Lattice-Ordered Groups: An Introduction

196 Pages·1988·6.01 MB·English

by  Marlow Anderson

Lattice-Ordered Groups
An Introduction
Reidel Texts in the Mathematical Sciences
A Graduate-Level Book Series
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
Library of Congress Cataloging in Publication Data
Anderson, Marlow, 195~
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: 978-94-010-7792-7 e-TSBN-13: 978-94-009-2871-8
DOl: 10.1007/978-94-009-2871-8
1. Lattice ordered groups. I. Feil, Todd, 1951- n. Title.
III. Series.
QA171.A553 1987
512' .22-dc 19 87-28598
CIP
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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 representa
tion 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
VI
describe free lattice-ordered groups, preparatory to Chapter 7, which is an extended expo
sition 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 dif
ficulty 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 math
ematicians 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
VB
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.
Marlow Anderson Todd Feil
The Colorado College Denison University
Contents
Chapter 1: Fundamentals
Section 1: Preliminaries and Basic Examples 1
Section 2: Subobjects and Morphisms 7
Chapter 2: Bernau's representation for Archimedean i-groups 15
Chapter 3: The Conrad-Harvey-Holland Representation ............ 21
Chapter 4: Representable .and Normal-valued i-groups
Section 1: The Lorenzen Representation for i-groups 26
Section 2: Normal-valued i-groups 29
Chapter 5: Holland's Embedding Theorem . . . . . . . . . . . . . . . . . . 32
Chapter 6: Free i-groups . . . . . . . . . . . . . . . . . . . . . . . . . . 38
Chapter 7: Varieties of i-groups
Section 1: The lattice of Varieties 47
Section 2: Covers of the Abelian Variety 57
Section 3: The Cardinality of the lattice of i-group Vari£ties 62
Chapter 8: Completions of Representable and Archimedean i-groups
Section 1: Completions of Representable i-groups 64
Section 2: Completions of Archimedean i-groups . 70
Chapter 9: The Lateral Completion ............. 77
Chapter 10: Finite-valued and Special-valued i-groups . . . . . . . . . . . . . 86
Chapter 11: Groups of Divisibility . 102
Appendix: A Menagerie of Examples
Section 1: Varieties of i-groups 109
Section 2: Torsion and Radical Classes of i-groups 115
Section 3: Examples of Lattice-ordered Groups 121
Bibliography . 151
Subject Index 185
Author Index . 189
Chapter 1: Fundamentals
Section 1.1: Preliminaries and Basic Examples
In this section we shall develop the basic definitions and notations used in the theory of
lattice-ordered groups, and provide a set of examples whose significance will be more fully
appreciated when we have proved the basic representation theorems for lattice-ordered
groups in Chapters 2 through 5.
Additive and multiplicative notation for the underlying group of a lattice-ordered group
are both used, depending on the particular application in mind, and we shall feel free to use
both. In order that the word "positive" have its customary meaning, we shall in our initial
definitions use additive notation. The reader should not infer from this that all groups
referred to are abelian.
A group (G, +) is said to be partially ordered if it is equipped with a reflexive, an
tisymmetric and transitive relation ~ (a partial order) which is compatible with + ; that
is, if 9 ~ h , then 9 + k ~ h + k and k + 9 ~ k + h , for any g, h, kEG. The set
G+ = {g E G : 9 > o} of positive elements is called the positive cone of G. It is easily
verified that the partial order is determined by G+ , in the sense that 9 ~ h if and only if
h - 9 E G+ U {a}. Furthermore, any set P ~ G which is closed under addition, normal (i.e.,
9 + P - 9 = P for all 9 E G), and satisfies the property P n -P = 0, gives rise to a partial
order on G.
Suppose now that the partial order on G is in fact a lattice (meaning that each pair of
elements a, b of G has a least upper bound (join) a V b and a greatest lower bound (meet)
a 1\ b). Then G is a lattice-ordered group (i-group) (some authors prefer the term lattice
group). In cases where we wish to speak of joins (or meets) of sets with cardinality greater
than two, we shall use notation such as the following: al Va2 V ... an; ViE1 ai (for some
index set 1); vA (for some set A ~ G); or the analogues for meets. Notice, however, that
the definition of i-group does not necessarily imply the existence of joins and meets for
other than finite sets.
If the lattice order on an i-group G is a total order (that is, for any g, h E G , either
9 ~ h or h ~ g), we then call G an o-group. Notice that in this case the positive cone
satisfies the additional criterion that P U -P U {a} = G. Just as a semi group theorist
considers his problem done when it is reduced to group theory, we shall adopt the view that
our understanding is sufficient when we can reduce i-group questions to questions regarding
o-groups. The subject of o-groups is itself a rich and difficult one, and the reader is advised
to consult [MR] or [KK] for an introduction to it.
It is easy to verify that the group operation distributes over both meet and join; that
is,
a + b V c + d = (a + b + d) V (a + c + d),
and likewise for 1\ . The relationship between the lattice operations and the group inverse
is given by what might be called DeMorgan's laws:
1
2 Chapter 1
Proposition 1.1.1 Let a, bEG. Then:
(a) -(aVb)=-aA-b.
(b) -(aAb)=-aV-b.
Proof: We shall prove only (a). Since a,b ~ a V b then -a, -b 2:: -(a V b), and so
-a A -b 2:: -(a V b). For the reverse inequality, suppose that -a, -b 2:: c. Then, a, b ~ -c,
and so a V b ~ -c; that is c ~ -(a V b)./ /
We may infer from the proof of this proposition a further result. Namely, a partially
ordered group which is a join-semilattice (or meet-semilattice) is necessarily an i-group. A
useful identity (due to Birkhoff [42]) also follows from the proposition:
1.1.2: a- (aAb)+b=avb;
= =
this holds because a - (a A b) 0 V (a - b) (a V b) - b.
For 9 E G, the positive part g+ of 9 is 9 V 0; the negative part g- is (-g) V O. Then
9 + g- = 9 + (- g V 0) = 0 V 9 = g+ ;
and so we have that 9 = g+ - g-. Now,
If a A b = 0, we say that a and b are disjoint.
Thus, we have shown that each element of an i-group can be represented as the dif
ference of two disjoint positive elements; the reader may verify that this representation is
unique. In fact, this property characterizes i-groups: a directed partially ordered group
in which each element may be uniquely represented as the difference of disjoint positive
elements is necessarily an i-group (see [Fl).
Note that disjoint elements commute; in fact if a A b = 0 then a + b = a V b. For,
(a+b) - (a vb) = (a+b) + (-a A -b) = (a+ b - a) Aa = a+ (bAa) - a = O.
The absolute value Igl of 9 is g+ + g-; like ordinary absolute value, 9 = Igl exactly
= =
when 9 2:: O. The reader may verify that Igl g+ V g- 9 V -g. The absolute value satisfies
a weakened triangle inequality:
1.1.3a: Ig + hi ~ Igi + Ihl + Igl·
For,
Ig + hi = (g+ h) VO+ (-h - g) vO ~ (g VOl + (h vOl + (-h VOl + (-g VOl ~ Igl + Ihl + Igl.
Furthermore, since (g + h)+ and (g + ht are disjoint and so commute, it also follows that:
1.1.3b: Ig + hi ~ Ihl + Igl + Ihl·
Note that a slight modification of the proof above shows that if the i-group is abelian,
the usual triangle inequality holds. In fact, the converse holds: any i-group satisfying the
usual triangle inequality is necessarily abelian (Kalman [60]).