Lattice-Ordered Groups. An Introduction

Transcription

1 Lattice-Ordered Groups An Introduction

2 Reidel Texts in the Mathematical Sciences A Graduate-Level Book Series

3 Lattice-Ordered Groups An Introduction by Marlow Anderson The Colorado College, Colorado Springs, Colorado, U.S.A. and Todd Feil Department of Mathematical Sciences, Denison University, Granville, Ohio, U.S.A. D. Reidel Publishing Company A MEMBER OFTHE KLUWER ACADEMIC PUBLISHERS GROUP Dordrecht / Boston / Lancaster / Tokyo

4 Library of Congress Cataloging in Publication Data Anderson, Marlow, ~ Lattice-ordered groups: an introduction / by Marlow Anderson and Todd Feil. p. cm.-(reidel texts in the mathematical sciences) Bibliography: p. Includes indexes. TSBN-13: e-tsbn-13: DOl: / Lattice ordered groups. I. Feil, Todd, n. Title. III. Series. QA171.A '.22-dc CIP Published by D. Reidel Publishing Company, P.O. Box 17, 3300 AA Dordrecht, Hollafid. Sold and distributed in the U.S.A. and Canada by Kluwer Academic Publishers, 101 Philip Drive, Norwell, MA 02061, U.S.A. In all other countries, sold and distributed by Kluwer Academic Publishers Group, P.O. Box 322,3300 AH Dordrecht, Holland. All Rights Reserved 1988 by D. Reidel Publishing Company, Dordrecht, Holland Softcover reprint of the hardcover lst edition 1988 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner

5 Preface The study of groups equipped with a compatible lattice order ("lattice-ordered groups" or "I!-groups") has arisen in a number of different contexts. Examples of this include the study of ideals and divisibility, dating back to the work of Dedekind and continued by Krull; the pioneering work of Hahn on totally ordered abelian groups; and the work of Kantorovich and other analysts on partially ordered function spaces. After the Second World War, the theory of lattice-ordered groups became a subject of study in its own right, following the publication of fundamental papers by Birkhoff, Nakano and Lorenzen. The theory blossomed under the leadership of Paul Conrad, whose important papers in the 1960s provided the tools for describing the structure for many classes of I!-groups in terms of their convex I!-subgroups. A particularly significant success of this approach was the generalization of Hahn's embedding theorem to the case of abelian lattice-ordered groups, work done with his students John Harvey and Charles Holland. The results of this period are summarized in Conrad's "blue notes" [C]. Holland's proof in 1963 that every lattice-ordered group can be represented as a group of order-preserving permutations of a totally ordered set has been of decisive importance in the further development of the theory of nonabelian I!-groups, since it is the only tool available for studying arbitrary I!-groups. In particular, this approach has been essential in the study of varieties of I!-groups, a subject of major interest in the last 15 years, with important contributions being made by Holland and his students Stephen McCleary and Andrew Glass. In fact, Holland's theorem has led to considerable study of the permutation groups of totally ordered sets in their own right. This work is summarized in Glass' book Ordered Permutation Groups [G]. The present work intends to focus narrowly on a concise exposition of the classical theory oflattice-ordered groups. We consequently omit most work on totally ordered groups (for which, the reader may see the books of Kopytov and Kokorin [KK] and of Rhemtulla and Mura [MR]). We also do not consider other partially ordered structures, such as partially ordered rings or semigroups; some material on such subjects may be found in Fuchs' book [F]. In particular, we do not discuss lattice-ordered rings; they figure in Bigard, Keimel and Wolfenstein [BKW]. Our emphasis is algebraic rather than analytic; for the vast literature on partially ordered linear spaces, the reader should consult Schaefer [S], Vulich [V], Luxemberg and Zaanen [LZ] or Aliprantis and Burkinshaw [AB]. Finally, we use the techniques of lattice-ordered permutation groups only in so far as is necessary to obtain results in the theory of I!-groups; we consequently view Glass' book [G] as an essential but complementary volume to our own. After a chapter of introductory material on the lattice of convex I!-subgroups of a lattice-ordered group, we prove in the next several chapters the four important representation theorems of the theory, in increasing order of generality: the Bernau representation for archimedean I!-groups, the Conrad-Harvey-Holland representation for abelian I!-groups, the Lorenzen representation, and finally the Holland representation for all I!-groups. We next v

6 VI describe free lattice-ordered groups, preparatory to Chapter 7, which is an extended exposition of the theory of varieties of i-groups. Included in this chapter is a proof of Holland's theorem that the class of normal-valued i-groups is the largest proper i-group variety. The theory of i-group completions is the subject of the next two chapters; we include a proof of the existence of a lateral completion for any lattice-ordered group, a result of major difficulty and importance, first proved by Bernau in the early seventies. In the next chapter we describe the theory of finite-valued and special-valued i-groups, with emphasis on the root system of values of a normal-valued i-group, and analogies that can be drawn with the Conrad-Harvey-Holland theorem. The final chapter of the text provides an exposition of the theory of lattice-ordered groups of divisibility of commutative rings. What follows is an appendix, which is devoted to an extended compendium of examples of i-group classes and i-groups themselves, which illustrate the boundaries of the theory. Finally, we provide a comprehensive bibliography of papers in lattice-ordered groups, with special emphasis on the last twenty years. The text assumes only the usual graduate courses in algebra, analysis and point-set topology, and consequently should be accessible to graduate students and professional mathematicians in other fields; we hope it also offers something to the expert. The exposition is for the most part self-contained; we do occasionally discuss without proof further developments in the theory. The major exception to this is in Chapter 7, where we quote without proof major theorems from the theory of i-permutation groups; the interested reader may consult the proofs published in Glass' book [G]. Additive notation is used for the group operation into Chapter 5 where it is shown that each i-group can be considered as an i-subgroup of the i-group of ordered preserving permutations of a totally ordered set. From there until the end, with the exceptions of Chapters 8 and 11, multiplicative notation is used. It has become somewhat of a convention that abelian groups be written additively and non-abelian groups multiplicatively. This convention is by no means universal, however. We have tried to be consistent within chapters and believe that in this book, the change over to multiplicative notation makes sense where it occurs. Chapters 8 and 11 use additive notation since Section 8.2 concerns archimedean (and hence abelian) i-groups and Chapter 11 deals with abelian i-groups. References in the text to examples of i-groups which appear in the appendix are in the form E#; references to discussions in the appendix of i-group classes are by the script letter denoting the class; all such abbreviations appear in the index. The bibliography consists of a list of pertinent books, followed by a list of papers and dissertations; we refer to books by authors' initials [XY], and to papers by the authors' names, followed by two digits indicating the year. The reader should consult the appendix and the bibliography for more detail. We owe a considerable debt of gratitude to Paul Conrad and Charles Holland, on several grounds. Their mathematical influence can be seen on every page; note particularly that the earlier material in the book is heavily influenced by Conrad's "blue notes" [C]. Their personal examples as our thesis advisors have been inspirational for our careers as

7 mathematicians. Both men have also examined various portions of the manuscript and offered their comments. Charles Holland taught a course from a preliminary draft, and gathered many valuable corrections and comments. The extensive comments from Andrew Glass have improved the manuscript a great deal. We have also benefited from informal discussions of the project with many mathematicians across the country. As for any errors (typographical or otherwise) which remain, we of course take the only reasonable position, and blame one another. Our respective institutions deserve thanks for the financial support we have t;eceived to expedite this project, in the face of our considerable geographic separation. Their aid also enabled us to prepare this manuscript using 'lex, which has improved its appearance and readability, even in the hands of novice users. Finally, we wish to dedicate this book to Audrey and Robin. VB Marlow Anderson The Colorado College Todd Feil Denison University

8 Contents Chapter 1: Fundamentals Section 1: Preliminaries and Basic Examples Section 2: Subobjects and Morphisms 1 7 Chapter 2: Bernau's representation for Archimedean i-groups 15 Chapter 3: The Conrad-Harvey-Holland Representation Chapter 4: Representable.and Normal-valued i-groups Section 1: The Lorenzen Representation for i-groups Section 2: Normal-valued i-groups Chapter 5: Holland's Embedding Theorem Chapter 6: Free i-groups Chapter 7: Varieties of i-groups Section 1: The lattice of Varieties Section 2: Covers of the Abelian Variety Section 3: The Cardinality of the lattice of i-group Vari ties Chapter 8: Completions of Representable and Archimedean i-groups Section 1: Completions of Representable i-groups Section 2: Completions of Archimedean i-groups Chapter 9: The Lateral Completion Chapter 10: Finite-valued and Special-valued i-groups Chapter 11: Groups of Divisibility. Appendix: A Menagerie of Examples Section 1: Varieties of i-groups Section 2: Torsion and Radical Classes of i-groups Section 3: Examples of Lattice-ordered Groups Bibliography. Subject Index Author Index