Partially ordered groups

PARTIALLY ORDERED GROUPS

SERIES IN ALGEBRA Editors: J. M. Howie, D. J. Robinson, W. D. Munn Vol. 1: Infinite Groups and Group Rings ed. J. M. Corson et al. Vol. 2: Sylow Theory, Formations and Fitting Classes in Locally Finite Groups M. Dixon Vol. 3: Finite Semigroups and Universal Algebra J. Almeida Vol. 4: Generalizations of Steinberg Groups T. A. Fournelle and K. W. Weston Vol. 5: Semirings — Algebraic Theory and Applications in Computer Science U. Hebisch and H. J. Weinert Vol. 6: Semigroups of Matrices J. Okninski Vol. 7: Partially Ordered Groups A. M. W. Glass

SERIES

IN

ALGEBRA

VOLUME 7

PARTIALLY ORDERED GROUPS

A. M. W. Glass Department of Pure Mathematics & Mathematical Statistics University of Cambridge, England Fellow, Queens' College, Cambridge, England and Professor Emeritus, Bowling Green State University, Ohio, USA

World Scientific Sin9apore»ww

tersey

London* Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

Library of Congress Cataloging-in-Publication Data Glass, A. M. W. (Andrew Martin William), 1944Partially ordered groups / A.M.W. Glass. p. cm. (Series in algebra - Vol. 7) Includes bibliographical references. ISBN 9810234937 (alk. paper) 1. Partially ordered sets. I. Title. II. Series QA171.485.G57 1999 511.3'2--dc21 99-20174 CIP

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

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PREFACE In the last 50 years, research on partially ordered algebraic structures has burgeoned. This began with Paul Conrad's work on valuations and his subsequent papers on partially ordered groups and rings. Other inde­ pendent threads were Philip Hall's course at the University of Cambridge on Orderable Groups, Helmut Wielandt's crucial breakthroughs in the study of infinite permutation groups, Garrett Birkhoff's work on lattices, and L. Fuchs' seminal book "Partially Ordered Algebraic Systems" (the first comprehensive book on the interaction between algebraic operations and partial orders). As a consequence of their pioneering influence, the subject mushroomed to such an extent that such an exhaustive book would now be unmanageable. Instead, the focus of recent books has been restricted to lattice-ordered groups or right (totally) ordered groups. In contrast, this book is in the same spirit as Fuchs', but necessarily constrained to a study of the main results concerning partially ordered groups. I have selected topics which are both central to the subject and have the most overlap with ideas from other branches of mathematics; in particular, the geometry of the situation is stressed wherever possible. In Chapter 1,1 gather together a set of examples to illustrate the scope of the subject, and in Chapter 2 examine the interplay between the group operation and the order. Only some classes of partially ordered structures are definable group-theoretically; I pull together theorems concerning such definability questions that are either unpublished or dotted here and there in the literature. In Chapter 3, I define the basic subobjects for the subject and obtain initial "structure theory" results. These facts were obtained and studied by Paul Conrad and his students; they are fundamental and are used throughout the rest of the book. In Chapter 4, I set forth the special vii

viii

PREFACE

structure of Abelian and normal-valued partially ordered groups. I give representations of such groups as groups of functions (Hahn's Theorem and generalisations), though, to obtain such theorems, I have to confine attention to lattice-ordered groups. I also give a proof of the uniform decidability of the word problem for Abelian lattice-ordered groups and the undecidability of the full theory. In the next chapter, I examine the special subclass of Archimedean lattice-ordered groups. These are all groups of functions. I establish the Galois correspondence with simplicial complexes and then exploit it to the full. In particular, I use it to give a proof of the undecidability of the isomorphism problem for 10 generator 1 relator Abelian lattice-ordered groups. I examine in depth the special cases of nilpotent and soluble varieties in Chapter 6. Much of the material was first proved in Russia, though I have organised it to elucidate the link with Philip Hall's approach to group theory. Although much of that work is applicable, there are some interesting twists: for example, in Chapter 8 I show that if a and b are generators of a lattice-ordered group G and the subgroup generated by a and b is nilpotent class 2, then G is soluble (but not necessarily nilpotent of any class). I also provide proofs that many right orderable Engel groups are nilpotent and that a soluble group is right orderable if and only if it is locally indicable. In Chapter 7, I obtain the analogue of Cayley's Theorem for groups. This result (on permutational representations) is due to W. Charles Hol­ land. Its consequences are explored extensively in the remainder of the chapter. In Chapter 8 they are applied to restricted classes of partially ordered groups, to conjugation and to free products of lattice-ordered groups (among other things). I collect some results about various completions of lattice-ordered groups in Chapter 9. These are more general than topological comple­ tions (to which they bear comparison) and are due to Rick Ball. And in Chapter 10 I conclude by applying the results obtained (especially in Chapter 7) to varieties (equationally defined classes) of lattice-ordered groups. This is presently the "hottest" area of research in the subject. Residually finite groups play a very central role in the theory of in­ finite groups. In many ways, the analogue for the theory of partially ordered groups is residually (totally) ordered gorups. I emphasise the central importance of this class throughout the book.

PREFACE

ix

I have not restricted myself to a particular class of partially ordered groups ab initio, but instead searched for the most general class of groups for which I could give an uncluttered and intuitive proof of a particular theorem. In doing so, I believe that I am best serving the potential reader and yet providing something new in places (even for the expert). However, there are many places where I could not improve on proofs already in the literature and I have "borrowed" in these instances; after all, "imitation is the sincerest of flattery". In order to keep the manuscript to a readable length, I have left the reader to fill in some of the more straightforward details of proofs (the way I learnt the subject) and have avoided certain topics. For example, I have only briefly touched on orderability questions for classes of groups — there is no need to since an excellent book already exists on that subject ([Mura and Rhemtulla]). Consequently, this book is far from encyclopaedic; rather, it is envisioned as an expanded text of a topics course for graduate students. As one would therefore expect, it does provide the reader with the necessary background to attempt some of the main unsolved problems in the subject (some of which are collected at the end of the text) and I urge the reader to accept the challenge and work on their solution. Indeed, if I have imparted any of the excitement of the mathematics of the subject and encouraged others to successfully solve any of these problems, my efforts will be richly rewarded. I have been most fortunate to find only kindness and generous as­ sistance from teachers, lecturers, professors, colleagues and students on both sides of the Atlantic. So a complete list of mathematicians who have helped and inspired me over the years would be quite impossible. However, special thanks are due at this time to those who have influ­ enced my research in partially ordered groups: Richard N. Ball, William W. Boone, Paul F. Conrad, Graham Higman, Philip Hall, Valerie Kopytov, Angus Macintyre, Stephen H. McCleary, Kolya (N. Ya.) Medvedev, Bernhard H. Neumann, Akbar H. Rhemtulla, and Helmut Wielandt; to those who have written articles — and especially books — in the subject (I have learnt a lot from their strengths and occassional weaknesses); to those who have read some of the chapters in early drafts and made im­ provements and corrections: Richard N. Ball, Joachim Lambek, Akbar H. Rhemtulla, and Zheng Xiqiang; and to W. Charles Holland who first as Research Director, then as colleague and finally as "M^X" advisor

X

PREFACE

has played such an important role in my academic life; his lecture notes (by which I first became acquainted with the subject 30 years ago) have stood the test of time and have been incorporated (with thanks) into this text. I trust that this book (due 23 months before his 2 6 th birthday) will be perceived as an appropriate tribute and gift. I also wish to thank my past colleagues and students at Bowling Green State University, Ohio, USA for their encouragement and interest over the years, and the Fel­ lows of Queens' College, Cambridge for their warm hospitality. I am most grateful to Andrew Aitchison, Charles Applebaum, W. Charles Holland and especially Richard (R. G. E.) Pinch for the considerable time they have devoted to helping me with various computer problems that I have faced in writing this book and transferring files. On a more personal note, I wish to thank Paul Cohn and World Scientific Publishing Company for encouraging me to write a book on the subject, and a friend whose sarcastic remarks in the Cloisters of Gloucester Cathedral on 29.xii.96 prompted me to actually do so! Thanks also to my Mother for her constant encouragement and support, and for buying me this computer, and to Stuart and Hugh for their gift of a "Bugs Bunny" mouse pad which has seen much used. And to the one person whom it would be impossible to sufficiently thank and presumptious to try, I dedicate this book (and thereby spare her any further embarrassment !). Cambridge, 19.x.97.

CONTENTS 1

2

3

4

Definitions and Examples 1.1 Right partially ordered groups 1.2 Partially ordered groups 1.3 Examples .

.

.

...

Basic Properties 2.1 Basic group-theoretic properties 2.2 Orderability . . 2.3 Basic order-theoretic properties 2.4 Characterisations of classes Values, Primes and Polars 3.1 Values . . 3.2 Homomorphisms 3.3 Prime subgroups . . 3.4 Special values 3.5 Polars . . . 3.6 Closed subgroups 3.7 A limiting example . 3.8 Residually ordered groups . 3.9 Finite pairwise orthogonal sets

1 . 1 2 2 15 15 18 20 24

. . . .

.

.

31 31 34 36 39 41 44 49 51 53

Abelian and Normal-valued Lattice-ordered Groups 4.1 Simple Abelian lattice-ordered groups 4.2 Normal-valued lattice-ordered groups . . . . 4.3 Special-valued lattice-ordered groups . . . . 4.4 Archimedean lattice-ordered groups . 4.5 Hahn's Theorem . . . . .

55 55 58 . 6 3 65 69

XI

. . . , . . . .

.

.

xii

CONTENTS 4.6 4.7

5

The Conrad-Harvey-Holland Theorem Elementary theory (Abelian ^-groups)

Archimedean Function Groups 5.1 Free Abelian lattice-ordered groups 5.2 Finitely presented Abelian £-groups . . 5.3 The Isomorphism Problem . . 5.4 Free products of Abelian ^-groups 5.5 Kaplansky's Example . . 5.6 Bernau's Theorem ... 5.7 The Spectrum . . . . 5.8 Hyperarchimedean ^-groups

■ • ■ . . .

. . .

73 75

87 87 88 91 92 . 9 4 95 103 104

6

Soluble Right Partially Ordered Groups &: Generalisations 107 6.1 Nilpotent lattice-ordered groups . . 107 6.2 The Engel Condition . . . 109 6.3 The Word Problem . .... 113 6.4 Weakly Abelian ^-groups . .116 6.5 Divisibility 118 6.6 Conrad right orders 120 6.7 Local nilpotency 123 6.8 4-Engel right ordered groups 124 6.9 Local indicability . . . . . 127 6.10 Two sided right orders . . 133

7

Permutations 7.1 The Cayley-Holland Theorem 7.2 Amalgamation. 7.3 Convex blocks and congruences 7.4 Primitive permutation groups 7.5 Primitive components 7.6 The Wreath product

8

Applications 8.1 Normal-valued ^-permutation groups 8.2 Nilpotent relations and soluble identities 8.3 Conjugacy . .

.

139 139 144 145 146 154 156 163 163 165 169

xiii

CONTENTS 8.4 8.5 8.6 8.7 8.8 9

Free lattice-ordered groups . . . . Free products of £-groups . . . . Simple lattice-ordered groups . Finitely presented ^-groups . Undecidable Problems . .

. . .

Completions 9.1 Complete partially ordered groups 9.2 ^-convergence structures 9.3 The Order completion 9.4 Special closure 9.5 Iterated Cauchy closure . . . 9.6 The lateral completion 9.7 The distinguished completion

10 Varieties of Lattice-ordered Groups 10.1 Definitions and Examples 10.2 General facts . . . . 10.3 Minimal & maximal proper varieties 10.4 Socle 10.5 Powers of .4 . . . . 10.6 Dimension theory 10.7 Powers of 11 . . . 10.8 Covers of A . . . . 10.9 The number of varieties

. . .

.

170 175 177 182 184 191 191 195 . . 199 204 206 209 214 227 227 229 230 232 235 238 240 247 258

11 Unsolved Problems

261

REFERENCES

267

FURTHER SUGGESTED READING

275

LIST OF SYMBOLS

299

INDEX

301

Chapter 1 Definitions and Examples In this short chapter, we give the basic definitions and a host of examples to help set the stage. Some of these will be critical in understanding the limitations of subsequent theorems, and all will help illustrate their scope. Throughout the book we will write A C B as a shorthand for A C B and A ^ B, and N for {0,1,2,3,...}, the set of natural numbers.

1.1

Right partially ordered groups

Let G be a group and < a partial order on G. G is said to be a right partially ordered group if for all f,g,heG, f < g implies fh < gh. If, additionally, the partial order on G is total (i.e., for all f,g G G, either / < g or g < / ) , then G is said to be a right totally ordered group, or a right ordered group for short. Analogously, we can define left partially ordered groups and left or­ dered groups. Throughout, we will use 1 for the identity of a group (unless additive notation is used, in which case we will use 0). Note that, if G is a right partially ordered group, then the order is completely determined by G+ = {g G G : g > 1}, since f < g if and only if 1 £ gf~l We call G+ the set of positive elements of G. Similarly, if G is a left partially ordered group, then the order is completely determined byG+ 1

2

CHAPTER 1. DEFINITIONS

AND

EXAMPLES

Caution: the identity element is included as a positive element; if we wish to exclude it, we will just refer to the set of strictly positive elements of G. This is standard notation in the subject. However, it does lead to the following complication. When we refer to sets (as opposed to ordered groups) of positive real numbers, positive rational numbers or positive integers we will mean numbers that are strictly greater than 0. The standard response to this schizophrenia is that everything is clear from context; the reader can best judge how true this isn't!

1.2

Partially ordered groups

A group that is both right and left partially ordered (with respect to the same partial order) is called a partially ordered group. In this case, G+ is clearly closed under conjugation: if g € G+ and / € G then f~1gf € G+ A partially ordered set is said to be directed if every two elements have both an upper bound and a lower bound. A partially ordered set is said to be a lattice if every two elements have a least upper bound and a greatest lower bound. The least upper bound or sup of elements x and y will be denoted by xVy, and the greatest lower bound or in}of elements x and y will be denoted by x Ay. A partially ordered group whose partial order is directed will be called a directed group, and a partially ordered group whose partial order is a lattice will be called a lattice-ordered group, or l-group for short. If the order is total, then the group is said to be a totally ordered group, or an ordered group for short.

1.3

Examples

We now provide a list of examples. They illustrate the scope of the subject, as well as the limitations and strengths of the theorems that will be proved. Those marked with an asterisk are especially worth noting; the others will be of interest later. To verify that a partially ordered group is a lattice-ordered group, since f\/h = / ( l V / _ 1 / i ) and /A/i = / ( l A / _ 1 / i ) it is clearly enough to check that gVI and gAl exists for allg € G. Since g A 1 = (g~l V 1) _ 1 (Lemma 2.3.2), one needs only check the existence of g V 1 in G for all g eG.

3

1.3. EXAMPLES

*Example 1.3.1 The additive groups of integers, Z, rationals Q and real numbers 1 are all ordered groups under the natural ordering. We will use this notation throughout the book. E x a m p l e 1.3.2 Let £ be an arbitrary irrational real number. ThenZ (0,0) if and only if m + £n > 0 in K. * E x a m p l e 1.3.3 Let G be any group. Then G is trivially ordered if: / < g if and only if / = g. With respect to this partial order, G is a partially ordered group. Example 1.3.4 Let G be any group equipped with the trivial ordering and H be any right ordered group. Let L — G H. Define a partial ordering on L by: (gx, hi) < (g2, h2) if and only if hi < h2. Then L is a right directed group, and we write L — G® H. It is a directed group if H is an ordered group. In the particular case that G is the cyclic group C2 of 2 elements and H is the infinite cyclic group order-isomorphic to Z, we get "the infinite whalebone corset" It is a directed group that is not a latticeordered group: there is no least upper bound of (g, 1) and (1,1) if g is the generator of C2*Example 1.3.5 Let H be any subgroup of a (right) ordered group. Then H is also a (right) ordered group under the inherited ordering. The same is true for (right) partially ordered groups and their subgroups. Example 1.3.6 Let V be a rational vector space with basis {b; : i e / } . Let v, w € V; say v = J ^ t^b; and w = ] P r;bj, where I0 is a finite ie/o

i£h

subset of / and qt, fj G Q. Define v < w if and only if qt < rt (as rational numbers) for all i £ I0. Then V is a lattice-ordered group. * E x a m p l e 1.3.7 Let V be a rational vector space with basis {b, : i £ / } . Suppose that / is a totally ordered set. Let v , w e V with v ^ w; say v = Y^ qlhl and w = ^ r ; b t , where I0 is a finite subset of / and ft.rj G . Define v < w if and only if q} < r3 (as rational numbers) where j is the greatest i G IQ such that qz ^ r*. Then V is an ordered group.

4

CHAPTER 1. DEFINITIONS

AND

EXAMPLES

*Example 1.3.8 Every torsion-free Abelian group G can be embedded (as a group) in a rational vector space; call it V Totally order a basis for V By the previous example, V is an ordered group. With the inherited order, so is G. Thus every torsion-free Abelian group can be totally ordered so as to be an ordered group. A group G is said to be divisible if for every positive integer n and g £ G, there is h £ G such that hn = g. It can be shown that every group can be embedded in a divisible group [Lyndon k. Schupp, page 189]. We have just shown that every orderable Abelian group can be embedded in a divisible such. One of the biggest unsolved problems in the study of ordered groups is: Question: Can every orderable group be embedded in a divisible orderable group ? We will prove later (Corollary 8.6.3) that every lattice-ordered group (and so every ordered group) can be embedded (as a group and a lattice) in a divisible lattice-ordered group. *Example 1.3.9 Let X be an arbitrary topological space and C(X) be the additive group of all continuous functions from X into R (where R is equipped with the usual topology); i.e., (/ + g)(x) = f(x) + g{x) for all a; £ X. Then C(X) is a lattice-ordered group under the pointwise ordering (i.e., / < g if and only if f(x) < g(x) for all x £ X). A related example is: Example 1.3.10 Let X be an extremally disconnected topological space; that is, a Hausdorff space in which the closure of every open set is open. Consider the set D(X) of all continuous functions / from X into RU {±00} such that {x £ X : f(x) G(D) is the natural map given by v : a H-> aU{D), then x\a, b if and only if v(x) < v(a),v(b). Thus: G(D) is a lattice-ordered group if and only if D is an hcf (gcd) domain; G(D) is an ordered group if and only if D is valuation ring (with respect to v); G(D) = Z if and only if D is a discrete valuation ring (with respect to v). The directed group Ci ® Z of Example 1.3.4 provides a uesful coun­ terexample to several conjectures in this subject [Lattice-ordered Groups: advances and techniques, Chapter 4]. *Example 1.3.13 Let {Gt : i e 1} be a collection of right partially ordered groups. Then their Cartesian product

G = Y[Gl is a right partially ordered group where the ordering on G is given by: g < h if and only if g% < h% for all i G /. Similarly, for families of par­ tially ordered groups, directed groups and lattice-ordered groups. This particular ordering is known as the cardinal ordering.

6

CHAPTER 1. DEFINITIONS

AND

EXAMPLES

Let {Gi : i € / } be a collection of right partially ordered groups. Then their direct sum iei

is a right partially ordered group under the ordering inherited from the cardinal ordering. Similarly, for families of partially ordered groups, di­ rected groups and lattice-ordered groups. Another set of interesting subgroups arises in the special case that / is the set of strictly positive integers and each G1 is the ordered Abelian group of real numbers (under addition and the usual ordering). The product is then just the set of real sequences under the pointwise order­ ing and is a lattice-ordered group. Under the inherited order, so are the subgroups of all bounded sequences of real numbers, all convergent se­ quences, all sequences of real numbers convergent to 0, and all sequences of real numbers that are eventually 0 (the above direct sum). *Example 1.3.14 Let / be a well-ordered set; that is, every non-empty subset of / has a least element. Let {Gi : i € / } be a collection of right partially ordered groups. Then so is their Cartesian product

G = Y[Gt where the ordering on G is given by: g < h if and only if gj < hj where j is the least i (E I such that g+ ^ hi- Similarly, for families of partially ordered groups, directed groups and ordered groups, and for inversely well-ordered index sets (every non-empty subset has a greatest element). *Example 1.3.15 Let / be a totally ordered set and {Gi : i € 1} be a collection of right partially ordered groups. Then so is their direct sum iei

where the ordering on G is given by: g < h if and only if g3 < hj where j is the greatest i G / such that gi ^ hi. Similarly, for families of partially ordered groups, directed groups and ordered groups. We could instead use the ordering on G: g < h if and only if g* < hj where j is the least i £ I such that gt ^ hi. In the special case that I is well ordered, this is just the order inherited from the full Cartesian product of the previous example.

1.3. EXAMPLES

7

E x a m p l e 1.3.16 Let F be an inversely well-ordered set and V = V(T, E) be the set of all functions from F into E. For each g G V, let supp(g), the support of g, be the set {7 G T : 5(7) ^ 0}. Then V is an ordered Abelian group where the group operation is addition of functions, and / < g if and only if /(/?) < g(fi) where 0 is the greatest element of supp((? - / ) . More generally, if F is any totally ordered set, let V = V(F, E) denote the set of all functions from F into E whose support is either empty or inversely well-ordered. Then V = V(F, E) is an ordered group under addition and the ordering just given; it is known as the full Hahn group on F. For more details, please see Section 4.5. E x a m p l e 1.3.17 Let T be a partially ordered set such that every non­ empty totally ordered subset of F has a greatest element. Let V = V(r, E) be the set of all functions from F into M, and supp(