chapter 6 is new and serves to separate the fixed clique property from the more fundamental fixed simplex property and to give an idea about graph homomorphisms and their connections to and differences from order-preserving maps. Finally, the future importance of discrete Morse functions for the fixed point property for ordered sets is indicated in Appendix B. Overall, I have shifted the primary focus toward finite ordered sets, with results on infinite ordered sets moved to the back of each chapter whenever possible. The references [21, 23, 25, 31, 41, 42, 54, 56, 91, 98, 102, 112, v vi Preface to the Second Edition 120, 246, 247, 251, 252, 254, 311] should provide ample opportunity for further study of various aspects of ordered sets. Although the goal was and is to have a self-contained exposition, the first edition’s appendix on ordered Lp-spaces has been turned into Section 3.5 and Exercises 2-50, 3-32, 3-33, 3-34, 4-37, 8-11, 8-12, 8-13, and 8-18. If you are familiar with Lp-spaces, the exercises will be natural; if you are not, the appendix probably would have felt quite uncomfortable anyway. So the simple advice here is that if Lp-spaces are not part of your repertoire, then these exercises should probably be skipped. Other additions are technical in nature, but important nonetheless. The text is now available as an ebook with live links for the internal references. This can make reading easier at times, but the usefulness of a paper copy should not be underestimated. I have read and written thousands of pages on screen. However, when I really want to learn something well, I read a paper copy. I hope you will enjoy reading this text, in any of its forms, as much as I enjoyed writing it. Let me conclude with a final request/recommendation for readers who use this text as part of an effort to improve proof-writing skills. The transition to doing proofs is hard. To me, it is the hardest challenge that I have ever successfully met. Looking up solutions does not facilitate this transition. So, as you work the exercises, do not use the library, Google, or other resources to find solutions. (I maintain that I learned how to do proofs partly because I was too lazy to look up solutions, preferring hours of thinking over a half hour in the library.) Use your brain, your whole brain and nothing but your brain, aided by paper and pencil. Hattiesburg, MS, USA Bernd Schröder January 7, 2016 Preface to the First Edition Order theory can be seen formally as a subject between lattice theory and graph theory. Indeed, one can say with good reason that lattices are special types of ordered sets, which are in turn special types of directed graphs. Yet this would be much too simplistic an approach. In each theory, the distinct strengths and weaknesses of the given structure can be explored. This leads to general as well as discipline- specific questions and results. Of the three research areas mentioned, order theory undoubtedly is the youngest. The first specialist journal Order was launched in 1984, and much of the research that guided my own development started in the 1970s. When I started teaching myself order theory (via a detour through category theory), I was only dimly aware of lattices and graphs. (I was working on a PhD in harmonic analysis and probability theory at the time.) I was attracted to the structure, as it apparently fits the way I think. It was possible to learn the needed basics from research papers as well as surveys. From there it was immensely enjoyable to start working on unexplored problems. This is the beauty of a fresh field. Interesting results are almost asking to be discovered. I hope that the reader will find the same type of attraction to this area (and will ultimately make interesting contributions to the field). Yet there also is a barrier to entering such a new field. In new fields, standard texts are not yet available. I felt it would be useful to have a text that would expose the reader to order theory as a discipline without too quickly focusing on one specific subarea. In this fashion a broader picture can be seen. This is my attempt at such a text. It contains all that I know about the theory of ordered sets. From here, articles on ordered sets as well as the standard references I had available starting out (which are primarily Rival’s Banff conference volumes [246, 247, 252], but also Birkhoff’s classic [21] on lattice theory, Fishburn’s text [91] on interval orders, and later Davey and Priestley’s text [56] on lattices and Trotter’s monograph [311] on dimension theory) should be easily accessible. The idea was to describe what I consider the basics of ordered sets without the work becoming totally idiosyncratic. Some of the salient features of this text are bulleted below. vii viii Preface to the First Edition • Theme-driven approach. Most of the topics in this text are introduced by investigating how they relate to research problems. We will frequently revisit the open problems that are explained early in the text. Further problems are added as we progress. In this fashion, I believe, the reader will be able to form the necessary intuitions about a new structure more easily than if there were no common undercurrent to the presentation. I have deliberately tried to avoid the often typical beginning of a text in discrete mathematics. There is no deluge of pages upon pages of basic definitions. New notions are introduced when they first arise, and they are connected with known ideas as soon as possible. • Connections between topics. Paraphrasing W. Edwards Deming, one can say that If we do not understand our work as part of a process, we do not understand our work at all. This statement applies to industry, education, and also research.