Ordered Permutation Groups (London Mathematical Society Lecture Note Series 55)

LONDON MATHEMATICAL SOCIETY LECTURE NOTE SERIES Managing Editor: Professor I.M.James, Mathematical Institute, 24-29 St Giles, Oxford 1. 4. 5. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 20. 21. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60.

General cohomology theory and K-theory, P.HILTON Algebraic topology: a student's guide, J.F.ADAMS Commutative algebra, J.T.KNIGHT Integration and harmonic analysis on compact groups, R.E.EDWARDS Elliptic functions and elliptic curves, P.DU VAL Numerical ranges II, F.F.BONSALL & J.DUNCAN New developments in topology, G.SEGAL (ed.) Symposium on complex analysis, Canterbury, 1973, J.CLUNIE & W.K.HAYMAN (eds.) Combinatorics: Proceedings of the British combinatorial conference 1973, T.P.McDONOUGH & V.C.MAVRON (eds.) Analytic theory of abelian varieties, H.P.F.SWINNERTON-DYER An introduction to topological groups, P.J.HIGGINS Topics in finite groups, T.M.GAGEN . Differentiable germs and catastrophes, Th.BROCKER & L.LANDER A geometric approach to homology theory, S.BUONCRISTIANO, C.P.ROURKE & B.J.SANDERSON Sheaf theory, B.R.TENNISON Automatic continuity of linear operators, A.M.SINCLAIR Parallelisms of complete designs, P.J.CAMERON The topology of Stiefel manifolds, I.M.JAMES Lie groups and compact groups, J.F.PRICE Transformation groups: Proceedings of the conference in the University of Newcastle upon Tyne, August 1976, C.KOSNIOWSKI Skew field constructions, P.M.COHN Brownian motion, Hardy spaces and bounded mean oscillation, K.E.PETERSEN Pontryagin duality and the structure of locally compact abelian groups, S.A.MORRIS Interaction models, N.L.BIGGS Continuous crossed products and type III von Neumann algebras, A.VAN DAELE Uniform algebras and Jensen measures, T.W.GAMELIN Permutation groups and combinatorial structures, N.L.BIGGS & A.T.WHITE Representation theory of Lie groups, M.F.ATIYAH et al. Trace ideals and their applications, B.SIMON Homological group theory, C.T.C.WALL (ed.) Partially ordered rings and semi-algebraic geometry, G.W.BRUMFIEL Surveys in combinatorics, B.BOLLOBAS (ed.) Affine sets and affine groups, D.G.NORTHCOTT Introduction to H p spaces, P.J.KOOSIS Theory and applications of Hopf bifurcation, B.D.HASSARD, N.D.KAZARINOFF & Y-H.WAN Topics in the theory of group presentations, D.L.JOHNSON Graphs, codes and designs, P.J.CAMERON & J.H.VAN LINT Z/2-homotopy theory, M.C.CRABB Recursion theory: its generalisations and applications, F.R.DRAKE & S.S.WAINER (eds.) p-adic analysis: a short course on recent work, N.KOBLITZ Coding the Universe, A. BELLER, R. JENSEN & P. WELCH Low-dimensional topology, R. BROWN & T.L. THICKSTUN (eds.) Finite geometries and designs, P. CAMERON, J.W.P. HIRSCHFELD & D.R. HUGHES (eds.) Commutator Calculus and groups of homotopy classes, H.J. BAUES Synthetic differential geometry, A. KOCK Combinatorics, H.N.V. TEMPERLEY (ed.) Singularity theory, V.I. ARNOLD Markov processes and related problems of analysis, E.B. DYNKIN Ordered permutation groups, A.M.W. GLASS Journees arithmetiques 1980, J.V. ARMITAGE (ed.) Techniques of geometric topology, R.A. FENN Singularities of differentiable functions, J. MARTINET Applicable differential geometry, F.A.E. PIRANI and M. CRAMPIN Integrable systems, S.P. NOVIKOV et al.

London Mathematical Society Lecture Note Series. 55

Ordered Permutation Groups

A.M.W. GLASS Professor, Bowling Green State University, Bowling Green, Ohio

CAMBRIDGE UNIVERSITY PRESS CAMBRIDGE LONDON

NEW YORK

MELBOURNE

SYDNEY

NEW ROCHELLE

CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521241908 © Cambridge University Press 1981 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1981 Re-issued in this digitally printed version 2008 A catalogue record for this publication is available from the British Library Library of Congress Catalogue Card Number: 81-16996 ISBN 978-0-521-24190-8 paperback

To my wife, Rona, and my mother, Vicky; and in memory of my father, Jack (1912-1969) and my aunt, Ethel (1902-1980).

TABLE OF CONTENTS

Background Terms and Notation Explanation of Diagrams PART I

OPENING THE INNINGS

Chapter 1 1.1. 1.2. 1.3. 1.4. 1.5. 1.6. 1.7. 1.8. 1.9. 1.10. 1.11. Chapter 2 2.1. 2.2. 2.3. 2.4. 2.5. PART II

Introduction

,

A(Q) Actions of groups on chains Partially ordered groups. . . , Congruences Stabilisers and blocks Transitive actions Primitive components Dedekind completion and character Bumps and supports Multiple transitivity Identities

1

. . . .

Doubly Transitive A(ft) Geometry versus algebra Divisibility and conjugacy in A(ti) The normal subgroups of A(tt) The automorphisms of A(fl) Embedding in doubly transitive A(tt) . . . .

2 4 6 8 11 13 18 21 24 28 35 38 42 56 65 71 73

THE STRUCTURE THEORY

Chapter 3 3.1. 3.2. Chapter 4 4.1. 4.2. 4.3. 4.4. 4.5. Chapter 5

Congruences and Blocks

76

Transitive ordered permutation groups . . . Intransitive ordered permutation groups . .

78 80

Primitive Ordered Permutation Groups . . .

93

Transitive primitive ordered permutation groups 95 Totally ordered transitive ordered permutation groups 98 Periodic primitive I-permutation groups . . 105 Intransitive primitive ordered permutation groups 112 The proof of Theorem 4D 114 The Wreath Product

117

PART III APPLICATIONS TO ORDERED PERMUTATION GROUPS Chapter 6 Simple I -Permutation Groups 129 Chapter 7 Uniqueness of Representation 140 7.1. ^-Permutation groups 143 7.2. Ordered permutation groups 14-8 Chapter 8 Pointwise Suprema and Closed Subgroups . . . 151 8.1. Closed stabiliser subgroups 153 8.2. Pointwise suprema 157 8.3. Closed subgroups of £-groups 164 Chapter 9 Automorphisms of A (ft) 172 PART IV APPLICATIONS TO LATTICE-ORDERED GROUPS Chapter 10 Embedding Theorems for Lattice-Ordered Groups 183 Chapter 11 Normal Valued Lattice-Ordered Groups. . . . 206 PART V

THE AUTHOR'S PREROGATIVE Chapter 12 Algebraically Closed Lattice-Ordered Groups Chapter 13 The Word Problem for Lattice-Ordered Groups

221 231

APPENDIX I

240

APPENDIX II

246

SOME UNSOLVED PROBLEMS

248

BIBLIOGRAPHY

253

ANNOTATIONS

xi

INDEX INDEX OF SYMBOLS

xliii xlvii

PREFACE

In the past thirty years, groups of order-preserving permutations of totally ordered sets have been extensively studied.

The purpose of these notes is to provide a uniform,

systematic account of this research and its applications.

In the

first half of this book (Parts I and II), I attempt a streamlined (and I trust intuitive) presentation of the main results in the structure theory, taking full advantage of recent research.

In

Chapters 3 and 5, the study of such permutation groups is reduced to an investigation of the basic building blocks of the subject, "primitive'1 order-preserving permutation groups. classified and examined in Chapters 2 and 4.

These are

The second half of

the book is devoted to various applications of the structure theory; e.g., to the construction of infinite simple groups. Most of the chapters in it can be read quite independently of each other.

I have chosen the topics to illustrate the use of the

structure theory in a wide variety of settings, but readily admit that the selection is strongly influenced by my own quirks and prejudices.

The total order on a set naturally lifts to a lattice

order on its group of order-preserving permutations.

Since every

lattice-ordered group can be embedded (as a group and lattice simultaneously) in such a lattice-ordered group—the Cayley-Holland Theorem, see Appendix I—the theory can be used to study latticeordered groups.

For example, I will show:

every lattice-ordered

group can be emb-edded in a simple divisible lattice-ordered group in which any two elements greater than the identity are conjugate; and my favourite application:

there is a finitely presented

lattice-ordered group with insoluble word problem

(Whereas

analogues of both these results for groups can be proved without

recourse to permutation groups, I know of no proof for latticeordered groups which avoids using groups of order-preserving permutations of totally ordered sets). My main aim throughout has been to impart the intuition and excitement of the subject.

If any of this comes across, I will be

most satisfied. With the exception of Chapter 1, I have begun each chapter with an account of the goals and main theorems it contains (though often not stated in the generality in which they are proved).

I

hope that this will help the reader to get a good overall picture of the subject, as well as whet his or her appetite; it should also make it easier to locate results in the book.

I have

included several theorems which have not previously been published, so these notes hold something new even for the specialist.

Since anyone who has read Parts I and II and a few

of the applications is at the forefront of the subject, I have included a list of unsolved problems which I think are of especial interest. Caution:

I am sure that the reader will find others to work on. in order to give as uncluttered and intuitive an

account as possible, I have often put more stringent hypotheses on theorems than are really necessary. technicalities.

This eliminates some

The interested reader should consult the

literature to find out the full story (an annotated bibliography is included to make this easier). As a result of my earlier book of the same title (published in 1976 by Bowling Green State University and now out of print), David Tranah approached me to write a monograph on the subject for Cambridge University Press.

These notes are the consequence.

Unlike the 1976 version, they are not encyclopaedic but are designed instead for a much broader audience.

I have confined my

attention to those theorems in the structure theory which have been most fruitful for applications, and provided simpler proofs to some when more recent research has made this possible. Also, many of the applications given here are the result of work done in the last five years, and so did not appear in the 1976 version.

This book was obtained by photocopying the typed version of the manuscript.

Consequently, everyone reading it will appreciate

the excellence of the typist, Linda Shellenbarger, and the deep gratitude I owe her.

I am also beholden to Bruce Lyle for his

superb illustrations; Ashok Kumar Arora and Manfred Droste for reading the handwritten version and making suggestions for needed improvements; Todd Feil for reading the typed version; and Rona Glass for the index. When it comes to thanking mathematicians, I do not know where to begin.

I have had the great fortune to find, without

exception, only kindness and generous assistance from teachers, lecturers, professors and colleagues on both sides of the Atlantic. Since a list of names would be far too long, let me just humbly thank all I have come in contact with.

However, I would

especially like to thank W. Charles Holland who, first as a thesis director and then as a colleague, has so willingly shared many insights and speculations with me.

Stephen McCleary has also been

a constant source of encouragement and contagious enthusiasm.

In

addition, I would like to thank my colleagues in the mathematics department here for their friendliness and interest in my work. Like everyone in the States, they have made my stay here most enjoyable. It gives me great pleasure to acknowledge my thanks to them in print. Finally, I would like to thank David Tranah at Cambridge University Press for his encouragement, help, patience and good humour throughout the ordeal of writing these notes.

It is quite

accurate to say that without his urging, this book would never have been written.

I trust it will not be held against him for

too long!

Bowling Green, Ohio, U.S.A Whitmonday, 1981

BACKGROUND TERMS AND NOTATION

1. SET THEORY. Let

A

and

B be sets.

subset of B, and A 9 B will use A\B for into

We will write A G. B

if A

If f

is a function from A

the range (or image) of

let f\C_ = {(a,b) £ f: a 6 C},

f; and

if

C ^ A,

to

C. We will use 0_ for the empty set, (u for the least

infinite ordinal, wetc. o)1

with &.,

the restriction of f

for the least uncountable infinite ordinal,

So o)= {0,1,277..}.

and say A

is a

is a proper subset of B. We

{a £ A: a £ B}.

B, let Af_ - {af: a C A},

if A

We will often identify

etc. We will write

is countable if

u> with N Q ,

\A | for the cardinality of A,

|A| = W Q .

2. PARTIALLY ORDERED SETS. A set ft with a reflexive, antisymmetric, transitive relation

<_ defined on it is called a partially ordered set, or

p.p. set. As usual, we will write A p.o. set in which

a <_ 3

a < 3 for a <_ 3 & a ? 3.

only if a = 3 is said to be

trivially ordered. In contrast, a p.o. set ft is said to be totally ordered or linearly ordered (or a chain, for short) if for all a,3£ft, each

a<_3

or

3£a.

Ap.o. set ft in which

a,3 £ ft have a least upper bound (denoted by a v 3) and a

greatest lower bound (denoted by q A 3) is called a lattice. If ft is a lattice and a v ( 3 A

Y

) = ( a v 3 ) A ( a V Y ) for all

a,3,y € ft, then ft is said to be a distributive lattice; equivalently, if a A ( e v y ) = ( a A g ) v ( a A

Y

) for all

a,3,y € ft. If ft is a p.o. set and T Q ft, we say that dense in ft if whenever that

a < x < 3

T is

a < 3 in ft, there is T £ T such

(cf., dense in itself, page 83).

We will call ft

dense if it is dense in ft. If write

(a3 gj = { y € f t : a < y < 3 } we

Y € [a,£], Y £ (a,3), T

a < 3

[a3 3] = { y € f t : a < ^ y ^ 3 } say that

that

y

y

for the open interval. If lies between a

and

lies strictly between a

3; if

and

3. A subset

of a p.o. set ft is said to be an interval of ft (or convex T

-\_yT2 ^

in ft) if whenever T2,

then

y € T.

T

an(

^

a <_ 3

lies between

in ft* precisely when

ft. So ft* is also a p.o. set. is a family of p.o. sets, then (J{ft.: i € 1}

Ye ^

T-. and

Ifftis a p.o. set,ftj^_is the set ft with the

reverse ordering; i.e.,

or

in a p.o. set ft, we

for the closed interval, and

ordered by:

If

I

is a chain and

a >_ 3 in {ft.: i € 1}

(j{ft.: i € 1} is the set

a £3

if

a 6 ft., 3 € ft. and

(i = j & a <_ 3 in ft.); (j{fl.: i € I) is the same set but

ordered by: a £ 3

a < 3

if

a € ft~ 3 € ft. and

i > j

or

(i = j &

inft.). We will use %j

@_ and

R_ for the chains of integers,

rationals and reals respectively (under the natural order). Qj

i < j

and

R_

Z ,

will denote the sets of strictly positive integers,

rationals and reals, respectively. If ft., and ftp are lattices, then a lattice isomorphism is a one-to-one map of operations.

ft

onto ft^ that preserves the lattice

In the special case that

ft

and ft^ are chains,

lattice isomorphisms are called ordermorphisms, and the chains are said to be ordermorphic.

3.

GROUP THEORY.

If

G

write

is a group and

H < G

if

H

G;

H,

i.e., an extension of

subgroup of a group cosets of

C

in

G.

is a subgroup of

is normal. We will write

group with normal subgroup to

H

G,

H * G

having quotient by H

by

G.

If

R(C) = {Cg: g € G},

C

H

we will for a isomorphic

is a

the set of right

All isomorphisms are onto. %~ will denote

the two element group. We will use of any group.

G,

e

for the identity element

4.

MODEL THEORY.

Our language will be the first order language of groups, lattices with

e, or 5,-groups.

So our formulae will be built up

in the usual way from equality of group words, lattice words or £-group words by using (or),

V

(for all) and

-• (not), 3

&

or

/X\ (and),

(there exists).

or

if the sentence G E H

9

We will write

G fc 8 G,

and

e, £-groups) satisfy the same

sentences of the language of groups (lattices with

5.

A formula

holds in the group (lattice, £-group)

if the groups (lattices with

V

Note (3 integer n )

is not permitted; nor are infinite conjunctions, etc. without free variables is called a sentence.

or

e, ^-groups).

CONVENTIONS ADOPTED.

Capital Greek letters are used for chains, small Greek letters for elements of them (exception: a, K, A, y, v, £, n, C are sometimes used in other contexts, e.g., as ordinals, members of index sets; G, H

ip, cj), 6

are usually reserved for mappings).

will denote groups, lattices with

c, f, g, h, k, x, y, z

will denote elements of them.

are reserved for general index sets; index sets. k

C,

e, or £-groups and

i, j

I

and

J

for elements of these

Script letters are used for equivalence relations;

is reserved for the set of covering pairs of congruences (see

Sections 1.6, 1.7 and Chapter 3 ) , and We will sometimes use

//

K

for an element of

k.

to denote the end of a proof;

sometimes Q.E.D.; sometimes nothing.

EXPLANATION OF DIAGRAMS.

The picture

o;

(

1

)

... (_

T

)

...A-T»E

will be used to illustrate the chain fi obtained from replacing each

a € Z by a copy of the chain

fi = T x z = { ( T , O ) : T € T, o € Z}

if a with

a

2 °r

1 "2 * Ll partition of Z.

The picture

ordered by:

Let

T;

z by

i.e.,

(T-^CO < ( T2

Z, T , T 2

be chains,

will be used to illustrate the chain each

a. € Z. by a copy of

Q

obtained by replacing

T. (i = 1,2). That is,

n = {(T]_,a1): i1 6 T 1 , a 1 € 5^} u {(x 2 ,a 2 ): x 2 € T 2 , a 2 € Z2> ordered by:

(i,a) < ( T ' , O ! )

if

o < a'

(in

z)

or

(0 = 0'

& T < xr ). For such layered diagrams, we will use the horizontal (rather than the diagonal) to depict the identity map; e.g.,

T.'iZ

PART I OPENING THE INNINGS

CHAPTER 1 INTRODUCTION

The theory of permutation groups (especially in the finite case) has been developed over a long period of time.

One motiva-

tion for the study was geometric. Another was provided by Cayley's Theorem that every group is (isomorphic to) a subgroup of a permutation group. More recently, infinite permutation groups have been studied and used to provide examples (and counterexamples) in infinite group theory. On the other hand, ordered permutation groups have been studied for a relatively short time. Again, one motivation to examine them was geometric. Another was provided in 1963 by W. Charles Holland's analogue of Cayley's Theorem:

Every lattice-

ordered group is (isomorphic to) a subgroup of the lattice-ordered group of all order-preserving permutations of a chain (linearly ordered set). T

A further reason was provided by model theory:

is a first order theory having infinite models and

chain, then there is a model Of of each order-preserving permutation of

T

containing

If

is a

Q, such that

Q, extends to an automorphism

of Ot (Ehrenfeucht and Mostowski [56]).

Hence any study of the

group of automorphisms of models of such theories should begin with a study of groups of order-preserving permutations of chains. In this chapter we will devote ourselves to basic definitions and results. Many of these are routine translations of theorems concerning permutation groups to the ordered permutation group setting.

The presentation we give is self-contained and does not

require prior knowledge of the theory of permutation groups. However, anyone wishing to become conversant with the roots of the subject could do worse than refer to H. Wielandt's works [64-], [67] and [69]. In contrast, some of the theory we present here

2 (especially in Sections 1.8, 1.10 and l.ll) is peculiar to the ordered case, though we have included it in this chapter only if it is rather straightforward.

1.1.

AfftJ.

Let ft be a chain.

We define

AfftJ

to be the set of a l l

order-preserving permutations of ft; i . e . , if

g

if

a,ft€ ft and a < 3, then

is a one-to-one function from

composition:

if

f,g € A(ft)

ft onto

ag <

define

A(ft)

af £ ag

ft

if and only

that s a t i s f i e s :

is a group under

fg by: a(fg) = (af)g

(a € ft). Further, we can define an order if

g € A(ft)

<_ on A(ft) via: f <_ g

for all a € ft. This order on A(ft) is called the

pointwise order.

It is immediate that if

f <_ g,

then

fh £ gh

elements

g ,g~ €- A(ft)

and hf <_ hg.

f,g,h £ A(ft) and

Moreover, any pair of

has a supremum (least upper bound) and

infimum (greatest lower bound) in A(ft)—which we denote by and

g

1

respectively--namely:

- max{ag 1 ,ag 2 > (Since

is a chain,

and a ( g 1

1 «g 2 or

= min{ag 1 ,ag 2 > (a € ft).

EXAMPLE 1.1.1. natural ordering).

Let fi = Z ,

The order-preserving permutations of

just translations by integers. by

n,

the chain of integers (in the

then the map

n

t

If

t

(at

= a

In this example, for each such that

ag = 3>

are

n <_ m,

then

<_a + m = a t

a,3 € Q,

t

onto

<_ t

for all

in the

a € 2Z).

there is a unique

g 6 A(ft)

and the pointwise order is a total order.

EXAMPLE 1.1.2.

Let fi =

the natural ordering).

Let

However,

e ;

lg

%

denotes translation

is a group isomorphism of

that preserves order; i.e., if pointwise ordering

€ k{%)

= 2

(so

the chain of real numbers (in

g . a i-* 2a (a € l ) . and

(-l)g0 = -2

Then e /gQ).

(so

Hence the pointwise order is not a total order; so there is no ordermorphism of Then group. h

f

€ A(1) If

€ A(R)

1 and

onto f

Let

g f

o o*

0g0

a ,& ,a0, such that

A(ffi).

f^: a

Thus

A(l)

= a?

straight line passing through

is not an Abelian a ? < 3p, '2

with a h

a + 1 (a € I ) .

&

e-g-

3

(a-.,oip )

and

(B-,,8.

h_

there is is the

EXAMPLE 1.1.3.

Let 2 . (i = 1,2) be disjoint copies of

and let ft =22

have

#

•

21. number of elements of ft between them. images under property. a

g

Hence if

of any two points of Z

Therefore 22pg = Z. orZp.

If

g £A(ft),

the

still enjoy this a. €22. (i = 1,2),

then

S0S0 a a g g< a ga # g # I t I f to lfloolwlso wt sh atth a Zt Z = z l 2 i ^i ^

< a ;

) l 2 l 2 Consequently, by Example 1.1.1, there is an order-preserving group isomorphism between (m ,n_) <_

A(ft) and Z ©2Z if m

&

<_ m

n

if we order Z Q7L by: <

(in 2Z).

ACTIONS 0F GROUPS OT CHAINS.

Let ft be a chain and is a triple kernel of

(G,ft,e) where 0,

g € ker(e),

ker(Q),

a group.

An action

of

G on ft The

is called the lazy subgroup (if

a(ge) = a

for all

(i.e., the lazy subgroup is say that

G

9: G + A(ft) is a homomorphism. a € ft). If

is one-to-one

{e}), we call the action faithful and

(G,ft,6) is a representation of

(G,ft,6) is an action,

0 G.

It is clear that if

(GAer( 6 ),ft,0*) is a faithful action where

(ker(e)g)e* = ge (g e G ) .

EXAMPLE 1.2.1. Consider (a,r € I).

Then

representation of

(1,1,8) where

a(re) = a + r

(I,ffi,e) is a faithful action, and so a I.

Observe that the group

16

(the group

operation is composition) is isomorphic to the additive group of reals.

Hence

10 £ A ( I )

16 ^ A(l).

is total (if

Indeed, the pointwise ordering on r,s € I,

is the pointwise ordering).

re <_ s0

or

s8 <_ re

where <_

5 LEMMA 1.2.2. If then the map

(G3UiQ)

is an action of G

(a.3g) *-> a(gQ) of ft x G

(i) (cxje) •—• aj

(ii) the image of

of

($>g) whenever a < 3., and

as

($3g) where 3

on a chain ft.,

onto ft satisfies

(a^g) is less than the image

(Hi) (a>fg) has the same image

is the image of

(a>f). Conversely3 any map

from fi x {? into ft satisfying (i)> (ii) and (Hi) gives rise to a homomorphism of

G

into AfoJ

(and hence to an action of

G on

a). Proof: That any action gives rise to a map satisfying (i), (ii) and (iii) is utterly obvious. ot(ge) is the image of (a,g~ g) »—* a

Conversely, define

(a,g) (a € ft, g € G).

so by (iii) 3(g0) = a

by (ii), g6 € A(ft). Thus

where

6: G -> A(ft).

6 by:

By (i), 3 = a(g" e). Hence,

By (iii), 6

is a homo-

morphism. We will often write clear from context.

(G,fl) instead of

In particular, if

(G,«,0) when

6 is

G £.A(ft), 0 will be

assumed to be the identity (unless specifically stated to the contrary). In this case we will endow ing and say that

(G,ft) is an ordered permutation group. So an

ordered permutation group

(G,ft) is a subgroup of

with its inherited action on ft. If permutation group and for each and

g- A g

of

G with the pointwise orderA(ft) together

(G,ft) is an ordered

g^,g^ € G,

A(ft) actually belong to

the elements

g. v g

G, we will say that

(G,ft) is an t-permutation group. So for any chain ft, (A(ft),ft) is an ^-permutation group.

EXAMPLE 1.2.3.

Let ft = 1

preserving permutations of That is, g £ G such that

if

(0 <_ i < n ) . Then

G

be the group of order-

that are finitely piecewise linear.

g 6 A(l) and there are

g|(-°°,a0], g| [a^ou^]

not Abelian 1.1.2).

1

and

and

a Q < ... < a n

g| [an,»]

where

*\yg0

1

are linear

(G,E) is an ^-permutation group.

(f ,g £ G

in

Note

are defined in Example

G

is

6 If

(G,ft) is an action, we may think of ft as an algebra

with unary operations A given by

g (g € G), and binary operations v

v( a ,3) = max{a,$}

and A(a,3) = min{a,3).

subalgebra (or G-invariant set) will then be a subset such that set. aG

If

Ag = A

for all

g € G.

and

A A

of ft

Note that

0

is a G-invariant

a £ ft, the subalgebra generated by

a

is denoted by

and is called the G-orbit of

THEOREM 1.2.4.

Let

a.

So

aG = {ag: g € G}.

(G39.) be an action. Every G-invariant

set is the union of the G-orbits it contains. Moreover3 the set of

G-orbits of points of ft partitions ft. Thus the collection

of G-invariant sets forms a complete Boolean algebra under union and intersection. Proof: If 3G = agG = aG.

3 € aG,

then

3 = ag

for some

g € G.

Hence

So the G-orbits partition ft., The rest of the

theorem is .now obvious

(0

and ft are the

0

and

1

of the

complete Boolean algebra).

If ft = aG action of

G

for some (and hence every) a € ft, then the

on •ft is said to be transitive. Examples 1.1.1,

1.1.2, 1.2.1 and 1.2.3 are transitive ^-permutation groups but Example 1.1.3 shows that

(A(ft),ft) need not be transitive.

If

(A(ft),ft) is transitive, we will simply say that ft is homogeneous. (Caution:

This is not the model theory usage of the word;

homogeneous for us is nl-homogeneousn of model theory.) Throughout the book we will assume that IF

(G,ft) IS TRANSITIVE, THEN ft IS INFINITE.

This eliminates the trivial case that

G = {e} and

|ft| = 1.

1.3. PARTIALLY ORDERED GROUPS. A group of

G

together with a partial order

<_ on the elements

is called a p.o.group (partially ordered group) if whenever

f,g,h € G, £

G

f < g. implies

is a lattice (i.e.,

bound of

g.

and

g

gp) and

fh <_ gh v g

and

hf <_ hg.

If, in addition,

(the supremum or least upper

g. A g

(the infimum or greatest

7 lower bound of that

G

g-. and

gp) exist for all

G

is an £-group,

and dually for A ; f A g = (f

vg

also

for short.

If

(f vg)h = fh v g h ,

G

is positive; if

h( f v g) = hf v hg

f V g = (f~ A g" )"

) ~ . If

total order, we say that

G

G

and

i s a p . o . group in which

<

e < g € G,

e <_ g € G, we say that

we say that I

g

with the usual orders are o-groups,

In particular,

G

If

(G,ft)

A(ft) is an £-group for

every chain ft. Another example of an £-group is % ®% as in Example 1.1.3.

g

is strictly positive.

is an £-group under the pointwise ordering whenever

is an ^-permutation group.

then

is a

is a totally ordered group , or o-group

is a p.o. group and

As examples, 22 and G

we say

is a lattice-ordered group, or l-group for short. Note

that if

and

g-.,gp £ G),

ordered

(G,ft) is an ordered permutation group,

is a p.o. group.

If

G

is an £-group and

H

is a subgroup of

closed under the lattice operations (v to be an Z-subgroup of

G.

and A ) ,

G

that is

then H

is said

Hence an ordered permutation group

(H,ft) is an ^-permutation group precisely when H

is an

£-subgroup of A(ft). An o-homomorphism from a p.o. group is a group homomorphism from partial ordering.

If

G

and

G H

into H

G

into a p.o. group

H

which preserves the

are ^-groups and the o-homomor-

phism preserves the lattice operations, we will call the o-homomorphism an t-homomorphism. If \\>

and

^

is one-to-one and both

\\i are o-homomorphisms, we say that

o-embedding. If

\\)

is an

ip is a one-to-one ^-homomorphism, we say that

ip is an t-embedding. It is trivial to show that any ^-embedding is an o-embedding.

An (£-)o-embedding of

an ('t-)o--isomorphism between

G

and

G

g € G.

is an t-homomorphism if In Example 1.1.3,

Let If

(G,ft) and

cj>:ft•> T

is called

H. We will frequently use

the following fact (which is easy to verify): ip: G •> H

onto H

A homomorphism

(g v e)\|/ = g\p v e

A(ft) and TL © ^

for all

are £-isomorphic.

(H,T) be ordered (£-)permutation groups.

is an order-preserving map and

o-(£- )homomorphism, then

ip: G ->- H

(ip,<J>) is said to be an

is an

8 o-(l-)homomorphism of (a € ft, g € G).

(G,ft) into

If, in addition,

4

(H,T) if

(a+)(g*) = (ag)<|>

is one-to-one and

*

is an

o-( £-)embedding we obtain an ordered (Z-)permutation group embedding of

(G,ft) into

also onto.

(H,T), and an o-(Z-)isomorphism if

If H = G

and

$

be an o-(Z-)homomorphism if

is the identity, then

<J> and 4

i|; are

is said to

(i|>,$) is.

1.4. CONGRUENCES. Let (in

X)

X

if

Let

be a partially ordered set and y-L^y2 € Y

convex and

for

C

<xg£3g

Y c x. Y imply

?

x € Y

is convex (see p. vii).

(G,ft) is an

on ft such that each equivalence class is

whenever

a C3 (a,3 € ft, g € G).

is an equivalence relation on ft, we will write

{3 £ ft: a$3}>

and ft/C for If

y^ <_ x <_ y 2

(G,ft) be an action. A congruence of

equivalence relation

If

and

the equivalence class of C

a£

which contains

{aC: a € ft}.

(G,ft) is an action with congruence 5,

linearly order ft/£ by:

aC < 3? if

a < T

then we can

for all

a € af and

t € 3C

There is a natural action (a € ft, g € G),

and

(G,ft/£) given by:

(a£)g = (ag)C

(j>:ft->ft/i?is an o-(£-)homomorphism if

(G,ft) is an ordered (I-)permutation group. Moreover, if

L

is

a,

the lazy subgroup of the action (G,ft/£), then (G/L,ft/£) is an ordered (t-)permutation group and the natural map of (G,ft) onto (G/L,ft/5) is an ordered (Z-)permutation group horaomorphism. If (G,ft) and (G,T) are ordered permutation groups and <j>:ft-> T is an o-homomorphism, the kernel of <|> is the equivalence relation ^ defined by: a X £ if ct(f> = 3<J>. Clearly, ^ is a congruence of (G,ft). So if T = ft/<J and <J> is the natural map of ft onto ft/£, the kernel of is C. The following two theorems are straightforward—either by direct proof or universal algebra. THEOREM 1.4.1 (Homomorphism Theorem). Let (G3ftj be an ordered permutation group. (i) The kernel of an o-homomorphism is a congruence of (Gy ftj. (ii) Let 4>:ft-> T be an o-homomorphism with kernel *% and r\:ft-> 0,/ti the natural map. Then there is an o-embedding <j>': Q.M -> T such that n<|>' = $* (Hi) If <J> is as in (ii)y then Q$ is a G-invariant subset of T; and if $ is onto, $ ' is an ordermorphism. THEOREM 1.4.2 (Correspondence Theorem). Let (GyQ) be an ordered permutation group and $ a congruence of (G,Q). The natural map Q •> Sl/6 induces a one-to-one correspondence between the congruences of (G, u) which contain C and the congruences of (G39/S); it induces a one-to-one correspondence between the G-invariant subsets of ft which are unions of C classes and the G-invariant subsets of Q./C. THEOREM I.4.3. Let (G>Q) be an ordered permutation group and N be a convex normal subgroup of G. Define C by: a GN& if a L < M ahp for some h^h^ € N. Then C^ is a congruence of (G, oj and (G/Nj $./$„) is an action where (a&N)(gN) = (ag)GN (OL € ft^ g € G). Conversely, if C is any congruence of (G>n), let N = {g € G: agC a for all a € ft}. Then

10 N

is a convex normal subgroup of

G

that is an t-subgroup if

(G3Si) is an ^-permutation group. Moreover, in this case> (G/Nj $./£)

is an t-permutation group where Ng >_ Nf

if g >_ hf

for some h € N. Proof: Suppose a

— $ Jiaf>

a£ 3

and let

ctg <_ $g <_ ctfg = ag(g"1fg). Since ag£.3g.

Hence £^

hN = gN;

g £ G.

(without loss of generality).

so

(3
(G,ft). If

a^ 3

and

Nh = Ng.

Conversely, if If

f € G

a € Q, of

f

so

Nh = Ng, then Thus

It now follows

(G/N,fi/JN) is an action (use Lemma 1.2.2). C

is a congruence of

(G,fi) and

N = {g € G: ag5a for all a € ft}, then clearly G.

with

ah _£ 3h £ afh = (ag)g-1fg-g"1h.

But if

f € N

g~Xfg £ N;

N < G,

is a congruence of

g-l-h € N.

Let

Then

and

gf € N.

g € N, Hence

then as N < G.

N

1

is a subgroup of 1

(af" )g £( af" )

Since

(G, ft/£), (G/N,ft/£) is faithful.

N

for all

is the lazy subgroup

The "£-" part is now

straightforward.

We next give some examples. The justifications of the claims in each are routine and are left to the reader.

EXAMPLE 1.4.4. group of

G.

ordered by:

Let

Let

G

Hg 1 <_ Hg 2

if

hg

is a partially ordered set. ordering) then

be a p.o. group and

H

a convex sub-

R(H) be the set of right cosets of <_ g 2

If

for some

h € H.

in

G

Then

R(H)

R(H) is a chain (under this

(G,R(H),e) is an action where

(Hf)g0 = Hfg

(f,g € G). This action is faithful precisely when In the special case that

H

H = {e} and

f^ g" Hg = {e}.

R(H) is a chain (hence

G = R(H) is an o-group), the faithful transitive action is called the right regular representation of

G.

(G,G,e)

Examples 1.1.1

and 1.2.1 are right regular representations.

EXAMPLE .1.4.5. Then

Let

(G,fi) be any ordered permutation group.

(G,Q) has two trivial congruences, denoted throughout the

book by J

and U.: a
if

a = 3

(the singleton congruence)

11 and a213 if a, 3 € ft (the universal congruence). EXAMPLE 1.4.6. Let ft = I\&. Then ft is homogeneous and (A(ft),ft) has a non-trivial congruence Z defined by: a £3 if a,3 € (n,n + l) for some integer n. 1.5. STABILISERS AND BLOCKS. Let (G,ft) be an action and A Q ft. The stabiliser (or isotropy) subgroup on A, G= {g £ G: 6g = 6 for all 6 € A}. So G. is the set of elements of G that fix A pointwise. If A = {a}, we will write G for G, ,. On the other hand, let Gf \ = {g € G: Ag = A}; i.e., G/ v is the set of all elements of G which map A to A (not necessarily fixing A pointwise). Note that G and G, \ are subgroups of G and G <^ G/ y Moreover, if (G,ft) is an ordered (£-)permutation group, then G and G/ N are convex (£-)subgroups of G. In Examples 1 1.1 and 1.2.1, G^ = {e} for all a € ft, and G A = G, x = {e} if A J 0,ft. However, in Example 1.1.3, if a € ft, G =% and Z = (L, $. G/ x = G (i = 1,2). In Example 1.1.2, if A = 7L, i i G G aS is easi1 A ^ (A) ^ s e e n (fo € G (A) NG A^' I n E x a m P l e i-2if A = (0,1), G A 1 G ( A ) . A central fact which we will use repeatedly (and should therefore be engraved on the reader's mind) is: The Fundamental Triviality:

a — - — > ag af

commutes

-»- agf

if (G,ft) is an action. Hence f" G f = G f . By the fundamental triviality, f^\ g - 1 G g p. Hence if G~. g =G g€G (G,ft) is transitive, f\ g"xG g = G r = G = {e}. So if (G,ft) a ab g€G " is a transitive ordered (t-)permutation group and R( G ) is a chain, (G,R(G )) is also a transitive ordered (£-)permutation group.

12 THEOREM 1.5.1. Let group and phic to

a € ft. Then R(G )

G

a

fg~

g = G

Let

a^

So

^

G

3 € ft. Choose

G f = G g.

f

a

G

af <_ ag.

1

g € G

Therefore

(f v g)g"" € G . Thus

g

- a *

3<|> = Gag.

one and onto.

ag = 3.

<j> is well-defined. let

(j>:ft-> R(G )

Let

fg""1 € G^

so

It is clearly one-to-

g. € G with

so

4> preserves order.

R(G ) is a chain.

af = 3 = ag, then

3 X <. 32>

If

Consequently,

so that

If

a(g1 v g 2 ) = 3 2 = ag 2 ,

Hence

(G,Q) is t-isomor-

f,g € G. Without loss of generality,

v e € G ; i.e.,

fV g

be defined by

Now

is a chain and

(G,R(G )).

Proof: Let Hence

(GSQ) be a transitive l-permutation

3-^ = G ^

agi = 3. (i = 1,2).

<_ G o (g 1 v g 2 ) = 32*.

Finally,

(Bh)* = (agh)(j> = G gh = (G g)h = (B*)h (h € G). Consequently, 4> is an L-isomorphism. // Let

(G,ft) be an action and

(G,ft)) if

A

Ag O A = 0.

ff/ACfl,

is convex and for each

g € G,

A

is a block (of

either

Ag = A

or

Any non-empty convex G-invariant subset of ft is of

course a block.

THEOREM 1.3.2. congruences of

Let

(G^Q) be an action. The classes of

(G3Q.) are blocks and conversely, every block is

the class of a congruence of (G^Q). Proof: Let £ C-class, say

be a congruence of

A = <5£. Then

A

Ag = 6$g = 6gG, a C-class.

(G,ft) and

A

be a

is convex by definition and

Since the ^-classes partition

Q, A

is a block. Conversely, let congruence of define

5

Formally, A

A

(G,ft),

be a block.

then so is

if

is a block, the a,3 € Ag.

a = 3

is a class of a

for each

or

a,3 € Ag

g €. G. Ag

for some

Hence we

(g € G).

g € G.

Since

5 classes are convex and partition ft (so 5 If

a £3

and

In the first case, af = 3f;

af,3f € Agf. Hence 5 class.

A

so that its non-singleton classes are aC3

is an equivalence relation). or

Ag

If

is a congruence of

f € G,

either

a = 3

in the second, (G,ft) having

A

as a

13 Note that in defining

C

we gave a minimal congruence. one: a,3

a#3

if

a,3 £ Ag

in the converse part of the proof, Instead, we could have used a maximal

for some

belong to the same interval

A nU{Ah: h € G} = 0.

A

Thus the S

g £ G,

or

a, 3 & LJ Ah and h£G of ft maximal with respect to

defined in Theorem 1.5.2 is not

in general unique, "but it is unique if

(G,ft) is transitive. This

makes transitive permutation groups nicer to deal with, a fact borne out most strikingly in Chapter 3. Let

A

and

B

be chains.

{(a,3): a € A, 3 € B} or

ordered by:

(3n = 3p & a-, < ou). EXAMPLE 1.5.3.

where

a^cu tf £.

Note

Let

Let

A = {(n,0): n 6 2Z}.

m = n.

I.g.

(a 1 ,B 1 ) < (<*2,B2) if

A x" B

{a-,,^} be linearly ordered by

Then

A 2

is a block of and K

of

since

3X < 32

is a chain. ex.. < a 2

0 = T "* 22. Let

(A(ft),ft) that is a

(A(Q),Q)

m = n & (a = 3 or a,3 € Z)

Note C ^K

is the set

T = 7L X) {a..,ap} and

class of the congruences (a,m)£(3,n) if

Then A * B

and

(a.j,0)#(a2,0) but

given by: (a,m)#(3,n) if

(a1,0) < (a2,0) £ .

TRANSITIVE ACTIONS.

Recapping Theorem 1.5.2 in the transitive setting:

(G3ftj be a transitive action and a € ft.

THEOREM 1.6.1. Let

There is a one-to-one containw.ent preserving correspondence between the congruences of by

(G3Q) and the blocks that contain a, given

£ I—• a£. Of the same ilk:

THEOREM 1.6.2. Let

(G3n) be a transitive ordered

(I-)permutation group and

a € ft. Every block containing a

orbit of a unique convex (t-)subgroup of Conversely, if

(G,ti)

G

is the

containing G .

is an t-permutation group, the orbit of

under any such convex l-subgroup is a block containing a. This one-to-one correspondence preserves order.

a

14 Proof: A = aG, y

g £ G

Therefore

aG, N c. A.

such that

with

6 € A. So

h £ H.

so

We show that

By transitivity,

A O Ag / 0.

A = aG, N .

let

c K,

Conversely, let 1 —

—

ag = 3

H

h £ K;

f n ,f o € H,

d

Observe that

Then

ah = ak

i.e.,

H c K.

Ag = A.

G

Q G, s.

for some By symmetry,

v g) A f

v g) A f € H.

Thus

Therefore

<_ f

Hence

3 € aH O aHf

Consequently,

3 € aH,

h fh" 1 € G

c H;

aH

f

fo' i

so

f € G.

G.

aH

Then so

A

f

2

€

H

I

Hence

(G,ft)

is primitive if

(G,ft)

aHf = aH.

is called primitive if there are

(G,ft)—see Example 1.4.5. (A(ft),ft)

is.

We will

In Example 1.4.6 we

is an action with lazy subgroup

is static; i.e.,

ag = a

EXAMPLE 1.6.3.

({e},l)

EXAMPLE 1.6.4.

(A(a)),a>)

(Recall

H 2 G , for some

We will see

is primitive (Corollary 1.6.6).

If (G,ft)

If

= 3 = ah f

f € H.

But

'

is convex. ah

saw that a homogeneous chain need not be primitive. that

If

be such that

is a block.

no non-trivial congruences of ft

g € G

3 = a[( f-, v g) A f ] .

and as

for some

A transitive action

say that

let

_L (L

^ (f

h ,h 2 € H.

for all

G,

we will say

a € ft.

is a static ^-permutation group.

is a static ^-permutation group.

03 = {0,1,2, ...}.)

Note that if

(G,ft)

is static and

£

is any equivalence

relation on ft whose classes are convex, then of

Hence

be a convex ^-subgroup of

for some

(by transitivity).

f-L A f

let

so

a.

and the uniqueness is proved.

af, < 3 < af o

(f

Let

ag = 6 .

G C H,K,

hk" 1 € G

Thus

K 9 H

be a block containing

6 = ag Q. aG, y

aH = aK

k € K.

A

Clearly

there is

If

Let

(G, ft). So if

|ft| >_ 3,

(G,ft)

5

is a congruence

has proper congruences.

On the

other hand, static ordered permutation groups are in some sense basic building bricks.

Hence when we extend the definition of

primitive to the intransitive case (Section 3.2), we will do so in such a way that static ordered permutation groups will be primitive. Thus having no non-trivial congruences will not necessarily be

15 enough to be primitive in the intransitive case. An action if whenever

(G,Sl) is said to satisfy the separation property

A c. Q

is a non-empty bounded interval of

are distinct points of of

ag,3g

belongs to

SI, there is A.

If

g £ G

|fl| = 1,

SI and

a,3

such that exactly one

(G,Sl) enjoys the

separation property vacuously.

THEOREM 1.6.5. A transitive action

(G3Sl) is primitive if

and only if it satisfies the separation property. Proof: If A j SI with

|A| > 1.

6g € A

only if

a

3

and

(G,Sl) is not primitive, there is a bounded block Let

Ag = A,

a,3 € A with

Hence

ag € A

cannot be separated by

Conversely, let

a / 3.

if

(G,Sl) and

6' £ 6f. Hence £

So

a,3 € SI be distinct points which cannot be

(Vg)(ag,xg € A or og,xg §L A).

congruence on

6 € A,

$g € A.

A.

separated by the non-empty bounded interval OCT

But if

precisely when

a £3.

If

is not trivial.

A i SI. Define &

Then clearly

6 € A Thus

and

C

by:

is a

6' € Sl\h, then

(G,Sl) is not primitive.

For further equivalences of "primitive", see Section 4.1. Let there is

(G,Sl) be an action such that g € G

with

a g = a

and

a. < 3. (i = 1,2)

3-,g = 3p. Then

implies

(G,Sl) is

said to be a doubly transitive action) e.g., Examples 1.1.2 and 1.2.3.

Clearly any doubly transitive action satisfies the

separation property.

Hence, excluding the static case,

COROLLARY 1.6.6. Any doubly transitive action is primitive.

EXAMPLE 1.6.7. Let each a € l)}. Then

G = {g € A(l): g is differentiable (at

(G,l) is a doubly transitive ordered permuta-

tion group (hn € G where

h

is defined in Example 1.1.2).and

hence is primitive. However, it is not an ^-permutation group since gQ v e £ G

(gQ: a ^ 2 a ( a 6 I)).

Next we note (if

G j {e}):

16 LEMMA 1.6.8.

If

(G,Q) is a doubly transitive ordered

permutation group, then G G

is not Abelian. Indeed, the centre of

is {e}. Proof: Let

e i g € G.

transitivity, there is Now

agf = ag

2

For some

f € G

such that

i ag = afg. Hence

not belong to the centre of

a € ft, ag i a. af = a

fg / gf.

and

By double

agf = ag2.

Consequently

g

does

G.

u ex

ex q

ex

t

f

We complete this section with a preliminary study of transitive ^-permutation groups.

In Theorem 1.6.2, we saw that if

(G,ft) is a transitive ^-permutation group and £-subgroup containing

G , then

aH

H

is a block.

is a convex Actually, the

first part of the proof of the converse direction of that theorem shows: LEMMA 1.6.9. Let H

(G,ti)

a convex l-subgroup of

Hence &G

G

is convex for all

Hence the orbits of

G

be a transitive l-permutation group,

and

8 € ft. Then B# is convex.

a,& € ft. partition ft into convex subsets and

the set of orbits inherits a linear ordering from ft: A, < Ap 6 1 < 62 let

for all

6± € Ai (i = 1,2). If

A' = {ag: ag-1 € A}.

A

is an orbit of

if

Ga,

17 LEMMA 1.6.10. If a € to and

A

is a

G a

(Gjto)

is a transitive l-permutation group,

orbit, then so is A' and the map

ts an anti-ordermorphism of the set of Proof: Let af"1 € A. If -1 -1 ag = af h. Ga orbit. If

Then

3 = af

orbits onto itself.

for some

f € G with

with ag"1 € A, there is h € G a -1 -1 f hg € G a and 3f hg = ag. Thus

ag € A'

such that

Hence

Af

A-,,Ap

ag € A'\A'

3 € A'.

G

A »—* A'

(if

are distinct

G

ag = af €. A'

with

ag"1 = af'^fg" 1 ) € A 2

orbits, let

ag" € A..\Ap. Then

af" € & , then

fg"1 € G a ) .

since

Hence

A ^ / A^.

A' and A' are distinct G orbits, A' O A' = 0. let ag" € A 1 and ah" € A 2# Then g"Xh v e e G max{ag,ah} = ag( e v g" h)€ A'

is a

But

ah € A'

so

Since

A^ < A ? ,

If

so

ah < ag. Thus

A- < A-. Since

A" = A,

the proof is complete. Af = A

Note that, as a consequence,

We partition the set of orbits of |A| = 1 = |A'|;

G

only if

into four classes:

|A| > 1 = |A!|;

|A| = 1 < |A'|;

A = {a}.

Of course some of these classes may be empty.

|A|,|A'| > 1.

The set of

G

orbits under the induced order, partitioned as above, is called the a.-configuration of

(G,ft). We now show:

THEOREM 1.6.11. group* ^1^2

€ n

an<

^

Let

(G>to)

f ^G

su

be a transitive l-permutation

t at

°h

a

7-f =

^

a

9*

T

hen the map

A »—* A/ is an ordermorphism of the a?-configuration onto the a«-configuration of A/ and if

(G3to); i.e.3

|A| = 1 = |A'|,

then

A'/ is the u^-paired orbit of |A/| = 1 = \b'f\,

Proof: It is enough to show that

G orbit and a2 (Af)f = A!f. But if A = 6G , Af = 6G f = 6fG P = 5fG by a a a f a l l l 2 the fundamental triviality; i.e., Af is a G orbit. Moreover, if

6 = a;Lg,

1 6f = a^f = a2(f" gf).

(Af)1 = aof"1g"1fG *

ot

= a-g"^ 1

Af

etc.

^ Thus

-f = a.g"^

-l

is a

1

a

f = A ff

and the

18 proof is complete. Thus the a-configuration is independent of the choice of a 6 ft and will just speak of the configuration of If

A

is a

G

A > {a} (negative |A| > 1,

A

orbit,

if

A

(G, ft).

is said to be positive if

A < {a}).

If A

is called a long orbit of

is an orbit of

G . If

(G,ft)

G

and

is a doubly

0£

transitive ^-permutation group it has just one positive and one negative

G

orbit of group

orbit and these are long.

G

so is

(G,ft)

If whenever

THEOREM 1.6.12.

A

By Theorem 1.6.11, the definition of

If

a € ft and A

is a block of

is a long

A , we say that the transitive ^-permutation

is balanced.

balanced is independent of the choice of

groupi

A

f

(G>ftJis a transitive

is a

(Gyn).

a € ft.

G

L-permutation

| A | > 1 = | A f 13 then

orbit with

Hence a primitive transitive L-permuta-

tion group is balanced. Proof: G

a

c GD

p

be long with

ag € A. We show that

aG n

i s ) . Since

P

A' = {$}, say. Then

is a block by Theorem 1.6.2.

p

such that (since

Let A

and aG_

aGfig = A, whence

ag € A,

But

aG n g = agG_ = agG = A P 3g a the theorem is proved.

Let g € G be

ag

1

U

A

= {£}.

is a block

Hence

a = Bg.

by the fundamental triviality, and

For further results about configurations, see McCleary [72a].

1. 7. Let of

PRIMITIVE (G,ft)

(G,ft).

COMPONENTS.

be a transitive action with

Let A

be a X

class.

Then

C Q ~K congruences

(G/ N , A ) is a

transitive action in the natural way and £' = £|A X A congruence of action. ^/*i "

(A;

G

If

(G/ x,A). Thus

L

^ \ /

L

{A;

=

!

(G/ \,A/£ )

is the lazy subgroup of G

A

/A\I -

S O

(A;

T

(G/AN,A/£ )

action called the component of

(A;

(G,ft)

is a

is a transitive

(G/ N , A / £ ! ) ,

let

is a faithful transitive

relative to

(£,#) at A.

We first show that the component is independent of the particular

%

class chosen.

19 LEMMA 1.7.1. Let be congruences of components of

(G,Q) be a transitive action and £ £L *K

(G3Q).

Let

A-jAn

(G3Q) relative to

be #

(£ > X)

o-isomorphic; indeed^ t-isomorphic if

classes. Then the at

(G3 Q)

A7

and A 9 are

is an l-permutation

group.

Proof:

There exists

g G G such that A g = A . By the -1 g G/. \g = G/ \ = G/. \. Let

fundamental triviality,

£.' = 51 A. x A . (i = 1,2), and g - 1 L n g = L o where 1 1 1 s u b g r o u p of

1 (G/ .

(G/ A \ , A o / f ' ) . ^ A p ; ^ ; ^ (L f )ip = L g 1
^

1

N,A^/<$').

For

6

Then

€ An, 1

1

f g ( f € G/ x). \ A.. /

Lp

let

is


L n is the lazy

1 l a z y subgroup

(6.$')<J) = ( 6 _ g ) £ L 1 1 ±
Then

\\n G(

and

v = G, v and \ A-, y \ Ap j

Hence (G(A y^/S]) = (6(A )^2/^2)

1

of

aS

desired. Hence, from now on, we will drop the reference to

A

in the

component of a transitive action relative to a pair of congruences. Also, we will write

A/5

for

A/£ H ( A * A )

if

A

is a union of

& classes. Note that the definitions make sense in the intransitive case as well, and the proof of Lemma 1.7.1 still works provided Ap = A_.g for some

g € G.

This should be remembered' when reading

Section 3.2 as should the next theorem. If £ £ K

are congruences of a transitive action

we say 71 covers £ if there is no congruence U'> of

THEOREM 1.7.2. congruences & ^ K3

If

(G,ft), (G,fi) with

(G3tt) is a transitive action with

the component of

(G3Q) relative to

(C 3ft)

is primitive if and only if ^i covers C. Proof: Let block of (G/ A \ ^ ) T c A.

A

(G/ yA/f) If

g € G

Because

A

be a

^C class. By Theorem 1.4.2, any

has the form and

T/tf where T r: A

Tg O T / 0,

is a block,

then

Ag = A.

is a block of

Ag ^ A / 0

Hence

since

g € G/ y

so

20 Tg = T.

Thus

T

is a block of

(G,fi) and the theorem follows

from Theorem 1.6.1. If

(G,ft) is a transitive action and X

component of

(G,ft) relative to

(
covers C,

the

is said to be a

( G,ft).

primitive component of

We complete this section with two examples to show how to determine the primitive components of transitive ordered permutation groups and to see how the ordered permutation group is reflected in the set of its primitive components.

EXAMPLE 1.7.3.

Let

Q = A x B

primitive, homogeneous chains.

where

if

(G,fi).

Note that

(J * C ) component: Let Let

are

5 by:

if

= < s j j^(a,3)

&

and U. are

(G,ft) is transitive.

A

be a A

5

class, say

is ordermorphic to A,

s € A(A) and define

r(as,3 Q ) (a,3)g

Define

So J,

A = {(CX,3Q): a € A}. Note that A(A) = A(A).

B

3 = 3' (a,a' € A, 3,3' € B ) . Let

G = {g € A(Q): a £x <=> ogCjg}. congruences of

and

Let <J and TJL be the trivial

equivalence relations (Example 1.4.5). (a,3)C(a',3')

A

gg e G, A N by:

3 = 3Q . The map

if

^-isomorphism between

so

s H-> Lg s

is an

3 ¥ 30

A(A) and

component is £-isomorphic to

6/ y

Hence the

(J , S )

(A(A),A) which is primitive by

hypothesis. ((j 32l ) component: Q

is the ZL class and

L = {g € G/fix: Tg = T for all 6 classes T} = {g € G: Tg = T for all
s € A(B) and define

(a,3)gg = (a,3s) (a € A, 3 € B ) . The map ^-isomorphism between is £-isomorphic to

A(B) and

G/L.

(H..,A) and

groups and let

s »-* Lgg is an

So the

(C , # )

component

(A(B),B) which is primitive by assumption.

Hence the only congruences of Let

gg € G by

(G,ft) are V , £

and U.

(Hp,B) be transitive ordered permutation

G = {g € G: g induces an element of H_ on the

21 (J , C ) component and an element of H^ on the ( C ,21 ) component}. Then

(G,ft) is a transitive permutation group with

component

(H_,A) and

( £ , 2O

component

(J,£)

(Hp,B) as is easily

seen by modifying the argument above. Of course, there will be other components unless

EXAMPLE 1.7.4

(H-,A) and

(Hp,B) are primitive.

Let ft = A x B x r where

primitive homogeneous chains. Let J equivalence relations. if

Y = Y'

and

J G C - ^ — 2£ OKT

Let

and #

by:

(CX,3,Y)# (a' ,3' ,Y )

(a,3>Y)£(a',3',Y!) Y = Yf.

if

G = {g€ A(ft): affx <=> erg C ig

Then J,
(G,ft) is

and ^

transitive.

(J , <S ) component is Let

A

(A(A),A) and the

be a %

So

and

are congruences of

As in Example 1.7.3> the (# ,21 ) component is

(A(r),r) (to within isomorphism). We examine the component.

are

and 1L be the trivial f

3 = 3 ;

<=> erg#xg}.

(G,ft) and

Define 5

!

A,B,r

( £ , X. )

class, say

A = {(CX,3,YQ): a € A, 3 £ B}.

Now

A/f

= B.

Let

{

s € A(B)

"(a,3s,YQ) if

Y = YQ

(a,3,Y)

Y ¥ YQ

if

and

Here L = {f € G^y. (Va G A)(^3 € B)(3a» £ A)((a,3,YQ)f = (a',3,Y0))}> and the map s *—• Lg is an ^-isomorphism between A(B) and to

G/ x/L = G, y

The

(C , ^ )

component is £-isomorphic

(A(B),B) which is primitive by hypothesis. So the only

congruences of

(G,ft) are J,

2 , ~K and II.

We can modify this example as we did in Example 1.7.3.

1.8. DEDEKIND COMPLETION AND CHARACTER. Let A

and

B

11 be a chain. We say that

(A,B) is a out in ft if

are non-empty disjoint subsets of ft whose union is ft

such that (1) A < B

(i.e.,

a<3

for all

either (2a) A

has no greatest element,

a € A,

3 € B),

and

22

(2b)

for some

3 £ ft, A = { a € f t : a < 3 )

and

B = {a G ft: a >_ 31. Let of

ft.

£

be the set of cuts in ft, the Dedekind completion

We single out clause (2a) by considering the Dedekind

completion

1

of

has an element

Qj, and (2b) by considering the case that ft

3

with an immediate predecessor (to ensure that

the predecessor of

3

"occurs" .as a bona fide element of

see below; e.g., ft = Z ) . A Q r.

Order embed

ft

A_ = {a € ft: a < 3)

of

{(A ,B ): a € B } .

{(A ,B ): a € A}

Moreover, if

a

(a

(A , B )

does not signify the image

a
If

there is

We will identify ft

for the general element of ft of

a ) . So under this

identification, ft is dense inft\ft(i.e., if a £ ft with

and the infimum

(A,B) < (r,A),

(A,B) _< (A ,B ) <_ (r,A).

with its image in ft and write

is

if

3 »—• (A_,B.) where p P B. - {a € ft: a > 3 ) . Each (A,B) € ft P ~~

of type (2a) is the supremum of

a € ft such that

ft—

(A,B) <_(r,A)

in ft via:

and

P

Totally order ft by:

a < 3

in ft\ft there

a €.ft\ft,we say that

a

is a

hole in ft; thus a cut in ft is a point of ft or a hole in ft. Note that if {(A.,B.): i € 1} is a set of cuts in ft with (A.,B. ) <_ (A,B) for all i € I, then ( U A - ^ n B - ) is the 1 1 _ i G l 1 i € I 1 supremum of

{(A.,B.): i € 1}

subset of ft, then Let of

g

g € A(ft).

A

is a convex

A ^ ft. There is a unique continuous extension

g

to a permutation of ft, namely

ag = sup{ag: a € ft & a _<_ a}. and so

g € A(ft).

extension

g

( ( ) 5 )

g

EXAMPLE 1.8.2.

preserves the order on ft g

g € A(ft)

( A( Q),CQ)

with its unique continuous if

g€

A(ft).

is transitive but

is not.

Let ft = 7L *" I.

(n i ,r 1 ) € ft (i = 1,2),

(m € Z , s € R )

Now

We will identify

and just write

EXAMPLE 1.8.1.

(If

in ft. Also if

let

Then ft is homogeneous.

(m,s)g = (m + n 2 - n ^ s + r 2 -

(i.e., "local" translation by

n^ - n ,

and

23

IR

"global" translation by n

r^ - r ). So

g € A(Q) and

r

(n..,r..)g = ( 2^ 2^*^ However, we now show

Q

is not homogeneous.

Let C-jjCp b e t w o n e w s y m l D o l s a n d l e t T = ^ u ^ 1 ^ 2 ^ by c < n < c 2 f o r a 1 1 n-€Z. Then S = T xffi.Now an immediate predecessor but g € A(Q) such that Let each

(0,0)g = (c 2 ,0);

1 9 A G (],

A € A,

A

there is

i.e.,

X. € An ± 1

such that

is cofinal and coinitial in

in

A.

A,

is cofinal in A}

(inf{|A |: A

of (A) for the cofinality of

the coinitiality of

A.

is not homogeneous.

If

A

if for

A., > X (\- < X). If 1— l —

it is said to be ooterminal

The cofinality (coinitiality ) of

We will write

S

is cofinal (co-initial) in

A_

inf{|A |: A

(0,0) has

(Cp^O) does not. Hence there is no

A is is coinitial in A}). A,

and

ci(k) for

cf(A) = ci(.A) we will, say that A

has coterminality cf(A).

Let

and

cf(A) the initial character of

sup(A) = a,

we call

a € Q.

If

A = {3€fi:3
and

dually for the final character of a. If a has initial character H and final character V , we write c for the y _ ** v yv character of a. If a has initial character H but inf{3 € f t : $ > a } ^ a , Similarly we may have

we write c 7VV

of fi. Note that a hole in

or

ci ^ for the character of

a.

c,,^ as characters of an element "A "A"

0, must have character

c yv

for some

24 ordinals

y,v

since

inf{3 € f i : 3 > o t } / a

only if

has an

immediate successor, etc.

Observe that

as

Hence if ft is homogeneous, all its

a

for all

g € A(ft).

ag

a

has the same character

elements have the same character (cf. Example 1.8.2). E,

all of whose points have character

chain with ft = I.

cQQ

For example,

in ft if ft is any

Also note that if ft = T,

the character of all

points of ft (with respect to ft) is the same as the character of all points of ft = f and

T

(with respect to

are dense in ft and

T ) . This is because ft

f = ft respectively.

1.9. BUMPS AND SUPPORTS. Let

(G,ft) be an ordered permutation group and

support of

g

is

{a € ft: ag i a} = supp(g) . If

supp( f) H supp( g) = 0,

So if

e <_ f,g € G,

f

then we say that

and

g

f

and

g € G.

The

f,g € G and g

are disjoint.

are disjoint precisely when

f A g = e. The following trivial lemma is very important.

LEMMA. G

group ommute

.

Disjoint elements of on ordered permutation

25 Proof: Let (G,ft) be an ordered permutation group and f,g € G be disjoint. Let a € ft. Note that if a ^ af, then af / af 2 ; hence afg = af. Thus "af

if a € supp(f j~l

faf

if a £ supp(g)"]

a(fg) = <

> = < j> = a(gf). ag if a £ supp(f) if a 6 supp(g) vI ag — —/ — —/ Therefore fg = gf. // If A c_ft, let Conv (A) = {a € ft: (35 1 £ A)(^5 2 C A X ^ <_a £ « 2 )}, the convexification of A in ft. If for some a € ft, supp(g) = Conv {ag : n € 2S}, we say that g is a bump or g has a ) ; unbounded if supp(g) is coterminal in ft. A bump can have any of these four boundities (if ft is countably coterminal):

26

A bow* ieA levmfi

If g € G,

(G,fi) is an ordered permutation group, then

f

is a 2pi^?7p 0/ £

f|supp(f) = g|supp(f).

If

f

if

f

f € A(fi) and

is a bump and

is a bump of

g,

then its support

is an interval and is called a supporting interval of

g.

Indeed.,

supp(g) is uniquely a union of its supporting intervals, and so is open in the interval topology on Q. We next give an example to show that the bumps of need not necessarily belong to tive ^-permutation group.

G

even when

g € G

(G,fi) is a transi-

27

1.9.2, a € 1}.

So

gQ € G

with

Let

G = {g € A(l): (a + l)g = ag + 1 for all

(G,l) is a transitive ^-permutation group. Let supp(gQ) =

An example of such a

Then no bump of

g belongs to

is: agQ = (a - n ) 2 + n

gQ

G.

if

a € [n,n + 1] (n € 7L). Let

(G,fi) be an ordered permutation group.

Lemma 1.9.1, each Moreover, if If

g > e,

g

g € G

In view of

is the "product" (in A(fi)) of its bumps.

has only finitely many bumps, this is accurate.

then

g = V{f: f is a bump of g}

(This v

(supremum)

is possibly infinite and is taken in

LEMMA 1.9.3.

e < f3g £ G. If

Let /A

(G3 oj

gf

=

be an ordered permutation group and then each bump of f

e

is a bump

of g. Proo for all

Since

a = a( f A gf"" ) = min{af,agf~ }, a = agf"~

a € supp(f).

Hence

af = ag

for all

a € supp(f) and

the lemma is proved. LEMMA 1.9.4.

Let

(G3to)be an ordered permutation group and 1

e < fyg € G. If f A g~ fg = e,

then f1 <_ g

for all n € ff*.

28 Hence if

supp(f) C (a>$) and

ag >_ $., then f <_g.

Proof: By the fundamental triviality, supp(g~" fg) = supp(f )g. Hence if supp(h) < supp(h)g. all

Thus if

h

is a bump of

a € supp(h),

af € supp(h) < ag

n € TL . Since this holds for all bumps

for all

a € Q

and

f,

n

h

of

f,

af

n

for

< ag

n £Z .

1.10. MULTIPLE TRANSITIVITY. Let

m £2£ . We say that an ordered permutation group

is m-transitive if whenever n,

there is

g € G

such that

example, we will see that m€Z+.

Indeed, if

F

1

(G,fi)

and

< a a.g = 3. (l ± i 1 m).

For

(A(tt),l) is m-transitive for all

is a totally ordered field,

(A(F),F) is

29 doubly transitive and hence m-transitive for all m € 7L

as we will

show. Throughout the book we will assume that IF

(G,n) IS m-TRANSITIVE, THEN fl IS INFINITE.

This avoids the trivial case that

G = {e} and

|fi| <_ m

(cf.

transitive = 1-transitive). If

(A(Q),Q)

is m-transitive, we will simply say that

Q

is

m-homogeneous. This conforms to the usual model theory usage. Note that if all

(G,ft) is m-transitive, it is n-transitive for

n < m. LEMMA 1.10.1. If

is a doubly transitive

l-permuta-

tion groupj then it is m-transitive for all m >_ 2. Proof: We will only prove double transitivity implies triple transitivity (the same proof works to spread the m-transitivity disease by induction).

Let

double transitivity, there is

<X|

~ and 3 << 3p < 3o. < a~ g € G such that oug = 3-, and

ou <

By

30 a2g = 32-

If

c^f = e 3 .

Now

cutg A f) - min{a 1 g,a i f} = B i (i = 1,2,3).

B~ > a~g,

let

h € G

3o £ °^g,

let

f £ G

be such that

be such that

a?h = $

a f = 82

and

and If

aJi = £-.

ou(g v h ) = 3 i (i = 1,2,3). As a useful simplification in checking double transitivity for ordered permutation groups, we have:

LEMMA 1.10.2. group.

Then

Let

(£_, oj

be a transitive ordered permutation

(G3$i) is doubly transitive if whenever

a- < a 0 < a7., 1

there is

g £ G

Proof:

Let

such that a.. < a o L

such that

and

Bn < B o .

d

±

a.g = 8. (i = 1,2).

i

i

8 2 < a 2 f.

d

Let

We must find

Then

(G,fi)

a, < a^,

there is

(if

g 6 G

a, < a ? < a^, such that

Similarly if

is a pathological

then as

a,g = a

S T a ^ - {f G A(fi): f is bounded}.

transitive ^-permutation group and (G,ft)

p..

oug = & i (i = 1,2).

f € G

a2f < 8 2 , such that

is a doubly transitive ordered permutation

group, the condition holds,

Let

g = f. If h G Go

//

Note that if

and

g € G

By transitivity, there is

d

g = fh.

O


such that c^f = B . If a 2 f = &2> let then B-, = a n f < a o f < 8 O ; so there is a 2 fh = 3 2 -

a

aog = a7.

If

a, < a^

and

(G,n)

a 2 g = a^.)

is a doubly

G H B ( f i ) = {e}, we say that

^-permutation group.

In later chapters,

we will see that the name is well chosen.

EXAMPLE 1.10.3.

Let

G = {g € A(ffi):

(3 m 6 '%+)(Va € l ) [(a + m)g = ag -f m] }. m € 7L

for which

period of

g. )

a £ supp(g), where

m

(a + m)g - ag + m

(G,l)

then

let

e < g £ A( [a ,a

g € G,

for all

a £ 1

the least is called the

is a pathological ^-permutation group: /

(a + km)g = ag + k m 7 a + km

is the period of

a, < ctp < a^,

(if

g;

hence

m > a^ - a .

+ m] )

extends to an element of

such that G

for all

(JAB(f]) = {e}.

If

k £ 22

Also if

There is a g = a^.

of period

m,

Clearly say

g.

g Now

uniquely

31 e < g €. G

and a9g = ou. By Lemma 1.10.2,

(G,l) is doubly

transitive. We now reexamine Lemma 1.10.1.

Unfortunately, there are

doubly transitive ordered permutation groups that are not triply transitive so the next lemma is optimal. LEMMA. 1.10.4. If

(G3Q) is a triply transitive ordered

permutation group and G D B(Q.) ^ {e}^

then

(G^U) is

m-transitive for all m G 7L . Proof: Assume that

(G,ft) is m-transitive for some

Let a, < ... < a + .

and

is g € G

cug = 3 1 (l <_ i £ m).

such that

3-, < ... < 3 +-,. By hypothesis, there

are home, so assume not.

if = a m+1 g

a +] gf~ gQf = 3 +1 -

we

Without loss of generality,

a m+1 g < 3 m + 1 , let f € G be such that

and TgQf = 3 m + r

h = gf-^f.

If ^m+1S = 3m+1>

Let e i g~ € G OB(fi), say

a < supp(gQ) and x € supp(g^). T < TgQ. If

m >_ 3-

Hence

Similarly if

Then

3 m < supp( f ^ f )

of = 3 m , and

a.h = 3. (l <_ i <_ m + l) where 3 m + 1 < %+1g.

//

Note that the above proof shows that the conclusion holds provided

(G,ft) is triply transitive and contains an element

whose support is bounded above or below.

LEMMA 1.10. 5.

Let

(G39.) be a triply transitive ordered

permutation group and N < G with N n B(Q,) f^ {e}. Then (N H B(Q).,oj is m-transitive for all m € 7L . Proof: By induction on m. Let

e 1 g e N n B(oJ.

generality, yg^f = 3

For m = 1, let a < 3 in Q.

Let y £ supp(gQ).

y < ygQ- There is f € G

since

Without loss of

such that

yf = a and

(G,ft) is triply transitive. Now

f"1g f € N H B(fi)

and a f ' ^ f = 3. Hence

(N H B(^),fi) is

1-transitive. Assume that

(N 0 B(Q),Q)

the preceding lemma shows that

is m-transitive.

The proof of

(NOB((]),ll) is (m+1 )-transitive

32 (take

g,gQ£ N n B(n)).

//

An ordered permutation group

(G,ft) is said to have the

interval support property if for each interval e/g€G

such that

supp(g) £ A.

A ^ ti, there is

For example,

(A(l),l)

satisfies the interval support property but Examples 1.9.2 and 1.10.3 do not.

COROLLARY 1.10.6.

If

(G3Q) is a triply transitive ordered

permutation group with G n B(Sl) / {e}3

then

(G r\ B(n)3ti)

satisfies the interval support property. Indeed, if a- < By < 8« < ou., there is g € G and

b^g = 3«.

Hence if

such that supp(g) Q. (a-,a~J

a- <
that otn € supp(g) S. (o.^yo.^).

there is g £ G such

The above conclusions hold if

(G39.) is a non-pathological doubly transitive l-permutation group. Proof: Let

6Q,Bo € Q with

By Lemma 1.10.5, there is supp(g0) c (y,<S). yf = B Q ,

1

By Lemma 1.10.4, there is

B1f = 3 1 ,

B2f = B 2

and

oL < oL < OL~, let

a^ < B-, £ oip < B 2 < a^

B

^ W

g Q £ G n B(a) with

s u p p C r ^ f ) £ (B Q ,B 3 ) £(5 1 ,5 3 ) If

^

3-S'

B ^ Q = 82f € G

Say

such that

6f = B^. Then and

3 ^ " ^

= 32-

3-, ,3 2 €. fi with

and apply the above. //

We now prove a result (due to McCleary [78b]) which we will use heavily in the next chapter.

It shows that in the triply

transitive case the order is (almost) definable from the group action.

Write

[x*y] for

x~ y" xy,

the commutator of

x and

y. THEOREM 1.10.7.

Let

(G,Q.)

be a triply transitive ordered

permutation group containing a strictly positive element of bounded support. Let e ^ f>g3p $. G. Then [fy (h~2ph)~1g(h~1ph)] = e

for all

(1) e < p & supp(f) < supp(g) or

h € G

if and only if

(2) p < e & nupp(g) < supp(f).

Proof: Assume (l). Then supp((h"1ph)"1g(h"1ph)) = [supp(g)](h"1ph). Since

e < p,

33 e < h ph for all h £ G. Therefore, supp(f) < supp(g) implies ). So f and supp( f) < supp( " 1g( h " 1g( h ^ are disjoint. By Lemma 1.9.1, [f,(h"1ph)"1g(h"1'1ph)] = e. Similarly, (2) implies [ f ^ h ' S h r V ^ p h ) ] = e for all h € G. Conversely, assume [f,(h~ ph)"" g(h~ ph)] = e for all h € G. Suppose p i e. If there are a 6 supp(f) and -2 3 € supp(g) with 3 < a, then < 3 < a < af < af 2 without loss of generality (replacing f by f or g by g" if necessary). Let a € 0, with a < ap. Choose T-i>Tp £ ^ with a < T, < Tp < ap. By Lemma 1.10.4, there is h € G such that ah = 3g~S T2ph i af2.

, aph = a, x^ph = af and Now af(h" ph)- g(h~1ph) = 2 )h ¥ af2 = 1

1

a(h"1ph)"1g(h"1ph)f. Thus [ f ^ h ' ^ h T V h ^ p h ) ] i e, a contradiction. Hence p / e implies supp(f) < supp(g). Similarly crp

Z,

i

flr p / e

implies

impossible.

CX/

supp(f) > supp(g).

Therefore

p > e

or

So p < e

( p / e & p ^ e )

is

and t h e theorem f o l l o w s .

COROLLARY 1.10.8. Let (G,ti) be a triply transitive ordered permutation group containing a strictly positive element of bounded supporty and let p € G. Then (p > e or p < e) precisely when there are fyg € G such that [fJ(h~1ph)~1g(h~1ph)] = e for all h fc G. Furthermore3 p3q > e or pyq < e if and only if (Vh) ([f3 (h~1ph)~1g(h~1ph)] = e ++ [fjh^qhf" provided p > e or p < e.

34 Finally, we note an obvious result about homogeneous chains which we will use repeatedly. LEMMA 1.10.9 (The Patching Lemma) . Let Q be a chain and {A^: i € J}_,

{B.: i € 1} be covers of ft. Suppose that for each

i € T, there is an ordermorphism f.: A. = B. such that 1s

a

1s

1s

f -= a/- for all a € y4 . n 4 . and a.f. < a.f. whenever

a . £ A ._,

a . € J4 . and a . < a .. Then f =

particularj let A

and B

\^J f. £ A (ft j.

7n

be intervals of a chainft.,and

A = U{A.: i e 1} and B = \J{B.: i € 1} with sup(A)y sup(B) belonging to the same orbit of A(ftJ_, or A and B

both final intervals of ft; and inf(A)5 inf(B) belonging to

the same orbit of AfftJ., or A

and B both initial intervals

of ft. If f.: A . = B. (i € I) are ordermorphisms with af. = af. 1s

whenever

1s

Is

a € A . n A . and 1s

Q

a . £ A . and a. < a.., 0

3 J

^ 'i

1s

a.f.

is 1^

< a.f.

Q 0

then there, exists

whenever

a. G A . ^ 1s-

f € AfftJ

Is

Q

is

such that

J

i

We give one application of the Patching Lemma now (others appear in later chapters). T.KMMA 1.10.10.

Let ft be a doubly homogeneous chain. Then

any two k holes in ft of character cnn

belong to the same AfftJ

UU-

orbit.

Proof: Let a,$ €ft^fthave character

c ~. Let

{a : n € a)} and {3 : n € a)} be increasing sequences in ft, and {y : n £ ID} and {6 : n € u)} be decreasing sequences in ft such that

sup{a : n € a)} = a = inf {y : n € a)} and

sup{3 : n € a)} = 3 = inf {6 : n € co}. Since any two bounded intervals of ft (with endpoints in ft) are ordermorphic (ft is doubly homogeneous), there exist ordermorphisms

V

t W l l = tBn'Sn+l]

and g

n: tVl'^l = t6n+l'6J (n £ ^'

Also any two final (initial) intervals of ft with endpoint in ft are ordermorphic since ft is doubly homogeneous. Hence there are ordermorphisms

h Q : (-°°,aQ) = (-oo,3Q) and h^: (YQ>°°) -

^Q'^^-

35 f £ A(ft)by the Patching

Let

f = {J f U h Q U h U ^ j g . Then n £ a) n £ a) Lemma. Moreover, af = sup{a f: n € co} = sup{3 : n € a)} = $, as desired. 1.11.

IDENTITIES.

We now use Holland's Theorem to obtain some identities about I- groups. LEMMA 1.11.1. Every l-group is torsion-free. Proof: Let (G,ft) be an ^-permutation group and g € G. If g = e for some n € 7L and g J e, there is a € ft such that ag i- a. Without loss of generality, a < ag. Hence ag < ag 2 so a < ag < ag 2 < ... < ag = ae = a, the desired contradiction. Thus g = e implies g = e; i.e., G is torsion-free. T;KMMA. 1.11.2. Proof: Let Let

Every Z-group is a distributive lattice.

(G,ft)be an ^-permutation group and

f,g,h € G.

a € ft. a[(f v g) A h] = min{max{af ,ag},ah} =

max{min{af,ah},min{ag,ah}}

since

Q

is a chain. Hence

a[(f v g) A h] = a[(f A h) v (g A h)] . Since this holds for all a€ft,

(fvg)Ah=(fAh)^(gAn);

i.e.,

G

is a distribu-

tive lattice. Since

f(g v h) = fg ^ fh,

(g v h)f = gf v hf

and dually,

Lemma 1.11.2 gives: COROLLARY 1.11.3.

Every l-group word has a normal form

w

V A ' - where w. . are group words> and I3J are finite. ^J i€I j€J Z0 Warning: This normal form is not necessarily unique as we now show: LEMMA 1.11.4. g V g"

> e,

Hence

If

G

is an l-group and g € G, then

(g v e) A (g~ V e) = e.

Proof: Let (G,ft) be an ^-permutation group and Either ag <_ a or a <_ ag. In the first case a £ ag

a € ft. ; so, in

36 either case,

a <_ a(g V g" ). Thus

1

1

(g" A g) = (g v g " ) " (gve)A(g""

1

e <_ g V g" . Therefore

£ e. Hence

v e ) = ( g A g " * ) v e = e by Lemma 1.11.2.

We will write

\g\ for

g v g~ . So

|g| 1 e.

We close with a less trivial identity.

Recall that

[x,y] = x~ y~ xy. LEMMA 1.11.3. In any l-group G3 and f1^f2

= e=g1^g2

if f.,g. € G (i = 1,2)

then

Proof: By Lemma 1.9.1, it is enough to show that (f^

1

and

fp

)""^^

1

(f-.f" )" f

1

)

is disjoint from

f^g^^,^

1

].

Since

commute (by Lemma 1.9.1), conjugating both sides by 1

reduces the proof to showing

2 g2flg2 f2g2fl *

must show

S o su

PP°se

a < a

&i

g-

is disjoint from

(whence

ag 2 = a ) . We

a f ^ g ^ g ^ f ^ f " 1 = a.

Case 1.

af^ = a = af2-

Then

af^ S 2 f l g 2 f 2 g 2 f l

= a

ag9 = a. Case 2.

afn / a _L

(so

af o = a ) . Now d

a < af., so 1

since

37 g" Hence

1

< af^

1

1 afr

af^f-jg^g,/"1 Case 3. afp / a

af~

Since

Thus

1

^ A f 2 = e, a f ^ ^ ^ = a f ^ .

lg2lf2g2fl1

(so af\, = a).

1 af~ g 2 < ag 2 = a;

af~ 1 g 2 f 1 = of ^ g g .

= af

1

= af

l g 2 lg 2 f l" L = a '

Now

so af~ g 2 € supp(f2). «f2lg2flg2lf2g2fl1

= af

Hence

2 lg 2 g 2 lf 2 g 2 f l" L = a ' Q.E.D.

Note that Lemma 1.11.5 gives an identity as well as an implication (Replace g v e

and

g2

by

f1

by

g~ v e

f v e,

?2

by

f" v e, g1

and use Lemma 1.11.4).

by

Lemma 1.11.5

is due to Holland and Scrimger [72]. For further "disguised" identities, see Holland [77] and Franchello [77].

CHAPTER 2 DOUBLY TRANSITIVE A(ft)

This chapter is devoted to the study of

A(ft) when it is

well-behaved and has a lot of additional properties.

The assumption

that gives us nice theorems is that ft is doubly homogeneous. Under this extra hypothesis, we can apply the Patching Lemma (Lemma 1.10.9).

This helps us give algebraic conditions equivalent to

geometric ones.

Indeed, we begin Section 2.1 with a series of

lemmas in which we capture various geometric properties by algebraic statements.

These we will use repeatedly in this chapter (as well

as later in the book). Although they may appear quite technical at first, their use in proving non-technical results emphasises the need to examine them early on. We will prove the following theorem due to Michele Jambu-Giraudet [8la] (Recall ft* is ft with the order reversed).

THEOREM 2A. and

A(ft) and

Let ft7 and ft~ be doubly homogeneous chains3

A(ft^J

satisfy the same group-theoretic sentences.

Then A (ft ) and either A (ft ) or A (ftp

satisfy the same l-group

sentences.

We will also use the technique to prove two theorems of Gurevich and Holland [81]: THEOREM 2B. If ft is doubly homogeneous and A(ftJ satisfies the same l-group sentences as if

A(E), then ft = _?.

In particular,

A(ftj = A(S) as l-groupsy then ft = S. THEOREM 2C. If ft is doubly homogeneous and A(ftJ satisfies

the same l-group sentences as

K(Q)3

then ft - _? or $\Q. In

39 particular, if

A(Q) = A(Q) as l-groups, then ft = Q

or R J®.

As a consequence we obtain (by Theorem 2A):

THEOREM 2B* (2C*).

Same as Theorem 2B (2C) with "l-group"

replaced by "group, " Another application of our ideas is:

THEOREM 2D. If ft and T

are doubly homogeneous chains and

\\>: A(Q) = A(T) as L-groups (groups), then ft is ordermorphic (or antiordermorphic) to an orbit of

A(T) in T. If

(anti)ordermorphism3 then gty = \p g^

$

denotes this

(g £ A(Q)).

The £-group part of Theorem 2D is due to Holland [65a], the group part to McCleary [78b] and Rabinovich [75]. In view of Theorem 2A, we can give essentially the same proof for both parts; it is quite different from any of the original proofs and is due to Jambu-Giraudet [8la]. We next wish to examine closely the lattice of a doubly homogeneous chain. the set

If

G

A(fi) for ft

is a group, we will use

G with the group operation

ftg = gf = (f

g

)

G_ for . We

obtain the following analogues (due to Jambu-Giraudet [8la] ) of the preceding theorems. THEOREM 2A . Let ft- and ftg be doubly homogeneous chains3 and

A(o)

and

Afft ) satisfy the same lattice-theoretic sentences t with constant e. Then A f o ) and either A(Q^) or Afft^ satisfy the same l-group sentences. THEOREM 2Bf ( 2 ^ ) .

Same as Theorem 2B (2C) with "l-group"

replaced by "lattice with e. " THEOREM 2E> . If and

0. and T

are doubly homogeneous chains

\|;: A(Q.) = A(T) as lattices with e3

then Q, is ordermorphic

40 (or antiordermorphio) to an orbit of A(T) in ¥. If i£ denotes ~—7 * — 7 — 7 this (anti)ordermorphisms then g\p = \\T g^ ($~ g~ $) (g € AfftJJ.

We will next turn our attention to some immediate applications of the Patching Lemma (Lemma 1.10.9). We will give a geometric criterion equivalent to the algebraic notion that two elements of doubly transitive

A(ft) are conjugate. We will also prove:

k(Q.) is doubly transitive,, then it is

THEOREM 2E. If divisible.

and THEOREM 2F. If ft is doubly homogeneous3 every element of is a commutator.

Recall that

B(ft) = {g € A(ft): g has bounded support}.

= {g € A(ft): supp(g) is bounded above} set of all elements of

(i.e.,

Let

L(ft) is the

A(ft) that live on the left), and

R(ti) = {g € A(ft): supp(g) is bounded below}. triviality, these are normal subgroups of

By the fundamental

A(ft). We will use our

conjugacy criterion to prove, in particular:

THEOREM 2G.

If ft is doubly homogeneous, B(Q.) is a simple

group and is contained in every normal subgroup of AfftJ

(L(Q.)>

R(Q.)) except {e}.

THEOREM 2H. The only proper normal subgroups of k(R) are B(R)y

L(R) and R(B).

Theorem 2H fails if we replace

1

by an arbitrary doubly

homogeneous chain as we will show by example. We next consider the I-automorphisms and group automorphisms of doubly transitive obtain

A(ft). As a special case of our work, we will

41 THEOREM 21. The only I-automorphisms of ACS) are inner— i.e., conjugation by elements of A(S). The only automorphisms of the group ACS) are I-automorphisms possibly followed by <x(g*J = ~((-o.)g) Ca € S;

where

g € A(S)).

The proof we give here relies on our work in Section 2.1. An alternative proof will be given in Chapter 9 when ft is not necessarily doubly homogeneous. Finally, we prove that doubly homogeneous chains are universal.

Specifically:

THEOREM 2J.

For any chain T,

eous chain ft and an l-embedding of

there exist a doubly homogenACT) in ACQ) such that

CD

the restriction maps B(T) into B(Sl)

(2)

|ft| = max{\T\, tf^}.

and

As an immediate consequence of Theorems 2E, 2J and Holland's Theorem we have

COROLLARY 2K. Every l-group can be l-embedded in a divisible I-group.

and COROLLARY 2L. Every l-group G

can be l-embedded in B(Q)

for some doubly homogeneous chain ft with (If

G

is an £-group, it can be ^-embedded in

doubly homogeneous chain

T

We can £-embed

B(T x Z )

{

|ft| <_ max{ \G\ J N J .

A(T) in

A( T) for some

by Holland's Theorem and Theorem 2J. via:

f K f

where

(xf,O) if n = 0 By Theorem 2J, we can £-embed

(x,n) _»

B(T x" Z )

in

if n / 0

B(ft) for some doubly homogeneous chain ft. Clearly

the composition of these maps ^-embeds G

in B(ft).)

42 This whole chapter can probably best be expressed in the caption "Geometry versus Algebra." of central importance.

The interplay of these two is

The techniques are as valuable as the

theorems they prove.

2.1. GEOMETRY VERSUS ALGEBRA. This section is devoted to establishing the machinery used frequently when dealing with doubly transitive

A(ft). Specifically,

we will use it in this section to prove Theorems 2A-2D and their analogues. Typically, we will take a geometric (pictorial) concept and write down an algebraic statement which will be equivalent for doubly transitive

A(ft).

Recall the definition of a bump (p. 25), a most geometric notion. We begin our catalogue with an algebraic description of bumps. Write bump(g) for -> f

= e or f

(Vf^XVf^Xf^ A f 2 = e & f1 v f 2 = g ->

= e).

LEMMA 2.1.1.

Let

0, be any chain and e < g € AfoJ.

g has

one bump if and only if burnp(g). Proof: If g.

Then

g

has more than one bump, let

h € A(ft) and as

gh~" i- e,

h A gh~

f.. v f

= g

with

f., A f

the disjoint union of the open sets supp(g) is an interval of

supp(f\) = 0

h v gh~

bump(g) is false (cf. Lemma 1.9.3).

has one bump and

Since

= e,

for

i = 1 or 2;

h

= e,

be a bump of = g

and

Conversely, if g then

supp(g) is

supp(f-. ) and / ( -J.X X <J. supp(f O *-4/>' V -L9Q). / $7, it is connected, Hence

i.e.,

f^ = e

or

f 2 = e.

43

K*1

Write

bimp(h^g) for

bump(h) & gh

A h = e.

By Lemmas

1.9.3 and 2.1.1: COROLLARY 2.1.2. Let ft be any chain and e < g>h € AfftJ. Then bumpfh^g) if and only if

Let that

h

is a bump of g.

(G,ft) be an ^-permutation group and if for all

g-, is to the left of

i = 1,2), a x < a'2* Write -1 (•f >. e)(|g1| A |f" g2f| = e

^

if

g-j>gp ^ G*

a. € supp(g.)

LEMMA 2.1.3* and

e ^ g'

left of

i"/ (G*tt)

€ G3

is a transitive Z-permutation group

then oJ^g^ if and only if g^

g2.

Proof: If

g_

is to the left of g

fundamental triviality.

Conversely, if g

g2

there exist

a. € supp(g.) (i = 1,2)

Let

e <_ f € G

be such that

Let

i?T^J

(3y / e)(g/y),

then g € B(Q)

bv

the

is not to the left of such that ot

(3y ^ e)(y/g),

gfe; the formula

COROLLARY 2.1.4.

g^g2

ou u.

so not

be the formula and

then

ouf = an .Then

a 1 € supp(g1) n supp(f" g 2 f),

R(g)).

is to the

L^j

the formula

L(g) & R(g).

If fi is doubly homogeneous and g € A(ti)>

(g € LT^^ ^ £tffftJJif and onZi/ if Big) (L(g)j

Hence B(tt)> L(Q.)> and R(Q) do not satisfy the same

L-group sentences (and so are not Z-isomorphic).

Let if

g

€ A(fi). We say that

is to the left of

(e ft). We write

Adj(g^gJ

, and

g.

is adjacent to

sup( supp( g_ )) = inf(supp(g^

for: g-Xg0 & -'Of /

Q,adja ctnt to

COROLLARY 2 . 1 . 5 . 91>92

€

Prooj Adj(g 1 ,g 2 ).

^^->

^

J / ft i s doubly homogeneous and

e n

#7 ^ 5 adjacent

By Lemma 2 . 1 . 3 , i f Conversely, if g

to

g~ if and only if

g-, i s adjacent t o g 2 ,

is to the left of g 2 , let a = sup(supp(g1))

g

and

Since g.. is to the left of (but not adjacent

8 = inf(supp(g2)).

to) g 2 , a < 8. By Corollary 1.10.6, there exists such that

then

is not adjacent to g 2 but

supp(f) £(a,8).

Hence

g-,<£f & f«£g?>

e^f € so n

°t

Adj( gl ,g 2 ). LEMMA 2.1.6. is

If Q is doubly homogeneous and a < 8 (in Q)

e < g € B(Q) such that a = inf(supp(g))

and

8 = sup(supp(g)). Proof: By Corollary 1.10.6, there is e < g 1 € B(ft) with supp(g1) 9 (a,8). with

If inf ( supp( g 1 )) = a][ > a, let e < g 2 € B(fi)

supp(g2) c (a,oL);

e < g^ e B(ft) with

and if sup(supp(g1)) = (L < 8,

let

supp(g^) c (g ,g). Continuing in this way

(possibly transfinitely) and taking the supremum (in A(fi)) of the

46 resulting set of positive elements of

A(ft), we obtain

g € B(ft)

having the desired properties. LEMMA 2.1.7. J/ ft is doubly homogeneous, then it is Dedekind complete if and only if

AfPj satisfies:

(Vg / e)(B(g) -> Oh)Adj (g3h~2 gh)). Proof: If ft is not Dedekind complete, let 3 €.ft\ftwith H

A(fi) and

a < 3. g

Let

e < g € B(ft) satisfy Lemma 2.1.6.

is to the left of

h

such that

gh)). Since

By Corollary 1.10.6, there is

supp(f~) Q (3,ah).

Hence not

a = inf(supp(g)) and

kind complete, Clearly

g

Adj(g,h~~ gh).

Let Then

«

3 = sup(supp(g)).

a,3 £ ft and there is

is adjacent to

e i g € B(ft).

Since ft is Dede-

h € A(ft) such that

ah = 3.

h~ gh.

e J f,g Q. B(ft).

Define

/ ^^

is an equivalence relation on

THEOREM 2.1.8. Let

3 & ft and

e < f n € A(ft)

Conversely, if ft is Dedekind complete, let Let

If

gh, then

3 <_ ah = inf ( supp( g) )h = inf(supp( h ah € ft, 3 < ah.

a € ft and

if Vh(fcTh ^-> g ^ h ) .

B(ft).

By Lemma 2.1.6:

Si be a doubly homogeneous chain. For

each e ^ f £ B(Q.), let a~ = sup(supp(f)) £ ft. Then f »-> 5« maps

£fftj\{e} onto ft^ and

Moreoverj a~ < a

a~ = a

^/ and only if f ~g.

i/ and onZz/ i/j /or sows

e ^ h € BfftJ^

/z ^^
B(ft) is characterised by Corollary 2.1.4, this

"interpretation" of ft (together with its order) in the first order £-group language. this way?

If

and

-*-

a

respectively,

a

{h

fh: h € A(ft)} interprets the points of ft.

_-j

€ ft, h

fh « g

COROLLARY 2.1.9. e

oif and

—1

a f € ft, then

So if

Can we distinguish ft in ft in

e i f,g £ B(ft) interpret

—

A(ft) is in

for some

h € A(ft).

Indeed:

Let ft be doubly homogeneous and

^ f>9 € B(Q,). Then a* and

a

belong to the same A(Q) orbit

if and only if ti~ fh & g

for some h

Note that this last statement is a first order condition (in the language of ^-groups). Now if

A(fi) cannot

language) from

be distinguished (in the £-group

A(R), then

Q

is Dedekind complete by Lemma 2.1.7,

so there is only one orbit and no problem arises. However, if we replace

1

by

A(Q) does in

Q,

then

A(fi) has just two orbits in

1. fA(Q) satisfies

(
Q

since

(3f ? e)(3g ? e)(B(f) & B(g) & * f or

«g))).] In order to prove Theorems 2B and 2C we must still capture

countability. Let

We first need a technical lemma.

G

X c G.

be a group and

The centraliser of

CJX) = {f € G: (Vx £ X)(fx = xf)}. We will often write

X

in

G,

C(X) for

G-

CG(X)

if

G

is clear from context. n

C({g}), and <_g> = {g : n a } .

If

g £ G,

Note that

write

CYffJ

for

C() = C(g). So

C(C(g)) c c ( g ) . LEMMA 2.1.10.

Let

Q be doubly homogeneous and

e < g G k(tt) have one bump. Then = C(C(g)). Proof: For reductio ad absurdum, assume Let

A = supp(g).

and

{6 € A: 6f >_ 6gn+ } are closed (in

A

For each

n € Z,

is connected, there exists Case 1.

af $? A.

e

so

f £ C(C( g))\. {6 € A: 6f <_ 6gn]

A) and disjoint. Since

such that

ctf ^ {ag : n € 2£}.

By Corollary 1.10.6, there exists

e < h € A(fi) such that h A g =

a € A

the sets

af € supp(h) and

h€C(g).

Hence

supp(h) H A = 0. Now

f € C(h). But

ahf = af < afh,

a contradiction. Case 2. af € A. e < h hQ

So for some

€ A(fi) be such that

and from it construct

m € 7L, agm < af < ag m

af € supp(hQ) c (ag m ,ag m+1 ).

h € C(g) by taking the pointwise

supremum of the disjoint set of elements A*0); i.e.,

. Let We take

{g~nhog : n € 2Z} of

h|[5gn,agn+1] = g - n + V n - m | [ 5 g n , a g n + 1 ]

and

supp(h) C A . Then h € C(g) so f € C(h). But afh = afh Q > af = ag\og~mf = ahf, a contradiction.

Note that the condition f € C(C(g)) can be expressed by a sentence (Vh)(hg = gh •* hf = fh), but the condition f € usually cannot (f = g° or g 1 or g~ or g 2 or g" . . . i s not permissible). We can now turn to countability. LEMMA. 2 . 1 . 1 1 . Let Q, be a doubly homogeneous chain and A ^ Q, with a < inf(L) < sup(L) < T. A is countable if and only if theve exist a 6 ft and f3g € BfoJ such that A = ioLjg : m € w} an^ supp(f) = supp(g) c fa.,T,). Clearly {af g~m: m € oa} i s a countable subset. Conversely, enumerate There exists

Let

3g

and

and a < A < Bg

< A.

and

By double transitivity and Corollary 1.10.6, there

e < f £ A(ft) having one bump such that m

a = 6

e < g € A(ft) having one bump such that

supp(g) £= (<J,T). is

A = {6 : n £ to}.

m

m

af = Smg (m € w).

Hence

supp( f) = supp(g)<=(a, T ) , A = {afmg"m: m € a)}.

A = {aab~ : e < a € ,e < b €
Note that

LEMMA 2.1.12. Let Q be doubly homogeneous and A 9 f l , A is countable if and only if there are a ^ a . € Q and

Then

Kfo,frgo,g1 {cLh

€ Afoj

1^9-1

m

•'

n

Uj

A = {aQh2nfQg~™: m e u, n * Z} U

such that Z}.

Proof: Clearly any such set is countable, so assume countable.

If

Lemma 2.1.11.

A

A

is

is bounded, the lemma follows at once from

Assume

A

is unbounded and let

{a : n £%}, n { 3 : n € Z } , {a : n £2£} and {T : n £2£} be coterminal in A n n n (and hence in Q) with a < a < 3 < T < a +-, (n £2£). Let A = A H [3 ,3 1 0,n 2n-l 2nJ

E

/////////////////^/7////,/'///>//////////1 /////////////////////////////////////// I

and A = A ^ ( 3p , 3~ +-. ). By Lemma 2.1.11, there are e < f. ,g. € A ( f i ) ( i = 0 , l ) having one bump with supp(fQ >n ) = supp(g 0 ; n ) £ ( « 2 n _ r a 2 n + 1 ) , a 2 n _ l g Q ; n = T ^ , A

0,n

= {a

2nfO,ngO™n:

m 6 w}

=

{a

2n a O,n b o!n :

e

K a

0,n

£


0,n>'

e K b

0,n £ < g 0 , n supp(f1,n) =

\ n = {a2n+l^,nC: e < b.,

€
m & u } = {a

>}D(o

,T

+

2n + l a l,n b l!n :

) (n € S ) .

supremum of t h e d i s j o i n t elements respectively (i = 0,1). I

i la^

,

n

i

n

a h = a . n n n+1

g

n

:

and

m G. co > .

Then

A =

{f.

n

e < a

Let

^

and

: n € ^} a

f

\^J ^ 2 n o^Q

l,n €

and :


gi

l, n >' be t h e

{g^ ^: n € %

m t w} U

Let

x h = T -, (n € %). n n n+1

Let

h = \J h . n

Thus

50 A = {a0h

2n

desired.

m

2n

f Q g Q : m e a>,n € Z> U {axh f!jg~ : m € 00,n € Z } , Similarly if

A

(in S ) . there are

e < h £ A(ft) and

a 1 kh' 1 ,T 1 k

< a

e < ai £ < c >

c. (d.)

of

5 = ouka^T1 and

containing

is countable and unbounded,

belonging to the same = « n h,

< a

,a.kh)

f. (g.)

a

for some

=

A(Q) oh,

contains the support

with

(i = 0,1; k 6
e < b. € ouk

A

< T

(a.kh

€ supp(c i ) O supp(d i )

if and only if

(g.)

a

and each interval

of a unique bump

a subset of any orbit of

having one (unbounded) bump,

ci.,a.,T. £ Q

orbit (i = 0,1) such that = x h,

A

It also gives that if

e < f.,g. £ A(Q)

as

is bounded above or below.

Note that the proof works for

T

m

and

6 € A

i £ {0,1},

k € ,

c. (d.)

that bump of

with

in its support, and

a.kh"

So we can code countable subsets of an orbit of

< a.ka.bT A(Q)

f. < T.k.

by tuples

(a Q,aQ,x ,h,f^,g ,f ,g ) where the coordinates' satisfy the above conditions.

If

f- ,f- ,f-

a (p £ {a,a,x}), elements of

o

a

€. B(ft) with

sup( supp( f- )) = p

T

p

o o

o

then the octuple

A(Q)

codes

A.

(f- ,f- ,f- ,h,f ,g ,f ,g ) of a 0 0 T0 Moreover, the above conditions on the

coordinates of this tuple can be expressed In the language of ^-groups by Corollaries 2.1.2, 2.1.9, Lemma 2.1.10 and the remarks following Theorem 2.1.8.

What is more, we can determine (in the

£-group language of

A(^))

countable subset

of an orbit of

A

if two such octuples code the same A(fi),

we can use these octuples to detect subsets of an orbit of countable

=

and

and if Q

3 £ A.

Hence

between countable

A ( Q ) . Furthermore, "there exists a

A..." can be translated into our language by ''there

exists such an octuple...", etc.

From now on we will assume that

this translation has been performed. We now prove Theorems 2B and 2C. By the remarks following Corollary 2.1.9, orbits one of which is countable.

A(Q)

has two

By the above, this Is expressible

51 in the language of ^-groups and so holds for any doubly transitive A(ft) E A(m).

Thus

Q

or ft\ft is countable proving Theorem 2C.

To prove Theorem 2B, note that if doubly transitive then

fi

A(Q) E A(l),

is Dedekind complete (Lemma 2.1.7) and (by the previous

paragraph) has a countable dense subset

(A

(Va)(V3)[a < 3 -+ (3 6 £ A)(a < 5 < §)] ).

Hence fi = I.

is dense in

Q

if

Actually, we have shown that there are sentences

0TO and m. of the language of ^-groups such that doubly transitive A(fi) satisfies 0^ ( e_ ) if and only if Q = 1 (d or I\Q). In in, ^

0m Ly

Chapter 4, we will show that "doubly transitive" can be replaced by "transitive" in Theorems 2B and 2C. We now wish to consider the group language instead of the £-group language.

Recall that Corollary 1.10.8 (in conjunction

with Lemma 1.10. l) shows:

There is a first order formula

(j)Q(p)

of the language of groups which holds in a doubly transitive A ( Q ) if and only if

p

or

p

exceeds

1.10.8, there is a first order formula

e.

Moreover, by Corollary

<\> (y,z)

language) which holds in doubly transitive

A(u)

j <_ z

_>_ e ) .

(when

sentence

p _>_ e)

0

or

y >_ z

(when

p

in the £-group language (using

be

and call the resulting sentence

(3P)(<J>Q(P) & 0 ).

and only if sentence

A(Q)

0'.

or

if and only if Now given a

<_ in place of the

lattice operations), replace each occurrence of <j) (y,z) P

(of the group

0 . p

Clearly, the £-group

y <_ z Let

A(Q)

by

0r satisfies

0

if

A(tt*) satisfies the group-theoretic

This proves Theorem 2A.

Since ffi* = I

and

$* = Q,

from Theorems 2A, 2B and 2C.

we obtain Theorems 2B 1 and 2C'

Again, in Chapter l+, we will show

that "doubly transitive" can be replaced by "transitive" in Theorems 2BJ and 2 C . We now prove Theorem 2D. So suppose A(Q)

= A(T)

Q

and

(as groups).

By Lemma 2.1.6, there are

T

are doubly homogeneous, and Let ty be the isomorphism. e ?< f ^ f ^ £ B(ft)

Let

a £ Q.

such that

sup( supp( f )) = a = inf( supp( f )). Such a pair

(f^f^)

satisfies:

Moreover, if

( V p ) U 0 ( p ) & f1 £

f 2 -> Adj (f^f^).

52 sup( supp(g )) = 3 = inf(supp(g 2 )),

then a = 3 if and only if

h g (Vp)(* Q (p) & *"» ^ p h ] ) . Hence, p l^pg2 by Theorem 2.1.8 and Corollaries 2.1.4 and 2.1.9, ty induces an ordermorphism $ from tt onto T or f* such that the restriction of ij) to Q gives an orbit of A(T) in f (or T * ) . Hence fi is ordermorphic (or anti-ordermorphic) to an orbit of A ( T ) in T. Moreover, if ij) is an ordermorphism and 3 € T, let f \p € B( T ) with sup( supp( f i//)) = 3 . Since sup( supp( 1^) )i|/ = sup( supp( h ^ )) (^^ A(fi)

sup( supp( fx

= sup(supp((g" = sup(supp(f1i[;))(gip) = (g £ A(fi)). Hence $" g = gty as desired. If ij; is an antiordermorphism, the proof is the same except sup( supp( h^) )$ = sup( supp( hpip)) where (h..,hp) is an adj acent pair. Finally, we wish to consider the lattice language. that the formula we gave for bump(g) was in the lattice and did not involve the group operation. If g,h € A(Q) bump and g A h = e we say that f £ A( Q) lies between if supp(g) < supp(f) < supp(h) or

Observe language have one g and h

53 supp(h) < supp(f) < supp(g). tice formula:

L e t B(f>g,h)

be the l a t -

e < f & bump(g) & bump(h) & g A h

= e & f A g

f A h = e & (Vk > e)(bump(k) & k A g ^ e & k A n ^ e (Vf

f

> e)(bump(f)

S c f ' ^ f

LEMMA. 2.1.13.

Let

Q be doubly homogeneous and g>h £ k(Q.)

lies between g

inf(supp(h)) = 3.

If

has one bump with

kAg^e^kAh,

for some

and since

f

lies between

k

n

a £ supp(k ) = supp(k).

and

h,

Thus

there exists

a £ supp(g) and and

h

such that but

3 < ak

k

L A

contains Let

and

B(f,g,h).

= e

a.

So

ff

e < k £ A(ft) Then

f

so

B(f,g,h).

does not lie between a £ (a,3).

There exists

xk = x. a.

Let

Then

is the bump of

akn >_ 3

so

gAf=e=hAf,

a f[ (a,x).

Let

and

g

e < k £ A(ft)

k~ £ A(ft) be the k Q A g / e / k Q A h, f whose support

B(f,g,h) fails, the desired contradiction.

B(f) be the formula:

COROLLARY 2.1.14. If e < f £ Afftjj

and

a,3 £ supp(k),

If

Since

ak = a,

where

h

a £ supp(f).

a £ supp(f) such that

whose support contains

f

and

supp( f) c supp(k),

x £ supp(h).

each have one bump,

bump of

g let

a < 3 £ ak n < ak n

Conversely, we assume g

a = sup(supp(g)) <_

has one bump and

n £ 7L . Hence

Then B(f,g3h)

and h.

Proof: Without loss of generality,

a < a < 3

+

+ k A f ' ^e ) ) .

have one bump with g A h = e. Let e < f £ A(Q). if and only if f

= e &

^

(3g > e)(3h > e)B(f,g,h).

Then:

is doubly homogeneous and

then ~B(f) if and only if f £ B(9,).

We next seek a left-right orientation on e < a,b £ B(ft) have one bump and are disjoint. has one bump with

{a,b,f,g}

A(ft). Suppose Let

e < f,g £ B(Q)

pairwise disjoint. Then we can use

betweenness to express in the lattice language:

a

of b

This is done by

if and only if

f

is to the left of

g.

is to the left

taking the disjunction of six possibilities each of the form B( , , ) & B( , , ). Let

L(ayb3f,g)

formula and the hypotheses on

be the conjunction of this

a,b,f,g.

formula of the lattice language.

Note that

L(a,b,f,g) is a

If

e < f,g c B(ft) have one bump and are disjoint

necessarily disjoint from lattice language:

(but not

a and b ) , we can still express in the

a is to the left of b if and only if f is

to the left of g by: (3c )(3 d) [L( a,b, c, d) & L(c,d,f,g)]. Let Lr(a>bsf3g)

be this formula. More generally, if e < f f ,f ,f22

are disjoint, write

£(a3b3f' / ' ) for 1 f2)(bump(f|) & bump( f

(Ve < f ^ < f ] _)(Ve <

be doubly homogeneous_,

LEMMA 2 . 1 . 1 5 .

<

a, 2? having one bump and a A b =

a3b3f'jf'2

Then AfoJ

satisfies

left

and fn

of

of a

b

and f ~ Let

pair

if

()

£(a3b3f^fj

is to the left

e<

if and only if

is to the left of

of

f9)

or

(b

(a

is to the

is to the

left

/-).

€ B(fi). We say that 1

= in ir sup(supp(f ))) = f(supp(f9)) = inf(supp(f

f-^f- is an adjacent

or

sup(supp( f„

LEMMA 2 . 1 . 1 6 . e

< A j / n £ Bf°J.

(Va>b > e

Let

Q be doubly

T/^en fjsfp

homogeneous and

ts an adjacent

one bumps) (a A b = e ->- Xia^b^f*3f')

(Vh > e)(3e < ft £< fJ(3e f2)

< ff'2 <_<_ fj

[bump(f')

pair

precisely

or £(a3b3f'>f\)) &bump(fL)

^

when: &

Let a,3 € ft. Choose e < f-pfp £ B(ft) s o that sup( supp( f ^)) = a = inf ( supp( f2 ) ) , and e < g^g^ G B(ft) so sup(supp(g_)) = 3 = inf(supp(gp)). Then a = 3 if and only if (V bump

h > e ) ( h A f ^ e & h A f ^ e

h/\g 1 By Corollary 2.1.14 and Lemmas 2.1.15 and 2.1.16, we can interpret the set ft together with equality in the lattice language of A(ft) by adjacent pairs of elements of B(ft). Indeed, using parameters e i a,b € B(ft) with a A b = e, we can put an ordering on ft. If a is to the left of b, this coincides with the original ordering on ft; otherwise it is the reverse ordering Q . We now wish to obtain the action of A(fi) on Q. Note that supp(f) c supp(g) is equivalent to (Vh)(h A g = e - ^ h A f = e) if Q is doubly homogeneous and e < f,g €. A(fi). LEMMA 2.1.17. Let and e < h 6 A(Q). Then e < x € B(tt) having one e < y / /z uit/z supp(y)

ft be doubly homogeneous> a <_ 3 in ft ah <_ 3 i/ anJ onZz/ i/ /or aZ-Z bump3 if a^3 € supp(x)^ there exists c: supp(x).

Proof: If ah > 3^ let a € ft be such that 3 < ah < ah. Let x have one bump with a,3 € supp(x) SE (a,oh). Now if supp(y) <=; supp(x) and T € supp(y), xy < ah < xh; so y < h.

Conversely, if assume

ah <_ (3,

a,3 £ supp(x).

y = x

i_ h.

Then

let

e < x € B(Q)

Then for some

3 < ax m .

m €22 ,

supp(y) = supp(x).

have one bump and Let

/

By Lemma 2.1.17 and the remarks preceding it, we can express ah = 3

in the lattice language of

appropriate pair of parameters

prove an analogue of the lemma if h €. A(u)

since

ah = 3 (± a) f-g - h

ah = 3 (>_ a)

if and only if

A(Q)

a,b)

(with the aid of an

if

h > e.

h < e;

so for an arbitrary

if and only if a(h A e) = 3.

in the lattice language by:

However, had we chosen

a,b > e

a

a(h v e) = 3,

"a

is to the left of

to the right of

b

(we can say

are disjoint, bounded and have one bump; so

left or right of would give

a

b ) , the expression we used to capture

a = 3h.

and

Finally, we can define

(Va)0/B )( af = 3 -> ah = 3g).

The only problem is that we cannot express b."

Clearly we can

Hence we have proved Theorem 2A 1 .

is to the ah = 3 Theorems

2B f and 2 C f now follow. To prove Theorem 2D , we note that either A(ft) = A ( T ) 1

^-groups or

as ^-groups.

result follows from Theorem 2D. 1

ty: A(fi) = A(T) *, (fg)* 1

V

r

1

A(ft)

and

= f ]p A gij,

phism between

as

In the latter case, if

by

gipr = (gip)"1.

Then

= ((fg)ip)"1 = (f^tgip)"1 = [(g^)(fi^)]"1 = f ^ g V .

(f v g ) f

2D

define

A(ft) = A(T)

In the first case, the

and dually. A(T)

Hence

T

\p

Also

is a group isomor-

that reverses the order.

Theorem

now follows from Theorem 2D. 2.2.

DIVISIBILITY AND CONJUGACY IN AfoJ.

This section could equally well be entitled "Applications of the Patching Lemma." the results.

Throughout this section, therefore,

doubly homogeneous. A(Q)

Indeed, the technique is just as important as 0, will be

Our first application will be to prove that

is divisible (Theorem 2 E ) . Our next will be to give a

completely geometric description of when two elements of conjugate.

A(l)

are

This will give a much simpler proof than that of the

previous section to show that

A($) f A(l)

(actually,

in the language of groups, lattices or ^-groups).

A(Q) 1 A(R),

We generalise

our results to give a geometric description of when two elements of

57 A(ft) are conjugate

(ft any doubly homogeneous chain).

these results are due to Holland [63] or folklore.

All of

We give one more

example of the technique to simultaneously make several elements of A(l)

conjugate by the same conjugator (under certain hypotheses).

This particular example can be found in Glass and Gurevich [81] and illustrates the power of the technique.

Further examples and

applications can then be left to the reader's imagination. We first prove Theorem 2E: Suppose that ft is doubly homogeneous, m €. % . First assume So

A = ^J [ag ,ag n€^

g > e has one bump. ] . There are

a = $ 0 < 3-, < . . . < 3 are or dermorphi sms

g € A(ft) and Let a £ A = supp(g).

$n>$i> - " >&m €

A

such

that

= ag. Since ft is doubly homogeneous, there

f : [3 TJ 3 ] = [3 ;3 + T ] (l £ n < m ) -

Let

m-irleDefine

Now

A(tt) by:

Yh

= yg~ k f n g k

emfm=*

and

V i V ^ ^

m

if

y ^ [^^

for

3n g

some

k

and

1 <_ n <_ m

and

yh = y

otherwise.

Now

some

k £ 2£

and

1 <_ n <_ m

and

yh = y

otherwise.

By the

yg if Y e A definition of

f m,

yh

m

=

= yg. j

h. = g..

Note

patching lemma, there exists h|supp(g.) = h.|supp(g.) Then

h

= g

of

there exists By the

such that

h.| A = e|A,

where

A = oAsupp(g).

and Theorem 2E is proved.

and

h3 = g

that

g,

supp(h.) = supp(g.).

h €. A(Q)

and

g.

Note that the proof yields, for example: a 6 supp(g)

h m = g.

if

In the general case, for each bump h. € A(fi) with

Hence

and

a < 3

< 3^

< a

If

S ; there exists

ah = 3 , 3 h = 3p

and and

g € A(^), h €. A( 9,) such

3~

We now give another application of the patching lemma. Again the technique is at least as important as the result. Recall that a bump is either bounded, bounded only above, bounded only below, or unbounded.

We call this the boundity of the

bump. LEMMA 2.2.1. same boundity. Proof:

Let

e < g3h €. A(S)

each have one bump and the

Then they are conjugate in A(E). Choose

a € supp(g)

and

3 € supp(h).

Since

A(l)

is doubly transitive, there is an ordermorphi SIR f : [a,ag] = [£,3h].

For each

n € T,

f : [ag n ,ag n+1 ] = [Bh n ,3h n + 1 ]. and

h

g

and

h

n € 7L. We now show that

since

f

= g" nfQQhh n .

So

By the patching lemma (since

have the same boundity), there exists

same boundity as all

let

such that

f|[ag ,ag

f~ gf = h.

supp(g) = \J [ag n ,ag n+1 ] ne Z

and

g

f €. A(ffi) of the ] = f

for

supp(g)f = supp(h)

supp(h) = \J [3h n ,Bh n + 1 ]. n£^

59

flu™ Let

y £ supp(h). = yf

Then Yf"

-1

Yf-V

= yh

Then

yh = y

f = y = yh.

Let

and

yf

y € supp(h),

supp(g). say

= Yf;V m + 1 = for all

Hence

y € [Bh m, 3h 3hm+1] . = yh.

y €

i.e. ,

f

Thus

= h.

As a consequence of the proof:

COROLLARY 2.2.2.

Let

e < g 3h

one bump and the same boundity.

£ B(R)

(L(R)3E(E))

each have

Then they are conjugate in

B(S)

We can now give an easier proof of the fact that

A(Q) ? A(ffi): COROLLARY 2.2.3.

? A($)

as l-groups.

Indeed, as

l-groups, A(®) 1 A(%). Proof:

Let

cj> be the sentence which says that any two

strictly positive bounded one bump elements are conjugate.

(This

is expressible as a sentence by Lemma 2.1.1 and Corollary 2.1.4.) A(ffi)

satisfies

cf> but

A( Q)

does not since there are

60 e < ggjhQ €. B(Q)

such that

supp(g Q ) = (0,1)

and

If = /2;

so no such

If of

G

X ^G

g~

and

f € A(Q)

and

G

generated by

hQ

each have one bump,

supp(h Q ) = (0,/2).

If

then

exists.)

is a group,

X.

f" 1 g Q f = h Q ,

(if

<X>

X = {g},

will denote the subgroup

we write



for

<{g}>,

conforming to our previous notation. Let

g £ G

The orbitals of

where

(G,fi)

The non-singleton orbitals of bumps of

g

is an ordered permutation group.

g_ are the convexifications of the orbits of g

are just the orbitals of the

and are the supporting intervals of

g

The parity qf_ an_ orbital is that of the bump of supports.

The set of orbitals of

right; i.e., if (for all

A

and

Ap

6. € A. (i = 1,2),

THEOREM 2.2.4. conjugate in

A(S)

right preserving those of

h

Let

g

that it

is totally ordered by left-

are orbitals of

g, < A

< 6 )

g,h € k(R).

or

A

Then

g

either or

and

If

h

bump of

if and only if there exists a one-to-one left-

h = f~ gf,

then the map I

g

and

I \—* If

is such a map

ranges over the orbitals

g

h,

$

let

is such a map between the orbitals of I

be an orbital of

whose support is

be the bump of

h

I.

I<j>

g.

Let

is an orbital of

whose support is

(by Lemma 2,2.1) g.

If

1 u {±00}

I<j>. Mow

|l| = 1,

let

A

gT

^

such that

A^

=

let

and f,,

hT say

be maximal closed interval of g.

For each

1

{<5}<J>.

f|l = f |I

6 € A,

!

{6}<J> = { 6 } , where

induces a one-to-one order-preserving map 5f

h;

is either a positive or negative orbital

6' e ffi such that

defined by:

f € A(l)

I

containing only fixed points of

there exists §

if

g

g-. be the

have the same parity and boundity so are conjugate by

A(j>

are

correspondence between the orbitals of

Conversely, if

Then

< A = A?.

g.

and those of

of

A A

such that corresponding orbitals have the same parity.

Proof:

hT

( see page 26 ).

g

6

by the fundamental triviality, where of

.

6 h = 6'. f

of

A

onto

By the patching lemma, there exists for all orbitals

I

of

g

with

61 |l| > 1,

and

interval of f 1gf\l

(even if

=

6f = 5f I u {±00}

if

and

A

is a maximal closed

containing only fixed points of

fT1gTfT\l^

|I| = 1).

6 e A

Icf)' Thus

= h|l(f) for all orbitals

g. I

Hence of

g

V = h.

We observe that the above proof can easily be modified to prove Theorem 2.2.4 with

L(l), R(ffi) or

B(R)

in place of A(l).

As examples of applications of Theorem 2.2.4, we see that the following are conjugate

but none of the following are:

(0

(z)

62

(3)

What happens if we replace ffi by an arbitrary doubly homogeneous chain ft? As we saw in the proof of Corollary 2.2.3, Lemma 2.2.1 fails if we replace

1

by

Q). However, when we

consider Theorem 2.2.4, there is no one-to-one map of the orbitals of

to those of

gQ

orbitals of where

g,

h

which preserves left-right and maps

to those of

S Q ^ Q ^ A(Q)

of the same parity and boundity,

are given in the proof of Corollary 2.2.3.

(There is a least orbital of orbital—viz.:

h

g

to the right of its positive

{1}—but no least orbital of

its positive orbital.) points of ft (i.e.,

hn

to the right of

Indeed, since we are taking orbitals of

Conv {ag n : n £ %) (a £ ft)), a moment's

reflection shows that the proof of Theorem 2.2.4 generalises to give:

THEOREM 2.2.5. Then

g

and

h

are conjugate in

exists a one-to-one of

g

onto those of

same parity. belong to

Let ft be doubly homogeneous and AfoJ

g,h € AfftJ.

if and only if there

left-right preserving map from the orbitals h

such that corresponding orbitals have the

Moreover, the conjugator in this case can be taken to

B(tt) (L(Q.), R(Q,)) if

g

and

h

do.

The following corollary implies Theorem 2F.

63 COROLLARY 2.2.6. there exists

If ft is doubly homogeneous and g € Afftjj such that g2 = f~1gf; i.e., g = [g,f].

f £ AfftJ

COROLLARY 2.2.7. If ft is doubly homogeneous and there exists L(Q.) or

f € B(Q)

R(Q)

such that

g = [f,g]. Similarly with

in place of B(9.).

Note that the fact that (m £3£ ) in

g € B(Q,)3

g t A(fi) is conjugate to

g

A( ft) if ft is doubly homogeneous gives an

alternative proof of Theorem 2E.

THEOREM 2.2.8.

The group theory and lattice theory of

lattice-ordered groups are both undecidable. That is, there is no algorithm which determines whether or not an arbitrary group (lattice) statement is a consequence of the l-group axioms. Proof: By standard arguments in logic (see Tarski-MostowskiRobinson [53] ), it is enough to find an £-group g~ £ G

such that we can code into

G

(using

together with addition and multiplication. make this possible are 2.1.10 and 2.2.6. have one bump. if

Now

f € G

G g~)

the integers

The key results that Let

G = A(ffi) and e < g Q

will be declared an integer if and only

f ec(c(g )) (i.e., 0

(Vh)(hg

hf = fh). By Lemma 0 = g 0 h "* n : for some n € 22 . Define g @ m - JP

2.1.,10 , any integer is £^n n m _ p . This is standard addition since if n

and an element

g

o=

m

n+m W+ , there is = g n • By Corollary 2.2.6 , for each n € & 0 x € A(ft) such that x £ x g >o o * For this x, m \in ran m X x" £J0x = (x" g o x ) = g Q , Hence define g^ n w g0 = % if r

0x)(x

g£ & x- X g^C

natural way for

n < 0.

= gn>

if

n > 0,

and extend in the

Thus we obtain the standard multiplication.

The proof of the theorem is completed by applying Theorems 2A and 2A

respectively. The group part of Theorem 2.2.8 has been proved independently

by Mal'cev [71; Chapter 15] (torsion-free nilpotent groups are

64 orderable, see Fuchs [63, p. 48]—the translator of Mal'cev uses "metabelian" for "nilpotent class 2 r T ), Jambu-Giraudet [8lt>], Buszkowski [79], and Glass and Pierce [80c]. the simplest.

The proof just given is

The lattice part of the theorem is due to Jambu-

Giraudet [Bib].

For another proof of the lattice part, see

Gurevich [67]. In Chapter 13 we will give an example of a finitely presented £-group with

insoluble

group word problem.

We give one more illustration of the technique to obtain a stronger result than Lemma 2.2.1.

LEMMA 2.2.9.

a.

n 'Z-'f'J.

Let

O^T^)

supp(gi)=

e < g. £ B(S) (i = 0,1,2,3).

< a. < x. < a., 7 #., 7 Is

such that

Is

I'll.

Let

Then there is

?^,£~

-1

-1

h gJ% = g2 and h g?h = g be ordermorphisms

2*

[Tr

in the proof of Lemma 2.2.1, we can extend

f~

such that

Let

f

o=

and

f ^: f

h € B(S)

1s~l J.

-1 h gQh = gQ

Proof:

(i = 0,1,2).

each have one bump and Suppose that

f.

] = As

2 to

65 a

T

so that

f

f

=

Let

h

f: [ Q 1 , T 1 ] = [ 2^ 2l ~ £i S?' \' 2 morphisms with h : [a~,a ] ^ [ a ^ a ^ ] and h 2 : [x^a-g^] = [T 2 ,a^g^]. Let h = ^ u f ^ h ^ ; so h'. [a~,a~g ] = [a«,a~g,J. can extend

hQ

to

h | [ ^ 1 , T 1 ] - f, h|[VTQ]

be

order

As in the proof of Lemma 2.2.1,

h £ B(t)

so that

h^gJi = g .

h" g^h = f" g 1 f = g

~

we

Since

Also since

= f|[ao,TQ] = f o |[a o ,T Q ] = e|[a o ,T Q ],

h'^h

= e'\e

= g

Once the technique is fully grasped, other applications, can be handled as needed.

See, for example, Chapter 13 which depends

only on this section.

2.3.

THE NORMAL SUBGROUPS 0F_ AfoJ .

In this section we wish to consider the normal subgroups of A(fi)

for

Q

a doubly homogeneous chain.

A normal convex ^-subgroup of an £-group is called an t-ideal.

We simplify our problem by first proving the following

generalisation of Lloyd [64]:

THEOREM 2.3-1. subgroup of

AfoJ

If

^

ts doubly homogeneous_, every normal

is an t-ideal.

From this we will be able to easily prove Theorem 2G: is doubly homogeneous,

B(fi)

If Q,

is simple.

We further use Theorem 2.3.1 to give a quick proof of the following generalisation of Theorem 2H:

THEOREM 2.3.2.

If ft is doubly homogeneous arid has countable

coterminality_, the only non-trivial normal subgroups of LfoJ,,

R(Q)j

and

B(Q).

normal subgroup of either

Moreover, LfoJ

A(Q)

are

B(tt) is the only proper

or Rio,).

In contrast, we give an example of a doubly homogeneous chain ft such that L(ft)

or

A(ft)

R(ft).

has a proper normal subgroup other than

This will illustrate the need to impose

B(ft),

"countable

66 coterminality" in the hypotheses of Theorem 2.3.2. We begin by proving Theorem 2.3.1. Let A(ft). Now

Q

be doubly homogeneous and

We first show that fg

and

f

both

fg

and

that

h~ fh - fg

is convex.

be a normal subgroup of So assume

e <_ g <_ f € N.

have exactly the same intervals of support. are positive (>_ e ) , there exists

f

(by Theorem 2.2.5).

g = f - ^ h ^ f h ) £ N.

Hence

Next suppose that there exists

N

N

h £ A(ti)

N

f € N

Since

h £ A(ft)

Since such

f £ N,

is convex. and consider

such that

h

2

f v e.

= f v e.

By Theorem 2E,

Define

gU(fi)

3 ~

if by:

af < a

ag =

Then ah

if

g

and

f

have exactly the same

af > a

intervals of support with corresponding intervals having the same parity. fg

Hence

€ N.

But

g

and

fg

f and

are conjugate, so f v e

is an ^-subgroup of

A(ft).

Thus

have the same (positive)

intervals of support, and so are conjugate. N

g € N.

Hence

f v e €. N

This establishes the theorem.

so

67 Note that the proof shows (using the last part of Theorem 2.2.5):

COROLLARY 2.3-3. subgroup of

Let ft be doubly homogeneous.

(L(Sl), R(Q))

B(Q)

is an l-ideal of

Every normal

B(tt)

(L(Q,)>

Theorem 2G follows immediately from Corollary 2.3-3 and the following lemma:

LEMMA 2.3-4and

Let

e < g £ AfftJ.

ft

be doubly homogeneous>

Then there exists

h £ B(Q)

e <_ f £ B(Q)

such that

f £ h~2gh. Proof: exists ah

Let

h e B(ft)

gh = 3

so

h

supp( f) £.(a,$) such that gh >_ f

and

yh = a

y € supp(g).

and

ygh = 3-

There

Thus

by Lemma 1.9-4-

In order to prove Theorem 2.3-2 we need only prove the following lemma similar to Lemma 2.3-4-

LEMMA 2.3.5. coterminality.

or

ft

(i)

e <_ f £ R(Q)

(ii)

e <_ f € LTftj

(Hi)

e <_ f

there exist h^h2

Let

and

(H(Q))

Proof:

( i)

a € ft with

and and

e < g

h^3h^ € AfftJ

€ L(Q)

CASE 1.

be doubly homogeneous of countable

If either

if

Let

e < g <£ L(Q)3 e < g $ R(Q), is

unbounded3

such that g

f <_ h^ gh^ v h^ gh^

and

does.

a <_ inf( supp( f)).

There exists

T € ft such that

ag > a

for all

a > T.

There exists

h € B(ft)

fixes no element of ft above the same intervals of support. r

(and so

same boundity as

g

h^ 1 g'h 1 = g ^ f;

i.e.,

such that a.

Hence

Th = a. g

1

v f

Thus there is g)

such that

f <_ h~1h~1ghli1.

h

Now

gr = h~ gh

and

gr

€ A(ft)

have of the

68 Case 2_. g

fixes arbitrarily large elements of ft. a, < 3-, < a^ < 3p < • • • cofinal in ft such that

We can find [a ,3 ] ^supp(g) and

but there are

a

€ Q

such that

3 < a < a _ n n n+1 a^g = a (n €. % ). By double transitivity and the patching

lemma, there exists (n £22 ). Now

h €

g v h ^gh

such that a h = 3 and 3 h = a n n n n n+1 fixes no point of ft to the right of

a,.

By Case 1, there exists hn £ A(ft) of the same boundity as -2. -1 such that f < hn (g v h gh)hn . —

i

±

(ii) The dual of case (i). (iii) is unbounded.

that:

af

=

If

f € L(ft) U R(ft), Let

3 < o < x.

af

if a >_ a

a

if a £ 3

use (i) or (ii). So assume

Define

e < fn ,fn € A(ft) 1 2

such

and

< af if a € ( 3,a)

Then where for

f = f g.,g.r g1 = g2

v f

and by ( i ) and ( ii)

are conjugates of and

g' = g

g.

Hence

^

<

gi

v g » (i = 1,2)

We may clearly arrange <

c

\/ p*

f

g

69 Note that as a consequence of the proof of 2.3.1 and 2.3.5 16 we obtain f = Tf~g- whenever Q is doubly homogeneous of countable coterminality and A(ft) generated by g

.

g,

f

belongs to the normal subgroup of

where each

g.

is a conjugate of

g

or

Indeed, M. Droste [81] has recently shown that the conclusion

holds even if the hypothesis of countable coterminality is dropped. He has also proved that if ft is doubly homogeneous, then any normal subgroup of of

A(ft) is divisible and any subnormal subgroup

A(ft) is actually a normal subgroup of A(ft). We complete this section with an example of a doubly

homogeneous chain

ft

other than

L(ft) or

B(ft),

such that

A(ft) has a proper normal subgroup

R(ft).

This shows that the "countable

coterminality" hypothesis cannot be removed from Theorem 2.3.2. The example is due to Ball [74]. For the rest of this section only, let interval

[0,1)

(r,y) 6 I 1 Lf

of the real line.

if and only if

has a least element

Any initial segment of 1

r € IQ

(0,0). 1

IQ

be the half-open

Let

I' = I n x w1; U JL

and

y

is a countable ordinal.

l_ = L T \{(0,0)},

Let

i.e.,

is ordermorphic to

1.

a long line.

Consequently,

is Dedekind complete, doubly homogeneous, has coinitiality

and cof inality CQQ.

Let

only if

S-,, and every point of

e < g Q £ A(l) (y > 0

and

L

be such that

0 <_ r <_ | ) .

(r,y)g Q = (r,y)

Note that

will show that the normal subgroup generated by whence it is a proper normal subgroup of L(l)

or

R(L).

g £ A(n),

let

A(L)

if and

g Q £ R(l) U L(l). g~

is not

A(l),

other than B(l),

T O prove this we first need a lemma.

For

Fx(g) = {a £ ft: ag = a } .

LEMMA 2.3.6. g £ A(l)3

Fx(g)

is closed and cofinal in

L.

(1)

If

(2)

The intersection of a countable collection of closed cofinal subsets of

Proof: cof inality

L

is closed and cofinal in

I.

Since any supporting interval is open and has AL,

(l) follows since

L

R~

has countable character

has cof inality

H-, .

We

70 (2)

Let

{A : n € a)} be a collection of closed cofinal

f^\ A^ is closed. Let n~,n ,n 2 ,... be any n e. OJ sequence of natural numbers in which each natural number occurs

subsets of

L.

Then

infinitely often.

Let

of elements of

with

1

3 £ L.

Form a sequence

3 < 6fi < 6

< ...

6. £ A (j € w ) . Since L has n J j cofinality & and each A is cofinal in I, such a sequence exists. Moreover, so does sup{6.: j € ui} = 6 £ 1. Since each m J occurs as infinitely many n., 6 = sup{6.: n. = m } . But J J J {6.: n. = m} Q A and as A is closed, 6 € f| A . Thus A is cofinal in 1. n n e a)

n

We can now prove: THEOREM 2.3.7. other than

B(l)„

Proof: by

A(l)

L(l)

contains a proper normal subgroup

or

R(l).

We show that the normal subgroup of

A(l)

generated

g

is proper. Let f , ...,f € A(l) and consider l _ m g = f~ g Q f . ..f gQ11^ > a typical element of N the normal n subgroup of A(I) generated by g . F x ( g ) o H F x ( g )f.. i=l Consider A - {(0,y): 0 i y € w } £ Fx( g ). Let f = e and n consider Pi Af., the intersection of a finite number of closed i=0 X cofinal subsets of L. By Lemma 2.3.6(2), this intersection is n non-empty. Hence there exists A 0. oo such that (0,A) € (~\ Af. . 1 i=0 X m

Let

(0,X i ) € A

be defined by:

be such that

h^ fi N,

so

and let

(r,A) = min{(^ , X± )f±: 0 <_ i <_ n}.

Fx(g) 3 [(0,A),(r,A)] there exists

(O,X i )f i = (0,A)

e < h

a non-degenerate interval in €. A(l)

N 1 A(L)

such that

as desired.

Fx(h Q ) = A.

r € I

Since I.

r > 0,

But

Hence

71 2.4.

THE AUTOMORPHISMS

For any group

G,

conjugation by an element of g i—>- f

gf

called inner.

of the group

G;

i.e.,, if

is an automorphism of I(G),

G.

G.

f e G,

the map

Such automorphisms are

the set of all inner automorphisms of

forms a normal subgroup of

group

OF_ Afpj .

there are obvious automorphisms obtained by

Aut(G),

the group of all automorphisms

The group of outer automorphisms of

G

is the

Aut(G)/l(G). For any set

ft,

if

|ft| / 6,

every automorphism of the

symmetric group on ft is inner (even if ft is infinite). is a doubly homogeneous chain, then that is not inner; e.g., let (a € I; g € A(I)).

Now

If ft

A(ft) may have an automorphism

cj)_ be defined by:

cf> € Aut(A(tt))

but

(its square is the identity automorphism). I*

G,

$_

a( g<J> ) = -((-a)g) is an involution

The crux here is that

(I with the reverse order) is ordermorphic to ffi. Let ft be a doubly homogeneous chain.

ordermorphic to the group course,

ft*,

say by

A(ft) where:

i.

Then

<j>. is an automorphism of

a(g<j>.) = ((ai)gTi

(J>. does not preserve order

Suppose that ft is

(if

( a £ ft, g £ A( ft)). Of e < g,

e > gc|>.).

We

will prove the following-generalisation of Theorem 21:

THEOREM 2.4.1. I-automorphism of

If ft is doubly homogeneous, then every

AfftJ

is conjugation by an element of

and every automorphism of followed, possibly, by

<j>.

AfftJ

is an t-automorphism of

(if there is

Schreier conjectured that if the outer automorphism group of

G

G

A(Q), AfftJ

i: ft = ft*J.

is a finite simple group,

is soluble.

We will give an

example due to McCleary [78a] to show:

THEOREM 2.4.2.

There is an infinite simple group whose

outer automorphism group is insoluble.

The proofs we give rely heavily on the results of Section 2.1.

For an alternative method, see Chapter 9.

72 We begin by proving Theorem 2.4.1. e < f,g £ B(ft)

By Corollary 2.1.4, if

and ty is an ^-automorphism of

e < ftyygty £ B(ft)

and

f & g

\p induces an ordermorphism By Theorem 2D,

implies

A(ft);

i.e.,

g £ A(ft).

So

then

By Theorem 2.1.8,

ij) of ft onto ft; i.e.,

gip = \p g\j) for all

by an element of

A(ft),

f\\J « g\\>.

ij) € A(ft).

\p is induced

ij; is conjugation by an element of

A(5). If

\p were just an automorphism of

e < g Q £ B(ft).

If

A(ft),

let

e < gQ\\), then by Theorems 2A and 2.1.8, ty is

an t-automorphism;

and if

e > grA),

ib6. is an ^.-automorphism.

U I Theorem 2.4-1 now follows from the previous

Since <j>? = e, paragraph.

We note as a consequence of the proof:

THEOREM 2.4.3. &-automorphism

of

If ft is doubly homogeneous,, then evevy

B(Q)

is conjugation by an element of A(Q.)>

and every automorphism of followed possibly by

Caution:

B(Q)

(f>.

(if

We do not claim in Theorems 2.4-1 and 2.4-3 that

conjugation by an element of A(ft).

is an L-automorphism of B(tt) i: ft = ft >).

Often it is not.

A(ft)

is indeed an automorphism of

For example (Lloyd [64]):

COROLLARY 2.4.4. (cf. Theorems 2C and 2 1 ) . Every t-automorphism Proof: A(Q), Hence

of

and

is inner.

By Theorem 2.4-1, if

^

is an £-automorphism of

it is conjugation by an element Qij) is an orbit of

ordermorphic to Q

A(@)

tt\Q.

is inner.

Q.

Since

A(Q)

in

ij) of A(l). I,

and it clearly must be

But the only two orbits of I\Q f Q,

8$ = Q

so

A(Q)

%\% 6 A ( d ) j

in

1

are

i.e., ty

73 For an example of a doubly homogeneous chain A(ft)

has an outer Z-automorphism,

Q

see Chapter 9.

such that

(The interested

reader can refer to it now since it requires no results from any intermediate chapters.) Finally we prove Theorem 2.4.2. Aut(B(l))~ A(l) xi ^ 2 is

A(ffi)/B(ffi) xi ^ 2 .

subgroups of

By Theorem 2.4-3,

so the group of outer automorphisms of

B(l)

But by Theorem 2H, the only proper normal

A(I)/B(I)

are

L(l)/B(l)

and

R(l)/B(R).

Since

these are simple and non-Abelian, the group of outer automorphisms of

B(ffi) is insoluble.

Since

B(l)

is simple (Theorem 2 G ) ,

Theorem 2.4.2 follows.

For more on automorphisms of

A(fi)—even when

doubly homogeneous—see Chapter 9.

Q

is not

For other examples of infinite

simple groups having insoluble outer automorphism groups, see McCleary [78a].

2.5.

EMBEDDING IN_ DOUBLY TRANSITIVE AfoJ .

This section is devoted to the proof of Theorem 2J. we give here is due to Weinberg [67]. Chapter 10.

The proof

For an alternative proof, see

(The proof given there does not depend on the inter-

mediate chapters and can be read straight after this section. ) In order to prove Theorem 2J, it is enough to prove the following two lemmas:

LEMMA 2.5.1.

Any chain

totally ordered field

ft

LEMMA 2.5.2. ACT)

If

can be order embedded in a

in such a way that each member of

can be extended to a member of

of

T

T

AfoJ

with

is a subchain of

can be extended to an element of

be l-embedded in

AfoJ

A(T)

|fi| = max{N ^ \T\ }.

Q

and every element

AfoJ,,

(the restriction mapping

then B(T)

k(T) into

can B(Q)).

Before proving these lemmas, we need some facts from algebra.

74 Let

G

be a group and

R

be a ring.

The group ring

g

r i R[G] = { I r.x : r. £ R,g. £ G} where T 1 i=l X addition and

+

is the coordinate

• is polynomial multiplication.

ward to show that

It is straightfor-

R[G] is an integral domain if R

is an Abelian o-group.

If

R

is and G

is totally ordered, we get a natural

total order on R[G] in this case—lexicographic from the right. m gi (If a = I r.x with each 0 i r. £ R and g < ... < g , then I l m i=1 i a > 0

if rm > 0. ) We now prove Lemma 2.5.1.

Let T

be a chain and G be the

free Abelian group on { T : x € T}. Order G by: m [ n.i. > 0 if n. > 0 where T . = max{i.: 1 < i < m & n. / 0}. i=! ! ! J J 1 1^ This clearly extends the order on o-group.

Form the group ring

T

Q)[G]

fractions.

We totally order ft by:

a$ > 0

Q1[G].

in

and

G

and let ft be its field of a$

> 0

This extends the order on

a totally ordered field.

Note

is an Abelian

if and only if Q[G]

|ft| = max{|T|,N

and makes ft

}.

Let

f € A(T). Define : A(T) -> A(G) by: m m ( I n.x. )(f<J>) = [ n . d . f ) . Then f<|> € A(G) and extends f. i=l i=l m g. m g.(f(J>; Define <J/: A(T)<(> ^ A(Q[G]) by ( ^ q x J )((f<|>)i|;) = I q.x J

Then

fcf)i|; £ A(Q[G])

extend

and e x t e n d s

f<j>

(and hence

f<\>\\) i n t h e o b v i o u s way t o a member of

1

(a3~ )(fcf)ip) = a(f<|>i|/)(3(f<W))~

( a , 3 ^ Q[G]).

e a s i l y s e e n t o b e w e l l - d e f i n e d and

f). A(ft);

We now viz:

This extension i s

f<j>ij; 6. A(ft)

extends

f

proving

t h e lemma. We now prove Lemma 2.5.2. segments of such that

T

£(T) be the set of all left

in ft; i.e., A € £(T) if there exists

A = {x £ T: x < a}. If

cut in ft determined by For a fixed

a Q £ ft, let

a Q £ C(A Q ).

If

A-,.

Let

A;

A € £(T),

let

a € ft

C( A) be the

i.e., C(A) = {3 £ ft: A < 3

A o = {x £ T: x < a Q } . Thus

A 1 € £(T) and

A± i A Q ,

then

In the first instance, there exists

A_L ^ A Q

x £ A^\A-..

or Hence

75 C(A ) £ x < C(A Q ).

In the second case,

C(A Q ) < C(A-,) similarly.

{C(A): A £ £(T)} is linearly ordered in the natural way.

So

Define an equivalence relation C( A-. ) =* C(Ap)

£(T) by: A., £ A~ if

(as totally ordered sets).

each equivalence class. define

on

5

If

Choose an element from

A were chosen and

TT. = TT. .TT where A 1 ,A 2 A1,A A,A 2

A.£ A (i = 1,2),

TT. . is the identity on C(A), A,A

TT is an ordermorphism of C( A) onto C(A n ) and A, A-, 1 TT. . = TT. . S o TT. . is a n o r d e r m o r p h i s m o f C ( A n ) o n t o A ,A A, A A,A 1 C(An)

and if

2

Let

A

£A,

TT.

O 3

.

= TT.

A X ,A 3

. TT.

A £ £(T) and f £ A(tt) be an extension of

a £ C(A), then for all x

£ A

and

and

T

SO

= T

l ^< a ^ - T 2 ^ = 1

C(Af )f" ^ C(A). A £ Z(T);

T

2f

Thus

i.e., the

1


Hence for all x ^ £ Af

f. If

x o £ T\A, xn < a < x .

1

lf

. .

A X ,A 2 A 2 ,A 3

(C

x 2 f £ T\Af, a

^G

c

( A f )•

Similarly,

C(Af) = C(A)f.

Thus

A Z Af

for all

C ( A ) get mapped onto ordermorphic copies

under any extension of an element of A(T) to A(ft). Next define ty: A( T) -> A(fi) by: a( f i)) = a-n is the unique element of £(T) such that f £ A(T)). then Hence

Now for all a £ ft and

a(fi|0 G Af; f

so

a(f*)(gT|;) =

i s a homomorphism.

such t h a t

i f

< T Q g.

xQf ^ A o g \ A o f /

and t h u s

A^

Af^Af ^ A f g

f < g,

But f o r a l l

a(fif;) = air A ^ A f £ ^ A , A g Then

If

a £ C( A)

f,g £ A(T),

a7r

= a(glfj)>

A

= o

so

AQf % A Qg.

xQ(fi{j) < x Q (g\|;).

Hence

A 6 £(T).

I t follows t h a t

( f V e )ip = fij; v e definition,

and

ip

B(T)i|/ £ B( n ) .

^ ^

a £ C(A), A,Afg

^

fip < gip.

v e )

^*

so

£ T: x < x Q } . C(A Q f) < C(AQg)

Consequently, A( T)

into

A( f v e ) = Af U A for ( f

= a( f g

xQ £ T

Af c. Ag;

Therefore

a o n e - t o - o n e o r d e r - p r e s e r v i n g homomorphism of An easy computation shows t h a t

if = aiT

A

(a £ ft;

there exists

A € £(T), Let

„ where

ip

is

A(fi).

all

= ir^Af v ^ ^ .

i s an ^.-embedding as d e s i r e d .

Thus By Q.E.D.

PART II THE STRUCTURE THEORY

76 CHAPTER 3 CONGRUENCES AND BLOCKS

In Sections 1.4-1.7 we obtained results concerning the set of congruences of an ordered permutation group

(G,fi). We now wish to

make a far more extensive study of this set. We will first deal with the transitive case as it is both easier and more intuitive. Recall that in this case each class of a congruence is a block and each block is the class of a unique congruence (Theorem 1.6.1).

We

will first prove: THEOREM 3A. group and

Let

(G3Q.) be a transitive ordered permutation

a e Q. The blocks of

(G>Q) that contain a

chain (under inclusion). Hence the congruences of

form a

(G3 Q,)

form a

chain under inclusion.

As an immediate consequence we will have: COROLLARY 3B. If

(G3Q) is a transitive ordered permutation

group3 then the set of congruences of

(£_,tt)is closed under

arbitrary unions and intersections. Recall that if group and £ 6

if

and

(G,ft) is a transitive ordered permutation

£

are congruences of

€ §& & and there is no congruence

g Q g c^ fa. Let

Zc(A(n),fi). We write (i.e., ? 9 ^ ' ) . £ „

°f

covers

(G,n) such that

(G,ft). We will simply write (Gv^)

Let

a

k(Q) for

= K € Zc(G,fi) where IJK

By Theorem 3k, fe(G,fi) is a chain where K

$

then P

kfGjtt) denote the set of covering pairs of

convex congruences of

K

(G,ft),

and

1

K < K

if 5

covers t?^. c^,

3 be distinct points of fi. Let

be the union of all congruences

£

of

(G,fi) for which

77 3 £ a £,

and 5

(G,ft) for which

be the intersection of all congruences

Q

of

a £'3.

By Corollary 3B, £T. and £ are K (G,ft) and £ clearly covers £„. Hence

congruences of K K = ( 5 jr; £ ) £fc(G,ft)and is uniquely determined by this case we write

(a,3).

In

K = VaKa^ 3J , the value of (a.3&) . We have

therefore established half of the following theorem:

THEOREM 3C. Let

(G3 ftj be a transitive ordered permutation

group. For each OL3 3 € ft with a ^ 3., VaKa^) over, each K £ k(G3Q)

is

VaKoLjft)

£ k(Gyti). More-

for some a., 3 € ft.

We next show that fc(G,ft) generates the set of congruences of

(G,ft).

Specifically:

THEOREM 3D. Let group and £

be a congruence of "~ K

K°

fe =

(Gy ftj be a transitive ordered permutation

-

-

(G3ftj.Then

i

•

—

JJ

k(G>Q). Let

a £ ft and

(G,ft) be a transitive ordered permutation group, K £ /c(G,ft). Now

G /

convex subgroup of

G

^

= {g G G: ag C^a]

is a

/? iv\

(actually, a convex ^-subgroup if (G,ft)

is a transitive ^-permutation group) as is easily verified. G /

acts on the chain

v

a 6 / 6v

yo IV \

of Gv IV

classes contained in A.

(a ? ) a^

(ordered in the natural way).

act faithfully on

a C /£ ^.

Of course, G

&

may not

However, if we factor out the lazy

subgroup, we obtain a (£-)homomorphic image of denote by

„

as before.

G

„ which we a u> ) (The lazy subgroup is

rr

ft {G/ n ^ v 3 ^ 9 a S '}, a convex normal (£- )subgroup of the (, 3 (e v )

is.

K G ^ .) (G ,a S /^ ) is a transitive ordered (a 5 ) (at ) permutation group that is an ^-permutation group if (G,ft) is. (Z-)group

By Theorem 1.7.2,

K (6 v ,a£ /£v) (a Q )

is primitive.

(Recall that

78 a transitive ordered permutation group is primitive if it has no non-trivial congruences.) Moreover, (£-)isomorphic to

(G

v

,8 5 /CV)

(S£ K )

(6

y

if

,a{ /
3 £ ft by Lemma 1.7.1.

K

Hence: THEOREM 3E. Tf

(G,ft,)is a transitive ordered (l-)permuta-

tion group„ a € ft and

X £ k(G,Q.), then

(G

jaf / 6 J

is

a primitive transitive ordered (L-)permutation group independent of the choice of

Since

a €. ft.

(G

„ ,aC / 6 y ) y» IV \

/

is independent of the choice of

IV

a € ft, we will simply write it as

(G >& )

and get

{(GK,ftK): K €fe(G,ft)},the set of primitive components of In future chapters we will frequently prove results about

(G,ft).

(G,ft)

by examining its individual primitive components. We next wish to consider the intransitive case. Unfortunately, there is a profusion of blocks and congruences some of which are more useful than others.

The idea is to select sufficiently many

that we get enough insight into

(G,ft),

and yet not too many since

we want analogues of Theorems 3A-3E to still hold and yield a nice structure theory.

That we can indeed do this is somewhat technical

and we postpone a close analysis and discussion to Section 3.2. However, the intuition of this paragraph together with the discussion at the beginning of Section 3-2 is adequate for the intransitive applications in later chapters. 3.1. TRANSITIVE ORDERED PERMUTATION GROUPS. In this section we wish to complete the proofs of Theorems 3A-3E.

Specifically, we need to prove Theorems 3A and 3D and half

of Theorem 3C. difficult.

The proofs are quite straightforward and present no

To establish this claim, let us now provide the proofs.

79 Proof of Theorem 3A: Let —

a £ A flA ,

If A

A2

and

a £ ft and

An ,AO

be blocks with

1
we have

A2

inf(A1) < inf(A2) < sup(A1) < sup(A2) without loss of generality. 6 i € A i (i = 1,2) with

Let

the transitivity of

= 6 . So

(G,ft),

g H Ap / 0.

6± < inf(A2) and there exists

inf(A2)g = inf(Ap) and consequently A^g = A., . But

a € A-, and

g £ G

Apg = A2«

Hence

6

A g n A

> sup( A ). By

such that

Thus / ((, Therefore

ag = 5p ft A-., the required contra-

diction.

Proof of Theorem 3C: Let exist

a,3 £ ft with

Val(a,3) < K

K€fc(G,ft).

a 5 3 but not

by Theorem 3A;

(again by Theorem 3A).

Hence

Since C ^ ^ S 1 ^ , there

a 6^-3. Since

and since

3 $? o.Cv,

a £ 3, K < Val(a,3)

K = Val(a,3).

Proof qf_ Theorem 3D_: Clearly G = U { £ K : K 6 k & £K c £} c £. If if

fi/lj, U

let

a, 3 £ ft be such that

= Val(a,3),

C

we have

u

but not S If.

a£>3.

Since

Similarly,

= n { £ K : K £ fe

c ^ S f

Then

C U S. ff,

•v o

S ^ by the definition of fi. So absurd.

a £3

which i s

80 3.2. INTRANSITIVE ORDERED PERMUTATION GROUPS. We give a very full introduction to this section so that a trusting reader will have all the facts needed for any applications (of the intransitive structure theory) in future chapters. Recall that our goal is to select enough "good" blocks so as to be able to recapture

(G,ft) in some sense, and yet not to select too many

blocks as we want analogues of Theorems 3A-3E to hold in the intransitive case (so that we get a nice structure theory). EXAMPLE 3.2.1. replacing each n, < n~ < n~.

Let ft be the result of taking

n £ 22 by a three point set Let

G = A(ft). Then

(G, ft) is intransitive since

A^ = {r € I: n < r < n + 1},

A U {np,n ,(n + l)-,},

and

{n_,np,n~} with

0?G = {n?: n € Z } g ft. The following are blocks of easily checked:

1

{n.^np} (n £ Z ) .

(G,ft) as is

{n-^n^n^},

In some sense, the first

two sets of blocks are more natural than the second two. Note that Theorem 3A fails here since {0 ,02>

and

CL t {O^CL}, {02,CL}

and both

{02,0 } are blocks of (G,ft).

EXAMPLE 3.2.2.

Let ft be the result of taking

replacing each rational by a copy of

1

1

and

and each irrational by a

81

[R

copy of a two point chain

T. So

ft = {(r,q): r £ R,q e Q} u {(x,q): T £ T,q £ffi\Q}ordered by (a,s) < (e,t) if

s < t

s £ Q,

s e R\Q).

in

T

if

(in l), or Let

£ 1} (q £ Q}) and

Aq = {(r,q):

"natural" blocks of

(G,n).

and (a,s),(3,t) £ A with

s = t & a < 3

I if

T. = {(x,q): x £ T} (q £

Note that if

s i t,

(in

G = A(ft). Now

then

A

are

is a block of

A = ft.

We wish to examine the property that blocks enjoy in the transitive case for a possible analogue in the intransitive case. If

A

is a block of a transitive ordered permutation group

and

S,61,S2 £ A,

and

6RO

i.e.,

6G

there exist

such that

<_ 6.,

> <$„;

(A)

is

coterminal in

permutation group and

A

A.

If

is a block of

(G,oJ

is an ordered

(G,n), we say

A

is

6G/AN is coterminal in A. Hence {A; (G,Q) is transitive, every block of (G,fi) is extensive. In

extensive if for some if

€ G £ G, ^ g-,,g^ (A ) 1>g2

(G,ft)

6 £ A,

Example 3.2.1, the non-singleton extensive blocks are ft and A n

82 (ne^);

in Example 3.2.2 they are ft and

obviously blocks we will want to select.

A

(q £ Q). These are

Observe that if

extensive block of an ordered permutation group then

6G/. \ is coterminal in

For example, if

is an 6 £ A,

A.

The extensive blocks are not enough to understand general.

A

(G,ft) and

(G,ft) in

(G,ft) is given by Example 3.2.2 and

G1 = (g e G: (Vq £ Q)(Vq € R\Q)(AQg = A q & T.g = T.)}, then (Gf,ft) is an ^-permutation group whose non-singleton extensive blocks are just

A

(q £ 01), and we have lost any evidence of

{T-: q € fi\Q}. We must therefore find some other natural blocks. The definition of "extensive" is internal to a block. We seek an external counterpart. Let block of

(G,ft) be an ordered permutation group and (G,ft).

Let

A be a

F(A) be the maximal convex subset of ft

which contains

A

Example 3.2.2,

F( A ) = A

but is disjoint from all

Example 3.2.1,

F({n ,n2,iu}) = A n - 1

and

Ag i A.

In

F( T-) = T- (q £ Q, q £ UNI);

in

u

Ag

with

{ n ^ n ^ n } u A R which is

not a block.

LEMMA. 3.2.3. group

Let

A

be a block of an ordered permutation

(G3Q) such that {Ag: Ag > A}

F(A) is a block of that G(h) If

has no least element. Then

(G3Q) and is the largest block A 3 A such

=G(A). A

is a block of an ordered permutation group,

{Ag: Ag > A} has no least element, and is a fat block.

Note that

Q

F(A) = A, we say that A

is always a fat block (of

In Example 3.2.2, the fat blocks are ft, A q £ t\$);

T- (q € Q,

in Example 3.2.1, ft is the only fat block.

in Example 3.2.2, T-

and

each

A

(G,ft)). Note that

is both extensive and fat whereas each

is merely fat. Actually, the conclusion of Lemma 3.2.3 can be strenthened to: LEA/MA 3.2.3'. Assume the hypotheses of Lemma 3.2.3. Then

F(A) is a fat block of

(G,Q).

83 Another way to obtain fat blocks is as follows. then T

T

< X

is said to be dense in_ itself if whenever

If

T 9fi,

T-.,Tp £ T

and

there exists

l 2' T € T such that T 1 < T < T ? . If T = ft, we will conform to our previous notation (p. vii) and write ft is dense. LEMMA 3.2.4. Let a € ft. If

(G3 oj

be an ordered permutation group and

T = aG is dense in itself and A

of ft maximal with T n A = {a},, then A

is a convex subset

is a fat block of

(G, a).

Fortunately, the extensive and fat blocks are sufficient for our purposes. A block

A

of an ordered permutation group

is natural if it is extensive or fat. Hence, if

(G,ft)

(G,ft) is

transitive, all blocks are natural. We will prove the following analogues of Theorem 3A and Corollary 3B: THEOREM 3A*. and

Let

(G3ftjbe an ordered permutation group

a € ft. The natural blocks of

(G^Q) that contain a form a

chain under inclusion.

COROLLARY 3B*_. Let

(Gyftjbe an ordered permutation group

and a £ ft. The set of natural blocks of

(G5 Q) that contain a

is closed under arbitrary union and intersection.

be natural blocks of

(G,ft). We say

Ap

and there does not exist a natural block such that

A £ A JA^

Note that in Example 3.2.1

covers an extensive block (indeed, ft covers each A

A

(n a ) ) j

in Example 3.2.2. the fat block ft covers each

(q € Qj) and the fat block

{(T,q)} ( T e T).

T-

covers the extensive blocks

This phenomenon is quite general (and is used in

the proof of Corollary 3B*):

LEMMA 3.2.5. permutation group

Let

A

be a non-extensive block of an ordered

(G^Q) and 6 £ A.

Then Convf^(^GyA J

is °-

84 natural block maximal ^ A. blocks of

Indeed, if

f£,ftj_, then A~

A- £ A«

are GO final natural

is non-extensive3 A-

is extensive and

A-. We also have analogues of Theorems 3C and 3D: THEOREM 3C*. Let a., 3

(G3ftjbe an ordered permutation group and

be distinct elements of ft. There is a unique covering pair

A. ^ Ar> 0/ natural blocks of 3 € AAA-.

Moreoverj if

A- ^ A~

such that a € A- and

is a covering pair of natural

there exists $ € A«\A-.

blocks containing a3 THEOREM 3D*. Let and

(G,tt)

(£,,ftjZ?e an ordered permutation group

a €.ft.ffac/znatural block containing a

is t/ze intersection

of lower halves of covering pairs of natural blocks that contain it and dually. Unlike the transitive case, there is not a one-to-one correspondence between congruences whose classes are natural blocks and the natural blocks that contain a given

a £ ft. So Theorems

3A*, 3C*, 3D* and Corollary 3B* cannot simply be lifted to "natural" congruences.

(Clearly we will only want to bother with such

congruences. ) More precisely, a congruence permutation group

?

of an ordered

(G,ft) is called natural if all of its classes

are natural blocks. We will show: LEMMA 3.2.6.

Let

permutation group congruence 5

of

A

(G3n).

be a natural block of an ordered Then there exists a largest natural

(G3 Q) for which A

is a C -class.

In order to get an analogue of Theorem 3E with the "primitive" component independent of condition on the £

a € ft, we will need a homogeneity

classes. A natural congruence

is homogeneous if whenever such that

A g = A?.

A,,Ap

£

of (G,ft)

are £-classes, there is

In Example 3.2.2, define

C

and &

by:

g £ G

(a,s)C(3,t) if

s = t.

and

if

(a,s) = (3,t)

Then

E ?s £>.

congruence

£

g

of

not homogeneous g € G

with

and

&

Moreover,

or

(A

covers £ ;

and

and

( ct,s )£> ( $,t)

are natural congruences of

£

(G, ft) such that

A g = T-

s = t £ Q

T-

C

(G,ft)

i.e., there is no natural §. £ ^ /Q.

However, >£) is

are >€)-classes but there is no

( q £ J , q £ I\Q_)).

To overcome this problem,

we define natural partial congruences. We say that

£

is a naturaI partial congruence of

with domain ft' 9 ft if ft'G = ft', a £ $

£

implies

ag C $g (a, 3 £ ft', g £ G ) , and each

class is a natural block of

(G,ft).

In particular, we are

insisting that the C -classes are convex subsets of of 'ft').

Note that if

(G,ft)

is an equivalence relation on ft',

(G,ft)

Q, (not just

is transitive, ft' = ft so each

natural partial congruence is a congruence (in the sense of Section 3.1). r Cs on

if

In Example 3.2.1, the relation

(3n € 2£) r,s € (n,n + l)

\J (n,n + l) = ft' ^ ft. The n£^

(n £ 22).

In Example 3.2.2,

congruences on

U{A

where (a,s)£($,t) s = t € Let

5

: q £ (£} if

£

?

defined by:

is a natural partial congruence classes are just the

A n

and 3P and

s = t € Q

are natural partial

U{T~: q € ffi\Q} respectively, and

(a,s)3f($,t)

(G,ft) be an ordered permutation group.

if

Define

k(G3 ftj

to be the set of pairs of natural partial congruences ( C C ) K K for which (i) 6 covers Cv (i.e., L S 5 and there is no K natural partial congruence 5 with £ v Q 5 S ^ ) and K (ii) £ is homogeneous. The members of fc(G,ft) are called values. We will show:

3.2.7. If (G> ftj is an ordered permutation group and _LEMMA ____ ^ fU , J j £ k(Gjtt)s then ^v and C have the same domain, K K In view of Lemma 3.2.7, we will write

dom(K)

for the

rr

domain of G „ ( = dom( C

)).

We also have the desired analogues of Theorems 3A-3E.

They

will follow immediately from Theorems 3A*-3D*, Lemma 3-2.7 and the

86 transfer principle:

LEMMA and

3-2.8.

Let

(G3 Q) be an ordered permutation group K K ( Zr,, C ) *-+ (a & 7,3a % ) is an order-

a €. Q. The map

morphism of

pairs of natural blocks of THEOREM 3A . Let £

A

is.

{K € k(G3Si): a £ dom(K)} onto the set of covering (G3Q.) that contain a.

(G3tt)be an ordered permutation group and

be a natural partial congruence of

natural partial congruences of

(G3U).

Then the set of

(G3Q) that contain a form a

chain (under inclusion). Hence if ~K3Kr € k(G3Q)3 r

or K

< K

r

or dom(K) n dom(K') = 0, where K < K

COROLLARY 3B . If

then K <_ Kr if £K

<= £„,.

(G3 ti) is an ordered permutation group>

then the set of natural partial congruences of

(G, Q) is closed

under arbitrary unions and non-empty intersections. THEOREM 3C . Let For each a., 3 € Q,

(G3Q.) be an ordered permutation group.

with a ^ 3j

there is a unique K £ k(G3Q)

K

such that aC &

but 3 £ a {„. Moreover> for each K € K there are a., 3 € Q such that a C 3 but 3 £ a £„. K

If we write

VaKajft) for the unique

K £ k{d, Q)

guaranteed

by Theorem 3C , we obtain Theorem 3C with the word "transitive" deleted (but now

k(G,Q) has a more general interpretation in the

intransitive case).

THEOREM 3D » Let %

(G3tt) be an ordered permutation group and

be a natural partial congruence of

r\{C {C K € v:K
k = k(G,n).

K

(G3Q.). Then

= %= U { G : K e k & CK Q Z}>

£££„}

where

A

An ordered permutation group

(G,ft) is said to be primitive

if it has no non-trivial natural congruences. This coincides with the definition given in Section 1.6 if

(G,fi) is transitive.

87 THEOREM 3E . If

K £ k(G^)

(G3 Q) is an ordered permutation group,

a £ dom(C K),

and

then

(G

v K

,aSK/Q

J

K

is a

primitive ordered permutation group independent of the choice of (G3tt) is an l-permutation group> so is

a € dom($ ). If (G (aGK)

K

aC /CJ. K

As before, we write call

(GvJttJ A

for

A

(6

,aCK/C (aG ) /

v

to K \

)

iv

and

{(G^ft^): K € fc(G,ft)} the set of primitive components of

(G,ft). We will often say that a value

K

enjoys a property if

(GK,ftK) does. We have two kinds of ordered permutation groups that are obviously primitive (we will find others in the next chapter). Recall that

(G,ft) is static if

G = {e}

(see Section 1.6).

These are clearly primitive under our extended definition. (G,ft) is said to be integral if acts like ^

Q = T x%

on ft (i.e., if

(n £ 71y x £ T)).

for some chain

T

and

G

g € G, 0 m € Z)[(x,n)g = (x,n + m)]

These too are clearly primitive.

Let us return to Examples 3.2.1 and 3.2.2.

In Example 3.2.1,

K

0 Kn,K_ . The £ classes are A (n £ Z ) ; u l vn 1 classes are {r} (r € \J A ); the C class is The Z v £„ i s defined by a S y 3 if a = 3 or a, 3 €. A for someft; n 1 K K 1 n £ ^ . Then C ° ^ \ (G^ , a. ) = (A(tt),I) and (G ,ft^) is /c(G,ft) has two elements

integral where

T

is a four point set, with

T x {n} = {n ,n2,n^,(n + p") ^K }•

So

we have recaptured

(G,ft)

from fc(G,ft) and the primitive components.

In Example 3.2.2, KQ The 5 classes are A

fc(G,ft) has three elements K ,K ,K?. U q K -1 ( q £ U ) , the 5 classes are T^ (q £ I\Qj) and the Z q K. i classes are singletons with dom( C v ) - dom(^? ) (i = 0,1). The K K -i KQ Z class is ft and the C ,r classes are either C or C K 2 classes. So KQ,K < K 2 but K Q and K are incomparable. (G ,ft ) = (A(l),l), KQ KQ

(G

K]_

,ft ) = ({e},T) and K1

(G^ ,ft ) = (A(Qi),l), where T *2 K2 Again we have recaptured (G,ft)

is the original two point set. from

/c(G,ft)

and the primitive

components. Finally, in this section, we prove an analogue of Lemma 3.2.5.

LEMVIA 3.2.9. K^

Let

(G3Q.) be an ordered permutation group and

be a static or integral value.

values in

Then the set of non-static

k^ = {K €. k(G3Sl): K < KA

is cofinal in

ky

Hence no

static or integral value covers a static value.

This entire section is due to McCleary [76]. The proofs of the results follow from the definitions and are easy—the hard part was selecting which blocks and congruences would "be useful.

For the

sake of completeness, we provide the omitted proofs for the dedicated reader.

Proof of Lemmas 3. 2. 3 and 3. 2. 3 T: (G,ft)

such that

{Ag: Ag > A}

Let

A

has no least element and let

be the maximal convex subset of ft such that F(A) D Ag = 0 F(A)

if

there exist

3 >_ 6f.

Let

\,^2

Ag / A. 6 £ A G

G

Af > F(A)h 1 > Ah 2 > F(A) {F(A)h: F(A)h > F ( A ) } sup(F(A)). So

If further

Let and

be

3 > F( A ) . f € G

such

SO

"be a block of

that

A G F( A)

By the maximality of

such that Af

> Ah

3 > F(A)h 1 .

F( A)

and

i

> Ah

Af > F( A) 2

> A

*

and

Then

Therefore

has no least element and has infimum eg" 1 € F ( A ) ,

F(A)g > Ah 2 > F(A). Hence

then

Af <_ Ag

F( A )g O F( A) = 0

since

and

g > 5f.

F( A)

is a

fat block. If

A

is a block with

A £ A J?F(A), G

then NG

Ag H A J 0 ThuS

some g e G with Ag i A. Hence g £ ( A ) ( A ) * G/ . \ = G/ A N and A G A, then A C F( A ). But since (A; {A;

F( A)

block and A S F ( A ) , G ( A ) ~ G( Ff A )) * B y t h e d e f i n i t i o n G / W NN c G/ A N. So F(A) is the largest block An 2 A \r{A) ) { A; 1

for

lf

of

is a

F (A)^ such that

Proof of Lemma 3.2.4: itself and

A

T € T

Ag n A = 0. A < 6.

g € G,

with

Thus

A

a < T

A

be dense in

T n Ag = {ag}. Since

a < T < ag.

Now

is a block of

By the maximality of

such that Hence

Then for all

without loss of generality.

there is

Now

a € ft, T = aG

be a convex subset of ft maximal with respect to

T n A = {a}. a < ag

Let

so there is

A,

T' G T

with

(G,ft).

Proof q£_ Theorem 3A*:

Assume

A, <£ Ap

Let

?

a < T

inf(A,) < inf(Ap) < sup(A^) < sup( Ap)

so

3 & ft with with

< T.

so

a €. A and

is dense in itself,

T £ T

T ' £ Ag

is a fat block of

Ag j A,

A < T < Ag

(G,ft).

there is

ag = T ' (T = a G ) . But

natural blocks such that

T

If

A < T <_ 3.

Let

g € G

n &

where

A ,Ap

If

Ap

without loss of generality.

is not extensive,

largest element and supremum Apg 9 A .

But then

ag € A ;

Proof of Lemma 3.2.5: non-extensive block and a block and

so

By Lemma 3.2.3,

A

A g = A .

Let

then

G/^N

=

T

g £ G,

sup(An ) < sup(A o ).

where

A

is extensive. s

is

Hence

T = Conv^n (6G/ ^ Af±\) j Then

Ap

has no

Thus for some

contradicting

6 €. A.

T £ A 5 A,

{Apg: Apg < Ap}

inf(Ap).

sup(A n g) = sup(A n ) > sup(A o g),

are

Ap 52 A, . We have

The proof of Theorem 3A gives the desired contradiction if extensive.

be

A < Ag < x <_ 3.

&( ft) i

nce

G

(T)

If

=

is a A

is

^f A ) '

is not fat and it is clearly not extensive

( 6 G / ^ N = 6G/. \ which is coterminal in

T ^ A).

Hence

A

is

unnatural. Assume

A

§ Ap

inf(A 1 )g = inf(A 1 ) Thus

Ap

for all

g e G

is not extensive and as

(by Lemma 3.2.3). A

are cofinal natural blocks of

= Conv(6 G,

x)

Therefore if

6

A

£ A .

and hence all G,

s = G/^ y

g

(G,ft). €

G

A

( A )• is not fat

is extensive; indeed, Hence

A

covers

Then

A1>

90 Proof of Corollary 3B*: Let of natural blocks such that X = {A.: i £ 1} ordered and U3£

A. p A.

if

is clearly natural. But

containing

UU a

a €. A. (i £ I).

i < j. If X

I

is totally

has a greatest element,

If it doesn't, there is a chain 11 of

(G,ft) such that

U%

- ^y

("by Lemma

is extensive so the set of natural blocks

is closed under arbitrary unions. The same

argument shows that we may assume that

and each

By Theorem 3A*,

is a chain so we may assume that

extensive blocks of 3.2.5).

{A.: i € 1} be a collection

I has no least element

A. is extensive. W e will show I

A = ^3£

is fat.

Clearly A is a block. If sup(A) < 3, then for some j € I, sup(A.) < 3. Let i < j . By Lemma 3.2.5 applied to (G, N A . ) , J \ A . )f J J sup( A. ) < sup( A . ). Thus sup( A ) < sup( A . ). Let 6 . €. A . with - " - « . ' J J J sup(A) < 6.. Now

A. is extensive so 6 . £ ag

A is a block so A < Ag < 3. Hence

for some

g £ G / . \.

A is fat and Corollary

3B* is proved. Theorem 3C* now follows immediately as does Theorem 3D*> and this completes the block part. Proof of Lemma 3.2.6: Clearly we will want each Ag (g €. G ) to be a C class. If A is a convex subset of Q, maximal with respect to A n \J Ag = 0 and A J 0, then A is a block of g€LG (G,U). If A is a natural block, we will make it a Z class. If it isn't, let the 5 classes in A be {Conv ( A G / . \ ) : A € A } . These are clearly natural and, by Lemma 3.2.5, C is maximal. We now turn to the partial congruence lemmas.

If with if

Proof 3.2.7: Since S^ 9.)^, dom( 5 ^„) C dom( SK ) . — of Lemma _ K dom(£ K) J dom( 5 ), let £ be the partial convex congruence dom( £ ) = dom(C ) such that a, 3 €. dom(C R ) .

contradiction.

Then 5

a^3

if

or a Gv$ K £ I £ 5 , a

a = B,

is natural and C K

91 then

Proof of Lemma 3.2.8: We f i r s t show that i f a € dom(K), aC covers OLCV. Clearly a£Tyr 9 a ij , if a £ = a £ then XV

A.

J\

K K K since £ is homogeneous and dom(5T_) = dom( C ), $CTr = 3t for all 3 6 dom(K). Hence Cv = CK , a contradiction. Thus K K afiL p aC . If A is a natural block and a? K ^ A ^ a£ , let 5 be the natural partial congruence whose classes are Ag (g £ G) and the £ K classes disjoint from U{Ag: g € G}. Then £„ § C £ £ , a contradiction. Thus {QLVV,QLZ ) is a covering -1^

v-

T^-

pair of natural blocks. Therefore the map (£„,£ ) |-~v (aff ,a5 ) maps k = {K € /c( G,U): a € dom(K)} into the set of covering pairs of natural blocks of (G,Q) that contain a. If K K a G dom( K ) H dom(Kp), then a 5l ^ a52 without loss of K^ Kp K^ Kp generality (by Theorem 3A*). If a? = aC , then C = G K K by the homogeneity of C l and ^ 2 . Hence the map is an

ordermorphism (into). Finally, if (A-^Ap) is a covering pair of natural blocks containing a, there is a largest natural congruence C of (6/ )>^2^ n a v i n S A., as a C class by Lemma 3-2.6. Since Ap covers

A,, C is the largest proper natural congruence of

(G/. )>&2^*

^e^ ^2

have

^ A p^ : S ^ G} as its classes and C

have

{Ag: g £ G & A is a 5 class} as its classes. Then to (A-.,Ap). Thus the map is onto and (5-ijCp) €. fe and is mapped m the proof is complete. Proof of Lemma 3.2.9: Let K , ^ €fc(G,fi)with K < K 1 (i.e., 5 K c K ). Let a € dom(K). If K. is integral, a.K K X K l is not a fat block. Hence it is extensive. If K is static, K a5 l

is not extensive. By Lemmas 3-2.5 and 3-2.8, a&v is 1 i static, extensive. Thus if K^ is integral or' static and K is K a£ 5a^T/ and there exists g e G/ * \ with a£ ag. i K

92 Then

K < Val(a,ag) < K1.

Since

Val(a,ag) is not static, the

lemma is proved.

For further results on intransitive ordered permutation groups, see McCleary [76].

93 CHAPTER 4 PRIMITIVE ORDERED PERMUTATION GROUPS

Our goal in this chapter is to give a complete classification of primitive ^-permutation groups.

The classification will prove

extremely useful in later chapters and is one of the cornerstones in the study of ordered permutation groups. Recall that an ordered permutation group is primitive if it has no non-trivial natural congruences.

In the transitive case

this means no non-trivial congruences (since all are extensive and hence natural).

We will first deal with the transitive case and

then show how to derive the intransitive case from it. Let a £. Q

and

period

z)

(G,fi)

be a transitive ordered permutation group,

e < z £ A( ft). (G,ft) if

is said to be periodic (with

{az : n £ 2Z} is coterminal in ft and

C W - N ( G ) = .

A canonical example of a periodic ^-permutation

group is Example 1.9.2:

(G,l)

where

G = {g €. A(l):

(a + l)g = ag + 1 for all a £ I } . .In this example is translation by primitive (If

+1;

i.e.,

a < 3,

a < ag < 3 < 3g;

so

z: a »—> a + 1.

there exists (a,3)

g £ G

z,

(G,l)

the period, is

such that

is never a block of

(G,l).).

We will prove:

THEOREM 4A. tion group.

Let

Then either:

(i)

(GjQ)

is doubly transitive,

(ii)

(G,Q)

is the right regular representation of a sub-

group of or

(G,Q,) be a transitive primitive l-permuta-

(Hi)

Moreover,

(GjQ)

S;

is periodic.

(i) and (ii) are always primitive.

94 As an immediate consequence, we will have: THEOREM 4B. If ft is homogeneous and primitive3 then AfftJ is either doubly transitive or the right regular representation of a subgroup of M.

We will then show that the intransitive primitive ordered permutation groups are derived in a natural way from transitive ones.

Recall that static and integral ordered permutation'groups

are primitive ( see p. 87). Let (G,T) be a transitive primitive ordered permutation group with is transitively derived from ing to T example,

T not ordermorphic to %.

(G,ft)

(G,T) if ft is obtained by adjoin-

a subset of the set of orbits of G in T. For (A(Q),I) is primitive, intransitive and transitively

derived from

(A( ($),(£). This is, in some sense, canonical:

THEOREM AC_. Transitively derived3 integral and static ordered permutation groups are all primitive, and every primitive ordered permutation group falls into precisely one of these classes. Finally, as a hint of the power of Theorem 4B, we deduce: THEOREM 4JD. Let ft be homogeneous. Then the following are equivalent: (a)

A(Q)

is

(b)

AfftJ

is totally

(c)

AfftJ

is the right

of

Abelian. ordered.

is a sentence

such that

ft

(for

language with if

of a subgroup

<> j of the language of group

homogeneous)

doubly homogeneous. lattice

representation

E.

Moreover, there

A(Q) ^ \

regular

Similarly e

and only if

AfftJ V § there

such that ft

if

and only if

is a sentence (for

ft

x

of the

homogeneous)

is doubly homogeneous.

theory ft

is

95 Theorem 4D shows that it is possible, using group-theoretic sentences about A(ft), to distinguish between the classes of imprimitive, doubly transitive and "regular" homogeneous chains. Similarly for the lattice language. This was the strengthening promised on page 51. 4.1. TRANSITIVE PRIMITIVE ORDERED PERMUTATION GROUPS. In this section we obtain conditions equivalent to primitivity. They will be used in the proof of Theorem 4A. They extend Theorem 1.6.5 and are of interest in their own right. A subset A of ft is dense in ft if whenever a, 3 £ ft and a < 3, there is 6 £ A with a < 6 < 3 (cf., dense in itself, page 83)—see page vi. The following theorem is due to Holland [65a]. THEOREM 4.1.1. Let (G,Q.) be a transitive l-permutation group. Then the following are equivalent: (1) (G,Q) is primitive. (2) (G,tt) satisfies the separation property. (3) For each a £ ft., G is a maximal convex subgroup of G. (4) For each a £ ft., G- is a maximal convex subgroup of G. (5) If a € IV, aG is dense in ft or ft = ft = %. Before proving the theorem we first note (cf. Theorem 2.3.1): LEMMA 4.1.2. Let (G3 ftj be an l-permutation group, a e ft and C be a subgroup of G with C 2 G-. Then C is an I-subgroup of G. Proof: Let f €. C. If af <_ a, then Similarly, if af > a, f"1 v e € C. Hence So f e e implies f v e € C. If f,g € C, so f v g = (fg" v e)g e C. Since f A g = f A g e C. Thus C is an ^-subgroup of G. We now prove Theorem 4.1.1:

f v e £ G- £ C. f v e = f( f"1 v e) £ C. then fg" 1 v e £ C; ( f" v g~ )" ,

96 (1) *-* (2).

This is Theorem 1.6.5.

(l) *-> (3). U ) -> (3).

By Lemma 4.1.2 and Theorems 1.6.1 and 1.6.2. A fortiori.

( 3) -> (4).

For reductio ad absurdum, suppose

for some convex subgroup

C

of generality,

Let

e < f € Go. — P e < f < f v g — GQ ^. C £ G. p

^

a < ag^.

of

G.

Let

g^ € OG-.

8 £ ft with

G- ^ C § G Without loss

a £ $ £ ag^. Let

af < 3f = 3 < agn; so fg"1 v e £ G- 9. C. Now .,— — U U a = (fgn v e ) g n € C. Therefore f £ C. Hence u u u But g n € C\GC; so Go g C 9 G. This contradicts (3). Then

U

p

p ^

(l) -> ( 5 ) . If an element

^

3 € ft has an immediate successor,

then so does every element of ft (by transitivity). Moreover, some element of ft (and hence all) would have an immediate predecessor.

Define

C

on ft by: a C 3

finitely many points of ft between congruence of Hence

a£3

a

if there are only

and 3. Then

£

is a

(G,ft) whose classes have more than one element.

for all a, 3 e ft if

(G,ft) is primitive.

Thus

ft =" 22. If no element of ft has an immediate predecessor, it follows (by the above) that ft is dense. a £ ft\ft, define

£

by: a £ 3

ag

a

and 3.

lies between

clearly there exist primitive, the C

Hence (5) holds if a £ ft. If

if there is no g £ G

such that

£ is a congruence of (G,ft) and

a, 3 £ ft such that

3 $- a G . Since

(G,ft) is

classes are singletons; i.e., aG is dense

in ft. ( 5 ) -* ( 2 ).

If ft =*2£,

is finite and G = Z. If ft ? %,

each bounded non-empty interval of ft

(2) follows immediately.

then it is dense (take

a £ ft). If A

is any

bounded non-empty interval of ft, let a - inf(A) £ ft. Let a,3 £ ft with

a < 3. By ( 5 ) , there is g £ G

a < ag < 3. Hence

ag

3g~ f" £ A. Then

ag~ f~

(G,ft)

< a < 8g

• For some

<_ ag"

< a

satisfies the separation property. As a consequence of our proof:

such that e <_ f p- G,

so ag"" f~

£ A.

Hence Q.E.D.

97 THEOREM 4.1.3. tion group.

Then

Let

(G3 Q)

be a transitive ordered permuta-

(4) + (3) + (1) +-+ (2) + (5)3

where (l)-(S)

are given in the statement of Theorem 4.1.1.

In the proof of ( 5 ) -> ( 2 ) of Theorem 4.1.1, all we needed was: G

If

a < 3,

there is

= {g £ G: g >_ e}.

g G G

such that

permutation group3

where

Any transitive ordered permutation group

satisfies this condition is called coherent.

THEOREM 4.1.4.

ag = 3,

Thus we have:

(G3tt) is a transitive coherent ordered

If

then (1)3

(2) and (5) of Theorem 4.1.1 are

equivalent.

We next strengthen Theorem 4.1.1 in the special case that (G,ft)

is a doubly transitive ordered permutation group (and so is

primitive—Corollary 1.6.6).

THEOREM 4.1.5. permutation group3

If

then

The result is due to Lloyd [64].

(G,Q.) is a doubly transitive ordered G

is a maximal subgroup of

G

for each

a € ft. Proof: C pG

.

f £ G.

Let

Let If

af = a,

h € G

such that

Hence

fhg" 1 £ G

af < a, G

a € tt and

C

be a subgroup of

G

g € C\G . Without loss of generality,

then

then

f £ G

£ C.

and

ah = a

afh = ag

If

af > a,

with ag > a.

Let

then there is

(by double transitivity).

£ C. Since h € G E C and g G C, f € C. If a a _1 _i af > a; so f € C. Thus G = C. Consequently,

is a maximal subgroup of

G.

Finally we note: LEMMA 4.1.6.

If

permutation group and Proof:

By Theorem 4.1.3,

is a block, then so is for

(G,ft).

(G3 oj

Since

is a transitive primitive ordered

a € ft^ then so is

A

(for

aG

(G^aG).

is dense in ft. If

(G,ft)).

(G,ft) is primitive,

Hence

A f) Q

AOft=ft

or

A £. aG is a block |A H ft| <_ 1.

98 Thus

A = aG

or

|A| = 1

(by the density of

fore

(G,aG) is primitive.

aG

in

Tl). There-

4.2. TOTALLY ORDERED TRANSITIVE ORDERED PERMUTATION GROUPS. We now wish to consider those primitive ordered permutation groups

(G,ft) for which

G

is totally ordered (i.e., an o-group)

under the pointwise ordering. (G, ft) is said to be uniquely transitive if given there is a unique

g € G

such that

ag = $;

regular representation of an o-group. transitive, then we will say that

If

Q

a,$ €. Q,

e.g., the right

(A(ft),ft) is uniquely

is rigidly homogeneous (cf.

Example 1.1.1). We wish to prove the following theorems:

THEOREM 4.2.1.

If

(G,ft,)is a transitive primitive ordered

permutation group such that G

is an o-group3 then

(G3ti) is the

right regular representation of a subgroup of R.

Conversely > the

right regular representation of any subgroup of R

is primitive.

THEOREM 4.2.2.

If

Q

is rigidly homogeneous, then it is

ordermorphic to a subgroup of R

and

(A(oJ_,oJ

is just the right * 02

regular representation. Moreover_, there exist 2

pairwise

non-ordermorphic rigidly homogeneous chains. Theorem 4.2.2 shows that, in Theorem 4B, there do_ exist homogeneous primitive chains ft such that

A(ft) is the right

regular representation of a subgroup of ffi—in fact, they exist in abundance. An o-group n e ^

+

implies

G

is said to be Archimedean if

g <_ e (g,h £ G).

G

is o-isomorphic to a subgroup of


for all

is Archimedean if and only

if it has no proper convex subgroups. G

g

1

This is also equivalent to: (Fuchs [63, page 45]).

99 LEMMA 4*2.3. with G

Let

(G3ftj be an ordered permutation group

an o-group3 and let a £ ft. If

af < ag (f3g £ G) > then

f < g. Proof: If 3f >_ $g

G

for all

is an o-group and

$ €. ft. So if

LEMMA 4.2.4.

af < ag,

then

f >_ g.

Hence

f < g.

-^/ (G^Q) is a uniquely transitive £-permuta-

tion group> then G (G3G),

f / g,

is an o-group, and

(G3ftjis o-isomorphio to

the right regular representation. Proof: If

af < ag.

Now

f ,g € G

and

fg~" v e € G , and a _

transitivity.

Therefore

o-group.

a Q € ft and let

Fix

such that

aQg = a.

ordermorphism. (f,g e G).

f j_ g,

fg~

The map

Moreover,

Thus

then for some

a e ft,

G = {e} by the unique a

<_ e;

i.e.,

f <_g.

Hence

a £ ft. There is a unique <|>: G ^ ft given by:

G

is an

g € G

g<j> = aQg

is an

(g<J>)f = (aQg)f = aQ(gf) = (gf)

<|>: (G,G) -> (G,ft) is an o-isomorphism.

We now prove Theorem 4.2.1. In any o-group, if then If

C1 9 C2

e < c 2 €. C 2 ,

Let

C 2 c C-L then

a € ft. So if

Theorem 4.1.1, of

or

G.

If

G

G

and (if

let

and

C ^ C2, so

c

let

< c 2

G Q Go a

a € ft with

= {e}; so

1

are convex subgroups of

e < c1 € C ^ ^ .

i* Hence

or

G,

C ^ Cj.

GQ c G . By

p

p a

and is the unique maximal convex subgroup

p

af j a.

Thus

f $ Gg = G .

(G,ft) is uniquely transitive and

has no proper convex subgroups. a subgroup of

Cp

c., {_ c 2

3 ^ ft, then = Go

a

f / e,

Consequently,

C,

Therefore,

G

G

is o-isomorphic to

(G,ft) is the right regular representation

(by Lemma 4.2.4). Conversely, if then

G/ x = G

or

G/ s = {e}, then

G

is a subgroup of

{e}. If

G/ x = G,

|A| = 1. Hence

1

then

and

A

is a block,

A = ft; and if

(G,G) is primitive and

Theorem 4.2.1 is proved. We now give examples to show that Theorem 4.2.1 is sharp.

100 EXAMPLE 4.2.5.

Let

G

(see Theorem 6F) and let (G,R(H))

H

be a simple non-Abelian o-group be a proper subgroup of

G.

Then

is an ^-permutation group (Example 1.4.4) that is not

uniquely transitive (and hence not primitive).

EXAMPLE 4.2.6.

Let ft = Z £ z

regular representation.

(G,ft)

and

(G,ft) be the right

is uniquely transitive but

Z x Z

is not Archimedean and so not o-isomorphic to a subgroup of Indeed

{(m,0): m €. 2£}

A(Z x z )

is a block so

is not an o-group since if

f € A(Z "x Z )

(G,ft)

(m,n)f = (m - n , n ) ,

(0,l)f = (-1,1) < (0,1)

and

1.

is not primitive.

but

(0,-l)f = (1,-1) > (0,-1).

In contrast to Example 4.2.6 we prove the first part of Theorem 4.2.2. So assume

(A(ft),ft)

A(ft)

is an o-group.

then

a~ < ctQfn < a^g

Let

Let

is uniquely transitive. a Q £ ft. If

for all

a = sup{a f n : n € X+}

n € Z

e < f

n

By Lemma 4.2.4,

< g

n £ ^+,

for all

by the unique transitivity.

and define

h € A(ft)

by:

pxf if a < a ah =

s

.

[jx

Then

h

fixes

a^g

but not

aQ,

a

otherwise

contradiction

(G

= {e}). Therefore A(ft) is Archimedean and 0g the first part of the proof of Theorem 4.2.2 follows. a

For the rest of this section we will assume that rigidly homogeneous subgroup of

1.

Z

ft

wish to construct dense rigidly homogeneous subgroups of If

p

is a prime number and

group, let over

Z ,

Y (G)

group of Y (G) >_ 1 G fixing

be the dimension of

the Galois field of

P

G

p

is a

is an example but we now I.

is a torsion-free Abelian G/pG

elements.

as a vector space If

G

is a sub-

(U

I,

then

y (G) <_ 2 °

(Otherwise the map 0).

and if

g *—> pg

G

is rigidly homogeneous

would be an element of

101 Let

{y : p prime}

be a set of cardinal numbers with

N 1 <_ y

< 2 ° and

sets with

{I : p prime} be a set of pairwise disjoint

|l | = y . Let

I = U{I : p prime}

be a linearly independent set (over each prime

p,

let

Q

Q)

and

{£.: i G 1}

of real numbers. For

be the set of rational numbers which, in

simplest form, have denominators relatively prime to if that

q 1 p

YqUp) = {

• Let if

G = Z{G : p prime}.

p. Note

G

= Z{qQ : i € I } and

q = p

Then

Y (G) = y

for each prime

p

and

G

is a dense subgroup of ffi. THEOREM 4.2.7. 1 <_ y

numbers with

Let

{y : p prime} be a set of cardinal

<_ 2 ° for all primes p.

Then there exists

a dense rigidly homogeneous subgroup ft of 1 y (ft) = Y

such that

for all primes p.

Proof: First assume

y

< 2 ° for all primes

above, there exists ftQ, a dense subgroup of for a

Y (^Q) = Y

H primes

p

with

1,

p.

By the

such that

| o l < 2 °. We wish to

enlarge fiQ, preserving the invariants

(Y : P prime},

so that

the only order-preserving permutations of the enlarged group will be translations.

To do this we must kill all elements of A(tt)

which are not translations.

f €. A(l) is

Note that since each

completely determined by its action on Qi,

1 AC a) | ^ Im^l = (2*° )K° = 2 N °. Let

if

• 1 < X < 2

0

}

be an enumeration of t h e elements of

A

A(R) which are not translations. divisible closure (in

1)

Let

E Q = {0}

of fi . We define

and

«x,

DA

DQ

be the

and

EA

inductively so that: (l) DA

is a divisible subgroup of

A

1

containing ft . A

( 2 ) | D^ | <_ max{ | X | , |ftQ| }. (3) ft is a direct sum of ftn and the divisible subgroup A

h

U

of ffi.

102 4)

If

X

1

< X,

then

Ex, £ EA

and

Then ft = U{ft : X < 2 °} will be the required dense rigidly A homogeneous subgroup of R. (By ( 3 ) , yift.) = Y (^n) = Y^ for all P X < 2 °,

so

Y (ft) = Y

a translation,

for all primes

f = f

for some

X.

A p.

p If

U

p

f € A(l)

is not

By (5) and ( 4 ) , there exists

-1

a € ft 9. ft such t h a t af o r af € DA ft. £ D \ft for a l l X X X y y y > X. Now af ( o r af" ) £ U { f t : y > A } = ft, so ft i s r i g i d l y — y — homogeneous.) Suppose

D., E. and ft have been defined for all X < y A A A so as to satisfy (l)-(5). Let X* = U { X , : X < y} where X = D, y A E or ft. Then D*, E* and y' y and ft* y are subgroups t> r of I with D* y E* divisible, and ft* = E* © ftA. Let f = f . We wish to find y y y 0 y £ £ D*

such that

the induction. Let

cf (or £f~ ) £ ft* 0 %x>, in order to continue

This is rather tricky.

A = {a € I: af = qa + a 1

(i) If I i A u ft^f u D*, E^ Q ft^ 9 D*, E = E* © aQ

Since

]i

ft = E

\i

© ft .

\i

If

a € ft*. Hence let

D

\i

q G Q

and

a t ft*}. 1

let a € R\(A U ft*f" u D * ) . is indeed a direct sum as is

\i

af € ft ,

a € A,

for some

then

af = qa + a

a contradiction.

be the divisible subgroup of

1

for some

q 6 Q,

Thus (l)-(5) hold if we generated by

af

and

D* © a(J. (ii) such that

I = A U ft*/"1 U D*, we wish to find 6 € A\D* y 6f = q6 + a some a £ ft* and q £ Q with q or — If

To do this, we partition A

A = A

= {a £ R: af = a + a

A-.={a£R:af -1

for some

a £ ft*},

some

a £ ft*} and y

= -a + a

B = {a £ R: af = qa + a We now show

B\D* i 0.

u A _ u ft*f" u B

for some Let

a £ ft* and

k: a i-* af + a.

q £ Q, Then

continuous strictly increasing real valued function. is at most one

a £ R

such that

ak = a.

where

k

q J 0,±l}. is a

Hence there

Therefore

IA J < Ift^k"1! < ID*1I < 2 °. Let h: a H > a f - a. Then h is 1 -I1 - ' y '- ' y continuous and since f is not a translation, h is not constant.

103

Hence

Rh contains an i n t e r v a l .

since

Anh £ft*.

Since

But

(R\A )h 2 RhY/Lh 2 Rh\ft*

llhl = 2 °,

|R\Aj = 2 °.

But

i « « M |ft*f" | < |D*| < 2 ° and |A .I < 2 °, so |B| = 2 °. Thus B\D* i 0. If there exists q &%,,

a € ft*, let

6 € B\D* with a = 6.

6f = q6 + a

Otherwise let

for some

a = 6f where

an arbitrary element of B\D*. In the latter case, for some

a t ft* and m Q. Z,

m / 0,il.

af" 1 = 6 = (a - a)/m. But a € D*

6 is

a = 6f = m6 + o

Therefore

and 6 £ D*,

so

a £ D*.

In

either case we have a $. D* with af (or af ~ ) having the form — a + a' with a' £ D* and k,n relatively prime integers with n i 1. If

a' £ ft*, let E = E* © aQL ft =ft^© E , and D y y y y 0 y y the divisible subgroup of I generated by D* © aQ and 3 (= af or af" )—the indicated sums being indeed direct. straightforward to show that (l)-(5) now hold for all (If

3 € ft , then

Since

— a + a' = r a + T

If

such a

a' € ft*, let p

X <_ y

r £ffland T £ ft*

be a prime such that

— € D \ftQ (y

£ must exist).

Let

E

= E* © cQ, ft = ft0 © E

I

generated by

D*,

It is

and a 1 = T € ft*, a contradiction).

a $2 D*, — = r

£ € ft be such that

for some

5

> 0

and D

and

p|n. Let

for all primes

£ = a - £. Then

p, so

c £ D*. Let

be the divisible subgroup of

3 (= af or af" )—the indicated sums

being indeed direct as is easily shown.

Again a routine

verification shows that (l)-(5) now hold for all X <_ y. that

a € ft and y

some

r € (t and a £ ft*. Since

3£ 3€ft, M ft 1 —if ^ y M y '

y But

— £ £ DAft n Q. D*\ft*,

=Y

if

dense subgroup of

P 4 J I

a £ D*, -- = r.

a contradiction.)

In the general case, l e t Y'

- a + a'=3 n

=

and

with

(Note

a + rc for

So - £ 6 ft*.

^ y n

proof of the theorem if all y

Let

be

n ^ y This completes the

< 2 °.

J = {p: p prime and y p yf = 1 if

y (ftQ) = y1

P € J. and

=

ft 2 °}.

Let ftQ be a

|ftJ < 2 °.

Let DQ

104 be the divisible closure of v

a limit ordinal and

E* = E v

x

p € J,

and let

E n = {0}. T £ I\D

© T Q , ft* = E* © ftQ and

of ft* V D .. y v+p-1 E*

ft

For all

py

D*

.

+

y = v + p

for

Let

be the divisible closure

not of this form, let ft*, D* \i' y

be defined as in the special case above

primes

If

(y

< 2 °

and

for all

p ) . Now proceed as before using the new ft*, D*,

and

E*

in place of the old ones. By varying

{y : p prime} P

in Theorem 4.2.7, we get

K

dense rigidly homogeneous chains. actually,

2

2

prescribed

2 °

o

22

We now sketch a proof for

—

pairwise non-isomorphic such chains for each

{y : p prime}.

In the proof of Theorem 4.2.7 we also wish to construct inductively A

OD A

of for n

A ,

= {0}, A

A 1

if

divisible subgroups of ffi such that |A.| = maxf^j ,|x|} A U

X' < A.

and

To do this, let

Let

A Q = Qn

|B| = 2 ° > |A*|,

.

At stage

we can select

vQ y

(Case (i) now becomes

v

and

D

the least

n 4. D v ^ y

v

A* = U{A

1

Now let

A y

v

= A* © Qn y v

with

: A < y} 1

1 i A V ft^f" u D* U A* U A ^ f " ) .

© A*. y

such that

S,a 6 B\(A* U A*f" 1 U D * )

are then obtained as before, we let

such that

be a basis

of the induction, since

the desired properties (as before), where

E

is a direct summand

{n v : v < 2 °}

as a rational vector space and

fL D Q .

A , A

If ft ,

be the least .

If

y

s

A = U { A • X < 2 °}, then A is a divisible subgroup of 1, |A| = 2 ° and A d ft = {0}. Let {^ : v < 2 °} be a basis for as a rational vector space and

X £.2 °.

A

Then ft © £{$£ : v € X}

is a dense rigidly homogeneous chain (as is easily seen) having the prescribed

y 's.

Thus the number of such dense rigidly

homogeneous chains is greater than or equal to

H of subsets of

22

(the number

H

2 ° ) . Since there can be no more

( |ft| £ 2 ° ) , and

as any order-preserving isomorphism between two totally ordered subgroups of

1

is obtained by multiplication by a positive real

105 number (so the number in any isomorphism class is 22

2 °),

there are

pairwise non-isomorphic dense rigidly homogeneous chains of

the prescribed

y 's.

Other modifications of the method are possible to prove, for example, that there exist rigidly homogeneous chains whose divisible closure is

I.

If

1

cannot be written as the union of

M less than 2 ° nowhere dense subsets (which is certainly true if we assume the Continuum Hypothesis—by Baire's Theorem), then all dense rigidly homogeneous chains have size

2 °. However, it is

consistent with ZFC (set theory) that the Continuum Hypothesis n

fails and for each uncountable cardinal dense rigidly homogeneous chains of size

K <_ 2 u ,

K

there exist

2

K. For these and other

results about dense rigidly homogeneous chains, see Glass, Gurevich, Holland and Shelah [81], and Roitman [81]. 4.3. PERIODIC PRIMITIVE l-PERMUTATION GROUPS. In this section we classify the remaining transitive primitive ^-permutation groups. Recall the example given in the introduction to this chapter. We showed it is primitive. However, it is clearly neither uniquely nor doubly transitive implies

lg = l).

(Og = 0

We wish to show this example is, in some sense,

canonical.

THEOREM 4.3.1.

If

(G, Q) is a transitive primitive

l-permutation that is neither uniquely nor doubly transitive, then

(G,Q) is periodic; i.e., there exists e < z £ A(Q) such

that

{az : n € %,} is coterminal in ft (for any a € oj and

Ckfo^G) = . Moreover, if

a, 3 £ Q and

A = 3£ ^ {3)j then

(G \k,k) is non-pathologically doubly transitive. This will complete the proof of Theorem 4A (and hence Theorem 4B). 4.1.6:

It also yields the following extension of Lemma

106 COROLLARY 4.3.2.

(G3Q.) be a transitive primitive

Let

t-permutation group and

a G Q>

then so is

(G^aG), and (G^aG)

is doubly transitive (regular* periodic) if

(G^Q) is.

Throughout this section (due to McCleary [72a] and [73c]), we will assume that

(G,Q) is a transitive ^-permutation group.

We will "build on the ideas of Section 1.6 to obtain the theorem. Recall Theorem 1.6.12:

If

(G,ft) is primitive, then it is

balanced; i.e., the paired orbit of a long orbit is long. For

a € fi, let

Tx(G ) = {a € fi: G S. G-}.

If

(G,fi) is

a transitive primitive ^-permutation group, Fx(Ga) = {a e fi: G a = G-}

LEMMA 4.3.3. tation group, and a € ~Fx(G ).

by Theorem 4.1.1.

(G3Q) be a transitive primitive t-permu-

Let

Z = ^A/Q)^*

Conversely, if

unique member of Proof: Let

g £ G ,

Conversely, since z •ft-> S

given by

3 € Q

f € G.

and

a € ISf£ J

A(Q) such that

ag = azg = agz = az = a,

G.

3

= 3

lg

so

If 2*

*\g*\"

3-,, $ 2

Let

^ G

f

G

with

= G-.

e

af

3 z Q = af-, < af.,g = $2 Q

=

i

Hence Z

< e

so

zQ let

2' ^i-

G;

COROLLARY 4.3.4. t-permutation group3 z *—* az o-group.

= G-.

i.e.,

Let

Hence the map

is well-defined. ag = 3.

Let

Then

commutes with every element g

Then

^

G+

af

ig

be SUCh that = 3

2

> 3

1

= af

l

af.,g ? af-,. Now z

n

^"S C"116"^0""0116

Therefore it extends to a unique element still commute with

Then

is defined as the

a € Fx(G ).

be such that

Thus i

s

a = az. Then Hence

(ag)zQ = ag (g £ G)

£ fi w i t h

1€

and

g € G-.

g €. G

a = as#

^

and

a & Fx(G ), G

Let

an

(ag)z = ag (g £ G)., then z €. Z.

z £ Z

so

$fzQ = (agf)ZQ = agf = 3zQf. of

z G 2

Let

z

an(i

of

preserves order.

A(Q) which must

z € Z.

(G3Q.) be a transitive primitive Z = CA(Q)^G^'

a € 0. and

is an ordermorphism of

Z

Then the map

onto Fx(G ). Hence Z

is an

107 Part (i) of the next corollary strengthens Lemma 1.6.8. COROLLARY 4.3.5. (i) If

(G^Q.)

is doubly transitive^ Z = {e}.

(ii) If

(Gjtt)

is the right regular representation of a

subgroup of M, Z = %

or Z = B.

Proof: If (G,ft) is doubly transitive in case (ii), R, ft = 1

Fx(G ) = {a}, and

Fx(G ) = ft. If ft is a non-discrete subgroup of

so the corollary follows.

We now come to the key lemma.

Recall that in our canonical

example of a periodic primitive ^-permutation group, first positive orbit, namely of

(a,a + l) = (a,az),

Ga

has a

the G a

orbit

a + p- (a € I ) . The paired orbit is (a - l,a) = ( a z " 1 ^ ) .

Both are long and the only point of ft between them is a. This phenomenon is quite general as we now show. is a transitive primitive t-permu-

LEMMA 4.3.6. If tation

group that

G

has a first

of

ft between

is neither

positive A7

and its

uniquely

long orbit paired

nor doubly transitive3 A-

and

orbits

A'.

a

is the only

then point

108 Proof: Hence

G

Since

(G,ft)

is not uniquely transitive,

has a long orbit

A.

1.6.12), we may assume that cofinal in ft. Let G

c. G-.

ag~

By Theorem 4.1.1,

£ A

and let

agQ 1 < a,

G

Thus

< a

g~ fh A 3 £ A .

Since Let

so there is € G .

e

is balanced (Theorem

A

is a

Choose

G

orbit,

gQ € G

so that

A_ = Conv (ag^G ). We now show that

(a,ag Q ) ^ 0.

ah"

(G,ft)

= G-.

ft

Now

Therefore

A_

is

is not discrete and

3 € ft H (a,ag Q ).

is dense in ft so we may pick

Then

so

Since

J {e}.

is a negative orbit and so not

a = sup(A) € ft. Since

the required long orbit.

aG

A

G

h £ G

By Theorem 4.1.1,

such that

f £ G + = G-

such that

a < ah < 3. ah"

<_ ag" f.

(ag o )(g~ fh A e) <_ afh = ah < 3 < ag Q ,

(a,agJcA

.

Since

a & A ,

inf (A ) = a

as required.

We are now ready to complete the proof of Theorem 4.3.1. assume that

(G,ft)

is a transitive primitive ^-permutation group

that is neither uniquely nor doubly transitive. G

has a first positive long orbit

ft since

(G,ft)

A

,azn).

E VA/?I\(G),

each

paired with

1-

(G,ft).

Since

A

(G,ft)

such that Note A

A.. = (a,az) is a

But if

az = a.

G

For each

and

orbit.

A = ^{An: n £ Z},

is primitive,

Fx(G ) = {az11: n £ ^ } , (G,ft)

so

Let

n £ 22,

= A,z

. A

let

Since is

A

is a block of

A = ft; i.e.,

{az : n £ 2£}

= C w - ^ G )

is periodic.

A

Moreover,

is coterminal in ft for some (and hence all)

Consequently,

By Lemma 4.3.6,

which is not cofinal in

By Lemma 4.3-3 and Corollary 4-3.4, we

e < z £ ^A(n)^^

= ft A (az

A,

is not doubly transitive (Lemma 1.10.2).

a = sup(A 1 ) € Fx(G ). obtain

So

a £ ft. Therefore

by Corollary 4.3.4.

109 Finally, we prove

(G |A,A) is non-pathologically doubly

A = $G (3)). Firstly by periodicity, G is G faithful on A. To prove double transitivity, it is enough to show Ga A n GGpQn such < 3o in A, there exists ee < g £ G.. that if transitive where

that 3pg = 3^ (by Lemma 1.10.2). There is e < h ) for some 3 2 h = 3 y A = (az ,az Now

az

< 3~z

e > f € Gn —

such that

Po

3og = 3 O (G | A, A) that

< az <

azmg

-1 < (3-.h < 3ph = 3o-

3_hf = 3_ . Let 1

and

and

3 g

P-,

bounded below (in A). But -1 Az ) so, by periodicity,

Consequently,

Next we can choose

> 3 . Then

all points in A n (az ,az g. )

eIA

g = hf v e €. GD . Then

ag = max{ahf,a} = max{af,a} = a.

< az m

such that

Thus there is

1

is doubly transitive.

a

g

v e ^ G

g £ G

such

and fixes

L

Therefore e i g1 v e|A -1 g. v e|Az~ is bounded above (in g- v e|A

has bounded support.

is too (in A).

is

Hence

Q.E.D.

We note that the proof given of Theorem 4A works equally well for coherent ordered permutation groups provided we change doubly transitive to weakly doubly transitive'. Given a,3-^$2 such that 3-^g >_ with a < 3-, < 3p^ "there exists g € G f eGo such that 3 n f < a. So 6

2

G

and

110 THEOREM 4.3.7.

Let

(G,Q) be a transitive primitive

coherent ordered permutation group. Then: either (i) (Gy oj or

is weakly doubly transitive^

(ii) (GjQ) is the right regular representation of a subgroup of R,

or

(Hi)

(GjQ) is periodic.

Moreover3 (i) and (ii) are always primitive. This classification may hold for all transitive primitive ordered permutation groups. We know of no counterexample. Howevery, we do know of examples of incoherent doubly transitive ordered permutation groups. Theorem 4A makes no claim that periodic ^-permutation groups are primitive. We now give an example to show that periodic does not imply primitivity. EXAMPLE 4.3-8.

Let

Q = B\Z

and

(a + l)g = ag + 1 for all a € ft}. Then

G = {g £ A(ft): (G,ft) is a transitive

periodic ^-permutation with period

z,

However,

(G,ft) for any

(n,n + l)

is a block of

(G, ft) is not primitive.

Note that

(i ,1) but this is not of the form

translation by +1. n € %,

so

G± has a first long orbit (i , i z).

We now obtain further information about periodic primitive ^-permutation groups. If period

z

is a least

(G,ft) is a periodic primitive ^-permutation group with and

a € ft, either

n €. Z

such that

az tf Q, for all m € Z

az m €. ft if and only if

n|m.

and value of such an

is independent of the choice of

n

In the first case we say that second, Config(n).

or there

az11 Q ft. In this latter case, By Theorem 1.6.11, the existence (G,ft)has

Con fig (<*>)

a € Q. and in the

Ill

Our canonical example, G = {g €. A(l): (a + l)g = ag + 1 for all a € R}

has Config(l).

It is straightforward to show that if

(O,to) is a periodic

primitive ^-permutation group, then any two long orbits have the same cardinality; and if j E k (mod n), If period G

and

(G,fi) has

Config(n),

A. = A v

if

Cwfi)(G) = .

(G,to) is a periodic primitive ^-permutation group with

z, then

G £ C J / \(Z).

If

G = ^i/^\(z),

we say that

is full. 1MMA 4.3.9.

let

(Gjti)

be a full periodic primitive

L-permutation group with period z. If there exist a € to* m € Z and g € G

such that ag = az >

Proof: If let

3 £ fl. Let

ag = az

m

then z

for some

f € G be such that

3z m = afz m = azmf = agf e n. Hence

€ G.

m £ Z,

a € 0,

and

g € G,

6 = af. Then z m £ A(fi). So

We conclude this section by proving: THEOREM 4.3.10.

If n £ %+ U {«>}, there exists a full

periodic primitive t-permutation group of Config(n).

112 Proof: Let and

T

{Q^: m € Z}

be distinct cosets of

Q

in

I

be the Dedekind completion of ffi x 7L. So

T = ({;} ul) * z. If +

n £ ^ ,

n = oo, let

^

let

ftn

= { ( ^ , 0 ) } U { ( a , m ) £ T: a £ 0 ^ } .

= { ( c , m n ) : m e ^ } u { ( a , m ) £ T: ( 3 k 6 Z ) ( l

& k = m (mod n ) & a £ Q}, ) } .

In e i t h e r

case,

K Let

ft

If < k < n

= T.

n

e < z £/I(T) be given by:

(a,m)z = (a,m + 1)

(a £ (c) U R; m £ Z ) . Let G = CAfn N(Z). In either case, it is quite easy to show that (G,ft ) is a full periodic ^-permutation group (with period

z) of

Config(n). //

4.4. INTRANSITIVE PRIMITIVE ORDERED PERMUTATION GROUPS. Recall that T

such that

(G,ft) is transitively derived if there exists

(G,T) is a transitive primitive ordered permutation

group and ft is obtained by adding to of

G

in

T.

T

some subset of orbits

We now wish to establish that every primitive

ordered permutation group is integral, static or transitively derived, and that any integral, static or transitively derived ordered permutation group is indeed primitive (Theorem 4C). As an immediate consequence we will have:

COROLLARY 4.4.1. group. Then

Let

(G3ftj be a primitive I-permutation

(G^tt) is static^ integral or transitively derived

from an l-permutation group that is doubly transitive, the right regular representation of a subgroup of #_, or periodic primitive with

Config(°°)

or

Config(n) (n £ % ).

113 Theorem 4C ( due to McCleary [76] ) shows that once we have classified all transitive primitive ordered permutation groups, it is straightforward to classify all primitive ordered permutation groups. Observe that if

(G,ft)

then, by Theorem 4.1.3, Moreover, Thus

(G,aG)

(G,ft)

aG

is transitively derived from is dense in

T

for all

(G,T),

a £ T.

is also transitive and primitive (Lemma 4-1.6).

is transitively derived from the ordered permutation

group on any one of its orbits. We now prove Theorem 4C. We already saw that static and integral ordered permutation groups are primitive (see page 87); so assume transitively derived from (G,ft).

Since

(G,T)

is dense in

T

If

G = {e},

Therefore there is

1.

G

then

all If some

> 1.

Then

n € 2S.

A

N,

n / m.

T = [aQ,a1).

(G,T),

is primitive.

is static; so assume

say

A n aG = 0

then

Assume A

so

h ^ G^ .

aG i- {a}.

G J {e}.

Then

Conv (aG)

aG = {a : n € zZ}

with

A = Conv (SG )

A £. (a ,a

Since

a n = a.

and

aG f Z.

Let

(G,ft).

Thus

= G

for

a

for some

m € Z.

amh = a n

for

A

(G,ft)

is an extensive block, G

= {e}.

Let

is integral.

T = aG.

would be extensive.

an extensive block of

a n

)

|A| > l ) . Thus and

G

with

affih £ aG,

A n Ah = 0; so

(a £ A

then

(G,ft)

is an extensive convex set and

Then ft = T x Z

Case 2. of

or

Conv (aG) = ft.

aG = Z,

Therefore

a contradiction

A n T = T

= {e}.

Hence

h € G\G/

is

be a primitive ordered permutation

(G,ft)

Thus

Thus

For reductio ad absurdum, assume |A| 1 1

(G,ft)

be a natural block of

A = ft or is a singleton

a €. ft such that

Assume

We claim that

Hence

(G,ft)

is an extensive block. Case 1.

A

by Theorem 4.1.3).

Conversely, let group.

Let

is transitive and primitive,

has cardinality at most (T

(G,T).

If

Hence

(G,T)

A

were a block

Conv (A)

would be

is transitive and

primitive. It remains to show ft c f ,

By Lemma 3.2.4, if

is a convex subset of ft maximal with respect to

3 € T

A H T

and

= (3l,

A

114 then

A

is a fat block of

Let

A

containing no point of A

Then

A = 0

T.

Let

or is a block of

If

A - {5}, then

If for some

± —

Hence

(G,T).

Now let A

Since

T

A

is

is an orbit of

Since

{6} U A

A,

is a convex subset of ft.

there exists

T, € A, l

(G,T) is primitive,

A U {6} = {s}, a contradiction; so for all T1 6 T

(A,B) be a cut in

A T

Af > A, then

such that

a contradiction.

sup(A) = inf{Af: Af > A}; i.e., Then

A i 0. We claim that

|A| = 1.

a £ ft, A < a < Af

by the maximality of A < Tn < a.

A = {£}.

A = {T £ T: A < x < Af for all Af > A}.

By the previous paragraph, A = 0.

Suppose

(G,ft) whence

is a block.

|A| <_ 1.

Hence

be a convex subset of ft maximal with respect to

a fat block of G,

(G,ft).

Consequently,

is a fat block of and

(G,ft).

A = { a £ f t : A < a < B}.

is a convex subset of ft maximal with respect to

containing no point of T = ft and

T.

Thus

|A| = 1

or

A = 0. Hence

(G,ft) is transitively derived from

(G,T).

We can now locate the non-extensive natural blocks of an ordered permutation group by combining Theorem 4C and Lemmas 3.2.5 and 3.2.8.

COROLLARY 4.4.2.

The non-extensive natural blocks of an 7/

ordered permutation group are precisely the C -classes for which K

is static. For more technical consequences, see McCleary [76].

4.5. THE PROOF 0F_ THEOREM 4V_. Throughout this section, let ft be a homogeneous chain. We now exploit Theorem 4B to deduce Theorem 4D.

This proof will be

the first of very many we will give heavily using Theorems 4A and 4B. By Theorem 4B and Lemma 1.6.8, the only primitive homogeneous chains that are Abelian are rigidly homogeneous chains. Moreover, if ft is imprimitive, let

A

be a non-trivial block.

115 ),s and h $ A(Q),y Then Let pp g and Ah O A = 0, so h" gh A g = e. Hence A(ft) is Abelian if and only if SI is rigidly homogeneous if and only if A(ft) is totally ordered. If fi is imprimitive, let A i (i = 1,2,3) be distinct classes of a non-trivial congruence of (A(ft),ft). Let e ^ g. e A(fl) with supp(g.) S A . (i = 1,2,3). Then for all 1 -1 "*" "*" g~ and g«. h €. A(ft), h g_h is disjoint from at least one of Thus A(ft) * ( 3 g ] / e)(3g2 e)(3g3 J e)(Vh)( ] = e or 9 denote this sentence, then clearly A(ft) h""»0 if (and only if) ft is doubly homogeneous, and A(ft) I* 0 if A(ft) is Abelian. If * E # 0 & (3f )(3g)( [f,g] ^ e ) , then A(ft) t= ty if and only if ft is imprimitive. [h g h,gj = e).

If we let

If ft is imprimitive and g-, ,g^ > e are defined as above, then there does not exist e < h £ A(ft) having one bump such that A ( f t ) £ h A

cr

^ e & h A A g;L

e & f2

e & (3f1)(3f2) g

2

/ e & h A f = e

h A f = e).

(If h has one bump and g-. A h ^ e i g.. A f but f, h = e, then AJi O A / (2(; hence A ^ = A., and so h A g^ = e. ) Since "h has one bump" is expressible by a sentence in the lattice language (Lemma 2.1.1), we have that if ft is imprimitive, then A(ft) £ x where x i s the lattice sentence

116 (3g1 > e)(3g2 > e)(g1 A g 2 = e & (Vh > e)[bump(h) & h A h A g 2 ^ e -> -i(3f1)(3f2)(f1 A

Clearly,

A(fi) N x

gl

g]L

/e &

/ e & f2 A g2 / e &

if and only if ft is imprimitive; and

A ( f t ) t = - » x & ( 3 a > e ) Q b > e ) ( a A b = e) if and only if ft is doubly homogeneous.

This completes the proof of Theorem 4D.

A group is said to be an 0-group if there is a total order which can be put on

G

making it an o-group.

The following

theorem due to P. M. Cohn [57] is similar to Theorem 4D.

Let ft be a (not necessarily homogeneous)

THEOREM 4.5.1.

chain. Then Afftj is an 0-group if and only if it is Abelian. Proof: By Appendix I Corollary 10, A(ft) can be right totally ordered.

Hence if

A(ft) is Abelian, this order makes

A( ft) an 0-group. A(ft) = TT"A(A.) (as a group), where X i£I is the set of distinct orbitals of A(ft); i.e.,

Conversely, {A.: i € 1}

{A.: i £ 1} = {Conv (aA(ft)): a £ ft}. Since

A(ft) is an 0-group if

A(A. ) is, we may assume ft = A.

and only if each

1

That is, we assume that ft is an extensive block of e < g €. A(ft) has bounded support, say e < f € A(ft) with exists). 2n

Let 2n

h € A(ft) with

h|(af ,6f ) = r (n € Z ) . implies

Then

af >_ 3 2

V

some

i n €. I. U

1Q

n

f~ hf = h

and

A(ft). If

supp(g) £:(a,$),

let

(ft is extensive so such an f supp(h) Q U{ supp( f ~ n g f n ) : n ^7L) and h| ( af 2 n + 1 ,6f 2 n + 1 ) = f"( 2n+l )g-lf 2n+l so

A(ft) is not an 0-group

f~ hf V e ) . Consequently, if

(h > e

A(ft) is an 0-group, it has

no elements of bounded support. Hence ft is primitive or has exactly one natural non-trivial congruence (Lemma 3.2.9).

£

that is static

By Corollary 4.4.1* the unique non-static value is

integral or transitively derived from a rigidly homogeneous chain (A(ft) has no elements of bounded support). is clearly Abelian.

In either case, A(ft)

117 CHAPTER 5 THE WREATH PRODUCT

Let

(G,fi) be a transitive ordered permutation group. By

Theorem 3A, the set of congruences of (G,fl) form a chain. As in Section 3.1, let k = k(G,fi) congruences of (G,fl).

be the set of covering pairs of

If K = (Cv^)

e

k, then

{(G K ,^ K ): K G fc} is the set of primitive components of (G,fi). We will define an ordered permutation group called the Wreath product of

{(G^ft ) : K a } Jtv

(written

Wr{(G77JQj: K£ kfG^Q)})

J\

so that:

K K

THEOREM 5A.

Let (G3 Q) be a transitive ordered (I-)permuta-

tion groupj k = k(G^U)Ji and {(G^Q, ): K € k} be the set of primitive components of (G3ti). Let (W3Q) = Wr{(G1/:>Q.J: K € k}. Then

(W3Q.) is a transitive ordered ('t-)permutation group whose

set of primitive components is { f ^ f i j : KG. k). Moreover3 if (HjA) is any transitive ordered (t-)permutation group having { (Gj,* &•»): K e k} as its set of primitive components3 then can be o-(Z-)embedded in

(H^h)

(W3Q).

We first define the Wreath product of two (not necessarily primitive) ordered permutation groups. The following lemma is easily verified (where we write as a shorthand for LEMMA $.1.

{g }

{g : T € T}):

Let (G3Q) and (EST) be ordered permutation

groups. Let A = n x" T and W = {({g },h): h £ H & g e G for alt

TQ.T}.

Define a binary operation on

({fThh2)({grh1i2) = (iajthfo), Then and

W by:

where a% = fxgxh

(x e T).

W is a group with identity ({e }3e)—where e = e (T £ T)-({g },h)~1 = ({f },h~2) where f = (g T

T

T

J'1 (T € T).

__ i^ ~ 1

(W,A)

118 is an ordered permutation group if

(a3a)({g },h) = (ag 3oh)

(a €. n3 o € T). Moreover3 (i)

(W3A) is transitive if

(G3ti) and

(ii) (W3A) is an t-permutation group if are;

(H3T) are, and (G39.) and (E3T)

({g }3h) v ({e },e) = ({f },h v e) where

I ag

if oh > o

afa = < a [^ a(g v e)

if oh < a. if oh = o

(W,A) is called the Wreath product of and is written

(G,fi) and

(GyQ.) Wr (H3T) . We will also write

(H,T)

W = G Wr H

we wish to emphasise the group part. If w = ({gT>,h) e (G,n) Wr (H,T), component of

w,

and

then

{g : x €. T}

h

is called the global

the set of local components

x

of

•

w. EXAMPLE 5.2.

Let

(G,fl) = (H,T) - (Z,Z),

representation of 22. Let

1 -1

if n = 1 if n = 0

. Let

f =

w- = ({f },l)

and

In

^ 0

if

the right regular

w.

<:

n i- 0,1

e = ({0n>,0) < w x ; w 2 v ({0n},0) = ({mn),0) where 1 if n = 1 m

n

=

1W2 = ({kn},l) ¥ ((k^},l) = w 2 w 1

Lo

if

n

P

if

n = 0

if

n = -1

if

n J -1,0

0

1

if

LO

if

n = 1

and n

¥ 1

where

if

119

C

o

'I

(To obtain

w^w«,

o

superimpose the

(the global component of WpW , superimpose the component of

w?)

Wp

on the

Wp

and

G

7T I T€T T€T (T € T, h e H ) .

G Wr H

w., picture.

and

+1

To obtain

+0

(G,fi) Wr (H,T) is (G,fi) and

({g },h) € W = G Wr H

for all but a finite number of Note that

o

O

(the global

picture.)

the [restricted) wreath product of

G

o

X

-I

picture moved left by

A useful (£-)subgroup of comprises those elements

O

picture moved left by

w.. ) on the

w,

0

(G,Sl) wr (H,T),

(H,T),

and

such that

g

=e

T € T.

G wr H

respectively, by

are splitting extensions of H,

given by

(g(he)) = g T

-, xh"

120 Let C tion group

be a congruence of a transitive ordered (t-permuta-

(G,ft).

Let L( £ ) = {g €. G: (Va £ ft)ag£a}, a convex

normal (£-)subgroup of G. So L( 5 ) is "the lazy subgroup of the action

(G,ft/5 )—recall that

action.

We call

( 3 5 )g = 3gE (3 £ ft) gives an

(G/L( 5),ft/C) the global ordered permutation

group of (G,ft) relative to C, or the for short.

Let A be a C

class. Recall that

(J ,C ) component of (G,ft),

(G/ \,A), the

is an ordered permutation group

(independent of the particular isomorphism), where

5. global part of (G,ft)

£ class chosen—to within

6/ . \ = G/.\/G . It is called the ff local

part of (G,ft). THEOREM _5__3. Let C (I-)permutation group

be a congruence of a transitive ordered

(G3Q,)3 and A be a C class. Then

can be o-(I-) embedded in

(

}

(W,A) = ^(^)^

Wr

(£_, oj

(G/L( G )3Q/G ).

()

Proof: For each 5 class A, choose c. £ G such that Ac. = A, with c = e. These are the coordinate maps and will A

A

'

remain fixed throughout the proof. by: ad> = (ac"-,a5). We show that at#

_____

Define <(>:ft->A= A x (ft/fi)
one-to-one (it is clearly onto). If

a < 3,

either

trivially, or aS = 35-

a£ i 3C in which case In this latter case,

a<j> = (ac"^,at) = (ac~ ,3C) < ($c~ Let for each

£1?) = 3
g h > g be the natural map of G onto 5 class

ip: g 1—> ({g«},g). If

a<j) < 3(f)

f < g,

G/L( £ ), and

A, let g. = e.gc" |A € G/ x. Let We now show that

then

aUf <_ a?g

Tp is an o-( t- )embedding. for all

*•

a £ft.Therefore

~\

"i

/\

f <_ g. If Af = Ag, then af. = ac.fc" = ac.fc" <_ ag. for all a £ A (so fip <_ gip). Moreover, if 3f < 3g, either 35? < 35g or

3fo* < 3go-, by the previous sentence. Hence pi*

fip < gip,

is one-to-one and preserves order.

C \p f Cis C ha0group

so

f v e exists.

so

^

PQ

=homomorphism. h \

Consequently,

If

h £ G,

Clearly

then

fh^ =

\

c

^^

(f v e )ip = fip v e if

ip is an o-(£-)embedding.

ip _-, c

121 Now if

a £ ft and

g €. G,

(ag)<J> = (agc~ -,agC) =

(ac~,alj)( {gA},g) = (a<j>)(gi|;). Thus we have an o-(£-)embedding of (G,ft) in

(G/Av,A) Wr (G/L( 5 ),ft/£).

In the above proof, observe that responds to a local shuffle inside the represents a shuffle of the g £ G

C

<j> is onto. Also £

class

A,

g.

cor-

whereas

g

classes. Hence we are thinking of

as a set of local pieces, and one global piece. We now obtain some information about the ^-ideals (i.e.,

convex normal ^-subgroups) of the wreath product. LEMMA 5.4 (The Hourglass Lemma). Z-permutation groups with N

of

G wr E

L-ideal of

Let

is comparable (by inclusion) to

G Wr E

not contained in

Proof: Write = e

if

Ga

~\ j G

for the set of

x / a (a £ T).

If

N

£

G £N.

G3

and every

\ G. T ( {g },e) £ G wr H with T£

is an £-ideal of

I G, there is ({g },h) £ N with h > e. T x£T ah > a. By convexity, G° C N. But if x £ T, where

\

contains

N g

GT = ({e},f)"1G°({e},f) 9.N,

(E3T) be

(E3T) transitive. Then every L-ideal

T£T

g

(GjQ) and

f £ H

and

G wr H

Let

with

a € T with

af = x. Hence

The rest of the lemma is proved similarly.

The following lemma (the associative law) is straightforward verification: LEMMA 5.5.

Let

(G,Q,), (E3T) and

(I-)permutation groups. Then o-(l-)isomorphic to

We will write

(L9B) be ordered

[(G3ti) Wr (E3T)] Wr (L3B) is

(G^ti) Wr [(E3T) Wr (L3B)]. (G,Q) Wr (E3T) Wr (L,B) for either of the

ordered (£-)permutation groups obtained by Lemma 5.5. We now wish to extend our construction to the Wreath product of (possibly infinitely many) ordered permutation groups.

In order

to motivate it, we give another way of viewing the Wreath product

122 of

(G,ft) and If

by:

(H,T).

(W,A) = (G,ft) Wr (H,T),

(a,a) = ( 3 , T ) if

(W,A).

So

(W,,A) ^ s

^

as a congruence. let

choice of x

Each

T € T

(a,a) = (a,x)f. afT = 3

an

=

=

on

A (= ft ? T)

is a congruence of

ordered permutation group having =

f £ W

induces an element

and choose

Since

W_

if

a £ ft. Then

respects

a £ ft. Also each

under

then define

Clearly

W 9 W x = {f € A(A): ( V A ^ V A ^ ^ f = X^ if and only if anc

X- = ^2-1^ follows:

a = T.

=, a

f €, W_

induces

(a,x)f = (3,xf)—where we write

?.

(f,f).

f K

is independent of the f: T -> A(ft) f

be any chain and

group for each

K €. k.

Let

via:

for the image of W

and

Hence

(W,A) = (G,fl) Wr (H,T) = {f 6 W ^ f 6 H & M k

xf = a where

Thus we obtain an o-(£-)isomorphism between

A(ft) Wr A(T) given by Let

f € A(T) as

G

for all

x £ T}.

(G^,^ ) be an ordered permutation

Q

= TT{fi • K 6 fc} and choose an j\

—

arbitrary fixed reference point in ft***, call it

0_. For each

a € fi+, let supp(a) = {K e k: av i O,}. Let fi = {a € fi : supp(a) is an inversely well-ordered subset of /c}. Note that if c

a,3 € 6

are distinct, 0 /fe(a,g)= (K € fe: a R / B

supp(a) u supp(3),

ordered.

and so fc(a,G) is also inversely well-

It therefore has a gr greatest element say

a chain via:

a < 3

if

a^ <

o

K

KQ.

We make

o

We next define some "natural" equivalence relations on K Q. k, if

For each a

a =KB

Hence i f

a if

if

¥3

define = 3K, a

K

o

and

=„ on ft by:

for all

a =„ 3

have convex classes (all so let

and

K1 > K

= 3 K , for all K! >_ K. is the largest element of

and

K > K

=

if

K > Kn .

Clearly

/c(a,3)> E

K

then

and =

K £ k). We wish them to be congruences,

W2 = {g £ A(ft): (VK £fc)(Va,B£ft)[(a= K 3 <-> ag E K 3g) &

(a ^^IV 3 <=> ag =XV 3g)]}.

Then TT

K

=

and

=XV

are congruences of

_xv

(W1,ft). K £ k.

Observe that

a" /=

is just ftK for each

a £ ft and

123 For each

K € k

and

a £ ft, let

K

K

(a ) K , = a K , (K! > K); K K a

= B

g

i.e., K

precisely when

For each

of a

a_ € TT % i with K >K ' a above K. Note that

g £ W_,

a

is

a = 3. a € ft and

K € k, g

induces an element

A(ft^) as we now show. Let a € ft and define f where af E a and a' = a. K = (a g) K £ Q K , K,a LEMMA 5 . 6 .

J/£t/z t h e a b o v e n o t a t i o n

a

3

g

„ €A ( o ) ,, A

and K £ fc. K Proof: If a = 3 and

by:

A

a € O, ^ € J/7

!

g € W1,

3g = K a g;

so

$T. = a, f

(3g) K = (a g) K

af =

then and

g K is wellK,a a with y^. = x,

K

defined. then

If

y =

Therefore maps ftK

x e ft^ with

r

a

but

a < x,

a' < Y^. Hence

and a

?

y = < y

3. Since

(a f g)^ < (YS)^*

and

g „ is one-to-one and preserves order. K,a onto ft^ proving the lemma.

It clearly

Let

W = {g € w 1 : g K € G K for all K G fe & a € a}. Then K,a (W,S) is called the Wreath product of { ( G ^ O : K € ^} and is written

Wr{(G 3Q, ): K € k}. We may think of the elements of W

k x ft matrices

(gTyr

} wTth

gT,

= gv

if

n

a

as

= 3 . Note that

the group operation on these matrices is given by: (*)

(fv

Ji,a

)(g

) = (h

Ji,a

) where

Ji,a

h

iv,a

= f

g

«

K Jtv,a is.,at

i.e.,

h v = f vg ^ (K £ k, a £ ft)—not the usual matrix K,a K,a^K,(afT multiplication (cf. the definition of multiplication for the Wreath product of two ordered permutation groups). LEMMA 5.7. If each is

(G7^Q7J

is an l-permutation group3 so

(W9to) = Wr{(Gv ft)- K € k}. Aj

Proof: Let

A

g £ W.

We must show that

g v e

exists in W.

A routine verification shows that the pointwise supremum of

g

and

124

e

is

h

where

%. a v e h £ W,

as desired.

K

if i f aa\ g << aaK . K K a g = a

= < e

h^

aKg > a

if

a

Clearly

//

Our definition of the generalised Wreath product depended on the choice of the reference point (Gj,,^)

0_. We wish to show that if each

is transitive, then the dependence is illusory.

Specifically:

LEMMA 5.8. so is 0_r

Assume that each

(W,fo) = Wr{ (G^Q.j.): K £ k}. 9, = TTittj,-' K £ k}

{(GV,Q ): K € k] r

(W',Q )

are

such that g

Then

Moreover, if

(Wf,Qf)

and

is transitive.

is the Wreath product of r

with reference point

0,

then

(W,Q)

and

o-(l-)isomorphic.

Proof:

if

(G7.,tt7.) K K

Let

a

a,B £ fi. For each =

ir&ir

3™- where we take

is defined by:

Moreover, if

K £ k,

(yg)K

y £ Q,

=

gv = e

there is if

K $• k(a,$).

Y K g K (y £ fi; K £ k),

supp(yg) c supp(y) U k(a,&)

inversely well-ordered.

Hence

g|fi

maps

gK £ G

then

Now

ag = B.

which is

0, into

Q.

It is

straightforward now to show that g\Q £ A(fi). Since (glfi)^ = (gl^)^ = gyr £ Gv, g|fi £ W and (W,fi) is transitive, iv, y

K

For each Define

<J>:fi -+Q by:

Define

ip: W

( atf>)( gip) = (ag)<j>, W

there is

1

ordermorphism and if S'.

(since

fv £ 0v

IV

A.

K £ k,

f

£ G

such that

(a<j> )„ = a^f^ (a £ fi). Now a<j) £ fi!. Indeed,

a £ S,

-> W and

1

by:

gip = <j>"gc(>.

ip|W

for all

0Kf $

<|> | S

is an maps

fi

onto

Clearly

is an o-( I- )isomorphism of K £ k).

= 0'.

Hence

dp,*)

W

onto

is the

desired o-( £-)isomorphism.

We will prove the following strong form of Theorem 5A:

THEOREM 5.9. tation group,

Let

k = k(G,Q),

primitive components of

(G,ti) be a transitive ordered (I-)permuand

{(G ,Q ): K £ k}

(G,Q,). Let

be the set of

(W,Q) = Wr{(G^,Si^): K £ k}.

125 There exists an o-(t-) embedding (^3$) of (GyQ.) into (W^Q) such that (i) each = class contained in (0$)= meets ft<j> (K £ k> a € fijj and (ii) (o$)w E a<j> implies there is g £ G with %(gi>) =K $w for all $ £ (o<$>)=K (o £ ft,, w £ W). In Theorem 5.3, the <>j part was actually onto; in Theorem 5.9 4> need not be onto. However, what i s claimed in ( i ) i s that ftcj) i s dense in ft except possibly whole segments may be added to ft in those places where k1 <=~ k i s not c o i n i t i a l in k, has no l e a s t element, and there are (?„ £ ft (K £ k1 ) with K K K ov C £

IV ^. /V

shows that (W,ft) has the same primitive components as (G,ft). Before we can prove Theorem 5.9, we need a technical lemma. Let L be a subgroup of a group H. T(L) is a transversal of L in H if T(L) contains exactly one element of each right coset of L in H; i.e., |T(L) D Lh| = 1 for all h £ H. LEMMA 5.10. If H is any group3 there is a set of transversals L J

{T(D: L a subgroup of H] with T(LJ c. T(LJ if

an

L- ~ 9 ^ T(L) D L = {e} for all subgroups L. Proof: Well-order the set of elements of H with e first. Let L be a subgroup of H and h €. H. Choose the least element of Lh for the unique member of T(L) in Lh. Clearly, {T(L): L subgroup of H} has the required properties. K Proof * of * Theorem 5.9: Fix a_Q £ ft and take ft^ Is.= anUg /£„ Is. (K £ k). Let T be a transversal function for the subgroups of G guaranteed by Lemma 5.10. For each a £ ft, let c(a,K) be the rr

unique element of T( G

„ ) such that

a c(a,K)6 a. These

c(a,K) (a £ft,K £ k) are the coordinate maps ( cf. proof of Theorem 5.3). Note that c(a Q ,K) = e by Lemma 5.10. Define <\>:ft->ft1*= T T ft^ by (a(J)L = a c t a ^ K ) " 1 ^ £ttv(a £ft,K £ k). y £ -I A.

Let

IV

0 = o~
—

(J

—Iv

U Ji

Vv

and l e t

ft

IS.

be t h e s u b s e t of

ft

126 comprising those elements whose support (with respect t o inversely well-ordered.

We now show t h a t

0)

ll|C(i

and t h a t

Choose

f 6 G

is < > j

is

one-to-one and o r d e r - p r e s e r v i n g . Let anf

= a,

0 / fe1 £ supp(acj>).

a € fi and and l e t

H =

0

tower of subgroups, say.

Then

fg"

1

U G „ . K£*c< (a Q S K )

H

£ H

Since

i s a subgroup of

so

fg"

1

€ G

{G : K € k'} v (a Q g K )

G.

Let

is a

T(H) fi Hf = {g},

for some

v

with

K- G fc' . 1

K

We

(«0C X ) now show by reductio ad absurdum that K-. is the largest element of k} whence supp( a<|>) is inversely well-ordered (and so a<j> £ fi). If

K

< K € kT,

g € T(H) £ T(G

then

a g

).

But

K

£ a £ fg

a "1'C K a Q

Hence

ao( a ,K)"

Therefore,

C H

-1 € G

S. Gr

K (a C

-

ar5 Q

=

— J-t©

!„ = CL. IV

diets the fact that

K £ supp(a).

Now let a,x €. fi with K

so

v

K°

SO \

and, by the choice of T,

This contra-

IV

Consequently,fi<j>£ fi.

a < x.

Let K = Val(a,x).

Since

S O ( O * ) V = o c ( a , K ) \ < Tc(a,K)"1STr = iv iv iv T c d X J " 1 ^ = (T<|)L. I f K' > K, t h e n K^,2tJK. Thus (acj))^, = a c ( a , K ' ) C^,= TC(O,K') 5 ^ ,= T C ( T , K ' ) ^ t = (T<J>)T,,.

a? T,

c(o,K) = C ( T , K ) .

Iv

Hence

aff T

IV

IV

completing t h e claim. K We n e x t show t h a t aE T i f and o n l y i f i f and o n l y i f a<j> = ^ T * (K € A:). K

Suppose

obx. T

c(a,K') - C ( T , K ) Hence

a* =

Then

K? > K

^^KrT

Then

if K' > K.

iv

f o r a

J\ rr

T

r

= ac(x,K )" ^,

We now show t h a t So l e t a K € fi^ = a ^

a

< > j

a € fi and /ffK

so

1 (xcj))^,.

IV

The p r o o f f o r

^-

T

is

Hence

IV

similar.

satisfies a € fi w i t h

a^ = x^ K

so

Thus

1

IV

T<j>.

Kr > K;

H

(a<|>L, = a c f o j 1 ) " ^ , = ( T C ^ L , . K jg a5 , let K r = Val(a,x).

and c(a,K f ) = c(x,K f ).

(a<J>L, = a c ( a , K ' ) " ^ f

K a<|> E TC{>, and

Thus

T4>. Conversely, if

1

acj) f

IV

a<J> < x<J>

(i). acj> E

f o r some

a.

x € a^S .

Let

127 T

1

= Tc(a,K) € ft. Then

f

K

K

x £ oQc(o,K)£

= aC . Thus

x'cf) = a<|) E a by the preceding paragraph. But = T ' c d S K ) " 1 ^ = T'c(a,K)" 1 ^ K = T £ K = S R . Hence T'<J> = K 5, as required. We must now define ty: G -* W. Let a G ft, K £ 7c and g € G. (T ! <|>) K

if

CJ<> |

=„ a for some

a € ft .

This i s

a

well-defined (since 06 E x<j> implies

aff^x

IV

and so

JY

xv

whence (ag)<j> E (xg)<j>). Moreover, by the definition, (a z a. For all K ! > K, (o(g#)) K I = ((ag)4>)K!. Hence 0 / {K» €fef: K' >_ K} £ {Kr €. k: Kf >_ K & K' G supp( ( ag)<{>)} c Supp( ( ag)cf>). Since supp( (ag)(j>) is inversely well-ordered, it follows that k1 has a greatest element. Therefore supp( a( g\\>)) is inversely well-ordered, so a(gij;) €. ft. It is now straightforward to check that g\p € A(ft) and respects each = and = (K €. k). But (giH v - € G^ for IV

each

K € k,

I V , Ot

a £ ft. Consequently, (((af)g))„

if

IV

g^ £ W.

acf> E^ a

IV

Now

f o r some a € ft"! '

1\

otherwise (a((fg)![;))„, so j\f i s a homomorphism. Moreover, i t i s easily verified that fip < gip precisely when f < g and that (g v e)\\> = gij; v e. Hence (i|;,<|>) i s an embedding of (G,ft) into Finally, we must prove (ii). Let a €.ft,w € W and (ad))w E ad>. Then wT, , € G v . So w^ , = f v for some IV, O(J)

J\

A., Q(|)

IV

f € G „ , where fv denotes the image of f in Gv. Let ^IV IV IV (a Q g ) g = c(a,K)~ 1 fc(a,K) € G „ . If B =K a<(., let T € Q with /

\

(5 K )

128 T(f> E K g ( $W j-r^- ~

(by (i)). ( ( T (J) JW )T^-

xgc(a,K)

=

Hence

c(a,K) = C(T,K).

( T Cy, (7 , K /

=

Cf ^ jWy.

Thus

T C\ O , K )

CT, = ( ( T g )()))„ = ( ( T<J> )(gip) ) i r = ( B ( g ^ ) V . IV

gw = B(g^)

IV

for a l l

Note that the

J\

^v-f v

=

Consequently,

IV

B =K o$. \\)

Q.E.D.

given in the proof is "by no means unique.

However, the one given in the proof is called a standard embedding of

G

in

W.

This chapter covers some of the material to be found in Holland [69], Holland and McCleary [69], and McCleary [70].

PART III APPLICATIONS TO ORDERED PERMUTATION GROUPS

129 CHAPTER 6 SIMPLE ^-PERMUTATION GROUPS

An £-group

G

having no proper £-ideals is called I-simple

(Recall that an £-ideal is a normal convex ^-subgroup).

Such

^-groups have a faithful transitive representation (Appendix I Corollary 8).

In this chapter we wish to determine which

transitive primitive ^-permutations are ^-simple and which are simple (no non-trivial normal subgroups; i.e., simple as groups). By Theorem 4A, there are four cases to consider if we split the doubly transitive case into two, pathological and non-pathological. The regular case is trivial. They are subgroups of

I

and hence

are ^-simple but not simple. The non-pathological doubly transitive case is also quite straightforward. doubly homogeneous, every element of (Corollary 2.2.7) and H,

write

[#>#]

B(ft) is simple (Theorem 2G).

for the subgroup of

{[hl9h2]: h r h 2 £ H}

Recall that if ft is

B(ft) is a commutator

(Recall

H

For a group

generated by

[h^h^] = h ' ^ " ^ ^ - ^ ©

[B(ft),B(ft)] = B(ft) by Corollary 2.2.7. We generalise Theorem 2G. Note first that tion group

GAB(fi)

is an £-ideal of

G

for any £-permuta-

(G,ft).

THEOREM 6A.

Let

(G3Q) be a doubly transitive l-permutation

group with G n B(Q) / {e}. Then G

is I-simple if and only if

G E THEOREM 6B. group with

Let

(Gyti)

G C\ B(Q.) / {e}.

be a doubly Then

transitive

[G C\ B(Q.)3G A B(Q)]

l-permutation is

simple.

The periodic primitive case is also quite straightforward. Recall that if

(G,ft) is periodic primitive with period

z and

130 G = C^/ j(z), we say that (G,ft) has and

(G,ft) is full.

Config(m) for some

m < «,

In this case, if

zm £ G

is a non-trivial normal subgroup of

by Lemma 4.3.9 G.

Hence if

(G, ft) is a full periodic primitive ^-permutation group, G simple only if

(G,ft) has

THEOREM 6C. If

is

Config(oo).

(G,ftj

is periodic primitive3 then G is

Z-simple.

THEOREM 6D.

Let

is simple if and only if

(G3 ftj be full periodic primitive. Then G (G3ti) has Config (<*>).

In the regular and periodic cases, G fiB(ft) = {e}. Hence, in view of Theorems 6A and 6C, we might hope that if transitive primitive ^.-permutation group, then and only if

G C\ B(ft) = {e}

to be ^-simple whenever

THEOREM 6E.

or

G.

G

(G, ft) is a

is £-simple if

Were this true, G would have

(G,ft) is pathological. However:

There is a pathological Z-permutation group

(G3Q.) such that G

is not Z-simple.

Finally, we prove: THEOREM 6F. There is a simple o-group that has no faithful primitive representation.

The proofs of Theorems 6A, 6B and 6D exploit similar ideas to those we used to prove Theorem 2G. expressed "bounded support" in that if

In Corollary 2.1.4 we

A(ft) for ft homogeneous. Recall

(G,ft) is a transitive ^-permutation group and

e i g 1 ^ 2 ^ G' t h e n g l ^ g 2 i f a n d ° n l y i f gl i s t 0 t of gp (Lemma 2.1.3). We now generalise Corollary 2.1.4.

131 LEMMA 6.1. If e ^ g € G,

and

(G3ftjis a transitive Z-permutation group

then

g

1

(3h)(Vf >_

has bounded support if and only if

1

e)(gX(f h" ghf)).

Proof: If

supp(g) £ ( a ,3), let

ah = 3. Hence if 3

f >_ e,

h £ G be such that

3 ± ahf < supp( f ~ h~ ghf). By Lemma

2.1.3, Conversely, if g«£(f~ h~ ghf) for all f >_ e, to the left of h~ gh by Lemma 2.1.3. bounded below. Hence

g Let

if

Thus

then

g is

supp(h~ gh) is

By the fundamental triviality, so is

supp(g).

has bounded support. // G

be an £-group and

e / g €. G. We call

(i?h)(Vf _> e)(gJ!(f"1 h~ 1 ghf)).

g insular

Insularity is an £-group concept

not an ^-permutation one. We now prove the following strong version of Theorem 6A due to Holland [65b]. THEOREM 6.2. Let G ^ {e} be an l-group. Then

G is

Z-simple and contains an insular element if and only if for some chain G

^

(G3 Q)

is a doubly transitive Z-permutation group with

9L

We first need a lemma: LEMMA 6.3»

Let

tion group and L

(G3 Q)

be a transitive primitive Z-permuta-

an Z-ideal of

transitive Z-permutation group. so is

L ^ {e}3

(L^Q) is a

(G,n) is doubly transitive^

(L3Q) and L 2BfoJ C\ G. Proof:

Lemma 1.6.9, If

G. If If

g £ G

Since A

L/{e},

L

3 £ A. So

3g = akg = ahg'1 (f"1 k)g. Hence

3g € A. So

Since

(G,ft) is primitive

3 = ak

f~\ €. L

If

f,h G L

for some

a £ ft. By

such that

k £ L.

so g"1(f"1k)g € L

Ag = A. Therefore

(L,ft) is transitive.

for some

is a convex ^-subgroup of G.

and Ag O A J 0, there exist

afg = ah. Let

a < 3 < y

A=aL/{a}

is convex since

A

(L < G ) .

is a block of (G,ft).

A = ft; i.e., aL = ft. Consequently, (G,ft) is doubly transitive, let

in ft. Let e < g £ G

be such that

3g = Y.

Choose

132 e < f £ L

such that

3f = Y.

Then

g A f £ La

By Lemma 1.10.2,

(L,fl) is doubly transitive.

Let

say

e < g e B(fl) 0 G,

supp(g) £ (a,3).

is transitive, there exists

e < f € L

Lemma 1.9.4,

g £ L.

e
so

h = (h V e X l f 1 v e ) " 1 .

Hence

G CB(ft),

G

If

(G,fl) is doubly transitive contains

G OB(fi) = G

G

is ^.-simple and contains an insular

By Appendix I Corollary 8, there is a chain

T

such

(G,T) is a transitive ^-permutation group. By Lemma 6.1,

GAB(T)/{el. However,

Since

GHB(T)

is an W d e a l of

(G,T) may not be primitive.

supp(f)S (a,$). Now

af = 3. By

is £-simple.

Conversely, suppose that

Since (L,fi)

But in any £-group

then as each £-ideal i {e}

(by Lemma 6.3), element.

such that

3(g A f) = y.

B(ll)nGCL,

We now prove Theorem 6.2. and

and

Let C

GCB(T).

e < f £ G,

say

be any non-trivial congruence of

L( (J ) = {g £ G: (Vy)yg^y}

L( C ) = te}. But

Let

G,

is an £-ideal of

3 £ a5 (Otherwise

G

(G,T).

and hence

e / f £ L( 5 )).

Since the

union of a family of congruences is a congruence (Corollary 3B), (G,T) has a maximal proper congruence 5 Q , all non-trivial congruences.

Let

namely the union of

Q = T/£Q.

U C Q ) = (e} so

(G,ft) is a transitive primitive ^-permutation group. 6.1,

GHB(fi) / {e}

so

G£B(fl).

By Theorem 4A,

By Lemma

(G,fl) is

doubly transitive. To prove Theorem 6B, it is enough to show (by Lemma 1.10.1): LEMMA 6.4.

If

(G3Qt) is a triply transitive ordered

permutation group and G A B(Q) ^ {e},

then

([G f\ B(tt)>GC\B(Q)]>tt)

is a triply transitive ordered permutation group. Moreover> if ^1*^2 ^ ^ ^ B(Sl) and e / g £ Gy

h'h^vh

€ G nB(a) such that

ri1g'1f1'r21gf2'r31g~1f2'f'41gfr

there exist

h h

^ r 2^ = Thus

[G n B(Z)3G HB(Q)]

contained in every proper normal subgroup of

G

is

and is simple.

133 Proof: Let H = G nB(fi). Since [H,H] < G and {e} i [H,H] £ B ( f i ) , ( [H,H],ft) i s t r i p l y t r a n s i t i v e by Lemma 1.10.5. Let tL-j_^h2 € H and e j g £ G. Let a € supp(g) and supp(h.) Q(O,T) There exists

(i = 1,2). Without loss of generality, f £ [H,H]

such that

a < ag.

of = a and xf = ag. Now

suppCf'-hijf) S. (a,ag) so supp(g"1f"1hifg) £ (ag,ag2) (i = 1,2). Since

f tuf and g~ f" h-.fg have disjoint supports, they

commute.

Thus

[ h ^ l ^ ] = h ^ f •f~1h":L f -

f ' Y ^ - ^ ^ - ^ V ^ ' ^ V ^ where f± = f"1^, f2 = f"1, f- = f" 1 ^

and f. = f " 1 ^ ^ .

normal subgroup of H

contains

Since

f. € H (j = 1,2,3,4),

It remains only to show that

[H,H] is simple. Since

([H,H],fl) is triply transitive, it follows (with place of H

^

- [H,H] in

in the previous paragraph) that any non-trivial normal

subgroup of H-. contains contains

every

[H,H].

H... Hence

H-

[H-jIL] . But

[H ,H ] < H

and so

is simple.

Lemma 6.4 is due to Graham Higman [54]. We now prove Theorem 6C by a standard compactness argument. Let

z be the period,

By Theorem 4.1.1, exists Let

h £ G

a € fl and e < f,g £ G. Let 3 £ [a,az].

BG is dense in S. So if y £ supp(g),

such that

g£ = hgh" . Then

y < J3h < yg. Hence p

P

6-,,..., 3^ £ [a,az] 1 _n

element of U of

3f <_ 3g ;

such that

[a,az].

so f <_ g

supp(hgh~ ).

P

m€Z , m

on

[a,az]

m

ag >_ azf.

[a,az]

of Q. Hence there

g = g- . ..g5 6 3 1 n

By the periodicity, +

S. Thus for some m

f

in

g

fixes no fixes no point

If 3 €. [a,az],

then

and hence on ft. Consequently,

belongs to the £-ideal of G generated by g. Therefore

is ^-simple.

there

3 £ supp(g-) so {supp(g-): g € [a,az]}

is an open cover of the compact subset are

3

£

G

134 We next prove Theorem 6D. periodic primitive with period

So assume that z

and

(G,ft) is full

Config(°°).

For each

a £ ft, let *t(a) "be the filter of neighbourhoods of

a

(with

respect to the open interval topology).

So

CL/ \ = {g £ G:

(3A € *t(a))g £ G A )

G.

Clearly,

is an ^-subgroup of

U{G / y. a £ ft} generates G

G

(as a group).

is simple, it is enough to show that if

(L/ \ Q N

for each

generated by

g.

a € ft, where

Fix

a £ ft and

We first show there is ag = a Since some

Config(«>),

ag

z £ (a,x).

a < ag

such that

a,x £ A

Since

af = a.

ag~ ,ag £ {azm: m e %}.

such that

(G |A,A)

If

But

agz

So for / agz11;

ag~ z

£ (a,x) but

is non-pathologically doubly

transitive (Theorem 4.3.1), there is

e < h € Ga

such that

11

agz €supp(h)H A £ ( C J , T ) . Let f = [g,h] - g^-h^gh £ N. nu m -1, -1 , -1 m, -1 . , m m az f = az g h gh = ag z n gh = ahz = az ; so af = a. However,

agz f = az gh / agz , so

Now let But

h e CL/ N and

ness, if

6 € (a,x),

f|A

(if

generated by

be such that (a,x)£it(a)

A(A) with

f.

Since

Hence

h £ M,

f £ N, M C N.

af = a. and

6h = 6

By the full-

G/ .\. Since

belongs to the normal subgroup of

by Theorem 2G.

Then

and the claim is proved.

x < supp(h| (a,az)) < az < az).

A = (a,az) we may identify

f £ G/ \, h | A by

f i e

e j f£ N

h|(a,az) has bounded support

for all

G

without loss of generality.

z ,agz € (a,az) = A.

hence there exist

then

is the normal subgroup of

n

m,n £ 2E, ag

e J g £ G,

g € G.

e i f £ N

we are home; so assume (G,ft) has

N

Hence to prove that

A( A) generated

the normal subgroup of Thus

h £ N

G

and the proof

is complete.

Recall Example 1.10.3.

(G,l) is pathological. A minor

modification of the proof of Theorem 6D gives:

THEOREM 6.5. ag + m}.

Then G

Let G = {g £ k(B) : (3m £ 7L+) (Va) (a + m)g = is simple (but (G^S) is pathological).

135 Theorems 6C, 6D and 6.5 are due to McCleary ( [72a] and [78a]) as is the next example [73b].

EXAMPLE 6.6. Let H = {h 6 A(l): (Va €. l)(3m £ ^ + ) ( V n e 7L) (a + mn)g = ag + mn}.

Then

(H,l) is an ^-simple pathological

£-permutation group.

We leave the details for the reader (or see McCleary

[73b]).

For other examples of ^-simple pathological ^-permutation groups, see Glass and McCleary [76].

It is not known if Example 6.6 and

the examples in Glass and McCleary [76] are indeed simple (as groups). We now prove Theorem 6E.

Let Q = I

(3m €. 22 )(Va £ I )(a + m)g = ag + m } . that

and G = {g £ A(I ):

It is straightforward to see

(5,1 ) is a pathological ^-permutation group.

R(ft) O 5

is a proper £-ideal of

5,

so

G

However,

is not ^-simple.

We give another informative example (Glass [74] and McCleary [81c]):

THEOREM 6.7. (K

Let F

a cardinal greater than

a faithful pathological Proof:

be the free l-group on 13

possibly infinite).

generators

Then

F

has

representation.

We will only prove this theorem when

proof can easily be adapted to the case that cardinal.

K

The proof for K

K

K = AL.

The

is any infinite

finite is more delicate—see McCleary

[81c]. We now prove that for some

\\>,

(F^ ,Ql,i|0

is a pathological

^-permutation group. Let |B| = Hrs,

there is

(J>: OJ -> B

& > = cj>(m). open intervals of [a ,3 1 U L

and

A = {: a , 3 £ Q & a < 3 )

[Y

Define a sequence

Ql such that

,<5 ] £ J

B = A x A.

Since

one-to-one and onto. Let

(m €. a)).

{J : m € oo}

of bounded

L C J, G J. Q ... and Let

m mJ m m m bounded non-empty open intervals of n < n' and I < J < I (m£2£ ). -m m m

{I : n £ Z]

be any set of

n if Q such that I n < I n , Now J is ordermorphic to Q m

136 so there is

f £ A( J ) such that a f = y m m m m m Corollary 2L, there is an ^-embedding of F

and in

B f = 6 . By m m m A(Q) and so in

S

A(I ) (since

I

is ordermorphic to

{x : m £ a)} be the generators of xm

in

A(l n ) (n € Z ) . if

a

Let

f f

the image of

. e A(Q) by

a £ J a £ l

A(Q) generated by

n| > m.

{g^: m £

pathological ^-permutation group. \p: F-. = W; W 0

and

^

Now define

m n otherwise

if

m,n

Q)) (nfcZ). (n

Let W

be the ^-subgroup of

By the coding of The map ty: x

B,

(W,{J) is a

»—> g

gives

so (R, ,CJ,ijO is the required ^-permutation group. 0

We conclude this chapter by proving Theorem 6F. Let

(G,l) be the ^-permutation group of all bounded piece-

wise linear order-preserving permutations of ffi (cf. Example 1.2.3). [G,G]

Then

(G,tt) is doubly transitive.

is simple and

will be declared greater than

exceeds

1.

G£B(R),

([G,G],l) is a doubly transitive ordered

permutation group. However, we may make g £ G

Since

Denote this o-group by

e G*.

G

an o-group via:

if its left most slope So

G

and

G*

have the

137 same underlying set and group operations, but different orders. Now the group [G*,G*]

[G,G]

is simple; i.e.,

[G*,G*] is simple. Hence

is a simple o-group and so has a transitive representation

as an ^-permutation group.

But it cannot have a primitive

representation (Since it is an o-group, this would have to be the regular representation; hence

[G*,G*] would have to be Abelian

which it clearly is not—Lemma 1.6.8). Actually,

G = [G,G] as we now show (Consequently, the group

of all bounded piecewise linear order-preserving permutations of is simple). We will write each

are such that

£ ,...,£

[£.,£. + ] with slope Let

X

e f g £ G

V " nn

G

h\

where

is linear on each interval

n. (0 <_ i <_ n),

be the subset of y

g

as

and

supp(g) Q (£Q,

comprising those elements of the form

-l . Note that

g

£ X

if

g £ X.

To prove that

G = [G,G] we establish and

(I) each element of

G

is a product of elements of

(II) each element of

X

is a product of two commutators.

Proof qf_ (I):

By induction on

n.

X,

Suppose

f 9 K

§..

L

1

138 h =

<

>

with

n

/ I .

n

L o ••• % j f = <

?

a

and

«

- ,

£ f = £ h

6 Y

and

n

n

JO

=

3

•=

= h. Since

f€ X

IX x

u

.

Since

n-1

y = —~,

mils fg .

n

0

•••

and g has no more than

V2"

n

the convexification of its support, induction product of elements of X. that

h~

,h

n

o-

Let

n

. Now

£ h - £

n

~l

z

.yj^i.„

£ h > £ .

n

x

g = r

€ f - £ r

Assume

n

\

linear pieces in

shows that

h

is a

If £ h < £ , the above argument shows

is a product of elements of X. Since

X

is closed

under inverses, (I) follows.

P P O Q / qf_ ( I I ) :

Consider

the

element

h

=

<

>

f

L n o ni J

c

X

.

Since the inverse of a commutator is a commutator, we may assume n Q > 1. We first suppose that

fa 0 a x a2">

_

n Q n 1 > 1. If

Fo

6

a f<

l l

6g = a., then a routine calculation shows that

R " 16

-i -i

[g \ f X ] = / = non-,^

let

5f

3Q = ^

So any such

f =

\ a \

a

1 Q

h

^ . Hence h = [g \ f x ] if

= Co^

,

-i -i

"I^I

Y

<5 = C-,

and

aQ = - 0 - 2 - ^

is a commutator.

B 0 ^ 2 \

with

B

o'Boni

a^

and

= £p.

This i s c o n s i s t e n t :

a, = ^

U±

- tQ)

f € X

i s a

If non-, £ 1> let

> x > B

o"r

So

+

aQ.

139 a

commutator.

a

gfg

= <

Let

a

0

g = <

lf

f

= ^

^

Thus

00

c^f" g~

a o

^

2g ^

^ > .

2g"

^ *

Then

a commutator too.

=

a commutator.

Note that

Sg > a,f as

hT = pfp~ -fr = CQ,

a^"

= ^1

and

6gf" g"

= £ 2 , we get

This is consistent:

let

— F

Bo- 1

6 = a0

a

a product of two commutators.

fir =

lg"

^

can be verified.

a

lg

a2i

0 1

If we put

a Q

w

0

h = hr,

a

g

6 Let

O

+

'

a

l

= Y

(1 -

0 S l • •a 0a (0Y(0Y 0 "

ni

1}

and

)

5

so (II) is

proved. //

This example is due to Chehata [52]. The only examples of £-simple £-groups that v/e know of that have no faithful primitive representation are o-groups!

Other

examples than the one above can be found in Dlab [68] and Feil [SO].

Indeed, Dlab produces uncountably many.

140 CHAPTER 7 UNIQUENESS OF REPRESENTATION

We saw in Chapter 2 that if ft and eous chains and orbit of

T

A(ft) is £-isomorphic to

A(ft) in ft (Theorem 2D).

are doubly homogen-

A(T), then

Appendix I. unique.

G

can be

(G,ft,i|/)—Holland's Theorem,

In this chapter we wish to consider whether

More precisely, if

(G,ft) and

groups, then are they £-isomorphic?

is an

We now wish to consider the

more general case. We know that every £-group represented as an ^-permutation group

T

(G,T,i|O

Q

is

are ^-permutation

In order to have any hope of

an affirmative answer, we must insist that the ^-permutation groups are transitive

(If

A

is any well-ordered set, then

is an ^-permutation group if A £ A,

g €. G). Consider

Let

?

(G, ft) is, and

Xg - X for all

Transitivity is not enough, however: (A(l),t) a doubly transitive ^-permutation group.

be any non-principal ultrafilter on 7L

P = {g £ A(ffi): (3X G *)(Vn €. X ) ~ g. = -}, A(R).

(G,A U ft)

Since £

Hence P S A ( l L °

is non-principal, each

Therefore

As in Example 7.1.8 below,

X £*

is cofinal in 7L .

(~\ g~\g o C\ g ^ A d L g = {e}. g£A(a) g£A(R) ° R(P) is a chain.

is an ^-permutation group. The orbits of non-trivial congruence of

and let

a convex ^-subgroup of

(A(l),R(P)),

Thus

(A(l),R(P))

A(l)^ give rise to a

so

(A(l),R(P)) is not

£-isomorphic to (A(l),l). In general, if 5 tation group then

(G,ft) and

is a non-trivial congruence of an £-permuL( C ) = {g €. G: agCa for all

L( 5 ) is an £-ideal of

group if and only if

G

L( 5 ) = {e}

the congruence defined, then

and

a €. ft},

(G,ft/C ) is an ^-permutation

(In the above example, if C

is

L( 5 ) = {e}). So, in order to obtain

any uniqueness results, we must further insist that

(G,ft) is

HI weakly primitive; i.e., L( 5 ) = {e} singletons.

only if the C-classes are

This is a sort of minimal condition on ft.

A transitive ordered permutation group is said to be locally primitive if it has a minimal non-singleton congruence. The classes of such a congruence are called primitive segments. We will say that a transitive ordered permutation group locally enjoys a property if it is locally primitive and the minimal primitive component enjoys the property. A transitive ordered permutation group

(G,ft) is said to

have the support property if for each congruence whose classes are non-singletons and for each e i g G. G

such that

£

of (G,ft)

a €. ft, there exists

supp( g) C a £ . Note that the interval

support property implies the support property but not conversely (Example 1.9.2).

Also if

(G,ft) is a transitive locally primitive

^-permutation group having the support property, then it is weakly primitive. If

(G,ft) is a transitive locally primitive ^-permutation

group, let 5 Q be the minimal non-trivial congruence of (G,ft). The local completion £ ^ for some

£Q-class

of ft is defined to be

{a 6.ft:a € A

0

A}. Observe that ft ^ ft, and if

(G,ft) is

primitive, the local completion is just the Dedekind completion. The main results we wish to prove are: THEOREM 7A.

Let

(G^Q) be a transitive locally primitive

t-permutation group having the support property. If not locally pathological and

(G^Q) is

(G3T,ty) is a weakly primitive

transitive t-permutation group, then for some a £ ft0., there is an ordermorphism <j>: aG = T and

(ty*§)

such that gty - g$ for all g £ G

is an t-permutation group isomorphism between (G,QLG)

and (G9T,ty). In the above example, generally:

(A(l),R(P)/ C ) = (A(t),l).

More

142 THEOREM 7B, If 7A and

(G3Ty^)

(G3Q) satisfies the hypotheses of Theorem

is a transitive Z-permutation groups then there

exists a congruence 5

of

(G3T3^)

such that

(G>T/£ 3\\>)

is

weakly primitive and Theorem 7A applies. Theorems 7A and 7B are unpublished improvements (due to McCleary) of results of Holland [65a]. associating points of

Q

and

T

They will be proved by

with certain algebraically

identifiable convex ^-subgroups of

G.

This is quite different

from the proof of Theorem 2D, but is in the same spirit. As immediate consequences we obtain: COROLLARY 7C. Let

(G3n) and

(G3T3^)

primitive t-permutation groups. Then

be transitive

(G3Q) and

(G,T3ty)

are

both regular_, both periodic., both pathological^ or both (nonpathological) doubly transitive.

COROLLARY 7D.

If

(G3Sl) and

(G3T3\\>)

satisfy the hypotheses

of Theorem 7A and one of the primitive segments of

(G3 Q) is

Dedekind complete3 then Q, = T. COROLLARY 7E. If

(G3Q) and

(G3Sl3ty)

satisfy the

hypotheses of Theorem 7A, then ty is the restriction of an inner automorphism A(Q°).

We will also prove an analogue for the ordered permutation group case due to McCleary [78b].

It necessarily has a more

restrictive hypothesis. THEOREM 7F. Let

(G3Q) and

(H3T) be triply transitive

ordered permutation groups containing strictly positive elements of bounded support. Then each group isomorphism ty between G and

H

is induced by a unique ordermorphism or anti-ordermorphism

<)>: Q = f

via gty = (J>~ g$ (g £ G).

reverses the pointwise order.

Thus ty either preserves or

143 7.1. l-PERMUTATION GROUPS. Let

C be a convex I-subgroup of an £-group

prime subgroup of g e e

or

g

G

€. C.

right cosets of

C

if

g-^'g? ^

G

and

g

A g

l

?

G.

= e

C

is a

^P1^

This is equivalent to R( C) , the set of in

G,

is linearly ordered in the induced

order (see Appendix I, Lemma 2).

If

C

is prime and

/^] g Cg = {e}, then C is a representing subgroup. In this g£G case, (G,R(C)) is transitive ^-permutation group; conversely, if (G, ft) is a transitive ^-permutation group, then representing subgroup (cf. Theorem 1.5.1).

G

is a

One of the key lemmas

we need is the following generalisation ( due to Holland [65a] ) of Theorem 4.1.1. LEMMA 7.1.1. group. Then

Let

(G^Q) be a transitive t-permutation

(G^Q) is weakly primitive if and only if

maximal representing subgroup of Proof: Since

G.

C 9 G . Define 5

g £ G.

Then

C

y£6

if

congruence of

is a

G.

Hence

L( JJ )

=

for some

L( C ) £ C. Then e

"f ^ since

C

is a

(G,ft) is not weakly primitive.

(G,ft) is not weakly primitive, let

5

be a

(G,ft)whose classes are non-singletons, with

L( ? ) = {e}. Let of

G

y,6 €. aCg

(G,ft) and

the I?-classes are non-singletons but

Conversely, if

is a

Assume there is a representing sub-

by:

is a congruence of

representing subgroup of

G

a £ ft.

(G,ft) is transitive, each

representing subgroup of group

G for all

a € ft. Then

(G,ft). Moreover,

G Q G, *\

(~\ g

a representing subgroup of

G

by the transitivity

G/ k?xg = L ( 5 )

=

properly containing

{e}

so

G/ y> \ is

G .

This algebraic identification of the stabilisers is exactly what we need.

LEMMA 7.1.2.

Let

(G,ti)

and

(H,T) be transitive

Z-permutation groups and suppose that \\> G

onto H. If

G-\p = H

is an l-isomorphism from

for some T € T

and

a G ft^ define

144 <j>: aG = T and and

1

by: (ag)<\> = TQ(gty). Then

gty = f g$

for all g £ G,

(ty>§) is an Z-permutation group isomorphism between

(G,aG)

(H,T)>

= x Q (fi|;);

so

there exists (af)<j> = x

< > j i s well-defined and one-to-one. h € H

and

such t h a t

x = xJi.

x(<J>~" g<|>) - x(gi^).

<> j~ g$ = gty (g €. G).

If

If

Hence

o ^ £ agp,

If

x £ T,

f = hty~ ,

then

< > j i s onto and

let

e <_ f £ G be such t h a t

otg,f = agp. Now so

< J > preserves order.

Thus

(^,<|>) is an ^-permutation group

isomorphism.

LEMMA 7.1.3» group, and C

Let

be a convex Z-subgroup of G

for all a £ ft. Then Proof: each Now

(G3ftjbe a transitive

Let

{supp(f-): a £ a £ 3}

\jsupp(f- ). Let .

1

say

e < f- £ C a

C $H G-

supp(g) C [a,3].

such that

For

a £supp(f-). a

is an open cover of the compact subset

of ft. Hence there are

[a,3] £

such that

B(Q) H G £ C.

e < g £ B(ft) A G,

a £ ft, there exists

[a,3]

Z-permutation

a.,...,a

£ [a,3]

such that

f = f- ---f- £ C. Then

Ot.

01-,

1=1 i fa,3] Q. supp(f). So for some e £ g £ f® £ C. Thus g £ C.

CX

1+ n m £ % , 3 £ ar . Hence

A routine modification of the above proof (cf. proof of Theorem 6C) gives:

LEMMA 7.1.4. tion group and C C ^G-

Let

(GyQ) be a periodic primitive Z-permuta-

be a convex Z-subgroup of G

for all a £ ft. Then

We know that

G

such that

C = G.

is a representing subgroup of G

a £ ft. We now extend this to show:

for all

LEMMA. 7.1.5.

If

(G,ftjis a transitive locally primitive a £ ft0., then G- is a representing

t-permutation group and subgroup of

G

and either aG is dense in ft or

Proof: If

A

aG/ x is dense in

is a primitive segment with A,

or equal to

A

primitive so apply Theorem 4-1.1).

if

Hence

aG = ft.

a € A,

then

A ^22 ((6/ \,A) is aG

is dense in ft0 or

equal to ft. G- is a prime subgroup and if e < g €. G-, a a exists 3 £ ft such that $g i 3. Therefore there exists such that Thus G-

G-

1

3 <_ af < 3g. Thus

1

a < 3gf" £ afgf" ;

contains no £-ideal of

G

is a representing subgroup of

other than

there f £ G fgf"1 £ G-.

so

{e}. Consequently,

G.

We now give a partial converse of Lemma 7.1.5 which, with Lemma 7.1.1, is key to proving Theorems 7A and 7B.

LEMMA. 7.1.6.

Let

(£,ftj be a transitive locally primitive

t-permutation group that has the support property. If

(Gjti)

is

not locally pathological^ then every representing subgroup of G is contained in some G-j a €. ft0. Proof: Let 5 Q (G,ft).

Let

N = {g £ L(5 Q ): supp(g|a$Q) = 0

finite number of aC Q

be the minimal non-singleton congruence of

for all

afiL}

an(i

for all but a

{g € N: supp(g|ot5L) is bounded in

a € ft}. By the fundamental triviality, M

are ^.-ideals of

G.

Let

and N

C be a representing subgroup of

reductio ad absurdum, assume all

M =

C

G.

For

is disobedient; i.e.,

C *jL G-

for

£(6/ - Ja

a € ft and

0

a £ ft . Case I.

(C n G/ - v)/G -

(aC)

a £ aS Q . Then

C

contained in

a$ Q

the fact that

C

that

aC

contains every element of (If

C Q. G, - y

N QC

M£C

if

if

then

G

whose support is

C£Gj

is disobedient; so there exists

a& 0 < a5^h. By the convexity of

Hence

for some

(°^

C,

which contradicts e < h £ C

such

the claim follows).

(G,ft) is not locally doubly transitive, and

(G,ft) is locally (non-pathological) doubly transitive.

By the support property,

N i {e}

(and M i {e}

if

(G,ft) is

146 locally (non-pathological) doubly transitive). are ^.-ideals, C Case II.

Since

is not a representing subgroup of

(C n G/ - N )/G - £ (G, - J-

N

and

M

G.

for all

a € ft and

a £ <X£Q»

If

(G,ft) is locally regular,

(Gfnif> x)- = {e}

has no proper convex subgroups for all C 2L(CJ,

a non-trivial £-ideal of

representing subgroup of If If

G.

Thus

G, ~ v

a £ a6L. Hence

C

is not a

G.

(G,ft) is locally periodic, then again

Lemma 7.1.4)* then

a £ ft and

and

C 2 L(C ) (by

a contradiction.

(G,ft) is locally (non-pathological) doubly transitive,

C 2M

(by Lemma 7.1.3) and again we get a contradiction.

The lemma now follows by Theorem 4A.

By Lemmas 7.1.5 and 7.1.6: COROLLARY 7.1.7.

If ft is homogeneous and locally

primitive3 the maximal representing subgroups of precisely AfoJ-j

AfftJ are

0

a £ ft .

We now prove Theorem 7A. So assume that ty is an ^.-embedding of that

(G,T,i|O

group.

Let

G

x n £ T.

By Lemma 7.1.1,

representing subgroup of

GiJ;. By Lemma

(G\\))

is a maximal

Lemma 7.1.2,

° 7.1.6,

Ga (G*) = G-* by the maximality of T a 0 __

there is

-1

(GiJ/) ij;~ Q G- for T a 0 is a representing subgroup of G (by

a £ ft0. Since

Lemma 7.1.5),

A(T) such

is a weakly primitive transitive ^-permutation T

some

into

cf>: aG -> T

group isomorphism between

making

(G,aG) and

kind. We now prove Theorem 7B.

(Gi|/) . By T 0

(i|;*(|)) an ^-permutation

(Gip,T) of the desired

147 Let

T £ T.

Then

(Gij;)

so

(Gi|/)Ti|r £ G-

(G,Q)

is a representing subgroup of Gip,

^

-l

for some

a e fi° (by Lemma 7.1.6).

Since

is locally primitive and has the support property, so does

(G,aG).

Hence

(G,aG)

is weakly primitive.

is a maximal representing subgroup of

G.

By Lemma 7.1.1,

G-

Now there is a one-to-

one containment preserving correspondence between the congruences of

(Gi|/,T) and the convex ^-subgroups of

(Gi|>)

(Theorems 1.6.1 and 1.6.2).

corresponding to

G-ip.

Then

GI/J which contain

Let $

"be "the congruence

(G,T/£,i|>) is weakly primitive by

the converse direction of Lemma 7.1.1 and the proof is complete. We conclude with an example to show that the conclusions of Theorem 7A are false if

(G,Q)

is pathological.

The example is

due to Glass and McCleary [76]. EXAMPLE 7.1.8. (a + m)g = ag + m}

G = { g U ( l ) : (3m6 1 + ) f t / a a )

Let

So ft = ft0 = Q = I.

(Example 1.10.3).

By

Lemma 7.1.1, it is enough to find a maximal representing subgroup C

of

G

with

C/G

primitive with

(a e R ) ,

C = G n0.

(i e T ); i.e., for

Let

for then

3. = (4

1

i > 1,

3.

1

(G,R(C)) is weakly

- l)/3

A

1-1

. Then

4^

00

A £ A 2 ...

and

id

f} A. = 0. i=1

there exists an ultrafilter ^

+

A. =1 3. 1

is the second smallest positive

1

element of

and

Hence

1

on

1

such that each

Let

P = {g C G: Fx(g)€ # } . Then

P

is a prime subgroup of

(If

g± A g 2 - e,

Fx(g o ) € f). let

Fx(g 1 ) U Fx(g 2 ) = R £ * ; so

Let

f € GA\Ga i (3€ffi)). Then

a £ R.

Choose

with period f £ P as

41

i£2

+

A. £ * .

Fx(g1)€^

so that

Fx(f)^Ai£^.

Thus

or

a ft A.

( 3 + 4 1 )f = Bf

(i.e.,

and

+

P £ Ga

G

41 for all

a € 1. Let

C 2 P be a prime subgroup of

to not containing 6.5), m

C

and let

z: a »—• a + 1.

is a representing subgroup. +

n £ %>

a prime subgroup of So

C

Since

be such that G

G

maximal with respect

G

is ^-simple (Theorem

Let

Og <_ n.

is equal to

G

e < g €. G Then

have period

g <_ z m + n .

only if it contains

is the desired maximal representing subgroup of

G.

Hence z.

Note

148 that, by Theorem 4.1.1,

(G,R(C)) is primitive and hence

pathological by Corollary 7C. 7.2. ORDERED PERMUTATION GROUPS. Throughout this section, which is devoted to proving Theorem 7F,

(G,ft) and

(H,T) will be triply transitive ordered permuta-

tion groups containing strictly positive elements of bounded support. m GZ g € G

Thus

(G,ft) and

(H,T) are m-transitive for all

(Lemma 1.10.4) and if +

u G"

implies

\p: G s H +

gty £ H U H~

(as groups), then

by Corollary 1.10.8

(G~ = {g € G: g < e}). Moreover, by Corollary 1.10.8, if e < g Q G. G

and

e < g-ti (e > g~fi) for some g Q € G,

then ty

preserves (reverses) order. Again we will use stabilisers to identify ft and

T

(The

techniques of Section 2.1 do not necessarily apply here since {a €,ft:(3g € G)sup( supp(g)) = a} may not even include ft). A convex subgroup whenever

g ,g2 £ G

and

C

of a p.o. group

g. A g

= e,

G

then

is called prime if

g. £ C

or

g € C.

This is a generalisation of the definition for £-groups except that C

is not necessarily a sublattice.

exist in

G.

If

C

Note

g-. A g^

may not always

is a prime subgroup of p.o. group

G

and

f} g~ Cg = {e}, we say that C is a representing subgroup of gCG Note that if (G,ft) is a triply transitive ordered permutation group containing a strictly positive element of bounded support, then

G

is a maximal subgroup of

G

for each

a € ft (Theorem

4.1.5). LEMMA 7.2.1.

Let

(£,Pj

be a triply transitive ordered

permutation group having strictly positive elements of bounded support. Then the maximal representing subgroups of

G are

precisely the stabilisers G- for which G- is a maximal subgroup of G. Proof: Let let

a € ft. Clearly

3 = min(af,ag) > a.

e < h € G

such that

Lemma 1.9.4.

So

G-

G-

is convex.

If

a < af,ag,

By Corollary 1.10.6, there exists

supp(h) ^ (a,3).

Hence

e < h < f,g

is prime. Moreover, since

aG

by

is dense in

G.

5

1-49 A g ^ G - g = (~\ G- = { e } . Therefore

(by Theorem 4.1.3),

each

a g€G g£Gag is a representing subgroup.

G-

We now show that if all

a £ ft C$gG-,

B(ft) O G < G,

Let

C

there is

c? > e

e < f,g € G

(Let

and for Since

G

and

C % G-

such that

6c^ > 6.

for all

For each We claim we may

6 < a < T < 6c^. There are

6 € supp(f) 9 [a,a] and

supp(f) H supp(g) = 0.

Now

Hence

f A g = e

since

f £ C

g £ C.

or

Thus

e < Cggc^ £ C\Gj establishing the claim).

{supp(c-r) O [a,3]: 5 £ [a,3]} subset

G

(cf. Lemma 7.1.3).

supp(b) £ [a,3].

<J,T € ft with

such that

or

say

c^ £ C

6cj € supp(g) Q fx,TCg]. e < f £ C\Gg

B(ft)OGC.C

is a prime subgroup of

b € G HB(ft),

5 £ [a,3], assume

then

is a prime subgroup of

the lemma will follow.

Assume a £ i

C

Now

is an open cover of the compact

fa,3]

of fl. Hence there exist a,,...,a £ [a,3] such n [a,3] £ (J supp( c- ). Let c = c- ... c- £ C. Then c > e a a a i=l i l n

that

and fixes no point of m £ Z , so b £ C

Ti in

[a,3].

Thus

c"m < b < c

for some

as required.

We next note that the proof of Lemma 7.1.2 shows: LEMMA 7.2.2.

Let

(G3Sl) and

(H,T) be transitive ordered

permutation groups and suppose ty is a group isomorphism from G onto H. If <J>: aG + T

G-\p = H

by:

for some x £ T

and a £ ft, define

(ag)<$> = i (gty). Then gty = f g<\>

for all g € G.

We now prove Theorem 7F. Let

\\>: G =* H

a={a£ft:G-

be a group isomorphism.

is a maximal subgroup of

defined similarly.

Note ft 9 fL and

Let

G}, and

T £ T^.

T^ be

By Lemma 7.2.1

-\p: a £ ?L} = {H-: T £ T^}. Moreover, distinct elements of ft (T) have distinct stabilisers by Corollary 1.10.6. define

<j>:ft-> T

by setting

a(j> = T where

Thus we

G i|> = H-.

By Lemma

150 7.2.2,

c> j maps t h e s e t

gty = <>| g<|> ( g € G). let

o n e - t o - o n e onto t h e s e t

By C o r o l l a r y 1 . 1 0 . 8 ,

o r d e r and hence so does <>j i 4>p,

ft .

Finally, if

B £ ft be such t h a t

g €. GAG

_^.

Now

(3^2/^^^/ -

(3<Ju)(gi|0,

a contradiction.

3<j>

TH and

ij; p r e s e r v e s or r e v e r s e s 4>_, <J>^

/ 3<|>9.

induce

\Ji

and

There e x i s t s

M

&^2

Hence the <|) inducing

i/; is unique.

CHAPTER 8 POINTWISE SUPREMA AND CLOSED SUBGROUPS

If

(G,ft) is an ^-permutation group and

a(f v g) = sup(af,ag) for all V g. i€.I X

exists in

G,

a £ n.

If

f,g £ G,

then

{gi: i 6 1} C. G

and

then is it still true that

a( V g.) = sup{ag.: i £ 1} for all a £ fi even when I is X i£I X infinite? If it is true whenever {g. : i £ 1} C G and \/ g. 1 i € I :L exists (in G ) , we will say that (G,fi) has pointwise suprema. It is not hard to show (using

/\ g. = ( V g~ )~ ) that (G,ft) i£I X i£I X has pointwise suprema if and only if it has pointwise infima. Unfortunately, even For example, let

(A(fi),fi) can fail to have pointwise suprema.

T = {x ,Tp} with

T

< T2

and

SI = T ? (£.

Let

A(fi) gg £ A ( f i ) bbee ddefined e f i n e d bbyy g : ( x , q ) •-* ( i , q - ^ ) ( T £ T , q £ ( S ) . nn n n + 1 W Then e = W g but sup{( T ,0)g : n £ X } = sup{(x ,- - ) : n n£2Z n£2Z n£2Z n

} = ( T ,0) (i = 1,2). Hence

( T ,0)( \J ,g) = (x ,0)e = * n£^ n

±

(x2,0) i sup{(x2,0)gn: n £ ^ }. However, THEOREM 8A.

If ti is homogeneous3

A(Q)

has pointwise

suprema.

Let G

H

be an ^-subgroup of

if whenever

{h±: i £ 1} 9 H

THEOREM 8 B . A f o J -

i s closed

G. and for

H

is said to be closed in

g = \J h± £ G, all

a £ ^

then

/ o r any

g £ H. chain

Q.

Pointwise suprema and closed stabilizers are very closely related.

152 THEOREM 8C. If

(G3 tt) is an l-permutation group that has

pointwise suprema^ then G-

is closed for all a €. ft. If (Gyti)

is transitives the converse is true.

Theorem 8A follows at once from Theorems 8B and 8C. In Section 8.1 we will establish Theorem 8B; and in Section 8.2 we will prove Theorem 8C. We will also investigate there when the converse direction in Theorem 8C holds (transitivity is not necessary).

We

will further examine the relationship between pointwise suprema, closed stabilisers, and the £-group concepts complete ^-subgroup and complete distributivity.

We complete Section 8.2 by examining

the primitive case. THEOREM 8D. If

(G, U) is a primitive l-permutation group,

then it fails to have pointwise suprema (closed stabilisers) if and only if it is transitively derived from a pathological l-permutation group.

In Section 8.3 we locate the closed convex ^-subgroups of an ^-permutation group £-group

G

g,e H

or

Recall that an ^-subgroup

is prime if whenever gp € H

subgroup of group.

(G,ft).

G

g.. ,gp €. G

(see page 143). Clearly

for all

a € S whenever

and G-

H

of an

g. A gp = e, is a prime

(G,ft) is an ^-permutation

We prove a "converse" of Theorem 8B.

THEOREM 8E. For any chain Q,, the closed prime subgroups of

AfpJ

other than A(Q) are precisely the stabiliser subgroups

k(Q.)-> a € 5.

Let Let

5

be a congruence of the ^-permutation group

L( 5 ) = {g € G: Ag = A

is an £-ideal of if « 1 £ 5 2 ,

G

for all $-classes

A}. Then

(i.e., a normal convex ^-subgroup of

L(51)SL(C2).

We will show:

(G,fl). L( £ ) G) and

153 THEOREM 8F. Let

(G3Q) be an l-permutation group and H

a closed l-ideal of G. Then H = L( G) of

for some congruence C

(G, Q). Finally, we generalise Theorem 8D.

extra condition.

To do this we need an

An ordered permutation group

(G,ft) has the

support property if for every non-singleton natural block

A with

G/ x i {e}, there is

(cf. the

e i- g € G

such that

supp(g) c: A

transitive definition given in Chapter 7).

THEOREM 8G. Let

(G3Q) be an l-permutation group having the

support property. Then G- fails to be closed if and only if a belongs to some pathologically derived primitive segment.

The results in this chapter are due to McCleary [69], [72b], [72c], [73b] and [76]. 8.1. CLOSED STABILISER SUBGROUPS. In this section we wish to establish Theorem SB, due originally to Lloyd [67]. Rather than give John Read's simplification of Lloyd's proof [71], we will employ Theorem 4C (or more specifically Corollary 4-4.1) after reducing the proof to the primitive case by using Theorem 3C . This approach is due to McCleary [76]. It has the advantage of being routine (given the results of Chapters 3 and 4) and, unlike Lloyd's and Read's proofs, does not require any outside results from the theory of ^-groups. Let

(G,n) be an ordered permutation group and

congruence of 3

(G,fi).

Define

5

by:

a£3

if

C

a = 3

a or

a and

belong to the Dedekind completion of the same C-class

(a,3 €. ft). This yields a congruence of (G,fl). The following lemma is of central importance in this chapter and is of a fundamental nature. LEMMA 8 . 1 . 1 . {g.: ^

If

i €. 1} G> G and

(Gs ftj is an l-permutation g = \J g • ^ * iei

(in

G) 9

group3

then for

each

a € O,

154 let a = sup{ag.: i £ 1} £ ft. If £ is a congruence of (G,ftJ3 a & agG and e ^ f £ G, then supp(f) <£ [inf(a$),inf(ag$ )). In particular3 no element of G has support contained in Proof: We prove only the special case when each 5 -class is a singleton—the general case is proved similarly. e i f 6 G

supp(f) £ [a,ag).

and

Since

1

f v f"

Suppose and

f

have the

same support, we may assume f > e. If Y [ag , a ) , then -1 , a ) , then >_ yg. for all i € I, and if 6 € [ag 1 " -1 for all i 6 1. Hence g. < gf -1 < g for <5gf -> a -> ag.l > &g. l I a contradiction. all i £ I. Thus g = \/ g. gf < iI X Let

(G,fi) be an ordered permutation group.

to be depressible if

to G;

i.e., if

g €. G

implies that every bump of n

A = Conv {ag

if h £ G where

(G,ft) is said

for some

g

belongs

a £ ft, then

3£A For example, (A(ft),ft) is

3h = if

6 Sf A

depressible for any chain ft, but Examples 1.9.2 and 1.10.3 are not. Note that if (G,ft) is depressible, then so are all its primitive components. Hence if (G,ft) is a depressible

^-permutation group, its components are either static, integral, or transitively derived from "regular" or (non-pathological) doubly transitive primitive ^-permutation groups

(Here is where we use

Corollary 4.4.1). The following theorem implies Theorem 8B (at least if a £ ft): THEOREM 8.1.2. groupj then G

If

(£,ftjis a depressible l-perrmtation

is closed for all a £ ft.

Suppose, for reductio ad absurdum, that and

g =

V

g.

exists in

G but

g $ G .

156 Since

ag >_ ag. = a (i € I),

ag > a.

we use Theorem 3C ). Clearly

K

generality, we may assume that i G I,

3 £ a£ . Let

T

We claim that

G

h £ G

^ K

are dense in

v

G/ T ,N = "te)-

we may assume that

and

Let

f

(ag. = a,

Thus

triviality.

G/T, x = {e},

K

(Theorem 4.1.1).

e < f € G/™, x. By depressibility,

so

l

i € I).

a£ . If

is not integral since aC /£

supp( f) Q T h 9 [a^Tr^agC^).

can exist

for all

such that

IV

no such

3g = 3

be a G -class contained in

The same is true if

then the orbits of

K = Val(a,ag) (Here

is not static. Without loss of

3g. = 3

is integral, then there exists Th ^ [a£K,agCK).

Let

a£^g. = a£Tyr for a l l j\ l

and so

But by Lemma 8.1.1,

IV

iv

G /rr \ = {e} by the fundamental

But this was true for any € -class

T ^ a£ . Since

any component of a depressible ordered permutation group is depressible, we have reduced the problem to the primitive case. So assume group with

G

a subgroup of

(G,ft) is a primitive depressible £-permutation not closed. BL

If

G

were an o-group, it would be

Hence its only convex subgroups are

{e} which are closed.

Hence

G

and

(G,ft) is transitively derived

from a (non-pathological) doubly transitive £-permutation group. Let

e < f & G

where

have bounded support, say

3 n ,3 9 £ otG. There is 1

32h = ag.

Hence

e < h

fh

h £ G and

supp(f) 9 ($-i>$2^

such that 1

supp(h

3nh = a

fh) £(a,ag).

and This is

the desired contradiction (by Lemma 8.1.1) and the theorem is proved. Note that if

(G,fi) is primitive and depressible, it

clearly has closed stabilisers (by Corollary 4.4.1 and Lemma 8.1.1).

Hence, in the proof of Theorem 8.1.2, our goal was to

reduce the general case to the primitive.

This we accomplished

by using the structure theory of Chapter 3. We will adopt this strategy elsewhere—especially in the proof of Theorem 11A.

157 Let

H

be an ^-subgroup of an £-group

{h.: i £ 1} C H we say that

has a supremum in H,

H

G

If

G

is a complete l-subgroup of

is closed for all a € ft.

{g.: i € 1} C G 1

g=

If whenever

is a complete Z-subgroup of G.

COROLLARY 8.1,3then

G.

this is the supremum in G,

and g = V g. (in G).

a

X

V g ,i (inA(^)). Since {g.: i £ 1} £ A(ft) I a , g€A(ft) a

by

iei

Theorem 8.1.2.

Hence

g € G O A(ft) = G ; i.e., G a a a

is closed.

8.2. POINTWISE SUPREMA. In this section we wish to complete the proof of Theorem 8B as well as prove Theorems 8C and 8D (due to McCleary [72b], [72c] and [73b]).

We will also thoroughly investigate the relationship

between pointwise suprema, closed stabilisers, complete £-subgroups and complete distributivity: Let

J

denote the set of functions from the index set I

into the index set

J.

An £-group

G

is said to be completely

distributive if

/\ \/ g. . = \/T A g*f( M whenever { J i € I j £ J 1J f € J i£I {g..: i £ I,j £ J} C G and all the indicated suprema and infima

exist (Equality does hold if

I

and

J

are finite).

Byrd and

Lloyd [67] have proved (Appendix II Theorem 2) an £-group completely distributive if and only if

G

is

D(G) = {e}, where D(G)

is the intersection of all closed prime subgroups of

G (see

Appendix II for a proof). We will prove the following strong version of Theorem 8C (at least for

a € ft).

THEOREM 8.2.1.

Let

(GyQ) be an l-permutation group.

of statements (1) to (4) implies the next.

If

transitivej (1) to (3) are equivalent: (1)

(G9Si) has pointwise suprema.

(2) G

is a complete t-subgroup of A(Q).

(G3Q) is

Each

158 (3) G is closed for all a € ft. (4) G is oompletely distributive. First we prove an easy technical lemma: LEMMA 8.2.2. Let a € ft be such that G

(G>Q) be an l-permutation group and is closed. If g = \J g. G. G and

a = sup{ag.: i £ I], then there does not exist h £ G such that a <_ ah < ag. Proof: Assume the hypotheses and suppose that for some h £ G, a <_ ah < ag. Without loss of generality, h < g (take

h A g). for all

Now

ag. <_ a <_ ah

i £ I. But

closed, gh"

for all

1

gh" v

e

v e 6 G . Hence

=

i £ I,

so g.h~

v e £ G

1

V (g.h" v e) and since G is i£I ^ ^ agh~l <_ a, a contradiction.

We can now prove Theorem 8.2.1 (and hence Theorem 8C for (1) -> (2). h >_ g.

for all

If g =

\ / S ' (in G) and h e A(ft) where i£I 1 i €. I, then ah >_ sup{ag. : i £ 1} = ag for all

159 a £ ft (since g = y

g±

(G,ft) has pointwise suprema).

Hence

h >^ g

so

(in A(ft)).

(2) + (3). By Corollary 8.1.3. is (3) •* (4). D(G) £ fi{G : a £ ft} = {e} since each G a a closed and prime. (3) •> (1). (G,ft) transitive: If g = \/ g. were not 16.1 X pointwise, there would be a € ft such that ag > a = sup{ag. : i € I}. Let transitivity, there exists

a <_ 3 < ag

h €. G

with

such that

3 ^ ft. By

ah = 3, contra-

dicting Lemma 8.2.2. In order to complete the proof of Theorems 8B and 8C, we establish: LEMMA 8.2.3.

Let

(Gjti)

be an l-permutation group. If G-

is not closed, let {g.: i € 1} ^ G- with g = is

G- is not closed for all

V 9 • & G-. Then » — -j. is

a

3 £ (o.,ag), and no

OL

e ^ f € G has

support contained in (a,ag). Proof: Without loss of generality,

e <_ g.

for all

i € I.

160 For reductio ad absurdum, suppose G^ is closed for some 1 - 1 3 £ (a,Sg). Then G_ ^ = gGgg~ is closed and 3g < a < gg"1 < 3 = (Bg"1) V g.. Since G is closed, g. .££GG J i ee ii 1 ig ig"1 for some j € I. Let y - 3g g. < ag. = a. G- is closed and

But

Ygig" gj < ag i g" 1 g

J

J

= a g ^ g . < 3g - 1 g. = y.

T

Hence

g, = V (g-g^g* v e) €. G-; i . e . , (3g~V.)g. = 3g" X g.. J i el J • JJ j g. € G 1 , the required contradiction. J 3g"X The rest of the lemma follows from Lemma 8.1.1.

Let

A^ = {a € Q: G~ is closed}.

COROLLARY 8.2.4.

Thus

Lemma 8.2.3 gives:

^or any l-penrrutation group

(G,&)9 A

is closed with respect to the open interval topology on 5.

Theorems 8B and 8C now follow from Theorems 8.1.2 and 8.2.1 and Corollary 8.2.4. We next wish to examine the converse directions (3) -*• (2) and (2) •> (l) in Theorem 8.2.1.

In the example we gave at the

beginning of the chapter, for each

q € Q , Q ={(T,q):x£T}

is

A

a fat block of (A(fi),n) that is dead; i.e. ( oj(o x = {e}. This turns out to be crucial. Suppose that

(G,fi) is an ordered permutation group and

fc(G,ft) has a minimal element K n that is static (i.e., for each U Kn G = G T h e n w e w i l 1 sa C -class A, ( A ) y t h a t K Q i s dead and A)that each

C

class is a dead segment. Define J9 by: a£>3 if

a = 3 or a and 3 belong to the same dead segment (if any). Clearly €

is a natural congruence of (G,fi).

In the passage from

161 Q to Q//9 , each dead segment collapses to a point of Q.//Q and each a £ ft becomes an unambiguous point of Q//9 (since distinct dead s e g m e n t s — b e i n g fat or s i n g l e t o n s — c a n n o t have a common endp o i n t ) . G acts faithfully o n Q//Q and b y Lemma 3.2.9, (G,ft/£ ) has n o dead segments* Note that (G,Q/£>) is just (G,ft) stripped of dead elements and there is n o change in the non-dead primitive components. Also observe that if (G,ft) is transitive, /© is trivial. In our example above, the /Q classes are just {ft : q G. $ } . W e n o w prove the following generalisation of ( 3 ) -> ( l ) of Theorem 8.2.1 (transitive): THEOREM 8.2.5. For any l-permutation group (G>Q,)3 the following are equivalent: (i) (G3 &/&) has pointwise suprema, k(tt/&), (ii) G is a complete t-suhgroup of (iii) The stabiliser subgroups of (G3 Q./& ) are closed^ (iv) The stabiliser subgroups of (£, oj are closed. Proof: (i) •» (ii) -> (iii). (iii) •*-> (iv).

If

By Theorem 8.2.1 and Corollary 8.2.4. a €. A

for some

dead segment

A,

then

G- = G/ N so (G,ft) and (G,ft/£) have the same stabiliser subgroups. (iii) •* (i). To save writing suppose that (G,Q) has closed stabilisers and no dead segments. Assume, for reductio ad absurdum, that for some a €. ft and {g. : i €. 1} C G, y g, i€I exists in G but a = sup{ag.: i € 1} < ag where g = V g., Without loss of generality, we may assume g = e. iei x Choose 3 £ Q so that a <_ 3 < ag = a and let K = Val(a,$). aG H [a,a) = 0 by Lemma 8.2.2 (G is closed). If K were transitively derived then, by the density of orbits, there exists h £ G such that $ S V < ( a C v ) h < a ! y , contradicting aG O [a,a) = 0. Thus K is integral or static. Hence a 5 ^ cannot be fat and so is extensive. Hence if a C ^ / {a}, then for j\

some

h € G,

diction.

a > ah 6 a £ _. > 3 5 v

Therefore

ls.

J\

and again we obtain a contra-

a 5 T , = { a } . If K were integral, then a is.

162 would have an immediate predecessor y in aG. Now y < a since aG D [a,a) = 0. But each g. £ G and so ag. <_ y. This contradicts the fact that sup{ag. : i £ I) = a > y. It follows that K is static. K K We prove that 6 C T/r = {6} for each 6 £ a 5 whence a 5 IV

is a non-trivial x9 class, the desired contradiction. Let 6 £ a £ K with 6 / inf a g K . Since ag± < a for all i € I and a Gv = {a}, a g . 5 ^ = {ag.} < <*£„ for all i £ I. Any element of IS-

1

ii

1

Ji

y.

fixes each (J^-class contained in a £

(K

is static),

K

so g. 4 G

„ . Thus

UC

)

a 5 g. < a 5 K

for all

i £ I. Now K

sup{6g.: i £ 1} <_ inf(a5 ) < 6 and we can find o £ a$ with a C ^ < S ^ ^ . Then K = Val( 6,a). Repeating the argument of the previous paragraph with 6,a in place of a,3, we obtain 6S^={6}. If inf(aC K ) = 6 £ a « K , then 6 £ K = a £ K is a non-extensive natural block covering 6 5^. Hence 6 $ ^ is K extensive by Lemma 3.2.5. Since 6 = inf(a 5 ), 6 & = {6} and the proof is complete. COROLLARY 8.2.6. Let (G,Q.) be any l-permutation group. (a) Any counterexample to (3) -> (2) in Theorem 8.2.1 must have a dead segment A such that k(k) ^ {e}. (b) Any counterexample of (2) -> (1) in Theorem 8.2.1 must have a dead segment A such that k(k) = {e}. Proof: (a) If (G,fi) satisfies (3) but not (2), then G is a complete ^-subgroup of A( Q.//9 ) (by Theorem 8.2.5) and there exists g = \/ g. £ G with g > h > _ g . ( i £ l ) for some 1 i£I 1 h £ A(ft). Now for every dead segment A of (G,Q), Ag = Ah; so gh~ €. A(fi)/ N for every dead segment A of (G,ft). Since gh~ I e, (a) follows. (b) By Theorems 8.2.1 and 8.2.5, if (G,n) is a complete ^.-subgroup of A(n) that doesn't have pointwise suprema, (G,fi/^)) has pointwise suprema. So for some dead segment A, Ag. < Ag for all i £ I and sup{Ag. : i £ 1} = Ag in (G,ft/£>) but

163 sup{ceg.: i £ 1} / ag If

for some

A(A) ^ {e}, there exists

a £ A, where

g =

V 6«

e < h £ A(A).

Let

f € A(fl) be

.

i

Then for all

£ Agi < Ag = A f 1 g ;

for all

i £ I.

.

Consequently,

g f

Hence

in

G.

3 £ A,

g± < f X g < g

\J g. (in A(ft)),

a contra-

1

diction.

Note that the structure theory helped us to analyse and prove the underlying problem with the converse direction of Theorem 8.2.1. It is now easy to give counterexamples to (3) -* (2) and (2) -> (l).

EXAMPLE 8.2.7.

Let

Q = T x" I

where

T = I,

and let

G = {g £ A(fi): (3g £ A(l))0/x £ T)(Vr £ t)((x,r)g = (x,rg))}. (G,ti/&) = (A(R),I) has closed stabilisers by Theorem SB. A(T) ^ {e}. Let i € (0,1) = I. but

g. £ G Then

be such that

g =

g>h>_g. (i£l)

\J g.

where

in

(x,r)g. = (x,r + i), G where

(x,r)g = (x,r + l)

h £. A(Q) is defined by

( x , r ) h = ( x - l , r + l). Thus (3) holds and (2) fails in the ^-permutation group EXAMPLE 8.2.8.

(G,fi). Let

n = T x" Q where

T = { T . T J with

x-, < x o . We already saw that Q = {(x,q): x £ T } are the dead 1 2 q segments of (A(fi),n) (q e $), A( n ) = {e}, and (A(n),fi) does not have pointwise suprema. Hence

(A(fi),fi) satisfies (2) but

not (1). For an example that (4) / (3), see McCleary [72c]. We close this section with a proof of Theorem 8D: If

(G,fi) is static or integral, then it clearly has point-

wise suprema, and so closed subgroups by Theorem 8C. So assume (G,fi) is transitively derived from

(G,T).

dead segments it is enough to decide if

Since

(G,Q) has no

(G,fi) has closed

164 stabilisers by Theorem 8.2.5. enough to decide if

G

Since

T

is dense in

is closed for all

a € T

Q, it is

(by Corollary

8.2.4). If

(G,T) is regular,

G

= {e} which is closed.

If

(G,T) is doubly transitive with

e < f £ G

of bounded

support, then by Corollary 1.10.6, Lemma 8.1.1 and Theorem 8C, G is closed for all If

a 6 T.

(G,T) is periodic primitive with period

were not closed, let Choose

e < h £ G

{g. : i e i } C Q

so that

with

z,

g =

and

G

a

V g. £ G .

supp(h) H [az~ ,a] £ [ag""^,a] . Let

x £ [az" ,a].

If x £ [ag~ ,a], ag" < xh~ , so xg. < a <_ xh~ g; -1 -1 ~~ -1 T £ [ag ,a], xg. < xg = xh g. Thus h g exceeds each 1 -1 -1

and if g.

on

[az

i € I,

,a]

and so on

a contradiction.

Now suppose e < g 6. G find

such that

x fL [ag" ,a].

Therefore

Consequently,

G

(G,T) is pathological.

be such that

gT £ G

T.

ag > a.

For all

Tg = Tg . Now

g. <_ h

g < g

for all

is closed. Let

a € T and

x fL [ag~ ,a],

g >_ g A g^

we can

for all

g >•_ h ^ g A g for all x £ [ag" ,a], then -1 -1 -1 xg >^ xh >_ xg if x g [ag" ,a]. So supp(gh ) c [ag~ , a ] and, consequently, gh" = e. Thus g = V -, (g A g ). But

g £ G

and

If

g A g £ G

for all

x.

Therefore

G

is not

closed. 8.3. CLOSED SUBGROUPS 0F_ l-GROUPS. In this section we wish to locate the closed convex £-subgroups of an £-group 8F and 8G.

G.

In particular, we will prove Theorems 8E,

The main result (a strong form of which we will use

throughout the section) is: THEOREM 8.3.1.

If

(G^Q) is an l-permutation group and C

is a closed convex t-subgroup of

G3

then C

is an intersection

of stabilisers.

From this we will deduce, besides Theorems 8E, 8F and 8G:

165 THEOREM 8.3.2.

If

G

is a completely distributive l-group3

then every closed convex l-subgroup of

G

is an intersection of

closed prime subgroups. We first set up some notation. ^-subgroup of an £-group

G.

Let

C be a convex

Since the intersection of a family

of closed convex ^-subgroups of

G

is a closed convex ^-subgroup,

there is a least closed convex ^-subgroup of

G

We call this the closure of_ C_ and denote it by Fx(C) = {aeil: (Vg e C)(ag = a)}

(Va € Fx(C))(ag = a)}. Then that

P

GCFX(Q)

c\

C.

Let

= {g € G:

G(Fx(C)) = /1{G-: a £ Fx(C)}.

G 5 = G(Fx(G5)) for all P

and

containing

Observe

$ £ ft.

The following theorem immediately implies Theorem 8.3.1 and is the key to this section. THEOREM 8.3.3.

Let

a convex l-subgroup of C = GiFxiO)

if

Proof: Let some

C

(G3ftjbe an l-permutation group and C G. Then C Q G(Fx(C)) c: c*. Hence

is closed.

H=G(Fx(C)).

a e Fx(C*) ^ Fx(C) and

a contradiction. Let that

Hence

e < ge H

A f a

a £ ft. There exists —

) € c*

as

3 ± a < 3h

h £ H = G(Fx(C));

e
(X

V (8 a£ft

So

for some thus

C


*

closed.

Thus

3 £ ft.

e < f £ C*

g A f £C*.

Ot

h € H.

ah = a,

Conv (&H) C Conv^(3C^) for all

and

ag < af . Since —

g =

Conv^(BH) ^Conv^(3C^) for

B £ ft, then (without loss of generality),

o = sup(Conv^(6C^)) €. Conv-(3H). But

If

such

Now

Ot

HCC^.

We will frequently use the following (Appendix II Theorem l) due to Byrd and Lloyd [67]: If

P

is a closed prime subgroup of an £-group

is a convex ^-subgroup of

G

containing

P,

then

Q

G

and

Q

is closed.

For a proof, see Appendix II. One immediate consequence of this and Theorem 8.3-1 is:

166 COROLLARY 8.3.4. C

If

(G3 Q.)

is an l-permutation group and

is a closed prime subgroup> then C

is an intersection of

closed stabilisers. By Theorems 8D and 8.2.1, if

(G,fi) is a transitive

primitive ^-permutation group that is not pathological, then

G

is

completely distributive.

COROLLARY 8.3.5.

If

(G,Si)

groupj then D(G) = G. Hence G

is a pathological l-permutation

is not completely distributive.

Proof: In the proof of Theorem 8D we saw that closed

(a €. ft). Hence

prime subgroups of

G,

COROLLARY 8.3.6.

is

G

Let

all convex l-subgroups C

(G,n) be an l-permutation group.

_

for

of G.

C QG(Fx(C)) £ C * , Conversely,

C* = G(Fx(O)

(G,ft) has closed stabilisers, then

G(Fx(C)) = H{G-: a € Fx(C)} Since

is not

(by Corollary 8.3-4).

(GjQ) has closed stabilisers if and only if Proof: If

G-

D(G), the intersection of all closed

is a closed convex ^-subgroup of

G.

G(Fx(C)) = C*.

G§ = G(Fx(G-)) = G^ p

P

so

P

G^

is closed for all

P

I e Q. Let

(G,fi) be an ^-permutation group and

congruence of

(G,ft).

for every £-class recall

A.

5

If ?

a partial G/.\

is a (full) congruence of

L(5 ) = {g £ G: Ag = A

H {G/ x: A is a l?-class} closed if £

5

is said to be closed if for all £-classes

is an £-ideal of

G.

So

is closed (G,fi),

A} = L( & ) is

is.

We can now prove the following generalisation of Theorem 8F: COROLLARY 8.3.7. H

Let

be a closed l-ideal of

(G3tt) be an l-permutation group and G. Let S

be the congruence of (G,Q)

whose classes are the H-orbits. Then H = L(tf). Thus )

is an l-permutation group. Hence there is a standard

167 l-embedding of

(G,Q) into

transitive, where H

(E,aH) Wr (G/H^/C)

if

denotes the closure (in G.

(G,&) is

J

of the

(OLnJ

restriction of H

to aH.

Proof: Since (G,fl).

H

is an £-ideal, tj is a congruence of

Clearly, it is natural and

by Theorem 8.3-3.

Since

L( C ) = G(Fx(H));

SO L( ij ) = H

L( 5 ) = ^ { G ( A y A is a £-class}, the

lazy subgroup of the action of

G

L( JJ ) = H.

on fi/g is just

We next prove the following generalisation of Theorem BE. THEOREM 8.3.8.

Let

(G3 n) be a depressible l-permutation

group. Then the closed prime subgroups of

G

(other than G)

are precisely the stabiliser subgroups G-, a € Q. Proof: By Theorem 8.1.2 and Corollary 8.2.4, (a €. Q)

is a closed prime subgroup of

Conversely, let We must associate

C with a point

\ £ Q

C = H{G-: a £ Fx(C)}.

so

Since

C

is prime,

G-

G.

C be a closed prime subgroup of

Theorem 8.3-3, Fx(C) J 0.

each

If

such that

G,

C / G.

C = G-.

By

Fx(C) = 0,

{G-: a €Fx(C)}

We must show this tower has a least element,

G-

then

C = G;

is a tower.

(Then

C = G-).

Suppose, for reductio ad absurdum, that the tower has no least element.

If

G- Q G T ,

extensive block of if

G-

$G-

Since g € GP

§ Gg,

then

A(a,3) = Conv^aG-)

(G,ft) (cf. Theorem 1.6.2).

is a non-trivial

We now show that

A(51,3) = A(52,B).

G- ^ G- , a- €. A(a,,8) (Otherwise, we could choose

such that

natural block

cLg ^ an and g is the identity off the _ -jl A(ou,3)—here is where we use depressibility.

Then

g 6 G- \G- , a contradiction). Thus there are f ,g €. G- such a 2 ai ^ that a,f <_ a 2 <_ a,g. Therefore a^g <_ a, <_ ouf and, as f" ,g~ €. G-, P

a. € A(ao,8). l *-

This establishes the claim. _ __ _

Consequently, we will now write is any point such that and

G- $ G-,

let

A( 3) for

a £ Fx(C) and

y € Fx(C) with

G- £Gg.

G-^G-§G-.

A(a) = A(y,a) S A(y,3) = A(B) and as

_

A(a,3) where If

a,3 € Fx(C)

Then

a € A(B)\A(a),

a

168 Cf = {A(ot): a € Fx(C)} extensive blocks of

is a strictly decreasing tower of (non-empty)

(G,S) having no least element.

sup(A(a)) < sup(A(B)) if

1(5)^1(5)

I = inf{sup(A(a)): a £Fx(C)}, since all

C C G-

g £ C.

I el(c)

for all

and as

G-

a € Fx(C).

But

sup(A(a))g = sup(A(a)) for

C £ G=;

A(l) is a

Hence if

I € A(a) for all

a £Fx(C),

Consequently,

in J.

By Lemma 3.2.5,

i.e.,

I € Fx(C).

Thus

orbit, A(|) = {£}. Therefore

Cf has a minimal element, the desired contradiction.

We next consider Theorem 8.3.2. Let a£B

if

(G,fl) be an ^-permutation group. a = 3

or

G-

is not closed for any

B.

C is a congruence of

if

G-

is. Recall

Define £, by:

A^

(G,fi) since = {B £ fi: Gg

G-

y

between

a

and

is closed if and only

is closed}

is closed

(Corollary 8.2.4). We now use Appendix II Theorem 1 to identify the closed congruences of (G,ft). LEMMA. 8.3.9. €

Let

(G3Q.) be any l-permutation group. Then

is natural and the smallest closed congruence of

£ classes are extensive. A partial congruence 5

(G^Q). All

of

(G^Q) is

closed if and only if each 5-class is a union of £ classes. Proof: If

A

is a £-class, then for all

a > sup(A) = 6,

[6,a) H A^

Corollary 8.2.4-.

Therefore (T is closed.

extensive (For if

6 £ A

/ 0.

and

Hence

a £ Q, if

G, N = G-r is closed by Moreover,

A = ConvJ 6G/ A x),

let

A

is

a = sup(A).

Q, {A) Then Hence

G- 2 G/ s = G-r and so is closed by Appendix II Theorem 1. a £ A.

Thus

A = A).

If each %-class is a union of <£-classes, then

C

is closed

by Appendix II Theorem 1. If where

tj is closed and

B = sup(A).

Hence

A

is a 5-class,

B jL A

G^ = G/ . \ is closed

for any non-singleton
Thus the g-classes are unions of (T-classes.

169 COROLLARY 8.3.10. P

a prime subgroup of

Let G.

(G,to) be an l-permutation group and Then ~¥x(P*) = fx(P) n A

and

P* = G(Jx(P) A A ^ j . Proof: If 3 € Fx(P*), then P* £ 0- so G 5 is closed. __ P P Hence 3 € Fx(P) A A- . Conversely, since G- is closed for all a € Fx(P) A A ^ , H = Hence

H^P*

fl{G-: a £ Fx(P) O A

Fx( P) A A

so

£ Fx( P*).

}2P

and is closed.

The rest of the corollary

follows from Theorem 8.3.3. By Corollary 8.3-4 and Lemma 8.3.9: COROLLARY 8.3.11. Then L( S )

D(G) = L(
Let

(G,Q.) be an l-permutation group.

Hence, if

G

is completely distributive,

is closed for every natural congruence 5

of (G,Q.).

We can now prove Theorem 8.3.2. If

G

is completely distributive,

D(G) = {e}

so

(G,fi/(T)

is an ^-permutation group with closed stabilisers by Lemma 8.3.9 and Corollary 8.3.11. (G,n/(t).

Then

Let & r

be the "dead" congruence for

( G,(fi/(T)/£> ' ) is an ^-permutation group with

closed stabilisers by Theorem 8.2.5.

Since stabiliser subgroups

are prime, Theorem 8.3.2 now follows from Theorem 8.3.1. // Finally, we prove Theorem 8G which generalises Theorem 8D. Recall that an ordered permutation group

(G,Q)

support property if for each non-dead segment there exists that

e / g €. G

such that

satisfies the

A

supp(g) £ A.

with

|A| > 1,

We must prove

<£" has as its non-singleton classes pathologically derived

segments.

We may assume that

(G,ft) has no dead segments.

is a non-singleton natural block, Lemma 8.2.3

G/ . \ = G. w A \

(Here we use the support property).

is closed by Hence

5 = {Ag: g £ G}

is a closed natural partial congruence of

By Lemma 8.3.9,


or

If A

G-

(G,fi).

are closed)

(T has primitive segments and is a minimal non-trivial full

natural congruence of

(G,ft).

is a primitive segment and and only if

A

So it is enough to show that if A

a € A,

then

G

fails to be closed if

is a pathologically derived segment.

K

Let K £ fc(G,n) be such that A = a S , Note that (Gv) not closed if and only if (GK,ft ) is transitively derived from a

170 is

pathological ^-permutation group (by Theorem 8D). Let L K = {g|a? K : supp(g) £ a£ K }, an £-ideal of G . If (Gj,,^K) is pathological, so is (LK,ft ) by Lemma 6.3. Therefore there exist g. 6 (l^) and g £ L j M L R ) with g = V §•• By the support property, we can find g.,g £ G (i £ I) i£I x \ such that supp(g),supp(gi) S A ( i G l ) , and g| A = g, g ± | A = ^ (i £ I), If f G G and g. < f < g for all i £ I, then supp(f) 9 A S O y g. <_ f < g, a contradiction. Thus g= V g. i£I 1 iG I X and &g. £ G ( i 6 I) but G is not closed. &g £ G\G . Hence i a a a Conversely, if G is not closed, let g. £ G (i £ I), g =

V g. £ G with

g f G . Since

G, x is closed, ag € A. If

(Gv,tiv) were not pathologically derived, Jv

(GT,)

is. a

IV

would be closed in

Gv. If (GT,,ftT,) is not doubly transitively derived, GT, is ^-simple (Theorem 6C); so G K = 1^. If (GR,fi ) is doubly transitively derived, L K 2 G K A B(ft ) ^7 Lemma 6.3. Let g = gk, g± gl i. Then g± £ (GK a and g jT ( G K ) a so

= u

g 1

V g.. Hence there would exist e < h £ LKT, such that

i£ I i gn"1 < g for all i £ I. By tthe support property, there would exist e < h £ G such that h|A = h and supp(h) £ A . Then g.I < gh~~ < g for all i £ I, the desired contradiction.

k-

Consequently, (G^,^) is pathologically derived. // The support property is crucial for Theorem 8G as the following example shows. EXAMPLE 8.3.12.

Let

(W,fi) = (ZS,Z) Wr (Z,Z) and let

+

G = {({gn },g) £ W: 3m £ ^ ( givv = g n if k = n (mod m))}. Then (G,fi) is a transitive locally primitive ^-permutation group whose local component is integral. Define g

({g

1} g

)£G(i£^)

g = ( {g },g) £ G

by ? ag

=

ag

=

a

and (a i

if n = 0 (mod i) i£ ^+),

ag

= a + 1, ag^ l ) = < a + 1 otherwise

171 ( M ^ ; i « all

+

) .

1 € % , but

Then

g=

g ^ G/0 Q\.

V ^ Hence

^) G

and (Q Q)

g(l)£G,nns

for

i s no t

" closed.

This example also has the disturbing property that it has no lateral completion in

(W,ft)—see the list of open problems.

For further results concerning complete distributivity, etc., see McCleary [72b], [72c], [73b] and [76].

172 CHAPTER 9 AUTOMORPHISMS OF A(fi)

In Section 2.4 we showed that if then every ^-automorphism of of

A(fi) (Theorem 2.4.1).

to show that if

Q

is doubly homogeneous,

A(Q) is conjugation by an element In Chapter 7 we extended the result

is homogeneous and locally primitive, every

I-automorphism of (Corollary 7E).

Q

A(ti) is conjugation by an element of A(fi°)

In this chapter we wish to pursue two directions.

The first is to try to extend Corollary 7E to the non-locally primitive case. by replacing

The second is to try to improve on Corollary 7E

Q° by

t-automorphism of

Q;

i.e., to attempt to prove that every

A(fi) is inner (cf. Corollary 2.4.4).

latter can only be hoped for if

Q

This

is homogeneous as the following

easy example (1.1.3) shows. Let T ^ we saw that S © ^

Q

be disjoint copies of %. and is not homogeneous and

ordered by:

(m,n) >_ 0

if

Q = %. u Z?.

A(Q) is £-isomorphic to

m,n >_ 0. Hence

A(fi) is

Abelian and has only one inner automorphism, the identity. is an &-automorphism of JL ® %, where Consequently,

Then

But

^

(m,n)i|; = (n,m) (m,n € Z ) .

A(Q) has an outer ^-automorphism.

We will prove the following theorem due to Holland [75a]: THEOREM 9A.

There exists a doubly homogeneous chain Q, such

that A(Q) has an outer l-automorphism.

The proof relies only on the results established in Section 2.4. We return to generalising Corollary 7E. Let

k = k(ti) be

the collection of covering pairs of congruences of the transitive ^-permutation group

(A(ft),ft).

Let

oj° = { a e ^ : a £

f^[ &

173 1

is coinitial in k, Ajr is a 5 class and AT^ Q A, where k 0 if K <_ K»}. Note that ft 2 fi and if fi is locally primitive, the definition of ft0 coincides with the one given in Chapter 7. We will continue to call ft0 the local completion of ft. Using stabilisers in a more subtle way than we did in Chapter 7, we will prove the following result due to McCleary [69] (It is an improvement of results by Holland [65a] and Lloyd [64]): THEOREM 9B. If ft is homogeneous_, every t-automorphism of A (ft,) is conjugation by an element of A(tt®). The proof incorporates some of the ideas of Sections 7.1 and 8.3. Finally, we will prove the following theorem about ^-characteristic subgroups (A convex ^-subgroup H of an £-group G is JL-characteristic if Hip = H for every t-automorphism ty of G): THEOREM 9C. (1) If 9. is doubly homogeneous3 every normal subgroup of Afoj is L-characteristic, (2) If 0, is homogeneous> every closed t-ideal of A(tt) is t-characteristic. Note that if every ^-automorphism of A(ft) is inner, every £-ideal of A(ft) is automatically ^-characteristic. Hence Theorem 9C is a weakening of our previous intention. The proof of Theorem 9C (l) relies on Theorems 2.4.1, 2.3.1, Lemma 2.3.5 and the patching lemma. Not surprisingly, the proof of Theorem 9C (2) is similar to that of Theorem 9B. Theorem 9C is due to Holland [75a]. We commence by constructing a chain Q satisfying Theorem 9A. By Theorem 2.4.1, if i|> is an ^-automorphism of A(ft), there is ty € A(S) such that gip = i^g* (g € A(ft)). Now ftiji is

174 an orbit of A(ft) ordermorphic to ft. Thus

A(ft) has an outer

^-automorphism only if ft is ordermorphic to another orbit of A( ft) in ft. We construct such an ft. Let y £ a) Let of

T = T T %'> i.e., x < a y € oo

such that

if and only if there exists

x(y) < o(\i) and T ( V ) = a(v) for all v < y.

*$* be an ultrafilter on CD, that contains all final segments a).. (i.e., if

T £ T,

y 6 a).,,

{v 6 w^: v > ^ y } € y ).

For each

let X ( T ) = {y £ OJ : x(y) is even}. Let

ft={3£T:X(3)£7},

a subchain of

T. We now show that ft

satisfies Theorem 9A. (1) ft is homogeneous:

Let

a, 3 £ ft. Define

(xg)(y) = x(y) + 3(y"> - a( y) (y £ u^). Then g |ft£ A( ft) since

X( xg) 2 X( x ) D X( a) 0 X( 3 ).

(2) ft_ is doubly homogeneous: a =

3

write that

for the equivalence class of

a

if

y £ uj-.,

define

a

under

= . Observe

is ordermorphic to ft via 3 »—>• 3' where

3'(v) = 3(y + v) (v € a), ) (Since 3 £ a

For each

if a(v) = 3(v) for all v <_ y (a,3 € ft). For any a €. ft, a

y

g £ A(T) by:

ag = 3, and

implies

y + v = v

if v ^ y

2

+ io,

3' € ft by the definition of 7*). Consequently,

a,3 6 ft, then

{a € a y : a £ a} is ordermorphic to

{x £ 3 y : T 1 3 ) . Let

Y < a < 3

find

g € A(ft)

that

y(v) + a(y).

(in ft). By Lemma 1.10.2, it is enough to

such that

ag = 3.

Choose

Let A = {6 6 ft: 6 <_ a

B = ft\A. By the previous paragraph, and hence is homogeneous.

B

But a,3 £ B

y € a)-L

least such

for all a £ a y } and

is ordermorphic to ft so there exists

pxg r such that

ag' = 3. Define

ag = 3

and

(3) T C j j ; f

yg

=

Let

inf{a : v € a), } where

Y

if x £ B

g 6. A( ft) by: xg = < LJ

Then

gr €. A(B)

(since

x £T. r

if X £ A

y £ A).

Then

x = sup{a : v £ oo } =

a ,a £ ft are defined by:

175 x(y)

if y < v

a (p) = < i(y) - 1 if p > v and x(y) odd (jr(y) - 2 if y > v and x(y) even

f(y)

if y £ v

o\(\i) = <[ i(y) + 1 if y > v and x(y) odd l(\i) + 2 if y > v and x(y) even Note that

a

c

or

< o]. < o1

< o V

V'

V

Since

|fi| = SL ,

c

c

io or

(4)

Thus

TCS,

every point of

c

Q, has character

nr| j

ir _

T = { a € . o J : a has character c . }:

every point of

T

has character

x = sup{a : n € a)} where n < m.

v < v!.

if

V

x > a

c _. € Q

We first show that

Suppose and

a

x € T < a

and

if

n,m £ OJ,

n n £ w such that a (0) = x(0). 0 nQ For all n > n~, a (0) = x(0). There exists a least y_ € oa such that a (y ) < x(y ). Let n.. £ OJ be least such that n 0 n x > n Q and a n (y ) = x(y ). So a n (v0 = T (v0 f or all n >_ n 1 and

There exists a least

y <_ y_ .

{y.: i 6 u}

Continuing in this way we obtain

respectively (where

yfi = 0 ) . Let

p e T

by:

and

03 and OJ.,

v = sup{y.: i € oo} < w

x( y ) define

{n. : i € w}

strictly increasing sequences in

and

if y / v

p(y) = <(

.

Let

m £ a).

There

i(y) - 1 if y - v exists

j £ a) such that

m < n..

if

y < y.

by construction.

so

a

for all

J

< p

m £ w.

desired contradiction.

Now

so a(y)
But

a

-

Since


^

So no element of

j

n

.

j

T

Q.

ly for infimum.

c _.

Now if of

Q,

{a : v £ a), }

then as

there exists

the

has character

Similar-

is a strictly increasing set of points

{a ( 0 ) : v £ a)-}

x(0) £ ^

T

j

is the supremum of a

countable strictly increasing sequence of elements of Thus each point of

j

J x ^ sup{a : m £ a)},

p < x,

such that

is an increasing set of integers, a (0) = a (0)

if

y,v >_ A(0).

176 Suppose

A( £) £ a)

has been defined for all

£ < c < w-, so that

A(O < A(n) whenever £ < r\ < c and a (£) = a (£) if — — y v y,v >_ A(C). N O W {a : v €. w, & v >_ sup{A(^): £ < £}} is uncountable so there exists £ < C

and

C 6 u),,

A(c) £ a)-,

a(c) = a(c)

if

if

such that

y,v >_ A(c).

^or

A( C) 2. *(O

a11

By induction on

{a : v €. a) } has an upper bound in ft, then

sup{a :v€.aj} =

€T

T

where

T ( C ) = a./ i O ( ^ u j .

VI

A\ C, )

all points of character

c.., belong to

T

Thus

1

(Observe that the

proof just given shows that the points of ft\T have character Now let

$ € A(T)

(x 6 T; y £ a ). Then then

be defined by:

n$ = T\fi and

$" gij) € A(ft) since

Thus conjugation by is outer.

(T$)(P) = x(y) + 1

(T\fi)$ = fi. If

T\ft is an orbit of

g € A(fi),

A(ft) by (4).

\p is indeed an ^-automorphism of

A(ft) that

This proves Theorem 9A.

We next turn to Theorem 9C (l) since its proof (like that of Theorem 9A) depends only on Chapter 2. Lemma 2.3.5.

The key to the proof is

To apply that lemma, we need doubly homogeneous

chains of countable coterminality.

We obtain such chains as sub-

chains of ft and extend from them to ft via the patching lemma. So assume that ft is doubly homogeneous and ^-automorphism of such that

A(ft).

By Theorem 2.4.1, there is

fi// = if" fij) (f 6 A(ftl).

Let

T = {a € ft: ag = a & aij) = a}. Let in ft\T and ft = ft n (a,3).

Let

6 £ (a,3).

Then

since

(a,3) does not have cofinality AL. 6

6 J$| > 6

= sup{60|ij)| : n €. a)} < 3.

Continuing we obtain and

If not, say ft., does not

6g f 6.

62

A

= 6

where So

6.$ = 6

But if

But

6 Q £ (a,3)

Sjg = 6 Q

so

|iji| = i(i v $~ . Let

{6 : n €. a)} £.(a,3) (n € oo).

So

= sup{6gn: n € a)}. Then

(a,3) has cofinality greater than S~.

6 $ i- 6 . Hence 6

(a, 3) be any maximal interval

Without loss of generality,

6 < Ig < Sg2 < ... . Let

ij) € A(ft)

e < g £ A(ft) and

So ft. is doubly homogeneous. We

claim it has countable coterminality. have cofinality N~.

\p is an

whence

5,g / 6_.

such that

6^ g = 5 ^

y = sup{6 : n € w},

then

177 Yg = Y

=

Y^.

Now gi/;|o of

Thus

Y = 3.

g|^ieA(^1)

Consequently,

and

(a,3) has cofinality

$|(a,g) € A((a,B )).

By Lemma 2.3. 5,

is dominated by the supremum of two conjugates (in

A(ft))

g|ft-,. Therefore, by the patching lemma (as ft. ranges through -1 -1 gty < f gf v f gf for some

all such maximal intervals),

— f

l' f 2 €

A fi

H e n c e if

( )-

L

is true for all ^-automorphisms of A(fi) is ^-characteristic.

1

1

ed

is any £-ideal of

ed

A(ft),

Li|/ <=: L.

This

A(Q). Thus every £-ideal of

Theorem 9C (l) now follows from

Theorem 2.3.1.

If

0, is rigidly homogeneous, the only £-automorphism of

A(fi) is the identity.

COROLLARY 9.1. Z-ideal of

Hence

If ft ^s homogeneous and primitive, every

A(ti) is t-charact eristic.

We now turn our attention to Theorem 9B. Recall that has the interval support property if for each a < 3,

there exists

e f g ^ G

such that

(G,ft)

a,3 €. ft with

supp(g) £ (a,3).

We

break the proof of Theorem 9B into two cases: THEOREM 9.2.

If ft is homogeneous, it is locally regular

or has the interval support property, Proof: Let % congruences of &

be the intersection of all the non-singleton

(A(ft),ft)—a tower by Theorem 3A.

By Corollary 3B,

is a congruence. Case _1. C

is not the singleton congruence.

Then it is

the unique minimal congruence and ft is locally primitive.

If

the primitive segments are regular, the theorem is proved; if they are not, they are doubly homogeneous (by Theorem 4-B) and thus ft enjoys the interval support property. Case 2.

5

is the singleton congruence.

So ft is not

locally primitive. Also ft is dense (otherwise ? non-trivial congruence where

a 7B

is the minimal

if there are only finitely many

178 elements of ft between with £ 6^

|A| > 1.

Let

a

and

6 £ A

3 ) . Let

with

6

A

be an interval of ft

not an endpoint of

is trivial, there is a non-singleton congruence Q A.

exists

Let

a £ 6#

g £ A(ft)

supp(g) £.6^ ^ support).

A

with

with

a ^ S.

ag = 6

(Take

g

A.

Since

"# such that

Since ft is homogeneous, there

and we may assume the bump of

g

with

a

in its

Thus ft has the interval support property.

In the homogeneous locally regular case we already know every I-automorphism A(ft°).

of

A(ft)

is conjugation by an element of

We can do better:

THEOREM 9.3.

If

every t-automorphism Proof:

of

ft

is homogeneous and locally regular>

Afftj

is inner. {A(ft)-: a £ ft0} is the set of

By Corollary 7.1.7,

maximal representing subgroups

of

A(ft).

a cut in some primitive segment, say regular,

A(ft)- = A(ft)g

where

A.

6 £ A.

But if Since

Hence

the set of maximal representing subgroups of and some

ijj be an ^-automorphism of 3 £ ft. By Lemma 7.1.2,

(g £ A(ft)), of

and

gip = (f>~ g(|>.

A(ft).

So

ip

ft

is

is locally

{A(ft) : a € ft} is A(ft).

Let

a n €. 0,

A(ft) ty = A(ft)

<j> €. A(ft) where

Thus

a £ ft°\ft, a

for

(

i s an i n n e r automorphism

A(ft).

We must now consider the interval support case.

By

Theorem 8E, we know that the proper closed prime subgroups of are precisely the stabilisers distinguish the points of

ft0

A(ft)

A(ft)- (a £ ft). We now wish to from arbitrary points of ft. The

next lemma is the key to proving Theorem 9B.

LEMMA. 9.4.

If ft is homogeneous and has the interval

support property3

then the minimal closed prime subgroups of

are precisely the stabilisers

0

AfftJ- (a £ ft

some doubly homogeneous primitive

component).

or

a

ATftJ

is a cut in

179 Proof:

Let

a €. fi "be the supremum or infimum of some

non-singleton block )-

A,

and let

is not a minimal prime.

regular

3 5 /5

(K

A(ft)o S A(ft)/o „

3

A(ft)-

\ = A(ft)a

a = 3.

a,3

Then

A(ft)a

is not minimal.

a £ ft0 or

Since

ft

a

is a cut in some

enjoys the interval

fit the above description,

By Theorem 8E,

so

3 £ ft), then

and again

3 £ / £ „.

support property, if only if

If

is minimal only if

doubly homogeneous

Affi). S A((l)o ^ a a €. ft is a cut in some

non-minimal,

(W¥J

Hence

6 € A.

A(ft)-

A(ft)- ^ a £ ft0 or

is minimal if

is a cut in some doubly homogeneous primitive component.

We can now prove Theorem 9B. a €. ft0, then ever, if

a

A(ft)-

Firstly, observe that if

is a representing subgroup of

A(ft).

How-

is a cut in some non-minimal doubly homogeneous T/-

primitive component

3 C /E

ir,

then

A(ft)- 2 L ( 5T_) / {e}

so is not a representing subgroup. By Lemma 9.4, if ^ ^-automorphism of A(n), {A(n)-i|;: a G fi0} = (A(fi)-: a

and

is an

Theorem 9B now follows from Lemma 7.1.2.

We can now prove Theorem 9C (2). Suppose that the theorem is false.

Then there is a closed £-ideal

£-automorphism

ty

not be inner.

By Theorems 9.2 and 9.3*

property.

U H^ m m £ 03

A( Q)

A( 0,) such that

of

A(Q)

Hip / H.

and an

Clearly,

Hence, without loss of generality, and

and hence

N = P | Hi|;~m. m 6. OJ N = L(C )

Then

N

H/N

can-

A(Q)

H §Hip.

Let

is a closed £-ideal of

for some congruence

Now fi/15 is homogeneous and

^

£2 has the interval support

By Theorems 8F and 3A, the closed ^-ideals of

form a tower. M =

of

H

5

is an £-ideal of

of

(A(ft),ft).

A(fi/5 )

that

is not fixed by the £-automorphism of A(fi/5 ) induced by ip. Hence we may assume that N = {e}. Let £ be the congruence such that

US

n

) = Hip

points of ft by (A(ft),ft)

(n £ Z ) . Hip .

£

£

classes are the orbits of

n

be the natural congruence of

whose non-singleton classes are the convexifications (in

ft) of the orbits of g £ Hipn.

Let

So the

Hence

Hipn.

If

a$ 3,

then

3ip = (ag)ip = (aip)(gip),

3 = ag

where

ip

for some is conjugation

180 by

ij) £ A ( f t ° ) a 1^3

(by Theorem 9B). implies

Thus

(otxj; )^"n+:L( 3*) •

(+)

Let A be a non-trivial 5 n ~class. By transitivity, there exists g £ A(ft) such that A ciA^g. Replacing i// by conjugation by ijig, we may assume A £ A$ £ A$ 2 c ## . m Let A = ^/ A$ . Then Aiji = A so $ induces an ^-automorphism if of m £ a) ^ A(A O ft) under which the closed £-ideal H corresponding to H is not fixed, and {J H $ m - A(A Oft). Since A H ft is m €. GO homogeneous, we may assume that M = A(ft). We next show there is an orbit T of A(ft) such that T\p = T. Let a,3 G ft with a < 3. There exists n £ Z such that a ^ n+] _B but 3 ft, a£ n . So aijT1!; B^" 1 by (f). Since $ n is extensive (for (A(ft),ft)), there exists a € ft such that a < a$ and a£ aijT . By homogeneity, there exists g € A(ft) such that ag = a. Then a = og < a\p g < 3^ g. But a£ 3^ ^ _-i

so

agC B^

i.e.,

^-1

g < 3.

a < T < 3.

As Thus

^

a^ 3$~ g.

/v—]_

3$

n

A

g;

a = og < a\\) g, 1

1

TijT = T g " .

Since

3 £ a£

^-1

$ So

g

and -

f i x e s some

a < 3, -

x € ft w i t h

1

TA(ft) $ " = x( A(ft)i|; )iji~1 =

- xA(ft) and T = xA(ft) is the desired orbit. Now if T is any orbit of A(ft) and Tij) = T, then xA(ft) is dense in ft for any T € T (If 5 is defined by: a 5 3 if a = 3 or no point of xA(ft) lies between a and 3, then £ is a congruence of (A(ft),ft). By Theorems 8B, 8.2.1 and Corollary 8.3.11, L( C ) is a closed £-ideal of A(ft). But N = {e} and M = A(ft) so L(5 n ) c L( C ) § L ( g n + i ) f o r s o m e n e ^ or ? is trivial. Since L ^ ) = Ei\jn and L(Cn+]_) = Hi|;n+1, L( C ) ^ L( 5 )i|; if g is not trivial. By definition, L( 5 )i|; = L( 5 ) so 5 is trivial. Hence xA(ft) is dense in ft). Let A = {a €ft\ft:aA(ft) is dense in ft}. By the previous two paragraphs, A j- 0. Clearly AA(ft) = A. If we can show that A is an orbit of A(ft) it will follow that A and ft are the only dense orbits of A(ft). But Tij) = T for some orbit T which is necessarily dense. Hence we will have AIJJ = A or ft$ = ft. Since ftiji is dense, in either case we will haveftij)= ft. Thus \p

181 is inner and consequently

Hip = H

the desired contradiction.

to complete the proof of Theorem 9C (2), we show that

A

So

is an

orbit of A(ft). Let that

3 > a.

such that jf

a, 3 £ A

and

Since

n£2.

a partition of Since each I

TT

A_m

.

a £ atf" g~

3 € £

transitivity, there exists and

a € ft and pick

3 £ a£

aA(ft) is dense in ft, there is

a < ag < 3. Hence

class. Similarly,

A Q = Q\1TO

Let

g

a non-singleton

and

Z "m

g €. A(ft)

non-singleton £

class. By

€A(ft) such that 77 g +

vSff"m (m

Define

= 7T ,

f by:

Then

af = ag

is a block of

so

if

= Z . Let

{Affl:m €

is

a € A_m (m € w).

A(ft) and

f\ TT

= {a},

n

= (3), it follows that

f €. A(ft) and

_ af = 3. Hence A

n

is an orbit of

A(ft) and the proof is complete.

For further results about ^-characteristic subgroups of A(ft) see Holland [75a].

In particular, an example is given there to

show: THEOREM 9.5. AfftJ

There exists a homogeneous chain ft such that

has an Z-ideal that is not l-chavactevistic. We outline the proof.

For each

doubly homogeneous chain with

n € 7L , let

|A | = H

A

be a

and coteriminality

182 S

• Let

X

T

= % "* A

and

= {(k,a) £ T : k is even} +

filter on % .

For each

T = "ff+ T . Let and %

x £ T,

and fi = { T € T: Y(T ) £ 7 }.

be a non-principal ultra-

let

Y( T ) = {n e %+: T € X }

It is easily seen that

n is

homogeneous. Define natural equivalence relations on a = x mm each £

) ^

a? T if

m < n £ % . Then 6 e g if n < m. Moreover, — m n — ' is a congruence of (A(fi),ft) if the A 's are

sufficiently dissimilar. +

Q by:

for all

Hence, by Theorem 5A, we can £-embed

in Wr{(A(T ),T ): n e (%+)*}, where

ordered by

0 > 1 > 2 > ... . That is, each

determined by a Z *Q matrix (n,a) 6 A(T n ),

{g(n,a)}

(°g) n = ang(n,a),

and

(^+)*

is the set

g £ A(ft) is , where

w+

g(n,a) = g(n,t) if

l For each

by:

m £ Z , choose

a e A + 2,3), if Bmm= a m

(k,B)fm = <

(k£2, (k,3)

X f = X . Define m m m ~e if

g < n = 1

or

a _£ X . n-1 n-1

otherwise

g|fi £ A(fi). For each

(T$)

3 £ A j . Then

(n € Z , a £ T). f - n

(k,3)iji

f €. A( T ) mm

, otherwise

[n,a) = {

Then

and define

Let L

m € ^+,

be the £-ideal of A(fi) generated by

let \\>

€ A( T ) be defined by:

= (k + 1,3), and let $ £ A(T) be defined by

= x $ . Then

Tip = T,

Q$ = T\fl and conjugation by

outer ^.-automorphism. Moreover, it can be shown that ~—1 gip = ip gty £ L; so L is not ^.-characteristic.

$

is an

PART IV APPLICATIONS TO LATTICE-ORDERED GROUPS

183 CHAPTER 10 EMBEDDING THEOREMS FOR LATTICE-ORDERED GROUPS

In Chapter 2 we proved that every £-group can be ^-embedded in a divisible £-group. theory result.

This result is analogous to the group

In this chapter we wish to prove two further

analogues of embedding theorems (as well as giving another proof of Corollary 2K cited above).

THEOREM 1QA.

Every countable l-group oan be l-embedded in a

two generator l-group.

THEOREM 10B. Every l-group oan be l-embedded in an l-group in which any two strictly positive elements are conjugate.

The standard tool for proving the group theoretic analogues of Theorems 2K, 10A and 10B (where "strictly positive" is replaced by "of infinite order") is the free product with amalgamation.

In

order for this creature to exist, we need the amalgamation property: A class %

of algebras has the amalgamation property if whenever and

0.: G -> H.

are monomorphisms (i = 1,2), there is

e,

H,

•*

an algebra

L € ft and monomorphisms

L

jh.: H. -> L (i = 1,2) 1

i

such

that the diagram commutes; i.e., for all

g 6 G,

g0-,i|>-, = gO-iK.

Both the class of groups and the class of distributive lattices enjoy the amalgamation property.

THEOREM IOC. amalgamation

However, surprisingly,

The class of all l-groups does not enjoy the

property.

The repercussions of Theorem IOC are frightful and justify extensive study of ^-permutation groups for anyone interested in ^-groups.

Anyone doubting this should examine the proofs of the

group-theoretic analogues of theorems in this chapter and Chapters 12 and 13, and compare them with the devices we are forced to use. In especial, we do not know if each recursively presented £-group can be ^-embedded in a finitely presented £-group.

(This is the

£-group analogue of Graham Higman's famous embedding theorem for groups.

All proofs of Higman's theorem for groups (or other

classes of algebras ) rely heavily on the amalgamation property or on facts that imply the amalgamation property. ) We begin with the proof of Theorem IOC. where

(m,n,0)

a = (1,0,0),

conjugated by

b = (0,1,0)

(metabelian) £-group. t-subgroup of so

H-. .

and

Let

Q = (S § Z ) U (^ § ^ ) ,

X

(Af )h = Af A( Q)

generated by

T $ 7L. Since

\\J ,

ipp

b0?

E1

is a

an Abelian

and

G

R(X)

is

i.e., A n B = {e},

h 6 A(Q)

h,

H2

and let

be given

be the £-sub0 * G -> H

be

We now prove, by reductio ad absurdum, that

such that

generated by

We write

0 • G -> A( Q,) is given by:

f,g € G ) . Let

G0~

5z

are prime subgroups of

(X = A or B ) ;

two copies of

there does not exist an £-group (i = 1,2)

Then

(Bf)h = Bfc 2 (f G G ) . Let

and

the inclusion map.

L,

G

is an ^-permutation group where

group of

such

B =

in

(Xf )(g0 2 ) = X(fg) (X = A or B; by:

(n,m,0).

c = (0,0,1).

and

= (Z @ Z )

is a chain (Appendix I Lemma 2) where

the set of right cosets of

(G0p,fi)

is

^

G = ( © ) x" ,

A =

0, - R(A) U R(B)

(0,0,1)

Let

L

g0-.ip-, = g0?ipp exist.

and

Let

C

with £-embeddings ty.: H. -> L for all

g G G.

So suppose

be the convex ^-subgroup of

k = h(c0 ? )~ .

Since

B(b0 ? ) = B

and

H

185 Bk = B, C C H

2

H A(fi) . Now B( a9 ) = Ba f B

Zorn's lemma, there is a convex ^-subgroup

so a0 £ C. By

P of L

containing

Cipp which is maximal with respect to not containing (= a e ^ ) . Since

c^bc = a

a0piK

and b e ^ = b e ^ £ C^2'

C

^l ^ P*

By Appendix I Lemma 3, P

is a prime subgroup of L; so T, the

set of right cosets of P

in L, is a chain.

be defined by:

(Px)(yx) = Pxy (x,y € L ) . Then

morphism and (Lx,T) G

is Abelian,

Let x : L -* A(T) x

is a n £-homo-

is a transitive ^-permutation group.

Since

Bfb = Bbf = Bf for all f 6 G . Thus

supp(b02) C R(A).

Hence

be A h = e (in A(ft)).

Therefore

Now

A

6_. ip., = aQp^p ^ P *

Hence

(Pcip.. Xhij^x) = Pcip, .

Consequently,

PciP1 = P( c ^ X h ^ x ) > Phi|>2x = P(kc 2 e 2 )ij, 2 x = Pc 2 6 2 ^ 2 x = PCciJ^)2, ci|;1 £ P

and

kip2 € Ci^2 £ P .

Thus

2

Pcij^ > P ( c ^ 1 ) .

But

as

ci^1 > e,

the desired contradiction. We next give a proof of Theorem 10A.

The proof mirrors one

given for groups by B. H. and Hanna Neumann [59]. For the definition of

[G,G] see page 129. We need the following lemma:

LEMMA 10.1. Let (G3Q.) be an l-permutation group and (W3R) = (G,Q) Wr CEJ^).

Then there is an l-embedding ip: G •* W so

that Gty S, [W,W]. Proof: "g

Let g € G. Define

if n = 0 Then

_e Let

fn = i

ip: g »—> gip is clearly an ^-isomorphism.

if n / 0

g € G and define g"1

g\p £ W by: gij; = ({g },0), where

f = f(g) = ({f },0) where

if n >_ 0 . Let z = ({e},l).

e

if n < 0

Then

z"Xfz = ({hn},0)

186 g" where

1

if

n >_ 1

if

n < 0

hn = <

, So _e

g ^

Proof of Theorem 10A: Let of the countable £-group

G

fg(i)" f(i) = <

be defined by:

n

(g(i): i €. Z } be an enumeration

and let 1

[f,z] £ [W,W] .

f (i) = ({f(i) },0) € W

if n >_ 0 . So if

[e

z and ty are

if n < 0

defined as in the proof of Lemma 10.1, g( i )\\> = [f(i),z] +

i £ ^ .

Let

(H,T) = (W,R) Wr (Z,Z) and define

f ^-embedding and b = ({e},l)

G

f"z

H,

and

e

if

n f 0

n = 2i - 1

and

Gtp6

b.

is clearly an

= g(i)i|;. Let

be defined by:

(i 6 ^ + ) .

We complete the proof by

is contained in the ^-subgroup of

First observe that

c(i)n = < z

if

I

2i—1 Hence

0

where h( i)

a = ({a },0) € H

"f (i + j ) if where

Then

otherwise

showing that a

n = 0

by:

n = 0

a n = < f(i) if

by

if

g(i)i^6 = ({h(i) },0)

if

[__e

w

9: W -> H

for all

b

~ ab

"

n = 2j & -i < j n = 1 - 2i

generated

= ({c(i) },0) a (i£K+).

otherwise

-(2i-l)

ab ~f(i)z

" " a = ({d(i)},0) if

f (i + j) if z f ( j) e

b

H

where

n = 0 n = 2j f 0 & -i < j € Z

if n = 1 - 2i if n = 2j - 1 (j £ % ) otherwise

. Therefore

187

[f(i),z] where

if

n = 0

if

n f 0

k(i) =

[b 1 ~ ab

1

"

;

i.e.,

,a] = g(i)i|;0, as desired.

We next provide a different proof of Corollary 2K which illustrates both a new technique, and a new class of chains. Let

a

be an ordinal number. A chain

n -set if whenever there exists

A,B Q fi with

a €. Q

cardinality #

such that

A < B

and

Q

is said to be an |A|,|B| < H ,

A < a < B. Any n -set of

is called an a-set; e.g., a O-set is just Ifi.

n -sets are precisely the H -saturated models of the theory of dense linear ordering without endpoints (Chang and Keisler [73], Proposition 5.4.2).

Consequently, we obtain the following (see

Chang and Keisler, op. cit. ): (a) Any two a-sets are ordermorphic (Theorem 5.1.3). (b) Any a-set is

^

- homogeneous (i.e., any ordermorphism

between subsets (of the chain) of cardinality < #

can be extended

to an automorphism of the chain) (Theorem 5.1.14-). (c) If then

T

Q

is an n -set and

can be order embedded in

T Q

is a chain with (Theorem 5.1.12 ).

|T| <_N ,

188 (d) Assuming GCH (Generalised Continuum Hypothesis) a-sets exist whenever

ft

(cf(« a ) - HJ

(Proposition 5.1.5).

is a regular cardinal (i.e., if

If ft1 is regular and ft is an a-set, then

(e)

|ft| = 2

(Gilman and Jerison [60], pp. 185-189). It follows immediately that (f) If character

ft

is regular, then any point of an a-set has

ft

is regular and ft is an a-set, then

c aa

(g) If

cf(ft) = ci(ft) = # a for

ci(ft) = &

(For (g), for

, take

For each cardinal is regular.

cf(ft) = fta, take

B = 0;

A = 0). K, K

(the successor cardinal of K )

Hence the class of regular cardinals is cofinal in the

class of all cardinals.

In view of (b), Theorem 2E, Holland's

theorem and Lemma 2.5.2, to prove Corollary 2K it is enough to show: THEOREM 10.2. If

fa

is regular and T

is a chain with

< W then T can be order-embedded in an a-set ft so that — *^ a3 each element of A(T) extends to a member of AfftJ. Proof: Suppose If no point of A

and

B.

than S

T

A,B C T

with

lies between

A

A < B and

B

Since the number of subsets of

and

|A|,|B|

a adjoin an a-set between T

of cardinality less

is at most ft , we will have adjoined at most S

a-sets to

T

as

A

and

B

run through all possible candidates.

Hence the resulting set ft has cardinality ft . We now show that ft is an a-set. A1 < B !

sup A = sup A'

and

loss of generality. and so

A

1

< a < B .

inf B = inf B', If no such

0 j- A = {af € A': a' > a} T

B

A

a € A'

and

A,B £ T with

|A|,|B| < ft without

a 6 ft with B

A < a < B;

exist, then (without

such that

is contained in an a-set

in constructing ft. Then

(Of course,

then

Hence there exists

loss of generality) there is to

A',Br C ft with

|Af|,|BT| < ft . If there exist

and

T

Let

may be empty).

|A|,|B| < fij where

Now

A < B

and

A^

adjoined

B = Bf O A Q

A,B c: &

so

189 there is

6 €. A

such that

subset of ft. Therefore

A < 6 < B.

A

!

< 6 < B

r

But and

AQ

ft

We next define an equivalence relation if

a1&o2 (a

< a

a

l

=

0T

°2 a

(°1 < a 2

a

& T 0 [ 2^ i^

=

$)•

& T

Tne

^ ^al'a2^

is a convex

is an a-set. (ft on ft by:

=

°r

^

equivalence classes of <#

are of

three types: (I)

(II)

{ T } for some

A U {T}

T €. T

for some

a-set); such a (ill)

T

(such a

T € T

x

has initial character

(i.e.,

A

is a "tagged"

has initial character < &

An a-set we adjoined to

T

(in T ) .

to obtain ft.

Clearly any two equivalence classes of the same type are ordermorphic (use (a) for (II) and (III)), and no two equivalence classes of different types are ordermorphic. A., U {T..},Ap U {Tp} ordermorphism

(J>

For each

equivalence classes of type II, choose an

*, A- = A^;

and for each

classes of type III, choose an ordermorphism

A ,Ap <J>

equivalence

• A

= A .

Note

that each equivalence class of type III is obtained by a cut in equivalence classes of types between

sup{x <£ : y €. M}

(y € M, v €. N ) ,

I

and

and

II;

viz:

infixtfZ.: v € N}

|M|, |N| < H

and, in

T,

A

is inserted

where

x ,x

€ T

sup{x : y €. M} =

inf{x : v G N} € T\T. Let

f e A(T). Define

xf = xf.

If

a^

xf<S = A

U {xf}.

fU(fl)

as follows:

is of type II, say o& = A For

6

G. A ^

let

If

u { T },

6 f = 6 cj>

x G T,

let

.

Let

A][

A-,

is inserted between

{x # : v G N } .

Let

A^

be the type III class inserted between

{x f<#: y €. M}

and

Clearly

f GA(ft)

{x f<£ : v 6 N } .

and extends

f.

For

{x <£: y G M}

be a

type III class, say

6 G A.,

let

and

6f = S * ^ *

This completes the proof of

Theorem 10.2.

Now suppose

is regular and

Q

clearly no point of ft has character

c

ft _

S

be regular with

B < a.

Let

P

sequence of points in ft. Then

is an a-set. where

Then

3,Y < a.

Let

{a : v € a) } be an increasing v p a = sup{a : v € a)g} € ft and hence

190 a

has character

character

c

co . Similarly, there are points of ft of pa

. But there are only $

subsets of ft of

J* H |ft| = 2 a (by (e)), there are . By (b), if ^ D is regular

cardinality less than ft . Since 2 points in ft of character c otot

with

3 < a,

p

then any two points of ft of character

cn

(c _)

pa

lie in the same orbit of c

, then

A(ft).

{o € tt: o < o}

Now if

and

{a €-ft:a < T } are a-sets and

hence ordermorphic by (a), say by ordermorphism g € A(ft) by

g_.

Similarly there is an

g • {a €.ft:a > a} = {a £ ft: a > T } . if

o
if

o >o

ag = ^

ft of character

c

. Then

og - T

lie in the same orbit

T

is an I-automorphism of

A(ft),

permutes the orbits of

$

and ft are the only orbits of or ft$ = ft. But

\T\ = 2

an inner automorphism of

LEMMA 10.3. If & orbits of c

aa

,

a

A(ft).

If ty

it is conjugation by some A(ft).

A(ft) of character

c

\p €. A(ft)

Since

T

, ft$ = T

> ft = |n|. Hence fl$ = ft and

if; is

A(ft). We have thus established:

is regular and ft is an a-set> then the

Afftj comprise ft., all points of ft\ft of character

and all points of ft of character co

pa

(c o) ap

l^n regular). Therefore^ any Z-automorphism of

with

Define SO any holes in

of

by Theorem 2.4.1.

Now

ap

<J,T £ ft\ft have character

is inner.

COROLLARY 10.4. If

to

\G\ <_ to *

can be t-embedded in an t-group in

then G

is regular and G

(3 < a and AfftJ

is an l-group

which every t-automorphism is inner> namely A(Q) where ft is an ex-set.

Theorem 10.2 is due to Holland [63], Lemma 10.3 and Corollary 10.4 to McCleary [73a]. In the proof of Theorem 10.2 we did a "weak tagging" to show that any element of A(ft).

A(T) can be lifted to an element of

Before proving Theorem 10B (again using a-sets), we digress

191 to give a further (and truer) application of tagging due to Scrimger [70]. THEOREM 10.3. Let (G3Q) be an ordered permutation group. Assume that whenever (7T,ftJ is an ordered permutation group with aG = OLH for alt a € ft., then H *=. G. Then there is a chain T 2 ft such that G is t-isomorphio to A(T). Proof: Let {ft. : i £ 1} be the set ofG-orbits of points of ft (So if (G,ft) is transitive, |l| = l ) . The idea is to enlarge ft by adding to each element of ft. a tag, so that points of ft. receive an identifiably different tag from those of ft. if ft. i- ft.. This will ensure that A(T) will respect the original orbits of (G,ft). However, we must be careful to ensure that no point of ft. gets mapped by an element of A(T) onto a point on a tag. +

For the purposes of this proof, we consider Z to be the set of non-zero finite ordinals. So n = {0,1,...,n - 1}. Since {£ is countable, there is a one-to-one correspondence between it and Z . Denote this correspondence by q •—> n . Let X = {(m,q): q £ Q & m £ n } ordered antilexicographically ((m,q) < (m T ,q') if q < qT or (q = q1 & m < m r ) ) . Since

Q2 is dense, there are infinitely many points of X between (m,q) and (m',q T ) if q / q 1 . Hence if f £ A(X) and (m,q)f = (m1 ,q' ), n < n r. . But (m? ,qr )f = (m,q), so n , <_ n . Thus n = n , q - q

192 Therefore

q = q

T

and

f = e.

For any ordinal then as yf = y

X

y,

y*f = y*.

Since

(inversely) well-ordered, Well order ft) and let

I

|ft| < y. < y.

if

(y.)

a

T =

UT(CX);

a € ft ft S T . Let af € T(3)

i.e.,

with

where unless

(y*)

G-orbits

For

i.e.,

i = j

and

a = 3;

So if

T(a) < T( 3)

f € A(T)

if

and suppose

Therefore

y.j_f 1

y* J

we get

a < 3

(in

j 6 1).

Now

If

G

y*f

contains

X( y. )

such that

then

X

y*f -*-

A(T)

Clearly

a € supp(f). af # y. y.f;

So

(otheri.e., a

contains an Also

y..

Hence

y. < y..

the desired contradiction).

ft;

ft).

y. = y..

i.e.,

Hence

A(T)|ft

By hypothesis,

f|ft

has the

A(T)|ft 9. G.

naturally extends to an element of A(T).

is £-isomorphic to

COROLLARY 10.6.

Let

as a final segment whence

of points of

G

ft).

which is impossible.

y. < y.,

(for points of

But every element of

disjoint chains3

af € X,

and by the above argument

G-orbits

same orbits as

Consequently,

of points of

a,3 & ft belong to differ-

is an initial segment of

af = 3,

preserves the

is

i. e., we have

a € ft. with

3 € ft. (some

(Otherwise

f

y

Hence

a € ft., let

is a copy of

X(y.).

infinite initial segment of

By applying

Xf = X.

they receive an essentially different tag.

for some

and

f € A(X(y)),

A(X(y)) = {e}.

i < j.

X (y.)

non well-ordered subset).

Xf = X

and

contains an infinite initial segment of

af £ y* J

If

be a collection of limit ordinals with

with a copy of

G-orbits,

y.

f = e;

i,j€I

X (y.) H X (y .) = 0

ent

A(X) = {e}

(the set indexing the

{y. : i € 1}

T(a) - {a} U X

wise

A(X) = {e}.

X(y) = y* u X V y.

contains no final well-ordered segment,

and

tagged

Consequently,

let

If

{ft.: i £ 1}

there is a chain

T

G

(via restriction to ft).

is a family of pairwise such that

k(T) =

TJA(T.). ^ i€I

Chang and Ehrenfeucht [62] used tagging to prove:

THEOREM 10.7. A(T)

for some chain

Every proper subgroup of T.

%

is isomorphic to

193 The idea is to give different tags to different cosets of G

in

1

in the same way as in the proof of Theorem 10.5.

The

details are straightforward. We conclude the chapter with a proof of Theorem 10B. Specifically, we will prove:

THEOREM 10.8. ____________

Let

fa

________

be a regular cardinal and

> Mn

Q£

G

(j

be an l-group with \G\ <_H . Then there is an t-group H 2 0 such that \E\ <_# , and if e < hnih9 £ Ey y 6 supp(h^) and Ot

$0 £ supp(h^)3

±

there is

f £ H

ci

U

such that

JL

y / = 6

and

f\f = h2. The key to the proof is Theorem 2.2.5. ^-permutation group and conjugate in bump.

A(T)

e < g € G

to any

If

has one bump,

e < f £ G

(G,T) g

is an

is not

which has more than one

Indeed, as we saw in both the proof of Corollary 2.2.3 and

the illustrations prior to Theorem 2.2.5, two elements may have the same number of bumps and yet not be conjugate in will have to enlarge

T

mapped to an element of

to

Q,

say, so each

A(f).

e < g € G

So we

gets

A(ft) which has a lot of supporting

intervals with whole intervals between any two distinct bumps of the image of

g.

Then, hopefully, any two strictly positive

elements of (the image of)

G

will be conjugate in

G

' __-A(ft).

If we can achieve this we will be home since we can then proceed to spread the disease, by induction, up a countable sequence of ^-groups

G = G

elements of H =

G

U G ^ m£u)

_= G '

£, ...

where any two strictly positive G^ m +

are conjugate in

' (m e co).

will satisfy the theorem (if | G ^ | £ #

homogeneous and

Q.

(G,T)

an ^-permutation group and

By Corollary 2L, we may assume

GCB(T).

embedded in an a-set. A( Q)

T

is doubly

An obvious candidate for ft is an

a-set since each chain of size not exceeding ^

g\p of

all m ) . a

So we will start with obtain such an

element

Then

As we want each such that

gip

e < g €. G

can be orderto extend to an

has lots of bumps and whole

segments of ft between any two distinct bumps, it would be rather

194 nice if we could arrange that the set of bumps of

gty is an

a-set and similarly for the set of "fixed point intervals." More precisely, if

(G,T) is an ordered permutation group and

a fixed point interval of with respect to

is a convex subset

A

of

T maximal

g € G . Recall that the set of supporting

intervals of an element of

g

g € G,

g

is just the set of supports of bumps

g. Our proof will involve transfinite induction for ft steps. In the first lemma, we will ensure two things:

x £ T

(a) If

is the infimum or supremum of a set of bumps of

then we can squeeze a bump of the image of

g £ G,

g between them and

x.

This will help obtain density of the set of supporting intervals of the image of (b) If

g

x£ T

(if we repeat the process enough times),

and

{x}

is a fixed point interval of

g £ G,

there is a whole segment (in the extension of

T) about

is a fixed point interval of the image of

Even though the

image of

g

g.

x

then

that

may still have fixed point intervals comprising just

one point, we will be able to remove this problem if we repeat the process sufficiently often.

LEMMA 10.9. Let there exists a

(G3T) be an l-permutation group. Then

T__ containing T

such that

\T \ <_ max( \G\3 \T\ )

and an t-embedding \p: G -> k(T ) satisfying: (i) For each T £ ^

there exist ^\^\ ^

such that

{x' £ T: x' < x} < x* < x < x* < {x ' £ T: x' > x}; (ii) For each g £ G> gi>\T = g; (Hi)

If

I

is a non-empty collection of supporting intervals

of g

and

A

G^> such that o < A < VI

°f

infyp(U I) = o £ T3

then there is a supporting interval (and dually for

sup^U I) = x £ T); and (iv) If

A

is a fixed point interval of g3

then A

is a

subset of a fixed point interval (of cardinality greater than 1)

of gty*

195 As a good guide to the proof of this lemma, keep in mind (B(QJ),I) to

g

h

which has two orbits

and

h

where

has support

g

(0,/2*).

(fi and

has support

IVfi.

Consider what happens

(-1,0) U (1,2) U (2,3)

We will sketch what happens to

g

and

in the

proof. Proof: K £ k,

C =

°K 6

Let

k

be the set of G-orbits of

T.

For each

choose

0^ € K. For each 3 £ K, pick c €. G such that K insisting that c K, let If 3,Y K ift c . Consequently, 6 C and 6Y = Y ' C Y6 = °6Y and =c if e,Y,<5 € K. Define ^-ideals 3,

o

= {g € G: supp(g) H [3,0 ] = 0

by dually.


for some

Using Holland's theorem (Appendix I ) , we can find

and

Tp(K),

and

(G

non-empty disjoint chains such that

and

R

T..(K)

(G Q /L K ,T (K))

K are ^-permutation groups where

K | T-. (K) |, | Tp( K) | <_ |G|.

Moreover, by Corollary 2L, we may assume

that all elements of these ^-permutation groups have bounded support and that T(K) = T n (K) 0 {^} symbols.

Define

T.(K)

is doubly homogeneous

U T O (K) i|> • G n

where

•> A(T(K))

{gv: K £ by:

(i = 1,2).

Now let

is a set of new

196 "x(gL K )

r(gR K )

if

x £ T X (K)

if

x = CK

if

x £ T 2 (K)

•

We have made sure that if

was a fixed point interval of some interval around

CU

in

T(K)

g £ G,

{CL.}

then there is a whole

that is a fixed point interval of

g\\jv (since every element of in

G~ /LT. (G~ /RT,) has bounded support K K This avoids (2) and (3) on pages 61,62 not being

T (K) (T 2 (K))).

conjugate—provided that

CL. is strategically placed.

Clearly

tyrr is an £-homomorphism. We now generalise the Wreath product to form the orbit Wreath product. which

3

If

belongs.

3 € T, Let

b

|T j <_ max( |G|, |T| ).

T

or

(in

We can order-embed

T)

T

K(3 ) € k

be that G-orbit to

T* = {(x,3): 3 €. T & x £ T(K(3))}. b

if

31 < 32

let

in

is totally ordered by: ^

= 32

T*

by:

and

B ^

T± < x 2

SO T

(x ,3-,) < ( 2'®2^ (in

(CwoV^)JH P )

T(K(3 1 ))).

So (i) holds.

Under this identification, we have tagged each point of tag dependent on the G-orbit (cf. proof of Theorem 10.5).

T

by a Now let

T T G^Y(Q^' (W,T*) is an ^-permutation {{i) 3£T^ (x,3)(h,g) = (xh^/ y 3 g ) and the group and lattice

W = {(h,g): g £ G & h e group where

operations are the same as for the Wreath product of two £-permutation groups (see page 117).

The only difference between the Wreath

197 product of two ^-permutation groups and what we have described here is that the global component is not transitive and the local components of the orbit Wreath product depend on the G-orbit of the global part. —1

Define ty: G -> W

by:

gip = (g,g)

where

-i

(cf. page 120).

(Note that

CUc o gc~

g o = (cDgc P

= CL, so

P

P pg

K(B)

ij> is well-

defined. )

If

g,h € G

and for some

^Y2 ] = h|[y 1 ,y 2 ],

then

Bg = Bh i.

K

IB

^

= (c gc

r

and for all

Hence

g '/" = Y CCO -^^cD C^* yc gc

-± \

B Bh

^ ^

g

Thus

K(B) K(B)

*

)T|/J-/ \ = (c.lic., )L/ f t \ = h .

determined by the action of of

(in T ) ,

")( c hc~ )~ € L w „ x A R w . v. ;

so

-1

g

y1 < 3 < Y 2

^

That is,

g

is

on arbitrarily small neighbourhoods

B. We now show that

i|/ Is an ^-embedding.

If

g,h € G,

fp-hL = (c_p-hc~ , )ib,w . N = (c_gc^ -c. he ^ , )}!)„/~ \ = £ ^ h _

^K(B)

is a homomorphism.

Thus

then since

(gh)i|i = giphijj and so ty is a

198

homomorphism.

Moreover, i f

1

eltSgjSg" ];

so

3g < 3,

(g"v"e) 0 = e fi . P

If

3 < 3g,

then

so

(g^e)

=g

e(g v e ) = 3g = 3 and hence

= c gc

V c

3 3g^K(3)

ec X

B 3 *K(B)

(g v e)i> - (h,g v e )

where

= g

V e

3

h =

gip v e.

Furthermore,

3(g^) = ( ^v( Q V 3 )(gij>) = so

Consequently,

v e

3g = 3 c

> 3( g

Ve

Thfirefore

*

is precisely

y 3 g ) = 3g,

If

(g V e ) g = c ^ g

e

V

(Cw«

(g v e ) | [ 3 g , 3 g ~ ] =

P

(g v e)|[3g" 1 ,3g] = g|[3g~ 1 ,3g]; then

then

if if

3g > 3 .

if

3g = 3

3g < 3 But this

is an £-homomorphi sm.

*

i|; extends the inclusion map.

Thus ty is

one-to-one and satisfies (ii). It remains to prove that (iii) and (iv) hold. Let

(iii) intervals of 3 > a,

g

I

be a non-empty collection of supporting

and let

there exists

K = K(a),

then

a = inf„,( U I ) .

T € T

such that

c gc~ € G^ \RV. o og U^ K

interval contained in

(T 2 (K),a).

Then

og = a

x €. [a,3]

and for each

and

xg j- T .

Hence

gty has a supporting

Since

a < (T 2 (K),a) < U I ,

If

(iii) follows. (iv) then

Let

A

6(gi|;) = 6,

be a fixed point interval of so

A

g.

If

A = {6},

is contained in a fixed point interval of

gifj.

Since each element of G n tyv, x JT,(K(<5 )) has bounded support in U K( 6 ) ^ 6 ^ J b T.(K(6)) (j = 1,2), there is an interval of T properly J containing {6} on which g\\> is the identity; so we have expanded the singleton fixed point interval to a non-singleton one. |A| > 1, then

let

A

= Conv .(A). We show that if

(T,6 )gip = (x,6) (whence

interval of

gty). If

be such that g U M ^

6 < 6

=

A

x € T J K ( 6 ))

(T,6).

or there exist

,

is contained in a fixed point

c6gc-l 6 R K ( 6 ) ; 6

(x,6)(gip) = ( g K ( 6 )* g) element of

A

(x,6) €. A

is the least element of

Now

= e|[6,5l], T

5

If

so

let

6. € A

and as T g K ( f i ) = T.

Similarly, if

6^,6^ €. A

A,

with

6

Thus

is the largest

&.,<&<

S~. //

199 We have still some problems remaining. Firstly, if we look at (2) on page 61>

we are not guaranteed that any two elements

fitting that picture are conjugate—a hole in

may belong to

T. Also, if we have a non-empty set

intervals of

g

and

inf=(U I) € f\T,

I

T

or be a

of supporting

we haven't necessarily

increased the density (this is just a more general setting of the previous sentence). e < g,h € G

have supports

then in passing to g\\) of

around

hty around

(G,T) = (A(ffi),Q) and

U

(0,1) U , 2 )

(G,T^,i|0,

and

(0,/2)U (/2,2),

there is a fixed point interval of

1 but still only a hole for a "fixed point interval" /2. So it seems that we may be further from our

goal than before. might have

Indeed, if we took

|T| = 2

If we had just passed to >&

and #

(G,T) first, we

steps might have been

insufficient—we quickly get into trouble and the method of proof fails.

Hence we must insert some cuts into

T

before proceeding.

Since we seek a-sets, we must get by with very few cuts.

This

will be our thrust in Lemma 10.11. We first require some more background. We now generalise the concept of an a-set. chain, ever

Let ft be a

r Q.ftis said to be an a-splitting subset of ft if when-

A,B c ft, |A|,|B|
such that

A < y < B.

a-set of degree K_ if

Let

and

A < B,

there exists

y €r

K be a cardinal number, ft is an

|ft| =K

and ft = (J ft where each ft

is an a-splitting subset of cardinality #

and ft A ft f = 0

y ^ y' . Note that if ft is an a-set of degree

K, then

if

K <_tf

and each

ft is an a-set that is dense in ft and ft is itself y an a-set. Conversely, any a-set is an a-set of degree 1. We now generalise (a) of page 187. LEMMA 10.10.

If A{

is a regular cardinal and

K is a

cardinal not exceeding ft then any two a-sets of degree K are ordermorphic. In faot3 if ft = \J ft and A = \J A y
ft (A ) and

y

y
where each

y

is an a-splitting subset of ft (k) of cardinality H^

yn € Q,^ 6^ € A^j then there exists an ordermorphism

200 n.' ft -* A

such that ft n = A

Proof:

Let

each

and

{y : v < ft }

orderings of ft and (n : v < ft }

for all

A

y < K

We construct a sequence

Y Q H Q = 6~)

satisfying, for

v < dy : (1)

{ y t : v ' < v } ^ dom(n ) £ ft.

(2)

{6 , : vf < v} c: image(n ) 9 A.

(3)

n

is one-to-one and order-preserving.

(4)

If

V

< v,

then

n ,c n .

(5) y £ ft H dom(n ) if and only if Then

y~n = 6 ..

(6 : v < # } be well-

respectively.

of functions (with

and

n = \^J n V<

yn

€ A .

will be required ordermorphi sm, the onto-ness

V

*a being guaranteed by ( 2). By (a) and (b) on page 187, we may assume n0

=

Y^ru = 5^;

I.e.,

{(YO,«O)>-

Assume

(n^: S < v}

has already been constructed satisfying

(1)-(5 ) as well as (6) If

In^l < W a v

for all

g; < v.

is a limit ordinal, define

n

=

LJ n r*

Tn

i s clearly

satisfies (l)-(6). If

v

is a successor ordinal, say

dom(n ) ^ ^,

by (6). Let

£

v = £ + 1,

y

£ dom( n ). By hypothesis,

y

€. ft . Let

t, <_ E,. Then for a unique

A = {y 6. dom( n ): Y < Y r )

B = {y €. dom(n ): Y r < y].

then

be the least ordinal such that

Then

A < y

and

< B,

so

An

|An |,|Bn | < St • Hence there exists a least ordinal that

6

AUB

= dom(n ), 6

Yc, = 6

T

£ A

1

fi^' .

and

If

An

ft

^ image (n ). If

6^ = 6^t,

such that

0n

let

Then

y^

6 6 A

and

y . $2f dom(n ). Now define n

such

6 € image( n ), let = y^ ,

If say.

We can obtain

< y , < ^n" 1 , where

Q = {5 €. image(n ): 6 < 6 } Again,

< Bn . But £'

< 6 , < Bn . Note that since

^ image(n ) U {6 , } , then

Y ,€

y^ < K,

$ = {6 € image (n ): 6

< 6}.

n v = n V {(Y.^ , ), (y , ,6 )}.

satisfies (l)-(6), completing the proof. //

201 Let ^

be a regular cardinal greater than K n -

be the union of

T

ci(B) < to • T # C f

cf(A) < ft or on

T

to one on

(A,B),

Let

T*

where either

so we can extend the order

T*.

LEMMA 10.11. \T\ <_ Hrf,

together with all cuts

If

then

(GyT) is an l-permutation group with

(G3T )

is an JL-permutation group satisfying:

(i) \T*\
A C T

and

|A| < H „

then sup A € T*

(inf A e T ) provided A (Hi)

is bounded above (below) in T.

Each fixed point interval of g £ G

is contained in a

fixed point interval of the extension of g. Proof: Since element of

A(T*).

T £ T,

we can extend each

Specifically, if

g € G

to an

(A,B) € T*,

(A,B)g = (Ag,Bg) G T*. Now the number of subsets of

T

of cardinality S

K

H

i-s

|T|

<_^P . Hence |T | <_ max(^ ,sup{^ : y < a}). But u a a a svpito u : y < a} = sup{/V : y < a} = to (we are assuming G.C.H. ). Thus (i) is proved. To prove (ii), let B1 =

{T

x >_ X Bp. of

= {T £ T: T <_ X

£ T: T > A}, A o = for some

Hence A

A

in

{T

X ^ A}. A

£ T: T < A}

Bo =

{T

X € A}, G T:

is cofinal in A., and coinitial in

(A^, B^ ),(Ap,Bp) £ T T*.

and

for some

and are the infimum and supremum

(iii) is immediate.

Passing from

T

to

T* will again permit us more freedom

to expand the supporting and fixed point intervals in accordance with our plan. Moreover, every element of

G

is bounded (in

T ).

In order to prove Theorem 10.8 we will proceed by induction interlacing # added that if T €. T\T

and

t> as mentioned on page 199.

has character

c

and there is

e < g € G

is a supremum and infimum of collections of JV intervals of # s

and

It should be

(G,T) is our original ^-permutation group and

g,

then in enlarging

G

and

T

such that T

supporting through subsequent

t?s we will never remove this phenomenon.

It seems,

202 therefore, that our program is doomed. enlargement of any

e < f € G

However, the "final"

(even if

f

has but one supporting

interval originally) will also enjoy this phenomenon as we will see.

Proof qf_ Theorem 10.8: Let group with

T

doubly homogeneous,

(Corollary 2L). sequences

Let

ipQ = e

{ft : y < yQ}

satisfying for each

(I)

(G, T)

be an ^-permutation

|T| <_ |G| and

G^B(T)

and ft~ = T. Assume that the

and

{ip : y < y~}

have been constructed

y < yQ,

|fty I —< N' . ot

(II) (G,ft ,\\> ) is an ^-permutation group with (Ill) If

G\\>

is a limit ordinal, ft = \jft. ^ v
£B((i ).

y

(IV) If

n

finite, then ft = (ft,~ )*. y X+2n (V) If

y = X + 2n + 2

for

(VI)

r

(VII) If

y' < y

and

g\h ,, then

A

n

r

y • yf

is a fixed point interval of v

A of g\h . is a limit y ordinal, let ft = \J ft and

yn

U U

a limit ordinal and

(A . ) is contained in a fixed point \

interval

yf

t

Conv

v

If

X

finite, then fty = i^x+2n+1)^ I f y 1 < p , t h e n Gip , = Gip I ft , .

* • Clearly (I)-(VI) are satisfied.

If

V

0

fixed point interval of

A

is a

V

gi|/

with

y < yQ,

and

6 € Conv

(A ), y

then

3 ^ ft , for some

y'

the induction hypothesis,

such that 3(g^

satisfied. If

y £ y' < y^.

) = 3(g^ , ) = 6.

0 Hence, by Thus (VII) is

° y

u

= A + 2n + 1,

let

ft

y^

= (ft,+Or,)^ and let A'l'^n

the obvious embedding given in Lemma 10.11. satisfied.

Similarly if

y Q = X + 2n + 2

ij; y

be

Then (i)-(VIl) are (using Lemma 10.9).

203

So we can construct sequences

{fi : y < jy } and

satisfying (I)-(VII).

Let

a = |^J n

and

= U We now prove: (1) ft is an a-set. (2) (G,ft,iJ;) is an ^-permutation group. (3) If

e ^ g E G,

(I(gty))

the set of supporting intervals of gty

together with the fixed point intervals of

gip (omitting the greatest and least) a-set of degree (4) If

e 1 g € G

Proof: (1)

form an

2. and

A € F*(gi|;), then

|ft| <_^ =ft . Let

|A|,|B| < # . Then there exists

A

A,B£fl,

y
A,B 9 ft,+2n for some limit ordinal Lemma 10.11,

(F*(gty))

A < B and

such that

X and finite =

sup A, inf B £ i^x+2n^

is an a-set.

^x+2n+l -

A,B £fi . So n.

Qm

By (IV) and

If

sup A < inf B, there exists a £ (fix+2n+l/ = ^A+2n+2 - fi suctl that sup A < a < inf B (whence A < a < B). If sup A = a1 =inf B, then a1 ^ A without loss of generality. So there exists T £

^A+2n+l^ is an a-set.

=

^X+2n+2 £

(2) Clearly a € ft. Then

fi

SUch that

gf eA(fl) for all

a £ ft for some

A < T < d

g € G.

Also

Let

ThUS

n

g,h € G

and

y < # . a(gh)i|; = a(gh)i|/ =

(a(g* ))(hi(> ) = (a(g*))(h\|;) = a((g*)(hi|;)). homomorphism.

- B'

(g v e)i|i = (gi|i v e),

Thus

*

is a group

similarly.

Therefore

i|; is an ^-homomorphism that is one-to-one since each ty is. (3) Note that whenever embedded in

y' < y,

and

F^(g^) =

U

I(gip) UF^(gip) is an a-set of degree enough to prove that if d < 23 and

I(gi|/ ,) can be naturally

I(gi|> ) and similarly for F*(g^

y <S

f ).

F*(6* )•

that

and d,(B £l(gi|; ) U F*(gi|; ),

A 6 I(gi|; f ) (A € F*(gi|; ,)) d < A < & y

So t0 show

2, it is easily seen to be

| CL \ , \ £ \ < & , then for some

clearly assume that

Thus

(^
fits case (IV). If &

yr > y

and

iB). We may is non-empty and

204 does not have a least element that is a fixed point interval of gty , &'

then there is a coinitial subset

£l(g*

).

Now

H

inf(U<8 ! ) = a €

10.11, exists

A

1

£ I(gi|/ + 2 )

$'

of ?B

> \dB I > ci(^9 ' ). ft

;

such that

By (IV) and Lemma

and by (V) and Lemma 10.9, there a < Af < U ZS ' .

such that

By (IV) and

Lemma 10.11 (and # a > c i ( $ ' ) ) again, we find T = inf(Ul8 ? ) € ft , where Thus

a < Ar < x < # f . f

< A < U$

Af.

If d

a < A

contains

f

But then and so

T € A

for some

d < A < k <&

where

proceed dually to the above.

gty ,

So assume that

element that is a fixed point interval of

e I(gi|/ )

such that

d<

1

A

<$.

Using

A'

that belongs to

and dually for di.

Q

instead of

Q

of the first case. that belongs to replace 6i

A £ I(gi|/ and letting

If

28 = 0

If d=

F*(gi|; ),

A A

in place of 0

and & A!

then

A < Af.

Working in

we revert to the dual

A € F*(gi|0 Then

y ^_ yo«

A

y

Finally, if

d = 0 = t8,

By (VII),

l(gi|* )

This completes the proof of (3).

and

= AHS]

A!

0. has a greatest element

F*(g\p ), proceed dually.

Let

has

is not a

with the set consisting of a single element of

ft ^ 0. yQ

for all

there is

gij;. So for some

such that

d = {A},

and

and proceed by the first case. (4)

)

gty ,


subset of the least fixed point interval of \i > \i, there exists

then we can

r

IS we can proceed as in the first instance. a least element

)

+

CL has a greatest

gip

By the maximality of fixed point intervals of A

A € I(gi|>

is non-empty and does not have a greatest

element that is a fixed point interval of

T

A £ F*(gi|; ~ ) .

v

y»

be the least ordinal such that

i s a fixed point interval of

A =

[^J

A .

A

gip

y

is an a-set by the

same proof as used for ( l ) . Let

e < h-,,hp £ G.

l(h.}p) U F*(ti.\Ji) and

)rdermorphism

<j> such that

(h.,ij;)(j> = F(hpi|;),

'

e < h-,i|;,h?i|; € Gip.

are a-sets of degree

r € supp(h.)

p

Then

6^ € supp(hp).

2

by (3)

Now (i = 1,2).

Let

By Lemma 10.10, there is an

I( h^ip )4> = I(hpi^)

and

and the supporting interval of

is mapped to the supporting interval of

h^ip

containing

hpijj containing

6Q.

205 Each element of

F*(h.i|;) is an a-set by (4) (i = 1,2). Hence if

A € FCh^),

there is

A € iCh^),

let

3p

=

<5Q if

A

f € A(fi) such that

3 € A

and

B

G A

Acj> = Af . If

<J> (where

is the interval of support of

3

= yQ

h^>

and

containing

VFor each

n € Z,

there is

f = f (A) € A(fi) such that n n

31(h1i|;)Jfn = 3 2 ( h ^ ) J (j = n,n + l) by (b) on page 187. By the patching lemma, is f € A(ft) such1 ]that = fA A £ and there f | [B^h^^B^h^f* = fjf|A [B^h^ (n £ Z, 3-,,32 € A, A £ 1(11^)). f

By construction

all A for e^h ^ )n+1]

y f = 6

and

(h,i|;)f = hpi^. The proof of Theorem 10.8 is now complete.

By using Theorem 10B and the patching lemma, the following partial amalgamation result can be proved (see Pierce [72]):

THEOREM 10.12. be t-groups. Let

Let

G

0.: G -> H.

there exists an l-group L such that 6-7^-7

=

be an Archimedean o-group and be Z-embeddings

H^H^

(i = 1^2). Then

and t-embeddingsty.:H. •> L (i = 1*2)

8 J..

The proof of Theorem 10B relied on a-sets for ty regular. Unfortunately, the existence of such sets is equivalent to 2

a

= N +-, . For those who, like the author, feel the nearness of

the Dies Irae every time they invoke the Generalised Continuum Hypothesis, there is a word of consolation.

By using a modifica-

tion of a-sets that doesn't depend on G.C.H., all the theorems in this section can be deduced provided we relax some of the cardinality restrictions on

0, (see Weinberg [80] and [8la] );

i.e., all the theorems in this chapter are independent of the Generalised Continuum Hypothesis (provided we relax some of the cardinality restrictions on ft).

206 CHAPTER 11 NORMAL VALUED LATTICE-ORDERED GROUPS

Let

(G,ft) be an ^-permutation group and

a € supp(g), of

g

is denoted by n

A(a,g) = Conv^(a) = {3 £ ft: a|g| such that

L(u.,g); i.e.,

< 3 £ a|g|n+1

(G,ft) is overlapping if there are A(a,f) and

inclusion).

If

then the convex subset of ft swept out by the images

a under powers of

n £ %}.

e f g € G.

If

for some

a e ft and

f,g € G

A(a,g) are incomparable (with respect to

(G,ft) is a primitive ^-permutation group, then it

is overlapping if and only if it is transitively derived from a periodic or doubly transitive ^-permutation group; i.e., in the transitive primitive case, if and only if it is periodic or doubly transitive.

We will extend this to the non-primitive case by our

structure theory (Chapter 3).

207 A seemingly unrelated concept from classical £-group theory is normal valued. Let G be an £-group and H a convex ^-subgroup of G. H is a value in G if there exists e / g 6 G such that H is maximal with respect to being a convex ^-subgroup of G not containing g. In this case we say that H is a value qf_ g_. Each convex L-subgroup of G that properly contains H contains g. Hence the convex £-subgroup H' ^ H, where Hr = 0{M: M is a convex ^-subgroup of G and M ^j>H}. H' is called the cover of H. An £-group G is said to be normal valued if H
208 Note that condition (4) shows that the class Jf of normal valued ^-groups forms a variety (i.e., is defined by a set of equations):

G £Jf

if and only if 2

[(x v e)(y v e ) A (y v e ) (x v

(Vx £ G)(Vy € G)

2

e

) = (x v e)(j v e)] .

Theorem 11A is due to Wolfenstein [68] and Read [75], Corollary 11B to Read [75]. In sharp contrast to Theorem 11A, we will show the following result due to Ball (unpublished): THEOREM 11C.

k{R) has no normal values.

We will use Theorem 11A to deduce the following theorem due to Holland [76b]:

THEOREM IIP.

If

1/ is a proper variety of Z-groupsy then

We give two more applications of our ideas. are varieties of ^-groups, define the class there exists an £-ideal It is easy to see that

N

of G

such that

If U

and

\J

(J«l/ by: G € K»l/ if N £ U

and G/N € I/.

U«U is a variety if we use the following

result of G. Birkhoff (see, e.g., Cohn [65]) which we state only for ^-groups: A class ii of Z-groups is a variety if and only if X is closed under Z-subgroups^ Z-homomorphic images and products. It is easy to see that for

I/11-!/ ( n t 2 ,

of all abelian £-groups, 5 ^-groups, and

X

• is associative. We write (/

\J a variety of ^-groups),

A_ for the variety

for the variety of all one element

for the variety of all ^-groups. The following

facts are well-known ((a) is easy by classical £-group results; e.g., Bigard, Keimel and Wolfenstein [77, Proposition 2.4-7 and Corollary 2.5.9]): (a) Jf*Jf = JT (b)

If

1/ f g

(see Martinez [74]). is a variety of ^-groups, then

A 9 1/ (see

Weinberg [65] or Glass, Holland and McCleary [80]).

209 THEOREM H E . Jf is the smallest variety of l-groups that \J An

contains

If

H

(and hence

is an £-group and

valued convex ^-subgroups of of

H

\In if

\J

1/ ^ Z,£ ).

{G.: i £ 1}

H,

then

G

is a family of normal-

(the convex ^-subgroup

generated by

{J G. ) is normal valued. This is easy using i€I x Bigard, Keimel and Wolfenstein [77, Proposition 2.4.7 and Corollary 2.5.9].

We can generalise this result by using Theorems H A and

11D. THEOREM 11F. If

is an l-group and

is a family of convex t-subgroups of H

{G.: i £ 1} G. € \) ,

V is a variety, H

then G 6 \J where G

generated by

with each

is the convex l-subgroup of H

\J G.. i*il'L

Theorem H E is due to Glass, Holland and McCleary [80]; Theorem 11F to Holland [79]. If P

G

is an £-group and

P

is a prime subgroup of

contains a minimal prime subgroup of

Paul Conrad conjectured that if

G

G

G,

then

(by Zorn's lemma).

is an £-group in which each

proper prime subgroup contains a unique minimal prime subgroup, then

G

is normal valued.

We complete the chapter with a proof

of the following theorem due to George M. Bergman (unpublished): THEOREM 11G. There is a non-normal valued l-group in which each prime subgroup contains a unique minimal prime subgroup.

Proof of Theorem 11 A:

We first prove that (3) is equivalent

to both (2) and (4). (2) -> (3). Assume

(G,Q) has a primitive component

(G^,^ ) that is not static, integral or regularly derived.

By

A. J\

Corollary 4.4.1, it is doubly transitively or periodically derived. To prove that a < 3 < Y

(in

(G,ft) is overlapping, it is enough to find Q) with

K = Val(a,y),

and

e < f,g £ G

v

(5K)

such

210 that

Y C K £ A(B,g)\A(3,f) and

al?K CA(3,f)\A(B,g).

the problem to the primitive case.

So we may assume that

is doubly transitively or periodically derived. (using Theorem 4.3.1), there exist such that

and

(G,ft)

In either case

a < 3 < y

f £ G , g £ G , af = 3

a G A(3,f )\A(3,g) and

This reduces

and

e < f,g € G

3g = y. Now

y £ A(3,g)\A(3,f),

so

(G,ft) is overlap-

ping. (4) -v (3).

Assume

(G,ft) has a primitive component

that is not static, integral or regularly derived. to find

a £ dom(K) and

e < f ,g € G

„

such that

() (5K)

(a?v)fg > (aC^)g2f2.

So, as in (2) -* (3), we may assume that

(G,ft) is doubly transitively or periodically derived. a,3,Y £ ft and af = 3

and

(G,ft)

It is enough

e < f,g £ G

3g = Y-

be such that

Let

f £ G , g £ G ,

afg = y = yf2 > af 2 = ag 2 f 2

Then

as

desired. (3) -> (2). exist

Suppose that

a < 3 < y

in ft and

a £ A(3,f )\A(3,g) and some

+

n £ %,

(G,ft) is overlapping.

e < f,g £ G

y £ A( 3,g)\A( 3,f).

n

n

yg" < af .

n

So

So there

such that

yg" £ ySv

Let

K = Val(a,y). n

or

af fL a 5

For

Without

is.

J\

loss of generality, we assume the latter. Then K = Val(a,otf). Hence K is not static. If 5 = sup A(3,f) = sup A(a,f), —

K

<S5V € (aC / C v ) .

But

f £ G/y- \\G/ « x.

Therefore

K i s not

regularly derived or integral. (3) •> (4). Then there is

Suppose

fg /_ g 2 f 2

a £ ft such that

K = Val(a,afg).

a <_ ag 2 f 2 < afg,

Since

a < af,ag < afg,

f,g £ G

regularly derived,

G^

Y

^K < Y ^K g f 0 r a 1 1 a contradiction.

for some

afg > ag2 f 2.

„ . If

U5K)

K

^"

Thus

a

^K f g

is not static, and as

(G,ft) were integral or

would be Abelian and

Y € a

e <_ f,g £ G.

Let

< a

y&, < Y^f

^Kgfgf

=

or

(a^K)g2f2>

Thus we have established (2)-*->(3)+-*(4). (3),(4) -»• (l). is a value

H

in

of right cosets of

G H

Assume

G

such that in

is not normal valued.

Then there

H ft Hr . Let ft = R(H),

the set

G. ft is a chain by Appendix I Lemmas 2

211 and 3. Let L = ft g ^ H g and M = G/L. Then (M,ft) is a g G (transitive) locally primitive ^-permutation group (M^ = H is H1

covered by

=

^{HP-PCH1})''

Since

IL. is not normal in its

cover, the local component must be doubly transitively or periodically derived.

fg {_ g 2 f 2

Since (A) •*- (3),

e <_ f,g € M = G/L.

2

Hence

fg {_ g f

2

for some

for some

e <_ f,g € G (the

natural map G ->• G/L preserves order). (l) -> (4-). Assume (4) fails for G.

G £ A".

For reductio ad absurdum, assume

Let iC be the set of values in G.

Totally

order /f and let ft "be the set of right cosets of the value in

G.

Then

U

°-TT«

ft =

H, < H 2 ft^ . H i

&„ i s a chain by Appendix I Lemmas 2 and 3. Let ft is a chain where: a < 3

if

a € ft^ ,

H

H£*

H

H

(in the total order on it),

or

There is a natural ^-embedding of

E = G

Hp

into

$ €

primitive component

Since

K

(3) -> {A),

such that

or periodically derived.

(G^,ftT,) J\ is.

(G,ft)

and

2

and

a < 3

A(ft)

(see

Appendix I proof of Holland's theorem) so we may regard an ^-permutation group.

ft H

l

(G,ft)

in

as

has a

is doubly transitively

Since each ft^ is an orbit of G,

there is H £. ft such that a = H G. dom(K). Thus there exist e < f,g £ G K such that a^g = a ^ , a ^K f ~g ~ < a ^K f ~ < a ^K (aC ) 1 is a value of f. Clearly and a f lg f f = a*

°(a? )

*K

But

g

€ G

*K f"V

a

(c

*K)

X 2

f

5

so

v

' =

so

= (f-Vf1

2

,

since

G ۥ^.

f V v' e € G(

f"" ^

v e )f~ 2 £ ^f2 € G

Further, -

N.

Hence

Consequently,

a contradiction •

COROLLARY 11.1. groups, then

"

€ Q,

2gf2

= a?Kf

2 e <_ f <_ f v f- gf 2

f e G/ -

f

If

(G,Q)

and

(G,T)

are l-permutation

(G,Q.) has only static, integral and regularly

derived primitive components if and only if

(G3T)

has.

212 We omit the proof of the following theorem (due to Wolfenstein [68], Read [75], and Glass and Holland [73]) since it is very similar to that of Theorem 11A. (3) and (4) of Theorem 11A is said to be plenary if and

(b) if L

(A set of values 36 of an £-group

(a) each

is a value in

THEOREM 11.2. Let

Again the crux is to use

G

e / g £ G

has a value

and H 2 2 H e r ,

(G,Sl)

G

H € X

then R €3f).

be an l-permutation group. The

following are equivalent: (i) G
[f>9}n 1 | / | v \g\ [f>g]n ±f v g for [f,g]2 ±f v g for e V (f2g-2 A f2g2)

(H) (Hi) (iv) (V)

(vi) G

for all f,g € G and n € Z+. all e <_f,g € G and n G %*. all e <_ f3g € G. <_ g fQV aH & <_ fjff €. G.

has a plenary set 3£ of normal values

(H €3C + H < H'). We now prove Theorem 11C. G,

let

If

H

is a subgroup of a group

N(H) = {g e G: g^Hg = H}, the normaliser of

H

in

G.

We will actually prove:

THEOREM 11.3. If P

is a prime subgroup of

k(R), then

N(P) = P. Since every value is prime (Appendix I Lemma 3), Theorem 11C follows at once from Theorem 11.3. The proof of Theorem 11.3 hinges on the following observation: g € N(P) and so

1

f A g" fg = e,

f £ gPg"1 = P).

So

then

g^fg £ P

Proof of Theorem 11.3: Let e < g

and

g

has one bump.

Let

If

P

f € P

is a prime subgroup, (Otherwise

and hence

g € N(P).

g"1fg € p,

f v g~ fg € P.

We assume first

a € supp(g) and

a,x €ffiwith

213

a < a < x < ag. can f i n d

e < f € A(l)

such t h a t

a g 2 n f = x g 2 n (n £ Z ) .

and So

By doubly t r a n s i t i v i t y and t h e p a t c h i n g lemma, we

f]_ = g

fg v f

e P.

Hence we may f i n d

2

e P

6 £ (ogm,ogm+1)

(whence

(ag2n+\ag2n+2 )

3 < T < ag < og < 3g

x g 2 n h = a g 2 n + 1 (n e Z ) .

hg 6 P.

g £ P).

f o r some

(ag2r\ag2n+1)

such t h a t

and

h, = h v g

(J

s u p p t g ^ f g ) cz y

3 <=• ( a , x ) ; so

e < h £ A(R)

hg = e , so

g < (f L )

Let

(J ( 3 g 2 l \ s g 2 n + 1 )

supp(h) S h Ag

Then

supp( f ) c

m 6Z.

Let Thus

Again

We now show 6 € supp(g).

Then

6(f h ) 2 >_ a g m ( f h

)2 =

TY-I-J-O

og

> &g. Hence If

g

(f^h^ ) 2 >_ g

has more than one bump, do the same as above for each

bump simultaneously to get |g|

as desired.

f_g£|gU

S0

g^P

e <_ |g| <_ (f-,h-. ) 2 6 P.

But

and the theorem is proved.

Observe that the same proof works for any doubly transitive

A(n). We now prove Theorem 11D.

Suppose that

be the variety of ^-groups generated by

G.

G €^

and let M

We must pro\re /f = £,

the variety of all ^-groups. By Holland's theorem (Appendix I), (G,ft) is an ^-permutation group for some chain ft. By Theorem

214 11A, there is a primitive component

(Gv,tiv) that is transitively

derived from a periodic primitive or doubly transitive £-permutation group.

Now

G,= G

v /L (eg*) is an ^-subgroup of G,

for some

a € dom(K).

Since

K

G

v

it belongs to #

(K) theorem.

by Birkhoffs

!

Hence

G^ € //, again by Birkhoff s theorem.

But if

IV

(GK,ft ) is periodic primitive with period

z,

then

((GK)/

v/L',A) is doubly transitively derived by Theorem 4.3.1,

where

A = (a&^a^z)

L1

and

(Gj,)/ s is an ^-subgroup of

is the lazy subgroup. G«,

it belongs to //. Hence Af

contains a doubly transitive ^-permutation group there is chain

H £^

such

that

Since

(H,T)—strictly

(H,T) is doubly transitive for some

T. Let

m w = \/

n A

p TTx. ., be a word in the alphabet

i=l j=l k=l X U X" 1 U {e}; Assume that

1J

i.e., each

w = e

X V X"1 U { e }.

x± -k belongs to

fails in some £-group.

Let

F

be the free

£-group on a countably infinite set of generators containing Xf = {x £ X: x. ., = x or x"1 for some ^-permutation group for some chain Since

w ^ e

For each

in

i,j,k}.

F , there exists

a € A

(i, j ) (1 <_ i <_m; 1 £ j £ n),

(F ,A) is an

such that let

aw ^ a.

a( i, j, 0) = a

and

p ) . So

j

a(i,j,k) = ax. .^...x. .,.

So

A by Holland's theorem.

For each

x € X1

and each

(i,j),

let

1

M. = {k/x1=<xi. <J m,and >. Let A n ^(x) = {a(i,j,k): 1 < N.^ j < .(x) n, 0 = < {kk: < x" p}, = axljk finite subset ( of A. Choose T Q c T ordermorphic to A Q and let x(i,j,k) be the image of of

x

maps

a(i,j,k) to T(i,j,k

a(i,j,k) under this ordermorphism. a(i,j,k - l) to

a(i,j,k) if

a(i,j,k - l) if

k U .

Now the action

k € M. .(x), and

.(x). So

- 1) •-> T(i,j,k) (k e M^j(x)) and

T(i,j,k) •-> x(i,j,k - l)(k e N. .(x)) yields a one-to-one order preserving correspondence.

Since

(H,T) is a doubly transitive

^-permutation group (and hence q-transitive for all there exist whenever

h(x) € H

such that

k € M. .(x), and

q € 7L ),

x(i,j,k - l)h(x) = x(i,j,k)

x(i,j,k)h(x) = x(i,j,k - l) whenever

215 r

k e N. .(x). Note that

T(1,J,0) = T(i ,j',O) since both are

a = a(i,j,0). So T = x(i,j,O) £ T is independent of P Now T( T T ^ ^ J ^ ) ) = Th(xi ).. .h(x.. ) = x(i,j,p). Hence

images of (i,j).

K.~ -L

m T( V

n A

p TTHx. ., )) = max min x(i,j,p) / T because

i=l j=l k=l

1JK

i

j

a ^ max min a(i,j,p). Consequently, w = e fails in H (under i J the substitution h(x) for x). So H satisfies no laws that are not true in all ^-groups. Thus tf = <£ .

COROLLARY 11.4. If

G

is an l-group and G £tf3

then G

satisfies only those identities whioh are true in all l-groups.

An analysis of the last part of the proof of Theorem 11D shows:

THEOREM 11.5. The free l-group on a finite number of generators has soluble word problem.

For the details, see Holland and McCleary [79]. Contrast Theorem 11.5 with Theorem 13A.

Note that Theorem 11.5 is not

trivial (as it is in the group case) since there is no uniqueness m n p p of normal form \J A "TTx. ..,. of of £-group \/ /\ ~ffx. . £-group words. i=l j=l k=l 1J We use a similar technique, together with earlier theorems, to prove Theorem H E . We first note that induction on n£I

n.

Wr{(E,l): l < K < n } 6 A

Since Jf*Jf= M

and

n n

( n e Z+)

A £ /fy A cjf

by

f or all

. So Jf contains the variety of ^-groups generated by n

{J + A . To prove equality, we must show that any law which fails m n p n€Z + in Jf fails in some A n (n £ I ). So let w = \J A TT*. -I i=l j=l k=l l j k and suppose w = e fails in some element of J/. Then it fails in some subdirectly irreducible element of M.

But any subdirectly

216 irreducible £-group has a transitive representation (Appendix I Corollary 6) and so can be ^-embedded in (W,ft) = Wr{(l,t): K 6. k] for some set k by Corollary 11B. Hence w = e fails in (W,ft). m n p Let g. . €. W and a € ft be such that a( V /\ Tfg. .,) j a. 1JK i=l j=l k=l 1Ji" Let a(i,j,k) be defined as in the proof of Theorem 11D with g... in place of \--^> and let ft = {a(i,j,k): 1 <_ i <_ m, 1 < j < n, 0 <_ k <_ p}. Then ftQ is a finite subset of ft. Let fcQ = {K £ k: (3a 1 € ftQ)Qa2 € ftQ)(a;L f a 2 & K = V a l U ^ ) ) } , a finite subset of k. Let (W',ft') = Wr{(l,l): K € k n }. So

|fe|

°

W € A °. We now show that w = e fails in W whence the proof is complete. Let a »—> af be the projection of ft onto ft'. Note that the restriction of this map to ft is one-to-one and order 1 preserving. defined by: For each g € {g. ., }, let g = (g' ,) £ W be 1JK

I

e

1 __e

I\f OL

if (33,Y £ fJ O ) ( Bg = K Y (in ft) and

3' = K a' (in otherwise

n1

))

EK62 in ft an 1 so B^g =K B 2 g if g € W. Hence gK is well defined. More,a' similarly over, if x ,x , = x, then g J abc; and abc

if e r e 2 e aQ and

g

ijk

g!

= ( g

abc

(i,j,k)

g = gj.jk,

r l

i f

= 3i in

X

ijk =

x

X

aL-

3 = a ( i , j , k - 1) £ ftQ and

p = (aTTg,,v)f. k = l 1JiC

6

Thus r e p l a c i n g

i s a bona f i d e s u b s t i t u t i o n .

3,3g = a(i,j,k) £ ftQ). Thus p a ' ( Tfg\^) k=l 1 J i i

=

then

Pick

K £ kQ.

3'g? = (3g)'. I t follows that

x..k

(i,j,k)

by and l e t

Then

Hence for each i,j, m n p a ' ( V A 7TgJ,1.) = i = l j = l k=l 1 J K

max min a ( i , j , p ) ' / a 1 since a H~> a' i s one-to-one order i J p r e s e r v i n g on fiQ. Thus w = e f a i l s i n W as d e s i r e d . //

217 We now prove Theorem 11F using Theorems 11A and 11D. The crux is the following lemma (if

G

is an £-group and

X Q G, we

say that

X

generates G

as an Z-group if the convex £-sub-

group of

G

generated by

X

is

G.

If X = {g}, we say that

g_ generates G_ as^an_ Z-group). LEMMA 11.6. Let

G

be a normal valued subdirectly

irreducible Z-group generated (as an Z-group) by Then G

{g^3...yg }.

is generated as an Z-group by some g. (1 < i < n). Is

Proof: By Appendix I Corollary 6,

(G,ft) is a transitive

^-permutation group for some chain ft. Let rechristening,

A(a,g n )£ ... C A l a ^ )

overlapping by Theorem 11A.

Now

a € ft. By

since

(G,ft) is non-

A(a,g-, ) = ft (if not, say

3 = sup(A(a,g-. )) € ft. By the non-overlapping of supporting intervals,

3g. = 3

there is no

f € G

transitive).

Let

for with

1 <_ i < n. af = a.

K = Val(a,ag-,).

(GT,,ftT.) is Archimedean.

Hence if

a £ ft with

This contradicts Then

K

a > 3,

(G,ft) is

is regular so

SinceftT/r= fi/5v, gn

generates

G

as

an £-group. Proof of Theorem 11F: Let {G. : i £ 1} be a family of ________ ____ __________ _____ ^ convex Z-subgroups of an £-group H and let G be the convex ^-subgroup of H generated by {J G. . If 1/ is a variety and i_I x G. € 1/ for each i € I, we must show that G € I/. If 1/ = £ there is nothing to prove. Theorem 11D. Let

Then

So assume

G. G -V for each

w(x-L,...,xm) = e

(//__*; i.e., i € I,

and hence

l/S-^by G € Jf.

1/ and let h1,...,hjn € G. Thus n there are g.n £ U G . such that h. = TTgn-v- L e t H n b e t h e Jk J i e i 1 k=i J i c ° ^.-subgroup of G g e n e r a t e d by { g . , : 1 < j < m, 1 < k < n } . I f JK — — — w(h, ,...,h ) i- e,

hold in

there is a subdirectly irreducible factor

HQ

of H n such that w(h-.,...,h ) / e where h M - h denotes the natural map of H Q onto H Q . Since L E G ^ y , H Q € ^ . Hence H^ t ^ and as it is generated by {g., : 1 < j < m, 1 < k < n}, (J JK ~ — — — there is a g. . which generates H n (by Lemma 11.6). But J k U 0 0

218 g. € G. for some i £ I and H n H G . is a convex ^-subgroup J 1 1 CTO __ ° of H Q . Thus H Q = H Q H G. e I/. Therefore w(E1,...,hm) = e, the desired contradiction.

Consequently

G £ \J.

Finally we prove Theorem 11G. We first establish a lemma due to Glass and Holland (unpublished).

LEMMA 11.7. Let

(GySl) be a doubly transitive l-permutation

group with G GLBCti). Assume that each element of finitely many bumps, and that no point of

Q

of a supporting interval of an element of a supporting interval of an element of subgroup of

G

is a convex ^-subgroup of then

g € GA

G.

Let

where

f

and

g

Clearly, each M-

f,g € G with

f A g = e.

so

g € M-.

with

If

Thus

f A g = e.

have only finitely many supporting intervals and

supporting interval of M-

G.

f,g €. G-\M-

move points arbitrarily close to

But as

G. Then each proper prime

A = (af~\af),

is not prime, there are

Since

G and the supremum of

a €. ft, let M- = {g € G: g € GA for some a A a € A}. We will prove that {M-: a € ft}

is the set of minimal prime subgroups of

if M-

has only

contains a unique minimal prime subgroup of G.

Proof: For each "— open interval A with

f £ G-,

G

is both the infimum

f A g = e,

f

a,

a

is the endpoint of a

and of a supporting interval of

this contradicts the hypothesis on

a.

g. Hence

is prime. If

with

P

is prime and

P Q G-,

let

e < g € M-;

g £ G . By Corollary 1.10.6, there is

a € supp(h) Q A. g € P; If

i.e., P

Lemma 7.1.3.

Then

h £ G-

so

h ^ P.

say

a £A

e < h €. G with Since

g A h = e,

M- c p .

is prime and Hence

{M-: a € Q} a

subgroups of G. Note that M- 5 G p

P §£G-

a

only if

interval support property. proper prime subgroup of

G

for all

a £ fi, then

P = G by

is the set of minimal prime 3 = a

since

(G,pJ

has the

By the previous two paragraphs, each contains a unique minimal prime.

219 Proof of Theorem 11G: functions from

u)

graphic order:

a < 3

into

y £ a)-, ,

p € 2

and

a £

l

pa

if

v < y

[q(D,ql]

if

the ordinal map

linear maps,

A partial map

f

f.. 1 Let

from

v >_ max{A 2 ,y 2 }.

f 0 agree on
[pO,pl]

and A

onto

3

=

poBn . 21

If

q2B-, ± q ^ B n .

af.. <_ af~ & Bf-, >_ 3fp>

between

a

and

3

A. with

[P2O_^P21_] ^ [PTOJP-I1.] • afn < af o and 1 —
are

such

A.

f^: [ p ^ p ^ ] -> [ q ^ q ^ ]

So

is the

*\>fo

pi €

y.

2,

qi £

(i = 1,2). If a,3 £ dom(f ) n dom(f ), then p c p P 2 c p^. Without loss of generality, p, ~Pp> a n d w e P 2 = p r.

A,y £ a^,

A + v «—* y + v

We now show that if

a,3 € dom(f ) n dom(f^),

and

Let

3 & ft

a £ ft, (pa)f = qa. Note

[A,o3-, ) •> [y,oa-. ) given by

then there is a non-trivial interval that

If

v = y + A (A £ a)1)

is said to be linear if for each

identity for all

0.

will denote the element

.

a(A)

q € ^OJ..

1_ and smallest element

2,

t a)-, and

the set of all

G?y £ uu )[a(y) < 3(y) & a( v) = 3( v) if

p(v)

p £

2,

{0,1}. ft is a chain under the lexico-

if

v < y ] . ft has largest element y

l

Let ft =

q..r c q

Bf o < 3fn > 2 — 1 Or

q

a = Pp01-]

Let

<

an(

2

or ^

1

qT-^^n Qo^ 1 1 — 2-L

c q r.

By the observation

about ordinal maps in the previous paragraph, in either case we get the desired conclusion as is easily checked. Also observe that any non-trivial interval of ft has size

Now let and

G = {g £ B(ft): for some

g| [p.0_,q.0]

m £ 7L , ft = \J [p A q J J

is the restriction of a linear map

(l <_ j <_ m)},

the bounded "piecewise linear" order-preserving permutations of ft. Then

G

is an ^-subgroup of

B(ft) and by Lemma 1.10.2,

is a doubly transitive ^-permutation group if a^ / 0,1. U

(G,a Q G)

a Q £ ft with

If

[p.O_,q.lJ c

g-, ,g^ are restrictions of linear maps on l d [T.0_,T.1] (i = 1,2) and for some

a,3 £ [PnO,q n l] O [ p o 0 , q o l ] ,

agn < ag o

and

3g o < 3gn>

then g

220 and

gp

agree on an interval between

paragraph). 11.7.

Hence

Consequently,

a

and

3

("by the previous

(G,aQG) satisfies the hypotheses of Lemma G

satisfies Theorem 11G.

PART V THE AUTHOR'S PREROGATIVE

221 CHAPTER 12 ALGEBRAICALLY CLOSED LATTICE-ORDERED GROUPS

A field

F

is algebraically closed if every polynomial

with coefficients in closed field

F

F

has a root (zero) in

F.

Any algebraically

satisfies Hilbert's Nullstellensatz: If

p.(x),...,p (x) are polynomials with coefficients in a common root in some field a common root in If G £ H. ever

G

F

H ^ F,

then

(it is assumed here that

is an t-subgroup of an £-group

An £-group

G

F

that have

p.(x) (l <_ i <_ m) have m € 2£ ). H, we will write

is said to be algebraically closed if when-

iL.(x,g),. .. ,u (x,g) are £-group words (with parameters

g € G ) , and for some

H >^ G

then there are

such that

f £ G

and

h£.H,

u.(h,g) = e (1 <_ i <_m),

u.(f,g) = e (l <_ i <_ m ) .

The following theorem is easy (and nothing whatever to do with ^-groups!). THEOREM 12A.

If

G

is any l-group^ there is an algebra-

ically closed Z-group G >_ GQ. A field

F

of polynomials F,

is existentially closed if for any finite sets

p.(x), q..(x) (i 6 I, j € J) with coefficients in

if there is a field

H 2 F

and

h €. H

such that

P i (h) = O / q . ( h ) ( i £ I, j £ J), then there are p i (f) = 0 ^ q.(f) (i U , fields) to

yz = 1

j £ J). Since

y / 0

f e F

such that

is equivalent (for

has a solution, any algebraically closed field

is existentially closed. An £-group finite sets

with parameters such that

G

u.(x,g),

is said to be existentially closed if for any v.(x,g) (i £ I, j £ J) of £-group words

g £ G,

if there is an £-group

H >_ G

and

h € H

u.(h,g) = e / v.(h,g) (i € I, j € j), then there are

222 f € G

such that

u.(f,g) = e ^ v.(f,g) (i € I, j 6 j).

~

1 ~ ~

THEOREM 12B. J/

J ~ /x/

6 ^ {e} ts an algebraically closed l-group3

then it is existentially closed. Our main aim (besides establishing Theorem 12B) is to use our previous material (especially Chapters 2, 5, 6 and 10) to establish some properties of algebraically closed ^-groups. An £-group

G

is said to be laterally complete if \ / g , 6 G i€I 1 {gi: i € I } 9 G with g± A g. = e if i f j. THEOREM 12C. Let

L-group. Then G

whenever

G ^ {e} be an algebraically closed

is simple and divisible. Every element of G

is a commutator—indeed, every element is conjugate to its square. Any two strictly positive elements of e < g 6 Gj e <

f-

<

there exist fjsf?

€ G

G

are conjugate. If A

uith f1 f9

= e and

9 d = 1,2). The t-group variety generated by

G

is o£.

is neither finitely generated nor finitely related and contains

G

no finite maximal pairwise disjoint set of elements. Moreover.,

G

is not laterally complete.

THEOREM 12D. and n

e < g € G n

A

Gx = e

There is an algebraically closed l-group G

such that -if X £ % , there is e < g^ € G with If and only if n e X.

Finally in this chapter we prove a result about representations of algebraically closed ^-groups. By Theorem 12C and Appendix I Corollary 8, any algebraically closed £-group has a transitive representation.

THEOREM 12E. If

(G3ti) is a transitive primitive l-permuta-

tion group with G ^ {e} algebraically closed^ then

(G>Q) is

pathological. Moreover, there are algebraically closed t-groups having pathological representations.

223 All the results in this chapter (with the exception of the last part of Theorem 12E which is due toO. H. Kegel) are due to Glass and Pierce [80c]. We first prove Theorem 12A: algebra.

The proof is cheap universal

List all finite conjunctions of

We define

G

(A £ K ) by induction:

A

u(x,g) = e, u

If

y ~

H > G , let G . be the ^-subgroup of H — y y+1 and a solution. For A £ K a limit ordinal, let

G

|GQ| ).

0 (x) has a solution in

some £-group by

an

g £ GQ: eo(x),9-L(x), . . . ,0x(x), . .. ( A £ K =

£-group word and

generated

= (J G . Now let G o = [J G . Then for any finite set of A
H ^_ G^,

u.(x,g) = e (i £ I) has a solution in some £-group a solution in

G*?. Let

G = 1J G(n),

then

G(0) = G

G >_ G Q

and

and G

G(n + 1) = G* . If

is algebraically closed (if

u.(x,g) (i £ I) is a finite set of £-group words with then

g £ G(m) for some

m £ CJ. Hence if

a solution in some £-group in

H >_ G (>_ G(m)),

G(m)1" = G(m + l) <_ G).

it has

7

g £ G,

u.(x,g) = e ( i £ l )

has

then it has a solution

This proves Theorem 12A.

We next prove all but the last four sentences of Theorem 12C. Let

G f {e} be algebraically closed.

By Corollary 2K, every

£-group can be ^-embedded in a divisible £-group. and

m £X , x

= g

has a solution in some extension of

Hence it has a solution in Thus

G

G

(x

g

= x~ gx

has a solution in

every element of of

G

G

is the same as

there exist

g £ G.

G.

x g~* = e).

A(ft).

G

Hence it has one in

f. £ G ( l < _ i < _ 4 )

generated by

By Theorem 10B,

G

g.

G;

i.e.,

Thus every element

Therefore, by Lemma 6.4, if

h - f71g"1f .f~1gf0.f:1g"1f0.f~1gf,. subgroup of

By Corollary 2.2.6,

is conjugate to its square.

is a commutator.

g ^ e,

= g

g £ G

is divisible. Moreover, by Corollary 2L, G <_ B(ft) for

some doubly homogeneous ft. Let 2

So if

g,h £ G with

such that

Hence h belongs to the normal

It follows that

G

is simple.

can be ^-embedded in an £-group

which any two strictly positive elements of

H

H in

are conjugate.

224 Hence if e < g ^ g ^ 6 G, x g .x x = gp g 2 has a solution in an extension of G and so in G. To complete the proof of Theorem 12C we must first establish Theorem 12B. We need two lemmas. LEMMA 12.1. Let G be an t-group^ e < gQ £ G and g € G. Suppose u.(Xjg) and v .(x9g) (i £ J, j 6 J) are t-group words> where I and J are finite. Assume there is an t-group H >_ G satisfying for some h € H (1) uAh^g) = e (i € I), and (2) v.(h,g) ^ e (j £ J). Then there is an t-group G >_ H in which (l)-(5) are satisfied where z2. = v.(Kg)~1z.v.(Kg) d

d

~

d d ~

(j € J) "

y~h.y. = a (3 £ j) d

d

d

y" zgfl = z. Proof: Assume the hypotheses and write v. for v.(h,g) (j 6 J). By Corollary 2L, we can £-embed G in B(Q) for some doubly homogeneous ft. Define e < f. 6 B(ft) by: af. = a if J J av. = a. If a < av., choose y 6 (a,av.). There exists J ^. J J a. e A(ft)J > such that y € supp( a.) 9(a,av.). JLet f. n j= a.j and define f. inductively (n £ Z ) to satisfy f? Ml = v. f. v.j J >n 2E, B(ft) is divisible so f. J >n+± inj B(ft)). j n j (By Theorem _ exists () _1 . JNow ,n+ifor 3 € A(a,v. v.f? v. For n < 0, we let f. _ = v.f? v. . Now for 3 € A(a,v.), — n j say 3 €. [av.,av. ), Jn+l let gf.j =j3f. . Similarly, if av.J < a. J J J J >n J This gives e < f. e B(ft) such that f? = v ^ f .v. (j 6 j) so (3) J J J J J^ holds.

By Theorem 10B, we can £-embed

B(ft) in G an £-group in

which any two strictly positive elements are conjugate. So if e < f €. 6, then fg Q > e and (4) and (5) have a solution in 6. The following lemma is obvious.

225 LEMMA 12.2. Let

e < gQ € G

and let the l-group L >_G

satisfy (3)> (4) and (5); then any solution satisfies (2). We now prove Theorem 12B: Assume

G ^ {e}

is algebraically

closed and that (l) and (2) are satisfied in some £-group By Lemma 12.1, there is an £-group satisfied in

G.

Since

(1), (3), U ) and (5). Hence

G

G

G ^ H

H >_ G.

such that (l)-(5) are

is algebraically closed, it satisfies

By Lemma 12.2, G

satisfies (l) and (2).

also satisfies (2).

Consequently,

G

is

existentially closed.

We now return to the proof of Theorem 12C. We may assume that

G

is existentially closed.

transitive representation G

in

a € supp(g).

H~

via

Let

Let

if

Let

a,b i- e, g-^ag = b.

p

pL = s

b

3/a

Thus, in (G

G,

. Then

a A b = e,

3 ^ ag

a v g = g = b v g).

there are

fn,fo with

Actually, f A f

=e

and

is existentially closed); let

(IL,T- ) = (G,ft) Wr (Z,Z) and £-embed Rag,O) where

where

u

B = ag

(J) if

a,b <_ g (i.e.,

e < f,,fp < g

if

and

e < g £ G and

a = ({a },e), b = ({b },e) € H n ,

3 = a ,

j) if

is simple, it has a

(H Q ,T0 ) = (Z,Z) Wr (G,oJ

g i—> ({0},g). p

1

G

(G, Q) as an ^.-permutation group

(Appendix I Corollary 8). £-embed

Since

if

G

in

via: g i-± g*

n = 0

(a,n)g* = <

. If

|Ja,n)

IL

{g. : 1 < i < m} I — —

is a

if n f 0

pairwise disjoint set of positive elements of

G,

let

h € H,

be

p. if n = 1 defined by:

h = 0,

hn = < [0

. Then h = ({nn),h) ^ e if

and h A g . =e(l<_i<_m).

Thus, in

f Ag. =e(l<_i<_m).

Hence

pairwise disjoint subset of

G.

n f 1 G,

we can find

{g. : 1 <_ i <_ m}

f ^ e with is not a maximal

226 Let

w(x) = e

f € G 0 H2, f' £ G.

fail in some £-group

w(£) j- e.

Since

G <_ G © H 2 ,

H^ = G © Z.

If

g1,...,gm €. G,

e / h € H. which commutes with each C({g1,...,gn})'^ {e}, where G.

g1,...,gn

Since

is £)

Then for some w(f' ) / e

Thus the £-group variety generated by

Let

But if

Hp.

G

G

then there is

(g.,0) (l <_ i <_ m).

Thus

C(X) = {f 6 G: (Vx 6 X)(xf - fx)}.

generate

G,

C({g1,...,gn> ) is the centre of

is non-Abelian (the £-group variety

and simple, its centre is

finitely generated.

for some

is £.

G

generates

{e}. Consequently,

Therefore, if

G

G

is not

were finitely related,

infinitely many generators of

G

number of defining relations.

Hence they would generate a free

£-group

F

which is a free factor of

free product of of

F.

Since

g2 = x

would not occur in the finite

gx.

F

g Let

and

H, . Let

say

g f e 2

is conjugate to L = H, © Z 4

G;

g ,

G

be one of the generators

there is

and map

is the £-group

G

onto

x € G L

via

such that \p, where

iJ>|H. is ~th.e identity function,

gip = (e,l) and

all other generators

This is an £-homomorphism so

(g^) 2 (g^)

=

2 =

f

of

(XI|J)~ (gi|>)( XIJJ ). g^;

i.e.,

F.

But

g\\t

(e,2) = (e,l),

fi|; = (e,0) for

is in the centre of

L. Hence

a contradiction.

To complete the proof of Theorem 12C, we need a lemma. LEMMA 12.3. If

G

is an l-group and

{g^: i £ 1}

is an

infinite pairwise disjoint set of elements with supremum f existing in {g.: i € 1}

G,

then there is an l-group H >_ G

in which

has a supremum x < f.

Proof: Let

C be the convex ^-subgroup of

{g.: i € I}. Since., f £ C,

G

generated by

there is a convex ^-subgroup

maximal with respect to not containing

f

P 5 C

(by Zorn's lemma).

Appendix I Lemmas 2 and 3, R(P) is a chain.

Now

By

(G,fi) is an

^-permutation group for some chain ft. We can extend the action of G

to ft u R(P) via:

(Ph)g = Phg (h,g € G). Hence

is an ^-permutation group. does not.

Each

g.

fixes

By Theorem 8B, A(ft u R(P)) p

is not the supremum of

{g.: i 6 1}

in

(G,ftijR(P))

P £ ft u R(P) but

is closed. A(ft u R(P)).

Therefore Since

f f

227 {g. : i € 1}

is a pairwise disjoint subset of

a supremum

A(ft uR(P)),

it has

x £ A(fi u R(P)).

We now complete the proof of Theorem 12C.

Note that lateral

completeness is not something we would expect to capture. let

e < fQ

with

{f~nfof^: n € %+}

Then

its supremum in f

supp(f ) = (0,1) and

llf2fl - f 2 *

(3g

0

f2 A f

> e)(3g

l

> e

= e,

f" fof-, <_ f~

)Gg2)(g2 A

g

llg0gl - g2 & gllg2gl - g 2 ^ h o l d s l n G ® A ( R ) algebraically closed, there are gn>g-i >gp > e g

2

A g

0

= e

g

'

l g0gl - g 2

induction on

and

g

l g2gl - g 2 *

n

n £ %>* that

g~ gog^ <_ g^.

is an infinite pairwise disjoint set. complete, it would have a supremum, is

H >^ G

Clearly

satisfying: (*)

x < f,

f

g

0

Thus

in

If

G

say.

gn < x

and

xgn = gnx.

y G. G

g

n € Z

= y < f

for al1

>

is

^

m

{g~ gQg^: m € 7L)

were laterally By Lemma 12.3* there

Since

G

x < f. is

X

-L

existentially closed, there is

and

> if G G with

It f o l l o w s b

Hence

f^ be

= e &

{g. g. ggg-i • m £ %>} has a supremum U —

ing0gl - g i n y g i

In A(l),

be translation by +1.

is a pairwise disjoint set. Let

A(l). Then

Hence

f.

satisfying (*).

Now

contradicting

f = V eTVe? ^ G. U

We next prove a lemma reminiscent of Lemma 2.1.10. f €.

is not a first order concept but

Note that

f £ C(C(g)) is:

(Vx)(xg = gx + xf = fx).

LEMMA 12.4. If

G

is algebraically closed and e < g £ G9

then + = C(C(g))+. Proof: The proof of Lemma 2.1.10 shows that if e < f £ C(C(g)), a £ supp(g), Suppose H >_ G

then

there is

supp(f) Q supp(g) and for each n(a) £ 7L

a,3 € supp(g) and

n

such that

n

f|A(a,g) = g a|A(a,g).

f n . By Theorem 10.8, there is

and x £ H such that x~ gx = g and ax = 3. Hence n n n n afx = ag a x = axg a = 3g a ¥ 3g = 3f = axf so xf / fx. Since

228 G

is existentially closed, there is

xf •} fx.

x G C(g) n G

f G C(C(g)).

This contradicts

Hence

n

such that = no

for all

dp

a,$ € supp(g);

i.e., f = g

for some

n € S .

We now prove Theorem 12D. We first establish:

LEMMA 12.5. Let X c^". such that if n £ 7L ,

9^9rd\

Tfren there exist e < gQ,g• € A(S) A

QQ = &

if ond only if n 6 X.

Proof:

Enumerate 7L \X

in increasing order, say m n < m U

oo

Let

< mo < J_

c*

e < g Q G A(l) with

supp(g ) = (J(2n,2n + l). Define n=0 1 x e < g-. G A(l) so that, for all n G W, (4n - pOgn = 4n - p- , m^ (4n)g1 > 4n + 1, (4n + 2)g1 > 4n + 3, (4n)g1 = 4n + 2 and m (4n + l)g 1 n = 4n + 3; if Z \X is finite so m n doesn't exist, change the last part then

to

(4n)g

>_ 4n + 3.

Note that if

k

g^ gQg^ A gQ|(4n,4(n + l)) = e|(4n,4(n + l));

g^kgog^ AgQ|(4n,4(n + 1)) / e|(4n,4(n + 1)) if g

ikgogi

A g

o ^

e

if and only if

k e

k/ m ,

but

k = mn.

Hence

^+XX<

By Theorem 12A, there is an algebraically closed £-group G >_ A(I).

Let

be such that

e < g €G.

f^fg^

A f

Given

= e 0

X £Z+,

lf and only lf

let n 6 x

e < fQ,f]L € A(l) -

But

229 e < f-^g, G G

so for some g"ng g^1 A g

g Y = h~ fnh,

h 6 G, = e

h~ f.h = g_ . Then, for

if and only if

n £ X.

Thus

Theorem 12D is proved.

We next prove Theorem 12E.

Let

primitive ^-permutation group with Let

e < g € G.

£-embed

G

Now, in

H,

£-embed

in

H, where

we can find such an

are conjugate,

x

G f {e} algebraically closed.

JT G

diagonally and thereby

in

G.

By Lemma 6.1,

G

(G,ft) is doubly transitive and

z,

and

ah >_ az

a € ft, f (so

H

(If

and

so

does not have

Since any two strictly positive elements of

periodic with period af = a

in

(H,T) = (G,ft) Wr (Z,Z). Let g e G. -1 -1 x > e such that x g. gg-.x A g ^ e;

we can find

bounded support.

where

G

(G,ft) be a transitive

G

(G,ft) is

h

are not conjugate

fixes no point of ft)). Thus

(G,ft) is pathological. We now prove the converse.

The concept of existentially

closed applies equally to ^-permutation groups.

The proof of

Theorem 12A can be easily adapted to show that existentially closed ^.-permutation groups exist.

Now if

(G,ft) is an existentially

closed ^-permutation group, it is clearly doubly transitive (use Corollary 2L).

Let

H

be an £-group with

H >_ G

such that

H

satisfies a particular finite set of £-group equations and inequations (with parameters from there is a chain group and

T ^ ft such that

(G,ft) <_ (H,T).

(G,ft),

(H,T) is an ^-permutation

(G,ft) can be ^-embedded in

(H,T).

Hence

Thus the finite set of £-group equations and

inequations hold in (G,ft)).

G). By Appendix I Theorem 11,

G

(since they hold in

(H,T) and so in

Thus, for any existentially closed ^-permutation group G

is an existentially closed £-group.

Thus

(G,ft) is

pathological and the proof of Theorem 12E is complete.

Finally we prove the following easy consequence of Lemma 12.4 which in model-theoretic language says that the theory of ^-groups has no model companion.

230 THEOREM 12.6. If

G £ {e} is an algebraically closed

there is an l-group H = G

such that H

is not

algebraically closed. Proof: Let

U

be any non-principal ultrafilter on %

and

H = 'J7~"+G/U, an ultrapower of G; i.e., H is the quotient of n£^ the £-group 7 T + G by the £-ideal N = {f € JJ~+ G: {n £ %+: f(n) = e} £ U}. By Los' theorem (see Chang and Keisler [73] ), H = G.

Let

e < g £ G

(n £ % ). Then generated by h'x = xhr

h

coordinate in

h = Nh1

where

h'(n) = g n

does not belong to the subgroup of

g, where

whenever

and let

g = Ng

gx = xg

~[T +^n€^

B

and (in

H

g(n) = g (n £ 22 ). However,

H)

since this holds at each

y Lemma 12.4, H

is not algebraically

closed. For further results about algebraically closed ^.-groups, see Glass and Pierce [80a] and [80c]; the Abelian case is considered in Glass and Pierce [80b].

231 CHAPTER 13 THE WORD PROBLEM FOR LATTICE-ORDERED GROUPS

In this last chapter, we prove a result due to Glass and Gurevich [81]:

THEOREM 13A.

There exists a finitely-presentedlattice-

ordered group with insoluble word problem, A presentation of an £-group with generators and relations £-group

F

group) of and

J

r.(x) = e (j € J)

on F

x. (i € I),

is the quotient of the free

{x.: i £ 1} by the £-ideal (normal convex L-subgenerated by the elements

{r.(x):j£J}.

If

I

are both finite we will say that the £-group is finitely (x. (i £ I); r.(x) = e (j £ J)).

presented and write it L

!

J ~

We will obtain a certain primitive recursive set of relations

R,

involving only a finite set of letters correlated

with elements of

A(R), such that the subset

those members of

R

which hold in

S

of

R

will then show that there is a finite set of relations hold in

A(t) and imply

in

R

A(l).

S.

Xn,

finite set of letters

SQ

comprising

A(ffi) is non-recursive. We Sn

which

will be constructed from a

consisting of the letters which appear

and new elements correlated with additional elements of A member of

if it belongs to

R

S.

will then be implied by

SQ

if and only

Hence the finitely presented £-group

(X0;S n ) will have insoluble word problem; i.e., there is no algorithm to decide whether or not an arbitrary £-group word in the generators

X~

We will use

is equal to %jd_

e

in

(X~;S~).

for the set of functions from

u) into OK

In contradistinction to our notation for permutation groups, we will write the elements of will be that member of

w

w

^u) on the left; so if defined by

f,g € U, fg

fg(n) = f(g(n)) (n £ w).

232 A function

f €

general recursion if

a> is obtained from (i) fg = u,

g,h,u,v €

co by

(ii) fh = vf, and

m € a) belongs to the range of one of the functions

(iii) each h g (k £ w).

We will need two results from recursion theory: (I) Julia Robinson [68]. The class of recursive functions is the smallest class of numerical functions which is closed under composition and general recursion^ and contains the zero function 6 (Q(n) = 0) and the successor function s

(sin) = n + 1),

For those unfamiliar with recursion theory, (I) can be taken as the definition of a recursive function. A set A <= w recursive if there is an algorithm such that given decide, on the basis of the algorithm, whether (II).

is

n £ co, we can

n € A

or

n £ A.

There is a recursive function whose range is not a

recursive set. See, for example, Hartley Rogers [67] Theorem VI page 62. In the ensuing, we will write down several relations involving simultaneously conjugating one set of elements to another by a single conjugator.

That this is indeed possible (in

the absence of the amalgamation property) without all the elements in the sets reducing to

e

hinges on the techniques of Section

2.2—in especial, see Lemma 2.2.9. We now discuss the interpretation in /f(R).

233

234 In

A(fi),

there are elements

a~,b^,cn > e

each bounded

and having one bump, such that: A b~ n a

(a)

aQ

(b)

t>omaObO

(c)

between

b*J = e = b

< b

and

0

C

A

0mb0C0

supp(b~ a^tyj)

non-degenerate interval of

1

f

CQ^C^O < C

r

a 1 1

f0r a11

0

and

°

n

m €

£

m

and

n

0

^

^

%

'

supp( b^a^b^)

whenever

z

there is a

are distinct

integers, and (d)

a

Qa^

= b~

and

dn = a " 1 ^ ^ ,

Let

n

n

l

k

W = {y_ .. .y, (x^yz of

supp( c~ Dp.c~) and

supp( c n D n c n ) .

Because of the bump structure, there is —1

—1 a~

similarly for

a~ "b^a^ = c~

such that

(by the techniques of Section 2.2).

x_ = a^dLa-

and

y

= a~ n x n a^

(n£ ^ + ) .

+ : k € 22 ,n_., . . . ,n, € X}

{x^#w: w € W}

Let

-1

is shorthand for

a^,

a« 6 A(l)

x*(yz)).

and write

x*y

for

y

xy

By (a), (b) and the definition

forms a pairwise disjoint set.

Let

d, £ A(l) be the pointwise supremum of it. (Note that = x^ /\ d., x,~ = e~—i. e., a o d,~ao A d (a o d a o ) e—so x^ v d^x"

= d.. . )

Let

c^ €. A(l)

be the pointwise supremum of the

pairwise disjoint set of elements

{do*d^w: n € 2Z,w € W } ,

b

€ A(l)

be the pointwise supremum of the pairwise disjoint set of elements {co*c-.d..w: m,n € 25, w € W } ,

and

a _ € A(l)

supremum of the pairwise disjoint set By construction, other,

c

a

a,,

a^ - j ^ = d.,,

b.,,

c.

and

a~ "b^a^ = c^

I = b o |supp(b Q ),

be the pointwise

{b^Kb., c,d_w: m,n,k € Z,w € W}. d..

and

a-,bQ = t>Qa-i •

each commute with each a" a_a« = b.. . Let

a^

Also, since

be the pointwise

supremum of the pairwise disjoint set {anxa.. : n € JL }. So -1 a 0 A a 2 = e a n d al a2al — a 2 " Let a. and a 7 be the pointwise suprema of the pairwise disjoint sets

{ao*c??: m € w}

a 6 A a ? = e,

c" 1 (a 6 v a ^

c ^ a ^ i a ^

c1aoc-1ia7

and

"fa^c-i : m € ^ }.

= afe v and

a?,

Note that

a Q A a ^ " 1 = e,

c1a7c-1<_a7.

Because of the bump structure, we can find

a,,a, € A(l) 4 y

such

235 that a

a ^ a o a 4 = aQ,

a^a^

4 cla4 = d l ; a5 aOa5

=

= b^

a^b^

a la a

V

= b

5 l 5

= o±

and

i'

and = c

^ l S

the techniques of Section 2.2; indeed, the existence of specifically Lemma 2.2.9 applied to each bump of We now code

f £ ^co into

A(l) via

supp(af ) c U{supp(c o *c™d^): m € u,n £ Z} a f |supp(co*c™d£) = b ^ m ^ | s u p p ( c 0 ^ c ^ ) Now that

a«

commutes with

aQ^c.a^ = aQ^b^

c^

an,

for all

b_

af,

l

(by

a^

is

c. ). where

and

for all and

dn,

m e

and

w

n € Z.

and

m t w.

Finally, for technical reasons, we need one extra element a £ A(I) and

supp(a) c U{supp( c^c^d^w): m,n € Z,w € W},

defined by:

a|supp(cQ^c^d^w) = b^ + | supp( c^c^d^w) for all

w € W.

Note that

a

= "(a v e) A a"^d naq,

s

j

m,n € %,,

s being the successor

u j>

function. We are now ready to prove Theorem 13A. In

A(I) we constructed five "levels" of Z.

We now write

down a finite presentation which will capture this information. Let aQ,

G

be the finitely presented £-group with generators

a 1 , ci 2 , a 3 ,

a4,

£l

5'

a~ 1 a o a 1 < a. 2' a

lbl

= b

]a ,

a c =

? i^

and relations : a o

and a«

Ola.

I'

a

a

<_ i

£ i~ a_a

ldl

0- V

da. L'

=

a

3ld0a

a

?

= e,

A =

ibo

A d.L(

= e, c 1(a V a )c

l 6

Wl

1

7 l = a6 V a7^ a0 A W

=e

' " i V i -a6^

- a7' ^ Y l 1 ^ a7^ where bj E a3laja3' Cj E a 3 S a r

d. , a; 3 a ja 3

(J = 0,1).

Observe that all of the above relations hold in

A(l) under

the natural interpretation. Note that we write

x <_ y

We will continue to use x*(yz).

Clearly

as a shorthand for x#y

(x^y)^z = x*yz.

for

y~* xy

and

x A y = x. x^yz

for

236 LEMMA 13.1. In G3 the following facts hold: (i) b2co = cobi; c1d0 = dQcr (ii) b1o1 = cfi^ c^ = d2c2; b^ = d^by (iii) an*ai ;S ao

for a^

n €%-

(iv) (a^a^) A (aQ*a~) = e unless n = m. (v) (bQ*b") A (bQ*b™) = e unless n = m. (vi) (on*Qi)

A

(°n*ci) ~ e

un ess

^

n = m.

(vii) (d*c?l) A (do*d?) = e unless n = m. (viii) ('a *aJ)^c-£J A (aQ*a2 iPl c^ $!. ) = e unless (m = m13 n = nf, k = kr and p = pr). (ix) an*c2 — a6 ^ oru^ onty if w € w. Proof: (i) and (ii) follow immediately by conjugating the appropriate relation of G by a^ or ao. (iii) follows from the relations a'^a^a. <_ a ? and a" &2S~\ — a 2 ^ in^uc"t^on o n n» (iv) follows from (iii) and the relation a Q A Q. = e. (v), (vi) and (vii) now follow by conjugating (iv) by a ? , a| and a^ respectively. (viii) Since a, commutes with b Q , a Q <_ b Q implies a a — b 0 f o r a 1 1 n ^ ^' Con J u g a 't in g ^y a o a n d a o gives ^ ^0^1 — c0

an(

*

aQ^a^b^c^. _<_ d

c

0^ c l — ^0'

respectively, for all n € Z. Hence

and aQ^a^ b^ c 1 <_ d . (viii) now follows by

successively using (vii), (vi), (v) and (iv). (ix) Similar to (iii) using the relations involving a, b

and

a 7 . // We write (a;r(a) = e) for the G described above. We observe that G contains the five "levels" of 2 , as desired. f £ ^w is said to be presentable if there are a finite number of generators x(f)—including a and a«—and a finite number of words s(f) in these generators such that the relations ^(f) = e hold in A(E) in the natural interpretation and, in G(f) = (x(f);r(a) = e,s(f) = e), a^ commutes with a n , bn and

237 a f (a f A d Q ) ~

Since

Ad

a f (a f A d )

= e

A d

and for all m e u>

= e,

for any ^-permutation group m/ \ m

aa f <_ adQ

for all

a € supp(dQ)

(G(f),ft). Hence, if

mew,

V c i (a f A d o ) = V°i a f Note that in

A(ffi), a«

satisfies these relations. Indeed,

all the relations that we will write down hold in

A(l) in the

natural interpretation as can easily be checked. LEMMA. 13.2. Every recursive functions is presentable. Proof: Firstly, to

G

the generator

9(n) = 0

a

is clearly presentable—adjoin

and the relation

u

a

= e.

u

Secondly, for s the successor function, adjoin to G the generators a and a , and the relations: a commutes with s _ a_ y b., and a

= (a ^ e ) A a~ dna^.

S

3

We now prove

{J

a. -jfc a 0

=

Then

U,

a •Kb

=

a~.#b-, j a (a A d ) A d~ — e,

commutes with

S

u

c!?1 for all m 6 w

1

d.Q*U^d.

c .

, s(m) m, 1 1 lcl

1

d.Q*d.

Then

a , bn -

L

and -

d_ .

L

l

by induction.

o( r\\ d. n *U-,

d^-frD-,

Cn .

iit>bUiil«

a

, s ( m + l ) m+1 . , 0 1 1 ' a s re(lllireci-1, -, / - 1 , N-1 e = a^ d Q a^ A d^(a^ d Q a o) > s o = a

d^ = a" dQa^ v d (a~ dQa^) definition of

a

s

f

and

g

G(h) have generators

relations: £(f) = e,

a^Kc^a

= aQ*b-]_

c

is presentable.

We next show that if h = gf. Let

. Hence, by the above and the

(= (a v e) A a^ d^a^),

for all m € a). Thus

d1,

a

aAu*c.,a = an^fb^

III

Now

ab., c ^ a^^a

J>

!

rOI

a

=

d-, \ c a

sjCg) = e,

a h (a h A dQ)""1 A dQ = e,

a h

a,

are presentable, then so is x(f) U x(g) U {a,}, commutes with

a^,

A dQ = (ag t a f ) A dQ

(a t a f )(a

t a f A d j " A dQ = e, where

xxaT y" a,.

Now, in G(h),

x f y

and b,

is shorthand for

Vc!ah = 1a -1^ f

and

and

f m = a Q^^g ( )cf(m)jm d ^ lanf l x

238 h(m), f(m) m -1

Val

D

l

„ h(m) m

C a

l f a5

- h(m) ..„

c a

0 1

l5

0 1

V

Hence the composition of presentable functions is presentable. Lastly, we must show that if g, h, u and v are presentable and f is obtained from them by general recursion, then

f is presentable.

have as generators

So assume the hypothesis and let G(f)

a« together with those used to show the

presentability of g, h, u and v, and relations those required for

g, h, u and v together with:

and

d1; (af t a ) A d Q = a^ A dQ,

a f (a f A dQ)" 1 A d Q = e,

a« commutes with

a..,

b..

(af t s^) A dQ = (ay t a f ) A dQ,

(a f f a g ) ( a f t a g A dQ)" 1 A d Q = e,

(af t a h ) ( a f + a h A d Q ) ~ A d Q = e and (a

t a f )(a

t a f A d n ) " A d = e.

range of some m € rg(h g), For

Now each

m e w belongs to the

h-^g. We prove, by induction on k, that for each aQ*c^a^ = a Q^h^

k = 0,

c^.

let m = g(n), say.

ao^c^au = aQ*b!f(n)c^ = a o *bj (m) c^

Thus

presentable and u = fg.

Since

a Ad

since

u is

= (a« t a ) A dQ, it

follows that n a

Hence

Q

1

1

n

c a

0* l u a n xb-, 0 1

( -1 -1 \ o r a f * a 4 ag a 5 ; # c n a r a a . = a n -)tc n a c a a.a^,. I p g 4 O 1 5 g 4 i a

o. Since

f

, , d ^ ^=afd1,

a

_ f(m) m o ^ i c i=

Now assume that if n £ rg(h g), and suppose But

m € rg(h

So

then

0

i

f

ao#c.,a^ = aQ^-i

g ) . So m = h(n) for some

c^.,

n G rg(h g ) .

aQ^c^(af t s^) = aQ^c^(av t a f ) , so

By the hypotheses,

ao^c^n^d^afa/a^ = a0^ci n ^ y ^ ^ f

1

*

Thus

m ,n -1 -1 vVf(n) f(n),n9 -1 -1 „ a^^c^a^a^a. a, = an^b^ c^ ^n -;a-p • Hence

V°T a f d l and

a

=

Vblf(n)cl(n)dla4laflaha4'

commutes with

b,, c

SinCe

and a«,

vf(n) = fh(n) = f(m)

239 m a O* c l a f Therefore

i f ( n ) n - 1 f(m) ,-n c a a Vbl l f al hVl *

=

a~#c,a~ = a~*cna, an U l i

m

a^Kc^ a« - a^^D^

cL

U l h l

, h(n)c an f(m) a

commute,

-i -i

,— n

y i •

a^c^a^ = a^^b^

a.dl .

So

4 1

s

n

/= m\

^n/

and

D^ ,

c^ and

c3?.

By (I), every recursive function is presentable, and the proof of the lemma is complete. We now prove the theorem. By (II),

there is a recursive function

f whose range, X,

is not a recursive set. Let h be the characteristic function of

.

By Lemma 1 3 . 2 , f

presentable. Adjoin to G(f) a new generator a_, K

a

a

0=

e

and

commutes with

(a,

t a f ) A a, - e. Let H be the resulting finitely presented

Vcl(ah

+a

Vci(n)ah A(I),

Now since

a

Q* c ? £ a A

f)= V c l '

S

=

Vbf(n)4(n)

Vci(n)

if m £ X,

=

(hf(n) = 0)

*

a Q *c m a h = aQ*b1c™ f aQ^c^.

H

^ora H

n G w,

° a0^cla5lafVh = V c l a 5 1& f a 4-

aQ^cma, = a ^c?1 if and only if m £ X. presented £-group

V)^~"'"

and relations: A d

a,

lattice-ordered group.

and
a

A C

is

ThuS

However

>

in

Therefore, in H,

Consequently, the finitely

has insoluble word problem.

Q.E.D.

240 APPENDIX I

The purpose of this appendix is to establish Holland's theorem and some immediate consequences of it.

THEOREM (Holland).

Every L-group can be l-embedded in

for some chain Q. Indeed, if

G

is an L-group, then

an L-permutation group for some chain Q with

(G, Q) is

\o,\ <_ \G\.

This is the natural analogue of Cayley's theorem for groups. In order to prove the theorem, we first need to establish some lemmas.

If

C

is a subgroup of

right cosets of

C

in

LEMMA 1_. Let

C

G,

write

R(C)

for the set of

G.

be a convex subgroup of a p.o. group G.

Then R(C) is a p.o. set if we define Cg <_ Cf if and only if eg <_ f

for some c € C. If

L-subgroup of

G3

G

is an L-group and

C

is a convex

then R(C) becomes a lattice with

Cg v ch = C(g v h) and

Cg A ch = C(g A h) (g3h € G).

Proof: By routine verification.

A convex ^-subgroup prime if

f,g € G

and

LEMMA 2_. Let

C

C

of an £-group

f A g = e

imply

G

f € C

is said to be or

g 6 C.

be a convex t-subgroup of an L-group G.

Then the following are equivalent: 1.

C

is prime.

2. R(C) is a chain. 3. The set of convex L-subgroups of forms a chain under inclusion.

G

that contain C

241 4. For f3g £ G,

f A g € C

Proof: We prove that

1

implies f € C

or g 6 C.

is equivalent to each of 2, 3

and 4. 1 -> 2. g(g A h ) " or

1

Let

g,h € G

h(g A h ) "

A

1

and suppose that 1

= (g A h)(g A h ) "

h(g A h ) " 1 £ C; i.e.,

C

Cg = C(g A h) or

2 -> 1.

so

Ch - C(g A h). Now, by Lemma 1,

Cg <_ Ch.

R( C) is a chain and let

Assume

Cg A ch = C(g A h) = C by Lemma 1, and hence i.e.,

g€C

or

1 -» 3. e < a € A\B

G

and

a A b £ A H B,

since

A

Ab(aAb) £ C £ B.

and = e

Hence

Assume 3.

\g\ A |f| £ C} 2 C |f | = f v f"

if

Thus

For all

is prime and 4 •> 1.

B

or

Then Ch = C;

are convex and

B £ A.

e<_aAb
Let

so

are convex. But

so, without loss of generality,

For each

Now if

h G p(h,C).

C

Now

A,B

A <2 B

a contradiction. f £ G,

p(f,C) = {g € G:

and is a convex ^-subgroup of

>_e .

1 -• 4.

C but

a e B,

e < g,h € G

without loss of generality, Hence

is prime and

e < b £ B\A.

a(aAb)"

1

C

that contain

a( a A b)

3 + 1.

g A h = e. Cg = C

h£C,

Suppose that

^-subgroups of

g(g A h ) " 1 € C

= e,

Assume the former, without loss of generality. Cg A Ch = C(g A h) = Cg

is prime. Since

and

g A h = e, then

p(g,C) C p(h,C).

h A h £ C;

i.e.,

But

h £ p(g,C).

h£C.

f (f A g ) " 1 A g(f A g ) " 1 = e. So

f,g £ G,

fAg£C,

G, where

f£C

or

g£C.

A fortiori.

LEMMA 3_.

Let

G

a convex t-subgroup of

be an l-group and e ^ g £ G. Let G maximal with respect to g £ C

(existence by Zorn rs Lemma). Proof: Let p(f,C),p(h,C)pC

f A h = e. since

C be

Then C If

is a prime subgroup of G.

f,h £ C,

h e p(f,C) and

then

f£p(h,C).

Let

H be

the intersection of all convex t-subgroups of

G

that prpperly

contain

G

and since

g £ H,

C.

C $ H.

C^p(g,C), g £ C,

Then

H

Now

then

is a convex ^-subgroup of HCp(f,C)np(h,C).

g£HCp(g,C).

a contradiction.

Hence

Thus

C c p (g,c) and if |g| = |g| A |g| € C so

C = p(g,C).

But

g € H so

242 \g\ A |f| 6 C

and

a contradiction.

|g| A |h| € C.

Consequently,

LEMMA K. Let map

C

0: G + k(R(O)

f,h € p(g,C) = C,

is prime.

be a prime subgroup of an t-group G. The

defined by

t-homomorphism of

Therefore,

C

G

(Cf)(gQ) = Cfg (f9g £ G) is an

onto the transitive L-permutation group

(Ge,R(O). Proof: By routine verification.

LEMMA 5_. If

G

is an Z-group> then G

is t-isomorphic to

a subeartesian product of L-groups {B : e ^ g £ G}, where (B jft)

is a transitive locally primitive L-permutation group

(and 0, is a chain) for each e ^ g € G. Proof: Let G

e f g € G.

maximal with respect to

subgroup of

G,

so

Lemma 4, the map B

= Ge

Choose

a convex ^.-subgroup of

g £ C . By Lemma 3, C

R(C ) = Q

is a prime

is a chain by Lemma 2. By

0 : G -> A(R(C )) is an £-homomorphism and

is an £-group such that

^-permutation group. ^-subgroups of

C

G

Let

H

(B ,Q ) is a transitive

be the intersection of all convex

that properly contain

Theorem 1.6.2, the H -orbits of

ft

smallest non-trivial congruence of

C . Then

C Q H . By

are the classes of the (B ,ft ). Hence

(B ,ft ) is

Order Now ker(0 r) = O h -\ /geG g h£G k > e. Then by: f > e if and only if f > e for all g e G, g locally primitive.

0: G ->

TT B is an £-homomorphi sm where (he) = h0 (h € G; g g e/geGg e / g 6 G ) , ker(0) = f\ ker( 0 ) E C\ C . Since g fL C if g g e/geG e/g G G g g^e, C\ C = { e } ; i.e., 0 is an ^-embedding. G g COROLLARY 6. then

If

G

is a subdirectly irreducible l-group3

(Gs Si) is a transitive t-permutation group for some chain ft. We now prove Holland's Theorem:

^-embedding

0: G ->

By Lemma 5, there exists an

JJ" A(ft ). Totally order g e/geo

G\{e}

by

<,

say,

243

and l e t ft = \j ft S ; i . e . a < $ (in ft) if a G ft , 3 e ft, e/g€G g n and g
g

as can easily be checked. G

Each

ft

g

is a set of right cosets in

|ft| <_ |G|2 = |G| .

so

Note that we have actually proved that any £-group

G

can

be represented on a totally ordered set of chains, and the action on any one of these chains is transitive and locally primitive. A prime subgroup that

C

C

of an £-group

G

with the property

contains no £-ideal (normal convex ^-subgroup) of

other than

{e}

is called a representing subgroup of

COROLLARY 7.

Let G

G

G.

be an SL-group. (£,ftjis a transitive

l-permutation group for some chain ft if and only if G contains a representing subgroup. Proof: If

(G,ft) is transitive, let

a prime subgroup of C\ g~ G g = f\ G a g£G g€Ga subgroup of G.

G

Conversely, if (a chain).

G

is

and, by the fundamental triviality,

= O GR = "fe}- Hence 3€ft^

g

a € ft. Then

C

G

is a representing

a

is a representing subgroup, let ft = R( C)

As in Lemma 4, the map

0: G ->/((ft) has

_-i

kernel =

f\ g Cg = {e}. Hence 0 is an £-embedding of g£G a "transitive" ^-subgroup of /((ft); i.e., (G,ft,0) is a transitive £-permutation group.

COROLLARY 8.

If

G

is an l-simple l-group, then

G

onto

(G3tt) is

a transitive l-permutation group for some chain ft.

Proof: Let Since

G

G

be an £-simple £-group and let

is £-simple,

C

e / g 6 G.

is a representing subgroup, where C

is defined as in the proof of Lemma 5. Alternatively, apply Corollary 6 directly.

244 COROLLARY 9. and

G

If

(G^Q) is a transitive l-pemrutation group

is Abelian, then G

is an o-group and

(G3Q) is its

right regular representation. Proof: By Lemma 4.2.4, it is enough to show that uniquely transitive. 6 € supp(f). 6g" fg = 6 Abelian.

Let

so

If

y € ft and

g 6 G

Hence

G

e / f 6 G , let

be such that

g~ fg / f. = {e}

(G,ft) is

yg = 6. Now

This contradicts the fact that for all

y € ft; i.e.,

G

is

(G,ft) is

uniquely transitive. // (Cf., Theorem 4.5.1. ) A group

G

can be right totally ordered if we can endow

with a total order ^

so that

fh ^ gh

whenever

G

f ^ g.

COROLLARY 10. Every l-group can be right totally ordered. Proof: We first show that

A(ft) can be right totally

ordered for any chain ft. Well-order ft by ^ , say, and define f -< g

if

to < ) of

af < ag, where

a

is the least element (with respect

{$ € ft: $f j- 3g}. Then <

rt(ft). Moreover, if

f <_ g

is a right total order on

(in the pointwise ordering), then

f «g. Now, by Holland's Theorem, every £-group of some /4(ft). The restriction of ^

to

G

G

is a subgroup

makes

G

right

totally ordered. We conclude this appendix with one more application of Holland's Theorem.

THEOREM 11. Let group and H

(G3Q) be a doubly transitive £-permutation

be an l-group containing G

there is a chain T 3 ft such that group and

as an l-subgroup.

Then

(H,T) is an l-permutation

(G9Si) can be l-embedded in it.

Proof: Let

a £ ft and

be a convex £-subgroup of to not containing

gQ

H

e < g^ € G with ag Q J a. containing

G

Let C

maximal with respect

(existence by Zorn's lemma).

Since

245 C ^ G (gQ € G\C) 4.1.5),

and

G

is a maximal subgroup of

C H G = G . For each

^-subgroup of

H

h € H\G,

let

Ch

G (Theorem

be a convex

maximal with respect to not containing

h.

Let

C

= C and let I = (H\G) U {g }. Totally order I. Let U 0 ^_ T = i / R ( C ) . Let 3 £ fi. Then 3 = af for some f € G. g

1 i£I Define cj>: Q -> T by 3* = Cf. Since C H G = G , <j> is a welldefined order-embedding. Define \p: H -> A(T) by (Cih1)(hiJ;) = C 1 h 1 h (i € I; ^ ^ h e H ) . As in the proof of Holland's Theorem, i|/ is an ^-embedding (If g € G and (Cf^g* = Cf x for all f G H, then g € H fT 1 Cf 1 O G = p | G = {e}). If 1 1 X al f ^ G f 6G l g € G and 3 = af € fi, then (3g)(j) = (afg)* = Cfg = Thus (i^,((>) is an ^-embedding, as desired.

246 APPENDIX II

THEOREM 1. subgroup of Q

Let

G. If

G

be an l-group and P

Q 2 P

a closed prime

is a convex l-subgroup of

G, then

is a closed prime subgroup of G. Proof: We first show that if

g =

V g.,

then

id 1 Pg = supfPg^ i € I}. Pg. <_ Ph Since

P

Clearly,

for all is closed,

I € I,

gh

so

Pg <_ Ph.

Now assume f

{g. : i £ 1} = Q

+

Q ^P. and

Hence

g =

g A f £ P, 2).

If

then

g G P9Q

g A f g p,

then

i € I.

If

ve€P(i€l).

v e) € P.

Thus

x

Pg = sup{Pg. : i 6 I}.

Clearly

1

g.h"

V (g.h" i £ I

<_ P,

for all 1

then

v e =

-i Pgh

Pgi <_ Pg

Q

is prime. Suppose

V g. . Let i£I 1 since

P

e < f € QVP. If

is prime (Appendix I Lemma

P(g A f ) " ^ < Pg = sup{Pg : i 6 I}.

Hence P(g A f)"" g < Pg. for some j £ I. Thus -1 ^ e < _ ( g A f ) g < h g . for some h 6 P Q Q. Since Q is convex, (g A f )~1g € Q. But e < _ g A f ' < _ f € Q . Consequently, g € Q and

Q

is closed.

THEOREM 2_. Let

G

be an i-group. Then G

is completely

distributive if and only if D(G) = {e}. Proof: First observe that we may assume Theorem 8.2.5 (iv) -> (i) since its proof did not depend on the characterisation of complete distributivity given by D(G). Assume that

G

is completely distributive.

Let

{M.: i € 1} be the collection of all prime subgroups of G (so

D M. = {e} by Appendix I Lemma 3). Let i6I 1 Then for each i £ I, g £ M ? ; so g = V g - 1 i £ I 1J+ {g..: j G J} is the set of all elements of M. 1J

-•-

e <_ g £ D(G). where which are less

247 than or equal to

g.

Therefore

g = +

V

f €

I1

A

g

J i € I

Thus

li ft (Ui V ;

But

g

e ±f(±) 1 ± U j

M

i1

/\ V g.. = i e i j e j 1J S0 A g N e i e i ll f ff lU j

\ / e = e; i . e . , D(G) = {e}. f €1J Conversely, suppose that D(G) = {e}

H M = {e}. i£I 1

g =

and let

{G : X 6 A} A be the family of all closed prime subgroups of G. Totally order A and let ft = R(G. ) (A € A ). We may assume that ft 0 ft = 0 A A Ay if A and y are distinct elements of A. Let ft = \J ft . A€A A Define if;: G ->/4(ft) by: (G.f )(gi|;) = G.fg (f,g e G; A e A). Since A A {e}, i|; is an ^-embedding. Moreover, A A£A (Gip)n „ = (f A G.f) a closed prime subgroup of G. Thus (Gi|j,ft)

V

has closed stabilisers.

By Theorem 8.2.5 (iv) -> (i),

has pointwise suprema (and infima).

(Gip,ft/A9)

But chains are completely

a £ ft//9, a( A V g. •) = i £ I j £ J 1J infisupiag^: j € J}: i € 1} = sup{inf {ag±f, ±,: i 6 I}: f £

distributive.

Hence for all

J} =

a( \/T A g«-p/-O> whenever the indicated suprema and infima in f£XJ i£I l U i ; G exist. It follows that /\ \/g..= V T A g.^/.\ wheni€ I j € J 1J f eXJ i € I 1 1 U ; ever the indicated suprema and infima exist. Consequently, G is completely distributive.

248 SOME UNSOLVED PROBLEMS

TEE ABELIAN GROUPABILITY PROBLEM. Let operation

(Q,<_) be a chain. Assume that for some binary o

on ft, (ft,<_,o) is an o-group.

binary operation

*

on ft such that

Does there exist a

(ft,<_, *) is an Abelian

o-group? This question was asked by Reinhold Baer in 1961.

If (ft,<_)

satisfies the hypothesis and is countable, then it has the form of a countable ordinal number of copies of 7L with possibly a copy of

IQ on top.

Hence the answer is yes in this special case.

ORDERABILITY PROBLEMS. 1.

Classify those p.o. sets ft for which 4(ft) is a

(transitive) ii-permutation group.

This has been done by M. A.

Bardwell [80a] if it is further assumed that only if

supp(f) n supp(g) = 0.

f A g = e

if and

This is a definite restriction

as is shown by letting ft = I <& %

ordered by:

(r,m) <_ (s,n) if

r <_ s & m <_ n. 2.

If

G

is a group of permutations on a set ft, find

necessary and sufficient conditions (on

G

can be totally ordered by a G-invariant

<j

and ft) so that ft i.e.,

a <_ 3

implies

ag <_ $g (a,3 £ ft; g € G)—also see P. J. Cameron [76], M. Droste [80], G. Higman [77] and H. Putt [77].

MULTIPLE TRANSITIVITY PROBLEMS. 1.

Can A(lfi) be embedded as a group (as an £-group) in

every doubly transitive A(ft)? 2.

If

(G,ft) is triply (four, five, etc.) transitive, is

it m-transitive for all

m 6 % 1

(Assume ft is infinite. )

249 If

(G,ft) is an £-permutation group, the answer is yes

by Lemma 1.10.1.

The proof of Lemma 1.10.4 shows that the answer

is also yes provided

G

bounded above or below.

contains an element i- e whose support is However, if

G

contains no such elements,

the problem remains open. 3.

Find two doubly homogeneous chains ft-,ftp such that

A (ft.. ) and A(ftp) satisfy the same £-group sentences but are not Jl-isomorphic. Since there are only such chains must exist.

2 ° complete theories of £-groups,

The problem is to prove specific

,

and A(ftp) satisfy the same £-group sentences. 4. groups.

Give further examples of pathological £-permutation Our current list is limited by Glass and McCleary [76]

and Chapter 12. Surely there are many others.

TEE PRIMITIVITY PROBLEM. If

(G,ft) is a transitive primitive ordered permutation

group, must it be weakly doubly transitive, the right regular representation of a subgroup of ffi, or periodic? Theorem 4.3.7 shows that the answer is yes if coherent but leaves open this problem if

(G,ft) is

(G,ft) is incoherent.

If we can completely classify transitive primitive ordered permutation groups, we can completely classify all primitive ordered permutation groups by Theorem 4C.

PROBLEMS 0N_ SIMPLICITY AND ^SIMPLICITY. 1.

Does there exist an ^-simple £-group that has no

primitive faithful representation and is not an 0-group? By Appendix I Corollary 8, every ^-simple £-group has a transitive faithful representation

(G,ft) for some chain ft. The

only examples we know of that have no primitive such representation are o-groups (see the proof of Theorem 6F, Dlab [68] and Feil [80]). 2.

If

(G,ft) is a transitive £-permutation group with G

ft-simple, then are all the primitive components of fc-simple?

(G,ft) also

250 3.

Are Example 6.6 and those £-simple pathological

examples given in Glass and McCleary [76] actually simple as groups? Theorem 6.5 suggests that they might "be. 4.

Does there exist a (primitive) £-permutation group

(G, ft) with 5.

G

both finitely generated and simple as a group?

Can every £-group be ^-embedded in an £-group with

exactly four conjugacy classes?

That is, can we strengthen

Theorem 10B so that, in addition, any two elements unrelated (by the order) to

e

are conjugate?

If so, algebraically closed

£-groups will enjoy this property.

EXISTENTIALLY CLOSED Z-PERMUTATION GROUPS. Determine further properties of existentially closed ^-permutation groups (see Chapter 12 and Glass and Pierce [80c]).

We conclude with two classes of problems from £-groups that might well lend themselves to an £-permutation group approach.

THE LATERAL COMPLETION PROBLEM. If

G

is an £-group, then an £-group

H

containing

an £-subgroup is said to be a lateral completion of (i) for each (ii) H of

H

e < h 6 H,

there is

is laterally complete, and

that contains

G

g €. G

with

G

as

G if

e < g <_ h,

(iii) no proper £-subgroup

is laterally complete.

any chain ft, A(ft) is a lateral completion of

For example, for B(ft).

S. J.

Bernau [75] proved that every £-group has a lateral completion and that it is essentially unique.

The technique is a formal construc-

tion that is simple in concept but messy in detail

(This is still

f

true of McCleary s improvement [8la] though to a lesser extent). An alternative proof was provided by R. N. Ball [80a] which uses a more sophisticated concept (a generalisation of uniform spaces) but has a simpler proof.

However, in both constructions, it is

not easy to describe the lateral completion of given £-groups. It would be nice to give an £-permutation

251 group construction of the lateral completion of an £-group. However, a simple computation shows that

G = {({g

},m)£ZIrZ;

(3 k £ % )(g = g if nn E no(mod k))} has no lateral completion n~| o inside Z Wr Z = A(Z x" Z )

even though

(G,Z 5 Z )

and has only the obvious non-trivial congruence.

is transitive This means that

there is no facile answer to an £-permutation group construction of the lateral completion in the general setting; this is a shame but, in the immortal words, "sic biscuitus disintegrat."

For a

nice I-permutation answer in the completely distributive case, see Davis and McCleary [81b].

WORD PROBLEM TYPE PROBLEMS. Since the word problem for £-groups was solved (negatively) using £-permutation groups (Chapter 13 )> it may be possible to use ^-permutation groups to answer some of the following sample problems (cf. McKenzie and Thompson [73] and Thompson [80]): 1.

Does there exist a finitely presented £-group (whose

relations only use group words) having insoluble group word problem? Without the parenthetic part, we saw in Chapter 13 that the answer is yes. 2.

Does there exist a finitely presented group with

insoluble word problem which can be ordered so as to be an £-group or an o-group? 3.

Can every finitely generated £-group defined by a

recursively enumerable set of relations be ^-embedded in a finitely presented £-group? This is the £-group analogue of Graham Higman's famous embedding theorem for groups [61]. All proofs of this phenomenon use either free products with amalgamated subgroup or the HigmanNeumann-Neumann construction.

Unfortunately, these two

constructions are not possible for £-groups (Theorem IOC and Glass [75b] ). 4.

R. J. Thompson [80] proved:

A finitely generated group

has a soluble word problem if and only if there is an embedding of

252 the group (such that two elements not conjugate in the original group will not be conjugate in the embedding group) into a finitely generated simple group which is a subgroup of a finitely presented group.

This is a strong form of a theorem due originally

to Boone and Higman [74] . Is the il-group analogue true? that Thompson's proof uses permutation groups.

Note

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Reine

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263 G. F. McNulty: [80] Classes which generate the variety of all latticeordered groups, in Ordered Groups: Proc. Boise State Conference 1978, ed. J. E. Smith, G. 0. Kenny & R. N. Ball, Lecture Notes in Pure and Applied Math., No. 62, Marcel Dekker, 1980, 135-140. R. N. McKenzie and R. J. Thompson: [73] An elementary construction of unsolvable word problems in group theory, in Word Problems, ed. W. W. Boone, F. B. Cannonito and R. C. Lyndon, North Holland, 1973, 457-478; MR 53 (1977), #629. N. Ja. Medvedev: [77] The lattices of varieties of lattice-ordered groups and Lie algebras, Algebra and Logic, 16(1977), 27-30 (English translation); MR 58(1979), #16456. R. B. Mura and A. H. Rhemtulla: [77] Orderable Groups, Lecture Notes in Pure and Applied Math., No. 27, Marcel Dekker; MR 58(1979), #10652. B. H. Neumann: [60] Embedding theorems for ordered groups, J. London Math. Soc., 35(1960), 503-512; MR 24(1962), #A176. B. H. Neumann and H. Neumann: [59] Embedding theorems for groups, J. London Math. Soc. , 34(1959), 465-479. T. Ohkuma: [55] Sur quelques ensembles ordonne*s line"airement, Fund. Math., 43(1955), 326-337; MR 18(1957), p. 868. D. Passman: [68] Permutation Groups, W. A. Benjamin, 1968; MR 38(1969), #5908. K. R. Pierce: [72] Amalgamations of lattice-ordered groups, Trans. Amer. Math. Soc, 172(1972), 249-260; MR 48(1974), #3835. H. L. Putt: [77] Orderable permutation groups, Ph.D. Thesis, Bowling Green State Univ., Ohio, U.S.A., 1977.

264 E. B. Rabinovich: [75] On linearly ordered sets with 2-transitive groups of automorphisms, Vesti Akad. Navuk Belaru'skai SSR (Seria FizikaMate. Navuk), 6(1975), 10-17 (In Russian); MR 55(1978), #8157. E. B. Rabinovich and V. E. Feinberg: [74] Normal divisors of a 2-transitive group of automorphisms of a linearly ordered set, Mat. USSR Sbornik (English translation), 22(1974), 187-200; Ml 49(1975), #394. J. A. Read: [71] £-sous-groupes compressibles du groupe-re*ticule* A(S), S6minaire Dubreil (Algebre), 6(1971-2); Secretariat mathe*matique, 11 rue Pierre et Marie Curie, 75231 Paris Cedex 05, France; MR 52 (1976), #10538. [75] Wreath products of nonoverlapping lattice-ordered groups, Canad. Math. Bull., 17(1975), 713-722; MR 52(1976), #5515. N. R. Reilly: [69] Some applications of wreath products and ultraproducts in the theory of lattice-ordered groups, Duke Math. J. , 36(1969), 825-834; MR 40(1970), #4184. [72] Permutational products of lattice-ordered groups, J. Australian Math. Soc., 13(1972), 25-34; MR 45(1973), #3278. [81] A semilattice of the lattice of varieties of latticeordered groups, Canad. J. Math, (to appear). N. R. Reilly and R. Wroblewski: [81] Suprema of classes of generalized Scrimger varieties of lattice-ordered groups, Math. Zeit. , 176(1981), 293-309. A. Robinson and E. Zakon: [60] Elementary properties of ordered abelian groups, Trans. Amer. Math. Soc, 96(1960), 222-236; MR 22(1961), #5673. J. Robinson: [68] Recursive functions of one variable, Proc. Amer. Math. Soo. , 19(1968), 815-820; MR 37(1969), #6178. H. Rogers Jr.: [67] Theory of Recursive Functions and Effective Com-putability, McGraw-Hill, 1967; MR 37(1969), #61. J. Roitman: [81]

The measure of rigid homogeneous chains, to appear.

J. G. Rosenstein: [81] Linear Orderings, Academic Press (to appear).

265 J. Schreier and S. Ulam: [33] liber die Permutationsgruppe der nattlrlichen Zahlenfolge, Studia Math., 4(1933), 134-141. [35] Ein Bemerkung tlber die Gruppe der Topologische Abbildungen der Kreislinie auf sich selbst, Studia Math., 5(1935), 155-159. [3V] Uber die Automorphismen der Permutationsgruppe der nattlrlichen Zahlenfolge, Fund. Math., 28(1937), 258-260. T. J. Scott: [74] Monotonic permutations of chains, Pacific J. Math. , 55 (1974), 583-594; MR 52(1976), #5516. E. B. Scrimger: [70] Intransitive lattice-ordered groups of order-preserving permutations of chains, Ph.D. Thesis, University of Wisconsin, Madison, Wis., U.S.A., 1970. [75] A large class of small varieties of lattice-ordered groups, Proc. Amev. Math. Soc. , 51(1975), 301-306; MR 52(1976), #5517. W. Sierpinski: [49] Sur une propriete des ensembles ordonne"s, Fund. Math., 36(1949), 56-67; MR 11(1950), p. 165. F. Sik: [58] Automorphismen geordneter Mengen, Casopis Pest. Mat., 83(1958), 1-22; MR 20(1959), #2290. J. E. Smith: [80a] The lattice of £-group varieties, Trans. Amev. Math. Soc, 257(1980), 347-357. [80b] £-group varieties, in Ordered Groups: Proc. Boise State Conference 1978, ed. J. E. Smith, G. 0. Kenny & R. N. Ball, Lecture Notes in Pure and Applied Math., No. 62, 99-105. [81] A new family of £-group varieties, Houston J. Math. (to appear). A. Tarski (in collaboration with A. Mostowski and R. M. Robinson): [53] Vndecidable Theories, North Holland, 1953; MR 15(1954), p. 384. R. J. Thompson: [80] Embeddings into finitely generated simple groups which preserve the word problem, in Word Problems II, ed. S. I. Adian, W. W. Boone & G. Higman, North Holland, 1980, 401-441.

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ANNOTATIONS

Unless otherwise stated, each numbered theorem in the annotations refers to a theorem in our book. A. K. Arora: [81] A quasi-variety of £-groups is a class of £-groups closed under £-isomorphisms, £-subgroups, cardinal Cartesian products and ultraproducts. Equivalently, it is defined by a set of implications of the form \/x(wn(x) = e ->• w n (x) = e)> where w-.(x),w2(x) are £-group words.

Canonical examples are varieties,

the class of all totally orderable £-groups, and the class of £-groups having unique extraction of roots. There are at most distinct quasi-varieties. Whereas >V is the maximal proper

2 °

ft variety of £-groups (Theorem 11D—Holland [76b]), 2 ° pairwise incomparable quasi-varieties are constructed containing X. Moreover, it is shown that there is no maximal proper quasi-variety. The ^-semigroup of quasi-varieties is also examined. W. W. Babcock: [64] The main results in this thesis are (i) Any homogeneous Dedekind complete chain other than 7L has cardinality 2 °,

and

(ii) There are at least N-.

pairwise non-order-

morphic such chains. McCleary [72a] notes that each of them supports a full periodic primitive Config(l) £-permutation group, and those that come from non-ordermorphic chains are not £-isomorphic. R. N. Ball: [74] The only part of this thesis not included in Ball [75] concerns the normal subgroups of /4(IL)—see Theorem 2.3.7. Indeed, if N denotes the normal subgroup of A(IL) defined on page 70, it is shown that N covers L(IL) and the set of normal subgroups of A(IL) that contain N is lattice-isomorphic to the lattice of all filters on the set <^(u), )/~, where (P(u>-.) is the set of all subsets of

u

closed unbounded subset

and T

of

A~B

if A H T = B O T

u), . Moreover,

for some

L(IL) covers

B(IL),

A(JL) covers R(IL), and the lattice of normal subgroups between B(IL) and R(IL) is isomorphic to the lattice of normal subgroups between L(lL) and A(IL). The example given at the beginning of Chapter 7 is also taken from this thesis, where it is shown that

xii P

has a cover which can be taken to be

choice of ultrafilter

^

A(l)n

for a suitable

(assuming the Continuum Hypothesis).

[75] An £-group H is an a*-extension of an £-subgroup G if the map C»—»-C n G of the closed convex £-subgroups of H onto those of G is one-to-one. If G is an o-group, this is the same definition as a-extension (see Khuon [70a]). It is shown that an arbitrary £-group has an a*-closure (a maximal a*-extension)—cf., Glass, Holland and McCleary [75]. The proof has the same spirit as Khuon [70b] but is complicated by the fact that if G is an £-group with (G,Sl,Q) an £-permutation group, the £-embedding 0 of G into A(Sl) may not preserve infinite suprema (whence G and G0 may have different closed convex £-subgroups). The difficulty is overcome by considering full convex £-subgroups (A convex £-subgroup C of G is said to be full if whenever c £ C and g and c generate the same closed convex £-subgroup of G, then g £ C ) . Any full convex ^ s u b group of G is a union of closed convex £-subgroups, so it suffices to bound |G| in terms of the cardinality of the set of full convex £-subgroups of G. This is done by Khuon's lemma and a lemma of McCleary. [79] A topology «-7 on an £-group G in which the group and lattice operations are continuous is called an £-topology, and (G,J) is said to be a topological £-group. Properties of such structures are examined. In particular, a continuous version of Holland's Theorem is established. If (G,Sl) is a transitive £-permutation group and >J is any Hausdorff £-topology on or A(Sl°), then G° = {f €.A(si°): f respects the congruences of (G,Sl)} is shown to be closed in >J . The topological closures (in A (si)) are found for (G,fi) a transitive primitive ^ p e r m u tation group. [80a] This is a generalisation of both Ball [79] and uniformities. It adapts the techniques of convergence and Cauchy structures to an £-group setting to obtain the existence of various completions of £-groups, notably the existence of a lateral completion of an arbitrary £-group (cf. Bernau [75]). [81] Let a < b and c < d in an £-group G. Write (a,b) ~ (c,d) if +(a,b) = +(c,d) and +(a,b) = + (c,d), where + (a,b) = {g £ G: g v a >_ b} and +(a,b) = {g £ G: g A b <_ a}. If (G,Sl) is an £-permutation group, (a,b) ~ (c,d) can be described pictorially by the "bubble" where a and b differ is the same as the "bubble" where c and d differ; i.e., aa = ab implies ac = ad and aa < ab implies ac = aa < ab = ad. When ba is a bump, "bubble" accurately describes the picture. We can order these equivalence classes by: ( a , b f < ( c , d T if* the "bubble" of some element of ( a , b ) ~ is inside the "bubble" of (c,d). This leads to a completion of G known as the distinguished completion. The shadow cast by a bubble also plays an important role.

Xlll M. A. Bardwell: [79] Let G be a right totally ordered group. The right regular representation (G,G) is examined. It is shown that any transitive primitive ordered permutation group can occur as a primitive component of (G,G) for suitably chosen such G. Moreover, an example is given of a right totally ordered group G where the right regular representation is primitive and uniquely transitive but not isomorphic to a subgroup of 1 (cf. Lemma 4.2.4 and Theorem 4.2.1). Some results are obtained about (G,G) when it is a "large" subaction of a Wreath product and some unsolved problems are listed. The main tool is the proof of Appendix I Corollary 10. [80a] The p.o. sets ft for which A(to) is an £-group having the disjoint support property (f A g = e if and only if supp(f) D supp(g) = 0 ) are classified. This last property is equivalent to each of (i) every orbit of 4(ft) is a chain; (ii) a(f v g ) = max{af,ag} for all a € ft, f,g £ A(ft). It follows that such A(ft) are laterally complete. The classification relies heavily on the congruence structure. There are six main cases when A(ft) has precisely two orbits, and all are shown to be non-vacuous. An example is given of a p.o. set for which A(ft) is a transitive £-group without the disjoint support property. [80b] Most of this article is an outline of Bardwell [80a], but there are a few new results. If ft is a p.o. set such that A(ft) is an £-group, then A(ft) has the disjoint support property if and only if A(ft) is prime for all a e ft. By using the classification given in Bardwell [80a], it is noted that any such A(ft) having two orbits has closed stabilisers and is completely distributive—cf., Theorems 8B and 8.2.1. S. J. Bernau: [75]

Let

G

be an ^-subgroup of an £-group

H.

Each

element of the ^-subgroup G of H generated by the suprema of all pairwise disjoint non-empty subsets of G is easily seen to be of the form

v A h . ., where I,J are finite and each 1 J . .1J h. . = 7 T ( V M . .. y^i'J'k) f o r S ome finite K, e(i,j,k) - ±1, 1J !J K M. ... a non-empty pairwise disjoint subset of G having a supremum in

H.

If

G Q = G,

G

. = G ,

and

G

and

= (J G if X v
xiv to get an £-group. At limit stages, take direct limits. The cardinality bound is provided by K, the next cardinal after |G|, since G must be "dense" in each G . G is a lateral completion of G. The construction gives uniqueness. The last part of the paper is devoted to showing that certain properties of G transfer to the lateral completion; e.g., complete distributivity (cf. Davis and McCleary [81b]). A simplification of the construction is provided in McCleary [8la]. For a less elementary construction of the lateral completion, see Ball [80a]. Note that the details of Ball [80a] are much easier than those of even McCleary [8la]; however, the degree of abstraction is considerably greater. [77] It is shown that if V is a variety and G £ f, then the lateral completion of G belongs to V. The proof proceeds by showing that G € 1/ implies G*£ V , where G* is defined in the previous annotation. Actually, the proof shows that if G is an ^-subgroup of an £-group H and G £ V, then the closure of G in H belongs to Kl. This is used a lot in Glass, Holland and McCleary [80]. A. Bigard, K. Keimel and S. Wolfenstein: [77] This is now the main collected work on £-groups. It is strongly influenced by Conrad's Tulane University lectures on "Lattice-ordered Groups" (no longer in print) and the authors' own work. It recounts research up to 1972 (approximately). For a more detailed review, see Zentralblatt fur Math. , 384(1979), 06022. G. Birkhoff: [46] For a given group G, a p.o. set is constructed with G isomorphic to the group Aut() of all order-preserving permutations of the p.o. set ; cf., Scrimger [70]. ~ W. W. Boone and G. Higman: [74] Using free products with amalgamation and Higman's embedding theorem (Higman [61]), the authors prove: A finitely generated group G has a soluble word problem if and only if there exist a simple group H and a finitely presented group L such that G can be embedded in H and H can be embedded in L. The semigroup analogue is also proved. R. J. Thompson [80], using permutation groups, improves on the group theory part, showing (among other things) that H can be taken to be finitely generated. The construction of such an H is difficult. Can it be taken to be finitely presented?

W. Buszkowski: [79] The theory of lattice-orderable groups is shown to be undecidable—Theorem 2.2.8. The proof given is as follows: Let d = ({d },1) € Z Wr Z, where d = n (n € Z ) . Let C(d) be the n n -. centraliser of d in % Wr2Z. Write x ~ y if xy"~ belongs to is constant). the centre of S I r S (i.e., x = y and x - y d n ~ dm

Then

only if

n = m,

and

x 6 C(d) if and only if

m

for some m e 7L. For x,y,z £ C(d), .write D(x,y) if x - d C(x) £ C ( y ) , and S(x,y,z) if xy ~ z. Then, when restricted to C(d), S and D respect ~ and give addition and divisibility (of Z via {dn: n € Z } ) . This guarantees the undecidability— just of the group part—of lattice-ordered groups; cf., Glass and Pierce [80c], Gurevich [67], Jambu-Giraudet [81b] andMal'cev [71]. R. D. Byrd and J. T. Lloyd: [67] The main result of this paper is Appendix II Theorem 2. Unlike our proof, theirs does not use I-permutation groups. Again the key lemma is Appendix II Theorem 1. P. J. Cameron: [76] A permutation group G on a set ft is said to be n-homogeneous if whenever r,A ^ ft and |r| = |A| = n, there exists g G G with Tg = A. A(l) is n-homogeneous for all n e %>+ (Lemma 1.10.1). However, it is not 2-transitive as a permutation group—there is no g € 4(1) with Og = 3 & lg = 2. The group ?^(R) of all monotonic permutations of R is still n-homogeneous for all n £ Z + ; it is 2-transitive but not 3-transitive as a permutation group. McDermott (unpublished) has shown that any 3-homogeneous but not 2-transitive group of permutations of an infinite set ft is a subgroup of A(ft) for some linear order on ft. Using a weak form of Ramsey's Theorem for infinite sets, this is extended to prove (l) Any ^-homogeneous but not 3-transitive group of permutations of an infinite set ft is a subgroup of either V((n) for some linear order on ft or the group of automorphisms of some cyclic order on ft. (2) If G is a group of permutations of an infinite set ft that is m-transitive but not (m+l)-transitive for some m £ Z + , and is n-homogeneous for all n e %,+, then m <_ 3 and there is a linear or cyclic order on ft that is preserved or reversed by all elements of G. C. C. Chang and A. Ehrenfeucht: [62] Theorem 10.7 is established, even with "proper" removed. The proof proceeds by tagging elements of distinct cosets of a proper subgroup of R with chains associated with distinct ordinals (as in the proof of Theorem 10.5). By Corollary 10.6, it follows that an Abelian group G is isomorphic to 4(ft) for some chain ft if and only if G is isomorphic to a direct product of subgroups of 1 (cf. Conn [57]—Theorem 4.5.1). This is the first use of the ingenious device of "tagging."

xvi C. G. Chehata: [52] The set of bounded elements of Example 1.2.3 is shown to be a simple group. The proof of simplicity is more computational than ours (see the proof of Theorem 6F) since it predates Higman [54]—Lemma 6.4. Also see Dlab [68] and Fell [BO]. P. M. Conn: [57] Theorem 4.5.1 is established--cf. Chang and Ehrenfeucht [62]. The proof is of an elementary nature and follows from considering the case that ft is an extensive /((ft) block. Our proof is shorter but uses Section 3.2 (which is more recent). The paper also includes a proof that A(ft) can be right totally ordered (cf. Appendix I Corollary 1 0 ) . P. F. Conrad: [70] The free £-group over a p.o. group G is shown to exist precisely when the partial order is the intersection of all right total orders that extend it. Moreover, the free £-group over such a p . o . group G is explicitly constructed. This extends the work of E. C. Weinberg [63] for the construction of free Abelian and representable £-groups. The construction is as follows: Let {G : X £ A} denote the set of all right total A orders on G extending the given partial order. Represent G on >4(G ) via the right regular representation (for each X 6 A ) . A

The ^-subgroup of

J 7 A ( 0 ) generated by this image is the free A X€A £-group over G. Since any free group can be right totally ordered, this yields a construction for the free £-group on any set X. Its relation to the free Abelian and free representable ^-groups on X is examined. For further results on free £-groups, see Arora and McCleary [81], Glass [74], Kopytov [79] and McCleary [Sic]. G. Davis and C. D. Fox: [76] The main result in this paper is that if ft is doubly homogeneous, then >4(ft) endures a CTRO (see annotation of Davis and Loci [76] for definition). This result was also proved independently by R. N. Ball (unpublished). Let T = {e < g 6A(fi): (3a 6 ft)supp(g) 0 [a,«>) is dense in [a,«)}. It is shown that if

cf(ft) = NQ.

T

is a maximal CTRO on such A(ft) P Dually for ci(ft) = NQ.

if and only

G. Davis and E. Loci: [76] The interval topology on an £-group is rarely Hausdorff. On the other hand, the interval topology on a tight Riesz group is (A p.o. group G is a tight Riesz group if the partial order is directed, and f ,f^ < g-,,g~ Implies there is h £ G such that

f-ijfp < h < g-ijgp*

Note, any Jl-group that is a tight

xvii Riesz group is an o-group). If G is an £-group that can be reordered to be a tight Riesz group with set of positive elements T in such a way that the two orders are compatible, then T is said to be a compatible tight Riesz order on G, or CTRO for short. More precisely, if G is an £-group and T Q G , then T is CTRO if (i) T is a normal subsemigroup of G, (ii) T-T = T and (iii) inf T = e, where the infimum is with respect to the lattice order. If ft is a totally ordered field, A(ft) contains positive elements that fix no point of ft. Since A(ft) is divisible, the set T Q of such elements forms a CTRO. Other CTROs are easily obtainable. Moreover, it is shown that there are prime subgroups of A(ft) maximal with respect to not meeting T^ which are not stabilisers. Other more technical results are proved. Like the above, they are e a s y — a lot simpler than the presentation given in the paper, G. Davis and S. H. McCleary: [81a] In an £-group G, write firg if (Vh)(f A h = e «-> g A h = e ) . A subset X ^ G is said to be ir-full if yirx and x € X imply y € X. The main theorem is: If (G,ft) is a laterally Complete £-permutation group with no block on which the appropriate restriction acts like 22, then (i) the ir-full prime subgroups are precisely the minimal primes, (ii) every ir-full tight Riesz order is a CTRO and (iii) there is a one-to-one correspondence between the ir-full CTROs on G and the ir-full £-ideals of G. Glass and Holland (unpublished) have shown that if ft is doubly homogeneous, any £-ideal of A(ft) is ir-full. Hence there is a one-to-one correspondence between the fr-full TROs on A(ft) and the set of normal subgroups of A(ft). When A(ft) is locally X , the ir-full TROs on A(ft)^ are shown to correspond to the £-ideals of A(ft)|ft, where ft is an appropriate subset of ft. [8lb] Let G be a completely distributive £-group. There is a chain ft such that (G,ft) is an £-permutation group with G a complete subgroup of A(ft). Let H = {h £/4(ft): (Va 1 ,...,a n e fi)0g G G)a.g = a.h (l < i < n)}. Then G is dense in H and H is laterally complete. Hence G has a lateral completion which is shown to be completely distributive (For the existence of the lateral completion of a general £-group, see Ball [80a], Bernau [75] or McCleary [8la]). The lateral completion is further restricted in certain nice cases. For an earlier proof of the existence of a lateral completion of a completely distributive £-group not using ordered permutation groups, see R. D. Byrd and J. T. Lloyd (J. London Math. Soc. ( 2 ) , 1(1969), 358-362).

V. Dlab: [68] Let ft = [0,1) and H be a subgroup of the group of positive real numbers. Let A~, be the group of locally right H-linear elements of A(ft);

i.e.,

g € A^

if

(Va € fl)0E = e(a,g) > o)(3h = h(a,g) € H)(Ve € (a,a + e)) (Bg = ag + h(3 - a ) ) . Let B R = ^ HB(fi), 1^ = A^ OL(ft) It, = A^ O R(ft).

Then

f,g £ A^

are conjugate in

and

A-, precisely

when there is a one-to-one left-right preserving map from the orbitals of f onto the orbitals of g such that corresponding orbitals have the same boundity and left-most slope (cf., Theorem 2.2.4). Moreover, if e f g € A-, and N is the normal subgroup of

A^

RJJ c N;

generated by and if

g,

then

B

f £ R^,

then

f € N

c N;

if

g £ L^,

most slope belongs to the cyclic subgroup of left-most slope of

g

(and

f € LR

if

L^

containing

B

H

generated by the

g £ L ). Hence

simple and is the only normal subgroup of normal subgroups of

then

if and only if its left-

FLj.

B

is

The lattice of

is isomorphic to the

lattice of all subgroups of

H.

and only if

has precisely two linear orderings

H

is; and

A^

The group

A^

is divisible if

if and only if H has rank 1 as an Abelian group. The collection of BTJ (as H ranges through all subgroups of the group of positive real numbers) provides an uncountable family of non-isomorphic simple orderable groups—see Theorem 6F. proofs are computational and similar to those in Section Indeed, recently, M. Droste has obtained a uniform method obtain the results of Dlab and Section 2.2. It too is the ideas in Section 2.2.

pairwise The 2.2. to akin to

M. Droste: [80] Let K be a cardinal, be a p.o. set and G be a subgroup of Aut(). Assume 2 <_ K < \Q\ . G is said to be K-homogeneous if whenever A,B Q Q have cardinality K, there is g £ G such that Ag = B. It is shown that if ft is infinite and <_ is not the trivial partial ordering, then is a chain and K is finite. Note that another definition (which agrees with the above if K is finite) is: If A,B Q ft have cardinality<|K+1| and there is an order-preserving one-to-one map of A onto B, then there is g £ G extending this map. Under this definition, Aut() can be ^-homogeneous without being a chain. We adopted this second definition—see page 1877 Also see Cameron [76] and HIgman [77].

B. Dushnik and E. W. Miller: [40] Using transfinite induction, a non-well-ordered subset of 1 of size 2 ° is constructed which is not ordermorphic to any proper subset of itself. This paper can be thought of as a precursor of Ohkuma [55] and Glass, Gurevich, Holland and Shelah [81]—see Section 4.2. D. Eisenbud: [69] Let fi be a doubly homogeneous chain of countable coterminality with all elements of ft having character co~. The key lemma is: f € A(ft) can be written uniquely as the product of commuting elements one from L(ft) and one from R(ft) precisely when_ f has two supporting intervals and exactly one fixed point in ft. Since L(ft) and R(ft) are the maximal normal subgroups of such A (ft-)—Theorem 2.3.2—ft is obtained from the group A(ft). It follows that A(ft) sA(ft) as groups only if ft = ti; and if A is also a doubly homogeneous chain of countable coterminality with A ( A ) =A(ft) (as groups), then A is ordermorphic or antiordermorphic to a dense subset of ft (cf. Theorem 2D). It is also shown that Aut(A(ft)) s ^J(o)^^^^ ^ e n o r m a l i s e r °f A(ft) in the symmetric group ^(ft); and if B(ft) £ G Q A ( Q ) , then Aut(G) = N. ././ Q))(G)This work is a generalisation of Holland [63] 2.3.

and a precursor of some of the ideas in Sections 2.1 and

T. Feil: [80] A previously known o-group C that is without proper convex normal subgroups is shown to be contained in the finitely piecewise linear elements of -A(R), as the set of such with rational "corners" and all slopes integer powers of 2. B = C O B(ft) is a normal subgroup of C and [B,B] is simple by Lemma 6.4. Does B = [B,B]? For other related work, see 1 Chehata [52] and Dlab [68]. [81] A tower of 2 ° varieties of representable ^-groups is given—the maximum cardinality possible. The construction of the tower is very elementary. Unlike the prior proofs of Kopytov and Medvedev [77] and Reilly [81], this proof of the existence of 2 ° varieties of ^-groups does not depend on any results about varieties of groups. I. Fleischer: [77] This gives an example to point out an error in a proof of Hickman [76]—at least if lexicographic sum is replaced by lexicographic union; then it is just Example 4.2.6.

C. D. Fox: [75]

In an orderable group,

[fn,g] = e

implies

n

[f,g] = e, and [f ,g,f] = e implies [f,g] = e. An example is given of an orderable group G such that, for any set X of positive integers, there are f,g £ G with [fn,g,f,f] = e if and only if n £ X. The group G is a natural subgroup of (H Wr Z ) w r l , where H is the (orderable) group given by (x,y: x yx = y 2 ) . G is finitely generated. It follows that there is a finitely generated £-group having a recursively enumerable set of (group) defining relations and an insoluble group word problem; cf., Glass [75a], J. D. Franchello: [77] Using the ideas of Holland [77] and the aid of a computer, it is shown that there are essentially 6 group words of length 16 that hold in the £-group free product of two arbitrary ^-groups (one of these is given by Lemma 1.11.5 and is due to Holland and Scrimger [72] ). Two infinite families of group words are found that hold in all such £-group free products. For other results in this thesis, see Franchello [78]. [78]

Let £

be the class of distributive lattices with

distinguished element £-groups and

G

e.

Let

{G.: i € 1} be a family of

be the £-group free product of

{G.: i £ I}. Using

Corollary 2L, the sublattice generated by the set of distributive lattices {G.: i € 1} is proved to be the /0 -free product of these lattices. This contrasts with the group theory results due to Franchello [77], Holland [77], and Holland and Scrimger [72]. This may be a bit surprising since, in Section 2.1, we showed that if A(ft) is doubly transitive, the lattice and group languages of A(tt) are equally powerful. E. Fried: [65] It is shown that if Q is a chain, then J{si), the symmetric group on Q, is a left partially ordered group. Furthermore, every left partially ordered group can be o-embedded in some such J ( Q ) . Necessary and sufficient conditions are also given for a p.o. group to be o-embeddable in some A ( Q ) . The footnote on page 17 is doubtful. Note "left" would be "right" if af were used instead of f(a). [67] It is proved that a p.o. group G can be o-embedded in A(Q) for some chain Q if and only if the partial order is the intersection of all left total orders on G that extend it. This tidies up the result in Fried [65]. Also cf. Conrad [70], Conn [57] and Appendix I Corollary 10.

xxi L. Fuchs: [63] This book used to take the reader to the forefront of the many aspects of partially ordered algebraic structures. However, the subject has mushroomed in the last twenty years. Nonetheless, the book is still valuable for its readability and bibliography, and is an excellent introduction to the material. A. M. W. Glass: [72] A directed p.o. set in which no two incomparable elements have a supremum or infimum is called an antilattice set. So any antilattice set that is a lattice is a chain. A generalisation of Holland's Theorem is proved; viz., any directed interpolation group (this class includes £-groups and tight Riesz groups) can be order embedded in the automorphisms of some antilattice set. [74] The first examples of non-£-simple pathological £-permutation groups are given. They include Theorem 6.7 for certain infinite cardinals K, including K = AL. They answer a question of McCleary [73b]: Is a transitive primitive £-permutation group £-simple if G GB(fl) or GHB(fi) = {e}? [75a] It is shown that there exist recursively presented £-groups (finitely generated with a recursively enumerable set of defining relations) having insoluble word problems. This paper is completely superceded by Chapter 13 (which is stronger and more difficult). Also cf. Fox [75]. [75b] Much combinatorial group theory is predicated on a theorem by Higman-Neumann-Neumann: If A and B are isomorphic subgroups of a group G, then it is possible to adjoin to G a new element t so that in a new group H generated by t and G, the isomorphism between A and B is realised as conjugation by t. This property is shown in groups with operators to imply the amalgamation property. Hence the property fails, in general, in £-groups. Conditions are given in which it holds. Also note Lemma 2.2.9. Unrelatedly, an example is given of an Abelian subgroup of A(R) that is contained in no Abelian divisible subgroup of 4(1). [76] The existence of CTRO's for general transitive ^-permutation groups (G,ft) is considered. Theorem 3.7 of the paper is false—see Glass [79]. For the definition of CTRO's, see Davis and Loci [76]. The work relies heavily on the structure theory given in Chapters 3-5. [79] Theorem 3.7 is corrected: A(Q) endures a CTRO if and only if A(n) j£ Z. Again the proof relies heavily on the structure theory. Also see Davis and McCleary [81a].

[81] This is a survey of the work to date on the problem: If ft,,ftp a r e homogeneous chains and /((ft,) and A(ftp) satisfy the same (£-)group sentences, what is the relationship between ft, and ftp? Some of the ideas used are established in Sections 2.1 and 4.2. A. M. W. Glass and Y. Gurevich: [81] This is Chapter 13.

It extends Glass [75a].

A. M. W. Glass, Y. Gurevich, W. C. Holland and M. Jambu-Giraudet: [81] Some of the material in Section 2.1 is developed including the proof of Theorems 2B and 2C. In addition, it is proved that if ft is a doubly homogeneous chain that satisfies the Souslin, Lusin or Specker property, then so does any homogeneous chain A such that A ( A ) and /l(ft) satisfy the same (a- )group sentences. In the same setting, other properties that transfer are: Archimedean groupable, Archimedean fieldable, and various long lines. Also see Gurevich and Holland [81]. A. M. W. Glass, Y. Gurevich, W. C. Holland and S. Shelah: [81] The material from Section 4.2 is taken from this paper. In addition, the existence of rigidly homogeneous chains of cardinality less than 2 ° is shown to be independent of the axioms of set theory. Other results about rigidly homogeneous chains are established; e.g., they cannot be covered by a set of intervals {I : n e %+} with y( I ) < 9~n; there are rigidly homogeneous chains whose divisible closure is R. For earlier results about rigidly homogeneous chains, see Ohkuma [55]. Also see Roitman [81]. A. M. W. Glass and W, C. Holland: [73] The equivalence of (i), (iii) and (iv) of Theorem 11.2 is established; cf. Wolfenstein [68]. A. M. W. Glass, W. C. Holland and S. H. McCleary: [75] For the definition of a*-extension and a*-closure, see our annotation of Ball [75]. If G is completely distributive, it is shown to have an a*-closure (which is not necessarily unique). Moreover, using the structure theory of Chapters 3-5, the a*-closures of such transitive £-groups are explicitly constructed (they have the same set k of pairs of convex congruences and if (GT.,ftT.) is a primitive component, the Ji i\

K-primitive component of

H

is contained in /KfL-)).

R. N. Ball

[75] subsequently showed that every £-group has an a*-closure. Also cf. D. Khuon [70b]. [80] This paper can be thought of as the sequel to Martinez [74]. Theorems H E and 8.3.7 are established as well as a dimension theory for varieties of £-groups, laws defining product varieties and certain distributive laws for the ^-semigroup of varieties.

xxiii Moreover, it is shown that every proper variety of £-groups other than >V can be written uniquely as a product of indecomposable varieties (A variety is indecomposable if it cannot be written as the product of two non-trivial varieties). The proofs rely heavily on Wreath products. Questions VI and IX of Martinez [74] are answered. A. M. W. Glass and S. H. McCleary: [76] Four new examples of pathological ^-permutation groups are given, three of which are £-simple. Two of these examples contain elements which fix precisely one point. This leads to a counterexample to a conjecture of Ball [74]. A. M. W. Glass and K. R. Pierce: [80a] This is an announcement of the results that appear in Glass and Pierce [80b] and [80c]. It also includes some informal remarks about finding an algebraic sentence that distinguishes between finitely and infinitely generic £-groups as well as a self-contained appendix (A Lazy Algebraist's Guide to ModelTheoretic Forcing) which gives the necessary background from model theory. [80b] This is the Abelian analogue of the material in Chapter 12 (Glass and Pierce [80c] ). It is shown that algebraically closed Abelian £-groups need not be existentially closed Abelian £-groups (contrast Theorem 12B), and that given an existentially closed Abelian £-group, there is an Abelian £-group satisfying the same sentences that is not existentially closed (cf. Theorem 12.6). Moreover, there is an V3TV sentence which is equivalent to the Archimedean property for existentially complete Abelian £-groups and holds in finitely generic Abelian £-groups but not in infinitely generic ones. [80c] This includes the results of Chapter 12 (with the exception of Theorem 12E). It also includes: (i) There are 2 ° pairwise non-elementarily equivalent countable algebraically closed £-groups and (ii) There is an yay sentence distinguishing between finitely and infinitely generic £-groups. A deeper examination of algebraically closed £-groups and £-permutation groups remains outstanding; see, e.g., problems on simplicity and £-simplicity No. 5. Y. Gurevich: [67] A class of Archimedean (Abelian) £-groups that are real vector lattices is shown to be hereditarily undecidable (The class includes sums of copies of l). Actually, the proof works for the lattice theory of the class—replace z i- u + v by z J u v v in the definition of P(x,y). It follows that the lattice theory of (Abelian) £-groups is undecidable; cf., Theorem 2.2.8 and Jambu-Giraudet [81b].

xxiv Y. Gurevich and W. C. Holland: [81] This is the first proof of Theorems 2B, 2B*, 2C and 2C*. Lemmas 2.1.1-2.1.9 are established. The proof is completed by heavily using the arithmetic of I. Specifically, Lemmas 2.1.102.1.12 are replaced by: Let ft be doubly homogeneous and G be a subgroup of A(ft) all of whose ^ e elements are unbounded bumps. Then each e ^ f € Cw 0 \(G) is ^ unbounded bump if and only if aG is dense in ft for all a € ft. Moreover, if H is a non-cyclic subgroup of G and G satisfies the above equivalent conditions, then so does H. Consequently, if ft is doubly homogeneous and there are f.. ,fp € A(ft) which commute and each

e ^ g € ^/^({fvfpl)

a countable dense subset.

^s

an

unbounded bump, then ft has

In A(R),

f-

and

fp

can be taken

to be translation by 1 and /2 respectively. This idea is used in the Archimedean groupable and fieldable part of Glass, Gurevich, Holland and Jambu-Giraudet [81]. J. L. Hickman: [76] Let T be any cihain. Then Q x_T is dense and A(T) can be ^-embedded in 4(H ^ T) via: g >-+ g where (q,x)g = (q,xg) (q € $; T € T). This proves a stronger theorem than that of this paper—and in ten pages less. What is worse, Lemma 8 is palpably false (See Example 4.2.6—which is clearly non-Abelian—or Fleischer [77] ). [77] This corrects Hickman [76] by recourse to wreath products thus making the trivial even more inscrutable. G. Higman: [54-] Let ft be an infinite set and G be a group of permutations of ft satisfying: For all f,g,h € G with h f e, there is k € G such that Akh O Ak = 0, where A = supp(f) u supp(g). Then [G,G] is proved to be simple (Lemma 6.4 and its proof are just a special case of Higman!s work). As a consequence, if K and X are infinite cardinals with K <_ X <_ 2 K , it is shown there is a simple group of size X having a subgroup of index K. [61] It is proved that a finitely generated group can be embedded in a finitely presented group if and only if it has a recursively enumerable set of defining relations. The proof of this superb embedding theorem uses the amalgamation property (free products with amalgamation). The theorem yields several important corollaries including an easy negative solution to the word problem for groups. Also see Boone and Higman [74] and Thompson [80].

XXV

[77] An n-ary relation R on a set ft is said to be homogeneous if whenever r,A Q. ft with |r| = |A| < Nn> then = . For example, if G is a group of permutations of ft that is n-homogeneous, take R to be the orbit under G of n-tuples of distinct elements of ft (see our annotation of Cameron [76] ). R is said to be derived from a binary relation S if for all a1,...,an,ei,...,0n € ft, R(a]_,...,on) ++ R(B r ...,B n ) whenever

(Vi,j 6 {1,... ,n} )(S(a±,a. )++S(&±,& .)).

By an

application of Ramsey's Theorem for infinite sets and the Compactness Theorem for first order logic, it is shown that: A relation R on an infinite set ft is homogeneous if and only if it is derived from some linear ordering on ft. If (a,$,Y,O 6 D when precisely one of y and 6 lies strictly between a and 3, D is called the derived separation relation. The main thrust of the rest of the paper is to classify the group of permutations of a set ft that respect a given relation derived from a linear order on ft. In the case that ft is infinite and the group is transitive, this reduces to one of five groups: 4(ft), the full symmetric group, the group of all permutations that preserve the derived betweenness relation, the derived cyclic order, and the derived separation relation. Hence: A transitive permutation group on an infinite set ft which preserves a non-trivial homogeneous relation preserves the derived separation relation for some linear order on ft. This generalises the main theorem of Cameron [76]. W. C. Holland: [63] The main results in this paper are Holland's Theorem (Appendix I), Theorem 2H (stated in a somewhat weaker form), Theorems 2E, 2F and 2J, and the first part of Section 2.2 (including Corollary 2.2.3 with a much longer proof). A most significant beginning to the whole subject of I-permutation groups! This paper lead to the study of £-permutation groups as a tool to develop the theory of non-abelian Jl-groups. [65a] Uniqueness of representation of transitive 5,-permutations is considered. First, Theorems 4.1.1 and 4B are proved. Next Theorems 7A and 7B are established (together with some of their consequences, Corollaries 7D and 7E). Theorems 2D and 3A are also proved. Note that this paper predates the classification of transitive primitive 5,-permutation groups given in Theorem 4A. Indeed, the form of Theorem 4B proved in the paper (If a transitive primitive ^-permutation group has an element whose support is bounded above or below, then it is doubly transitive) together with Example 1.9.2 (Holland [65b]) were the clues that lead to the classification of transitive primitive &-permutation groups (McCleary [72a] ) and the consequent Corollary 7C (McCleary, unpublished). Caution: there are several typographical errors in the paper.

xxvi [65b] Theorem 6.2 is stated and proved. (Example 1.9.2) of Theorem 6C is also shown.

A special case

[65c] Example 1.9.2 is shown to be a Hausdorff space that is both a topological group and lattice in its open interval topology. However, it is not an o-group. This answers in the negative a question of G. Birkhoff raised in the 1948 edition of [67]. [72] This is a readable exposition of some of the results about ordered permutation groups. It is mainly confined to facts about transitive £-permutation groups. [75a] Theorems 9A and 9C are established, as is Theorem 9.5. The existence of a homogeneous chain Q with A(Q) having an outer £-automorphism had been outstanding for ten years previously. [75b] This review gives an example of a doubly homogeneous chain Q having a coterminal set of intervals each of which is ordermorphic to itself with the reverse ordering, yet Q and Q* are not ordermorphic (This disproves the only new theorem of the paper being reviewed (Longyear [74])). This result was independently obtained by Rabinovich [75]. [76a] It is proved that no dense homogeneous subset of 1 is ordermorphic to its complement. Hence if Q, is a dense homogeneous subset of 1, A(Q.) has no outer £-automorphisms, cf. Theorem 9A and Holland [75a]. The proof proceeds by examining the two cases (i) 0, is doubly homogeneous and (ii) Q is rigidly homogeneous. The first case is dismissed as in Treybig [63]; the second is disposed of by establishing that is a coset of Q, in 1. [76b] Theorem 11D is stated and proved. and McCleary [79] and McNulty [80].

Also see Holland

[77] It is shown that if G and H are £-groups, then no reduced group word of length less than 14 reduces to the identity in the £-group free product of G and H. The proof is an ingenious combinatorial argument using wreath products. It also shows that there are reduced group words of length 14 that are the identity in this £-group, and indeed finds them all. This paper is the first in "combinatorial £-group theory." See Franchello [77] for length 16; also Holland and Scrimger [72]. [79] Theorem 11F (every variety of £-groups is a torsion class) is proved. [80] This is a series of remarks about doubly homogeneous chains Q, such that A(ft) and A(R) satisfy the same £-group sentences. The work is completed in Gurevich and Holland [81]—

xxvii see Theorem 2B (also Theorems 2B* and 2B1 ). Gurevich, Holland and Jambu-Giraudet [81].

Also see Glass,

I.C, Holland and S. H. McCleary: [69] This paper is basically Chapter 5 and builds on Holland [69], the construction of the generalised wreath product for groups. [79] By analysing Holland [76b] (the proof of Theorem 11D) it is first noted that to test whether an arbitrary £-group word reduces to e in all £-groups, it is enough to test it in A (I). An algorithm is then established to do this. Hence the free £-group on any number of generators has soluble word problem (cf. Chapter 13). This connection was recently underlined by the completion of the proof of Theorem 6.7 in the finite case by McCleary [8lc]. W. C. Holland and E. B. Scrimger: [72] The p.o. group free product of a family of p.o. groups is constructed as is the £-group free product of a family of £-groups. This £-group free product is shown to be the group free product only if each £-group in the family is an o-group. Lemma 1.11.5 is proved to show that the £-group free product of two £-groups need not contain the group free product (of the £-groups). For further results, see Franchello [77] and Holland [77]. Also see Bergman [81] for ordering the free product of o-groups. M. Jambu-Giraudet: [80] This is an announcement of the results in the author's , These 3 Cycle Cycle (1979). The proofs appear in her papers [81a], [81b] and [81c]. me n oeme

[81a]

Theorems 2A and 2A

(and consequently Theorems 2D and

1

2D ") are established. The proofs given in Section 2.1 are taken from this paper. - Theorem 2D had previously been established in the £-group case by Holland [65a], and in the group case by McCleary [78b] and Rabinovich [75], though via identifiable subgroups rather than directly using the elements of A(oJ. [81b] Using Theorems 2A and 2A , Theorem 2.2.8 and its lattice analogue are established. The proof is as follows: The set of bumps of f is a set of bumps of g if each bump of f is a bump of g. This latter is,expressible in the £-group language by Lemma 1.9.3. Hence g has a finite set of bumps if each non-empty subset of the set of bumps has a leftmost and rightmost member. We now interpret each n £ o> with the set of positive elements of A(ft) having n bounded non-adjacent bumps. Equality between elements of o> is easily interpretable. m + n = p if whenever g,h > e have m and n bumps, respectively, there is that g A x~ hx = e

and

g«x~ hx

has

p bumps.

xxviii m x n = p is obtained by putting conjugates of h under each bump of g and requiring the product of these conjugates have bumps. [81c] It was announced in Jambu-Giraudet [8lb] that order arithmetic can be interpreted in the group (lattice) every doubly transitive A(ft). If, in addition, ft has a i .. pairwise disjoint non-empty open intervals, then nth n—JL

p

second of set of order

arithmetic can be interpreted in the group (lattice) of A (ft). These are now proved. The idea is to interpret ",n," by a permutation with w u n u to* bumps, and " { " and " } " by permutations with u> + a> and (o> + o>)* bumps respectively. N. G. Khisamiev: [66] Y. Gurevich and A. I. Kokorin showed that any two Abelian o-groups satisfy the same universal sentences [Algebra i Logika, 2(1963), 37-39]. Let G be an Abelian il-group and n(G) be the supremum of the cardinals of pairwise disjoint subsets of G. The main result proved is that Abelian ^-groups G and H satisfy the same universal sentences if and only if m(G) = m(H), where m(G) = min{n(G),^{ ~ } . Hence every finitely generated Abelian £-group has soluble word problem. It is also shown that any free Abelian £-group is contained in a cardinal product of copies of % (also see Weinberg [65] ). D. Khuon: [70a] An il-group H is an a-extension of an Jl-group G if the map C —• C H G of the prime subgroups of H onto those of G is one-to-one. It is shown that if G is doubly transitive, so is any a-extension of G and A(ft) is the unique a-closure of A(ft) if ft is doubly homogeneous. The proofs rely on Wolfenstein [70]. [70b] This paper is devoted to proving: Theorem: Every jl-group has an a-closure (a maximal a-extension). If (G,ft) is a transitive £-permutation group and a € ft, then (G |ft ,ft ) , is uniquely transitive, where ft = { $ € f t : G o = G } . Ot

p

By a result of

CX

Conrad, this puts a cardinality bound on ft in terms of the prime subgroups of G. The proof of Holland's Theorem (Appendix I) now yields a cardinality bound on a-extensions of an l-group in terms of the cardinality of its prime subgroups as required. Note that the main theorem was first proved by McCleary (unpublished), and the proof was generalised by Ball [75] to cover a*-extensions. A. I. Kokorin and V. M. Kopytov: [74] This book is similar in scope to Mura and Rhemtulla [77], but is not so elegant and has several errors in statements and theorems. The books have considerable overlap but include a lot of material not found in the other.

xxix V, M, Kopytov: [79] The class of groups which can be made into ^-groups is not an axiomatisable class (Vinogradov [71] ). However, if (/ is a variety offc-groups,the class X(V) of all groups that can be embedded (as a group) in some member of V is proved to be a quasi-variety of groups. Moreover, if F^ is the {/-free I -group on

{x : n € w},

the subgroup

FQ generated by

the X(V)

free group on

{x : n 6 w}.

the X(U)

free group on

{xn: n € w}. Let

set of all right total orders on convex subgroup of

F*

F~.

{x : n € co} is

Conversely, let Let

Hn

F*

be

be the minimal

for which the Jl-subgroup A x

generated by the natural embedding of

FQ

{F*: A 6 A} be the

of A(R(H*))

belongs to V.

If

F*

is the cardinal product generated by

77" A, and F is the ^-subgroup A£A A (where (H*f)(x (A)) = H*fx ), then F is the

xn

V-free Jl-group on

{x : n € w}.

£-group

becomes a transitive I-permutation group

F

in V

As a consequence, if a free

for some chain ft, then there is a right total order for which group.

FQ

(F,ft)

on F Q

(F,R(H*)) is a (faithful) transitive I-permutation

By Theorem 6.7, if

generators, then

FQ

is the free group on any set of

(F,FQ) is a transitive ^-permutation group for

some A € A, where F is the free I-group on the same set of generators as F-.. Similar results are obtained for the £-group varieties >V, A, & (xve)Ay~(x"" (x v e )

2

and ~Yf, where $? is the variety defined by:

ve)y = e

A (y v e )

2

(or equivalently

= [(x A y) v/e]2)

and *W is defined by:

x"* |y|x v |y|2 = |y|2. The results in this paper extend Conrad [70] and use only the material covered in Appendix I. Other theorems are also proved. V. M. Kopytov and N. Ja. Medvedev: [77]

5

varieties

{3E^: i € 1} £ A .

generators, and

M.

Olshansky proved that there are N.

Let

2 ° distinct

F be the free group on AfQ

the verbal subgroup of

its second derived subgroup.

F

corresponding to 3C^,

It is well-known that

F/M.

admits linear orderings so the class of £-groups in distinct are distinct (i 6 I). Hence there are 2 ° distinct varieties of Jl-groups. For more recent proofs, see Feil [81] and Reilly [81]. An example is given of a variety of a-groups that is not generated by any finitely generated £-group. This shows that Theorem 5.1 of

XXX

Martinez [74] is false. The lattice of varieties of £-groups is shown not to be Brouwerian. The proof is the same as Smith [80a]. The last part of the paper contains an oversight. It is not at all clear that the defined W is closed under n £-homomorphic images. Hence Theorem 4 is not yet established. J. T. Lloyd: [64] First, /4(ft) is shown to be completely distributive and have closed stabilisers (Theorems 8B and 8.1.2)—see Lloyd [67] and Read [71]. Moreover, if ft is doubly homogeneous, it is proved that every proper normal subgroup of A(Q) is an £-subgroup that contains B(ft) (cf. Theorems 2.3.1 and 2G) and that stabiliser subgroups are maximal (cf. Theorem 4.1.5). This last fact, together with Holland [63] and [65a], lead to the central use of stabilisers in studying £-automorphisms (and £-isomorphisms) of A(ft). It was employed in the original proofs of Theorems 2.3.1 and 2G, and later extended by McCleary [69], [78a] and [78b]—see also Chapters 7, 8 and 9. Indeed, Lloyd uses it here to prove that if transitive A(ft) has the interval support property, then each £-automorphism of A (ft) is induced by conjugation by an element of A(Q)—cf. Theorem 9B. He thus obtains Corollary 2.4.4. Next it is shown that if for transitive A(ft), /4(ft) = {e} for all a € ft, then /?(ft) is an o-group (cf. Lemma 4.2.4). Corollary 2K is proved in special cases without using G.C.H.—Weinberg [67] manages this in total generality. Lloyd also gives a proof of Lemma 10.3 and hence Corollary 10.4. However, the statement at the beginning of the proof of his Lemma 1.29 is false, so the proof given fails. McCleary [73a] corrects the error. [65] It is shown that an £-group with a basis admits a minimal irreducible representation as a product of transitive ^-permutation groups, and that this minimal irreducible representation is essentially unique. [67] It is proved that /4(ft) is completely distributive for any chain ft. Using this, Theorem 8B is deduced (for the special case a € ft). For an amplification of the method used, see the annotation of Read [71]. Also Example 1.10.3 is shown to be il-simple (cf. Theorem 6.5) but not completely distributive. This last fact is generalised in Corollary 8.3.5 due to McCleary [73b]. J. Q. Longyear: [74] The only new theorem in this paper is unfortunately false—see Holland [75b] or Rabinovich [75]. A. Macintyre: [72] This is the group theory analogue of our Chapter 12. The results are obtained more cleanly by using free products with amalgamation (a luxury not afforded those who study ^-groups!) and

xxxi are stronger. For example, two algebraically closed groups are elementarily equivalent if and only if the sets of two generator groups that can be embedded in each are equal; in an algebraically closed group, f e if and only if f £C(C({g ,...,gn)))-cf. Lemma 12.4. There is an 3V3V

sentence,

equivalent for algebraically closed groups to a universal two generator group, which holds in some algebraically closed groups but fails in others. Indeed, there are 2 ° pairwise nonel ementarily equivalent countable algebraically closed groups (In an earlier paper, the same author showed that a finitely generated group can be embedded in all algebraically closed groups J {e} if and only if it has soluble word problem). A. I. Mal'cev: [71] Only Chapter 15 of this book is referred to. It gives an elegant interpretation of the ring of integers in the free nilpotent class 2 group on two generators (rrmetabelianrt used by the translator means "nilpotent class 2"). Since torsion-free nilpotent groups are orderable (see Fuchs [63]), it follows that the theory of orderable groups is undecidable, cf. Buszkowski [79], Glass and Pierce [80c], Gurevich [67] and Jambu-Giraudet [Bib]. J. Martinez: [74] This paper is the first on varieties of £-groups. It is more important for the questions it asks, the examples it gives, and the research of others it spawned than for the results it proves. Caution: there are many errors. 3.5 Proposition (b) needs 4.1 Lemma for its proof, 4.1 Lemma (e) should be: 1/ is idempotent if and only if [R(R(G,!/),(/) = 0 <=> G £ U ] and OO

00

Section 5 is wrong

(U - \/ \J, N, not \J - \J \J, s as asserted). n=l {n) n=l {n) For further results on varieties of £-groups, see Bernau [77], Feil [81], Glass, Holland and McCleary [80], Kopytov and Medvedev [77] (where Martinez's Theorem 5.1 is proved false), Medvedev [77], Reilly [81], Reilly and Wroblewski [81], Scrimger [75] and the works of Smith. S. H. McCleary: [69] Theorems 8.3.8 and 9B are proved—indeed, Theorem 9B is proved for certain depressible ^-subgroups of 4 (ft). The results generalise some of Lloyd [64]. [72a] Theorem 4.3.7 (and hence Theorem 4A) is established. So is Theorem 4-3.10. This paper is a polished version of McCleary1s Ph.D. thesis (1967). Its central importance is clear to anyone who has read this book. For prior results in this direction, see Holland [65aJ and [65b]. [72b] The transitive part of Theorem 8.2.1 is proved. Note that this work was done in 1969—the journal had a big backlog.

xxxii [72c] Theorem 8.2.1 is established and extended to Theorem 8C. Theorem 8F is also proved. Indeed, much of Sections 8.2 and 8.3 come from this paper (including Theorems 8.3.1-8.3.4, 8.3.6, 8.3.9 and 8.3.IO). The techniques used are considerable extensions of the ideas of McCleary [72b]. [73a] An error in Lloyd [64] is corrected, Lemma 10.3 being proved correctly. Hence, assuming G.C.H., every Jl-group can be ^-embedded in an Jl-group which has only inner I-automorphisms. (Weinberg [80] removed the dependence on G.C.H.) Further, a variant form of Theorem 7A is established: If Q is doubly homogeneous and i|; is an ^-embedding of /4(ft) into A(T) such that A(Q)ty is a complete ^-subgroup of /4(T), then there is a £ S and an ordermorphism <j>: a A (si) s T such that gty = <\>~ g(j> for all g £ G and (^,<j>) is a &-permutation group isomorphism between (A(tt),oA(to)) and (A(&)$,?). Truncated ot-sets are also considered but Proposition 18 is false if 3 or y equals a + 1 (see Holland [75b] or Rabinovich [75]). [73b] Theorem 8D is proved and Example 6.6 is given and shown to be ^-simple. Glass [74] answers a question left open in this paper. [73c] This is a sequel to McCleary [72a] and is rather technical. The only part we have used is the last sentence of our Theorem 4.3.1. Full periodic primitive £-permutation groups of each Config(n) (n € 7L u {«>}) are constructed using a-sets. Note that there are two minor typographical errors: p. 433 line -13: (u>ng - «g)/(o)n - a)) -> 0 or 00; p. 434 line 4 of (4): f A wG. [74] Some of the ideas of McCleary [76] are mentioned. Also included is the approach we have adopted in Chapter 11 (especially Theorem 11A)—rather than that of Read [75] which really predates it by three years. [76] This paper examines the structure of intransitive ordered permutation groups and brings order to the chaos that had previously existed. The classification of primitive i-permutation groups is obtained. This leads to straightforward proofs of prior results of Lloyd [64] and [67] and Read [71] and [75]. We have adopted its approach in Sections 3.2, 8.1 and Chapter 11 but have not included all the results to be found in this paper whose central importance cannot be overemphasized. For earlier work on the subject, see Scrimger [70]. [78a] Examples are given of infinite simple groups having insoluble outer automorphism groups (Schreier conjectured that this cannot occur if the simple group is finite—the conjecture is still open). The example given in Theorem 2.4.2 comes from this paper. The proof we give relies on the work of Section 2.1; McCleary1s, on stabilisers.

xxxiii [78b] Aut(G) is examined when (G,ft) is triply transitive. If ??£(&) denotes the group of all monotonic permutations of the chain 0, and G contains strictly positive elements of bounded support, then Aut(G) = 1 ^ x(G) is shown (where N denotes the normaliser); and no two of GnB(fi), GOL((]) and G OR(fi) are £-isomorphic (cf. Corollary 2.1.4-). Using a-sets, an example is given of a doubly homogeneous chain Q such that |Outer(/l(n))| = 4 but 10uter(7>( (n)) | = 2 (cf. Holland [75a] and Theorem 9A). The main tool is Corollary 1.10.8 which is established and used to obtain Section 7.2 and the above. This lemma is also centrally used by Jambu-Giraudet [81a] who obtained it independently. Also compare with Rabinovich [75]. [80] In Glass, Holland and McCleary [80] it was shown that the variety >V of normal-valued ^-groups is generated by { W A : n >_ 1}, where Wr 1 ^ = Z and Wr m + ] ^ = (Wrm2Z)Wr Z. Hence to decide whether or not an £-group identity Vx(w(x) = e) holds in every normal-valued £-group (w(^x) an £-group word), it is enough to check it in {Wr 22: n >_ 1}. An algorithm is explicitly given to do this; indeed, given such a word w, there is a bound m such that Vx(w(x) = e) holds in every normal-valued 5,-group W

*v

if it holds in

«%/

{Wr £Z : 1 £ n <_ m } (A gross bound for

obtained by taking 2 plus the sum of the lengths of the k

x

x

mw

is T

w.. s —

t n e n i-t n a s

if w... has form x~ x 2 o v length k + 1 + 3 + k). Hence the free normal-valued £-group on any number of generators has soluble word problem. Actually, an algorithm is given to decide whether or not a disjunctive sentence holds in a free normal valued £-group. [81a] This is essentially Bernau's proof [75] but with some considerable shortcuts. [81b] This is a considerable extension of the ideas in McCleary [80]. For example, it is shown that if w.,(x) and Wp(x) are group words, then Vx(w, (x) ^ w p(x) >^ e) holds in all normal valued £-groups if and only if there is a group word that

w^(x) and

w(x) such

w~ (x) are positive powers of w(.x).

[8lc] The main result of this paper is that the free £-group on a finite number (> 1) of generators has a doubly transitive representation on B. It was previously known—Glass [74]—that the free &-group on an infinite number of generators has a doubly transitive representation. Hence Theorem 6.7 holds and a free £-group on more than one generator is not directly decomposable. Also see Kopytov [79].

xxxiv G. F. McNulty: [80] This is a refinement of Holland [76b] (i.e., Theorem 11D): viz.: If an £-group G £ sf and 0 is a positive universal sentence that holds in G, then 0 is true in every £-group. Equivalently, the only class of £-groups that is closed under £-subgroups and £-homomorphic images and contains a doubly transitive £-group is the class of all £-groups. The last paragraph of the proof of Theorem 11D is shown to yield: any finite set of inequations in any A(A) can be realised in any doubly transitive G. R. N. McKenzie and R. J. Thompson: [73] This is an extremely ingenious construction of a finitely presented group with insoluble word problem. The proof proceeds by coding elements of ^co into the symmetric group H on ^o) in such a way that if A f denotes the permutation of ^w corresponding to f £ Wu), then in some finitely presented group £„, conjugation by the formal symbol A,, naturally corresponds to the function f if f is recursive, and the relations in £ f all hold in ft under a fixed interpretation. Heavy use is made of those permutations determined by their action on noo (n £ w) and of a definable element R of order 3 which cyclically permutes the top three levels of waj. For £-groups, the absence of such an element R is finessed by conjugating a set of elements simultaneously to another set by one element of A ( l ) (see Chapter 13). N. Ja. Medvedev: [77] Let E± m> 0

or

(m = 0 &

be X wr

S >oro)

z

({ ,m) > e if be Z wr IZ ordered by:

ordered by:

and H 2

i

({grr '

e

if

m > 0

(m = o

&

gn

> o)

where

9
and n = min{n £ X : g n 0}. Let 7T. be 2 l = max{n 6 S : g n / 0} the £-group variety generated by the o-group H. (i = 1,2). Let I\L be the free nilpotent class 2 group on x,y ordered by:

n

x

y [x>y] > e if m > 0 or ( m = 0 & n > 0 ) or (m = n = 08c k > 0 ) . Let 7^n. be the £-group variety generated by N Q . It is shown that if 1/ is a representable £-group variety that contains a soluble non-Abelian £-group, then U contains 7( , Tlf or yT^\ i.e., these three £-group varieties cover A and every representable £-group variety containing a soluble nonAbelian £-group, contains at least one of them. Also see Scrimger [75]. Are there any other representable covers of A?

XXXV

R. B. Mura and A. H. Rhemtulla: [77] An elegant and excellent account of various grouptheoretic conditions which imply various forms of orderability. B. H. Neumann: [60] The analogue of Theorem 10A for o-groups is proved (We have made only minimal changes in the proof to obtain Theorem 10A). Also a finitely generated o-group is constructed o-isomorphic to a proper o-homomorphic image of itself. B. H. Neumann and H. Neumann: [59] Several results first obtained for groups by Higman, Neumann and Neumann (J. London Math. Soc., 24(1949), 247-254) using free products with amalgamation are obtained using Wreath products. This method can be adapted to o-groups and £-groups (see B. H. Neumann [60]) whereas the original cannot. T. Ohkuma: [55] Theorem 4.2.2 is established. The proof is longer than the one we have given and does not yield the stronger form, Theorem 4.2.7, due to Glass, Gurevich, Holland and Shelah [81]. Dushnik and Miller [40] can be thought of as a precursor of Ohkuma [55]—though Ohkuma's work is considerably more sophisticated and difficult. K. R. Pierce: [72] Theorems 10B and IOC are proved as is 10.12. The proofs are essentially those we have given. For an application, see Chapter 12, or Glass and Pierce [80c]. H. L. Putt: [77] Let (G,ft) be a permutation group. A partial order <_ on ft is called G-invariant if a £ $ implies ag £ $g for all g £ G. Fix a 6 ft. A subsemigroup P of G is said to be an a-cone if P n G = 0 and PG = P = G P—P is allowed to be a a a empty. The set of G-invariant partial orderings on ft is shown to be in one-to-one correspondence with the set of a-cones. Necessary and sufficient conditions are given for each of the following: (1) Every G-invariant partial order on ft extends to a G-invariant total order on ft. (2) Condition (l) plus there are exactly two G-invariant total orders on ft. (3) There are just three G-invariant partial orders on ft, the trivial one and two G-invariant total orders. Moreover, (4(l),l) is shown to satisfy (3), where /4(R) is viewed as acting on the set I. Applications to right totally ordered groups and special G-invariant partial orderings are also considered.

xxxvi E. B. Rabinovich: [75] following:

Theorem 2D is established. The key lemma is the Let g £A(ft), where ft is doubly homogeneous.

If

n

M = {f £/((ft): g = f® for some m,n € 2Z}, then C(M ) = 77*{/4(A.): i 6 I}, where A. are pairwise disjoint intervals of ft maximal with respect to being disjoint from supp(g). It is also shown that if ft. are doubly homogeneous and G. are subgroups of /4(ft.) containing J isomorphism between

G,

and

Gp

B(ft.) (j = 1,2), then any group J extends uniquely to a group

isomorphism between 4 (ft ) and A (ftp).

Hence

L(ft-,) is a

characteristic subgroup of A (ft-,) unless ft. is antiordermorphic to an orbit of /((ft.,) in ft.; in this case, L(ft-) = R(ft-,) (as groups)—cf. McCleary [78b]. Proposition 18 of McCleary [73a] is shown to be false (also see Holland [75b] ). Further results are obtained for good chains—a doubly homogeneous chain ft is good if whenever A. and Ap are bounded intervals of ft having the same initial characters and the same final characters, then A., and Ap are ordermorphic. The orbits under /((ft) of holes in such ft comprise all holes of the given character. Let ft be the set of holes in ft of the same character as the points of ft. It is proved that if A is homogeneous, ft is good and ft ^ 0, then /I(ft) - /4(A) (as groups) if and only if A is ordermorphic or antiordermorphic to ft or ft. E. B. Rabinovich and V. E. Feinberg: [74] Theorem 2.3.2 is claimed without the restriction that ft has countable coterminality. As we saw in Theorem 2.3.7 this is false. The crucial error occurs in their Proposition 3 (which is false). It is claimed that there is s £A(X) such that s(cQ) = ao and s(do ) = b D (0 < 3 < a)—see p. 193 line -3 of the P P p P — English translation. This fails if a > a>, a = sup{b : n € OJ} and

c

OJ

> sup{d : n 6 OJ}. n

oj

n

J. A. Read: [71] The main theorem proved is Theorem 8.1.2. The proof is quite ingenious and is a simplification of that of Lloyd [64] or [67]. The key lemma (also in Lloyd, op. ait.) is: If e < g € A(ft) and a € supp(g), then either there is e < h € X(ft) with supp(h) £. (a,ag) or {h € /((ft): e <_ h <_ g & supp(h) c/\(a,g)) contains a basic element. The two cases are then examined separately using a criterion of Weinberg's from £-groups (the same is true of Lloyd's proof).

xxxvii [75] In this paper, non-overlapping I-permutation groups are examined. The study is motivated by the key lemma of Read [71] (see the previous annotation). Theorems 11A (parts (l), (2) and (4-)) and 11.2 (parts (i), (v) and (vi)) are established as is Corollary 11B. The proofs given are more ingenious than the ones given here (since they predate the intransitive structure theory of Sections 3.2 and 4.4). The work was done in 1970 and is independent of Wolfenstein [68]. Since the emphasis is on an ordered permutation group concept, the proofs are very different to Wolfenstein's. Warning: typographical errors riddle the paper; in special: Theorem 2.4 (l) should be hg <_ g 2 h 2 ; page 718 line -3 should end: let S be; page 719 line 9: change primarily to precisely. N. R. Reilly: [69] The major theorem is: Any o-group can be embedded in a simple o-group. The proof mirrors Philip Hall's for groups: Note first that if H is an o-group, then H wr 7L is an o-group under ({h },m) > e if m > 0 or (m = 0 & h > e in H where n 0 n Q is the least element of {n € Z: h f e})—cf. Medvedev [77]. Let

G

be an o-group,

H n = G and H , = H wr G so ordered. u n+l n Then the direct limit of {H : n £ w} is the desired group. [72] Using the analogue of B. H. Neumann's group theory construction "permutational products," it is shown that certain amalgamations are possible in the class of o-groups (I-groups)—cf. Theorems 10.12 and IOC. [81] An I-group G satisfies x" = x"

is representable precisely when it

(VxQ)(Vu)(x0 A U" X~U = e ) , where

v e.

If

F

x Q = x Q v e and

denotes the free group on

V i s a variety of £-groups, let

{x ,Xp,...} and

F(V) = {u £ F: x* A u" x"u = e

is a law in i/}, So if & is the variety of representable il-groups, F(<# ) = F. Also F( «£ ) = {e}. In general, F(V) is a fully invariant subgroup of F. For N a fully invariant subgroup of F, let R( N) be the variety of Jl-groups defined by: x~ A U"~ X"U = e

for all

u € N.

By making

F

an o-group and

considering (Z,Z) Wr (F,F), it is proved that R is one-to-one, FR(N) = N and V £ RF(l/). Moreover, many nice properties hold; e.g.,

D F(V. ) = F( V \J. ). In particular, since there are i€J ! i€I 1 M towers of fully invariant subgroups of cardinality 2 °, there is a tower cf varieties of ^-groups between ^ and «^ of size By a deep result of S. I. Adyan, there is a collection of 2 ° pairwise incomparable fully invariant subgroups of

F.

2 °. Hence

xxxviii there are

2

u

pairwise incomparable varieties of £-groups.

No

other proof of the existence of 2 ° varieties of £-groups captures this last fact. (Also see Feil [81] and Kopytov and Medvedev [77].) For a simpler proof for quasi-varieties, see Arora [81]. N. R. Reilly and R. Wroblewski: [81] The results in this paper are of the same ilk as those in Smith [81]. However, whereas Smith relies on the techniques in Scrimger [75], Reilly and Wroblewski use the more pictorial (^-permutation group) approach of Glass, Holland and McCleary [80], especially "mimicing." Indeed, this is the first application of these techniques to specific varieities of £-groups. A. Robinson and E. Zakon: [60] An Abelian o-group G is said to be regularly dense if for each non-empty open interval I of G, I n pG f 0 for each prime p. Any densely ordered Archimedean (Abelian) o-group is regularly dense. It is shown that two regularly dense Abelian o-groups are elementarily equivalent if and only if they are elementarily equivalent as groups; i.e., precisely when they are Abelian and have the same set of Szmielew invariants. Any two discretely ordered Abelian o-groups with all Szmielew invariants 1 are shown to be elementarily equivalent. Y. Gurevich (Elementary Properties of Ordered Abelian Groups, Translations Amer. Math. S o c , 46(1965), 165-192) has given a set of invariants which completely determine the elementary equivalence class of an Abelian o-group. He also shows that the theory of Abelian o-groups is decidable. J. Roitman: [81] The existence of dense rigidly homogeneous chains of arbitrary outer measure, of infinite inner measure, and of measure 0 are established. The construction is very similar to the one we gave in the proof of Theorem 4.2.7. The main difference is that the construction is achieved by constructing dense rigidly homogeneous chains inside divisible subgroups H of 1 with H O $ = {0} changed to:

and

|H| = 2 °.

Our condition (5) on page 102 is

1

(ft, f, U ft f~ ) n [(D \ft, ) u I\(H © Z ) ] ? 0. A A

A A

y

The proof

A

is then finessed by dealing with the case that for some a e (H ©Z)\D*,

af (or af" 1) £ ft* prior to (ii)—if

a € ( H @ I ) \ D * and af 6 fi^, l e t h € H be independent of D* © Qa, and l e t D = D* © Qa © Qh and ft = ft* © $h . y

yy

y

y

y

y

xxxix J. G. Rosenstein: [81] This book makes an extensive study of chains, collecting much of the work scattered in the literature. It will provide an invaluable source and addition to those interested in pursuing ordered permutation groups. Its overlap with our book is small. T. J. Scott: [74] For ft a chain, let T>{(ft) be the group of monotone permutations of ft. If H is a subgroup of ^((ft), let G(H) - H HA(ft) and H' = H\G(H). G(H) has index 1 or 2 in H. The first part of the paper is devoted to obtaining results about H when G(H) enjoys certain properties. The second classifies those (H,ft) for which Hf / 0 and (G(H),ft) is a transitive primitive ^-permutation group. Note the author uses "regular" for what we call "uniquely transitive." E. B. Scrimger: [70] This thesis is the precursor of Sections 3-2 and 4.4 (due to McCleary [76]). First Theorem 10.5 is established. Next, inverse limits of ordered permutation groups are considered. Even though inverse limits of p.o. groups and chains exist, the inverse limit of ordered permutation groups may not. A necessary and sufficient condition is given. The intransitive wreath product is studied next. Let (G,ft) be an ordered permutation group with congruence itj. Choose {a. : 1 £ 1} c= 0.so that {T.: i £ 1} is where the £set of orbits of T = n/e under H = G/L(5) and A x T.). T = (a.S)G. Let (Gi,ft = (G i i> (a.5)' # Form (W,A)== {(G.,ft W = ({W T >,W ) £ W if w £ H, ; i)}Wr (H,T) by: w € G i ( T £ T l ) and ( a,x )w = (aw ,T W ) (T £ T. ; a £ft.) . (G,ft) can be o-(I- )embedded in (W,A) (cf. orbit Wreath product). Indeed, if (G,ft) satisfies the hypotheses of Theorem 10.5 it is £-isomorphic to (W,A) and is an inverse limit of finite Wreath products of "essentially transitive" such £-permutation groups (except that ft is ordermorphic to a dense subset of the inverse limit chain). The last chapter gives necessary and sufficient conditions (on the £-group free product of G and 22) for an £-group G to be £-isomorphic to some /4(ft). Unfortunately, the free product is not sufficiently manageable to make this useful (see Holland and Scrimger [72] ).

= u (^

[75] Weinberg [65] has shown that the class A of all Abelian £-groups is the minimal proper variety of ^-groups. Let n € %+ and G^ = {( {gj,g) 6 7L Wr ZS: g = g if m E k (mod n)}. n in inm is. v Each is a two generator £-group. It is proved that if J

denotes

the variety of £-groups generated by G , then J f J^ if

xl n / m J

and

n A

{J : p prime}

~ A

if

n

and

m

is a set of covers of A.

Moreover,

are coprime.

F. Sik: [58] Let (G,ft) be an £-permutation group. Among other things, it is shown that each element of G has one bump if and only if: G is an o-group and if f is a bump of g € G, then r G G for some n e 2£. This paper foreshadows the importance of "bumps" in the more recent work of other authors (e.g., see Section 2.1). J. E. Smith: [80a] The main results are (1) M cannot be written as the join of a finite number of varieties of £-groups strictly contained in M, and (2) If W : n € X} is any infinite family of Scrimger varieties, then the join is the variety of £-groups generated by 22 Wr Z. As a consequence of (2), the lattice of £-group varieties is not Brouwerian. The proof of (2) can be accomplished more easily by the techniques of Glass, Holland and McCleary [80]; they also show that % W r Z (or 2 w r l ) generates the variety of 5,-groups A 2

and that

\J + A n = /f (Theorem H E ) — neZS

contrast with (l). [80b] This is a survey of some results on varieties of Jl-groups with special emphasis on Scrimger [75] and Smith [80a] and [81]. [81] n

Let X

n

[x ,y ] = e. that J

The Scrimger variety J

i- £

tfn: n 6 X}

be the variety of £-groups defined by the law:

if

n > 1

^£

(n € Z + ) .

It is shown

and is composite, and that if

is an infinite nested family of these varieties,

V Jf =Jf. Indeed, £-groups are constructed in X which do not n€X k belong to 4 *A, where k is the number of (not necessarily distinct) prime factors of n. These generate varieties of £-groups. Joins of these varieties are explored. Also see Reilly and Wroblewski [81]. Note that McCleary (unpublished) has shown that J ?£ if n is prime. L. B. Treybig: [63] let ft be a dense Dedekind complete chain. Then ft is shown to be homogeneous if and only if each pair of nondegenerate closed bounded intervals of ft are homeomorphic. The proof is elementary and relies on: if ag < a and $g > 3 for some a,6 € ft, then yg = y for some y € ft between a and 8.

xli As a consequence, a dense Dedekind complete chain is doubly homogeneous if it is homogeneous. A. A. Vinogradov: [71] Two groups are exhibited which satisfy the same sentences of group theory, only one of which is lattice-orderable. Hence the class of lattice-orderable groups is not definable by a set of sentences of the language of group theory. A. A. Vinogradov and A. A. Vinogradov: [69] An example is given of a group that is not latticeorderable, but is the union of a tower of finitely generated lattice-orderable groups; i.e., lattice-orderability is not a local property of groups. Holland (unpublished) has recently shown that lattice-orderability is not a residual property of groups; i.e., there is a group G having a family {N.: i £ 1} of normal subgroups such that lattice-orderable for each lattice-orderable.

D{N.: i € 1 } = {e} and

i € I, but

G

G/N.

is

itself is not

E. C. Weinberg: [67] In Holland [63], Corollary 2K was proved assuming G.C.H. (see the proof given in Chapter 10). Lloyd [64] proved Corollary 2K without assuming G.C.H. under certain restrictions. Weinberg's proof avoids the use of G.C.H. by using group rings— see the proof given in Section 2.5. [80] Whereas the original proofs of Corollary 2K (Holland [63]), Theorem 10B (Pierce [72]) and Corollary 10.4 (McCleary [73a]) depended heavily on G.C.H. (see Chapter 10), Weinberg shows that this dependence can be removed by using Harzheim [64-]. Harzheim produced a theory of "minimal" n -sets based on order-type rather than cardinality.

Let /V

be regular. A chain is said to

be order-small if every well-ordered or inversely well-ordered subset has cardinality less than bt . The union of a tower, indexed by small.

co , of order-small chains is said to be almost-order-

Almost-order-small n -sets exist (independent of G.C.H)—

they are special models of the theory of dense linear ordering without endpoints—and play the same role that a-sets played in Chapter 10, so removing the dependence on G.C.H. For the details in the case of Theorem 10B, see Weinberg [8la]. [Bla] The goal of this paper is to prove Theorem 10B without G.C.H. as promised in Weinberg [&0]. Almost-order-small n -sets of degree K are shown to exist (if K < N ) and to be unique (cf. Lemma 10.10). The existence is established in two ways. The first by an inductive construction; the second by a

xlii lexicographic product construction. The analogue of Theorem 10.8 is next obtained from this by essentially the same technique as employed in Chapter 10 and, as a consequence, Theorem 10.12. [8lb] Let G be an £-group. It is shown that (i) G can be ^-embedded in 4(H) if and only if G has a countable family of prime subgroups {P : n € u)} such that f} P = {e} and each n €u) chain R( P ) is countable; (ii) G can be ^-embedded in A(R) if and only if G has a countable family of prime subgroups {P : n G a)} such that r\ P = {e} and each chain R(P ) has n '' n n nG w a countable subset which is dense in R(P ). The proofs are by the techniques of Appendix I. Note that the o-group 1 cannot be o-embedded in 4(0}) by (i); however, it can be embedded in A($) as a group (or as a lattice) as is easily seen. S. Wolfenstein: [68] By classical £-group methods (as opposed to ordered permutation groups), Theorems 11A (parts (l) and (4)) and 11.2 (parts (i), (ii) and (vi)) are established. In particular, Jf is shown to be a variety. Also see Read [75] . [70] Let (G,fi) be a transitive &-permutation group and H any a-extension of G—see our annotation of Khuon [70a] for the definition. It is shown that there is a chain T 2 i ] such that (H,T) is a transitive ^-permutation group. Indeed, if Q is dense and Dedekind complete, T can be chosen to be Q. Hence if Q fits these hypotheses, G has an a-closure. The same conclusion follows if (G,ft) is doubly transitive—T can be chosen to be a subset of ft. Finally, /4(l) is shown to be the unique a-closure of A((fi) and ^ ( l ) . The proofs are straightforward using only the ideas in Appendix I. That every £-group has an a-closure was proved by Khuon [70b] and McCleary (unpublished).

xliii INDEX

A action 4 action, doubly transitive . 15 adjacent 44 adjacent pair 54 algebraically closed. . . . 221 amalgamation property . . . 183 Archimedean 98 automorphism, inner . . . . 71 automorphism, outer . . . . 71 a-configuration 17 a-set 187 a-set of degree K 199 a-splitting subset 199 B balanced 18 between vii between, lies 52 between, strictly vii block 12 block, covers 83 block, extensive 81 block, fat 82 block, natural 83 bounded 25 bounded above 25 bounded below 25 boundity 58 bump 25 bump, one 25 bump of 26 C cardinal, regular 188 centraliser 47 chain vi chain, primitive 14 character 23 character, final 23 character, initial 23 closed interval vii

closed ^-subgroup 151 closed partial congruence. .166 closure (of a subgroup). . .165 cofinal 23 cofinality 23 coherent 97 coinitial 23 coinitiality 23 complete ^-subgroup 157 completely distributive. . .157 completion, Dedekind . . . . 22 completion, local. . . .141,173 component 18 component, global 118 component, local 118 component, primitive . . . . 20 Config(n) 110 Config(oo) HO configuration 18 configuration, a17 congruence 8 congruence, closed 166 congruence, covers 19 congruence, natural '84 congruence, natural partial.' 85 congruence, singleton. . . . 10 congruence, trivial. . '. . . 1 0 congruence, universal. . . . 11 convex subset vii convexification 25 coordinate map 120,125 coterminal 23 coterminality 23 countable vi covers (blocks) 83 covers (congruences ) . . . . 19 cover ( convex 5,-subgroups ). 207 cut 21 D dead 160 dead segment 160

xliv Dedekind completion 22 degree K (a-set of) 199 dense chain vii,83 dense in itself 83 dense p.o. set vii dense subset vi,95 depressible 154 disjoint 24 distributive, completely . .157 distributive lattice . . . . vi domain (of a partial congruence) 85 doubly transitive 15 doubly transitive, weakly. .109 E embedding, standard 128 existentially closed . . . .221 extensive block 81 n -set 187 F faithful action 4 fat block 82 final character 23 finitely presented £-group .231 fixed point interval . . . .194 full (periodic) Ill fundamental triviality . . . 11 G G-invariant set G-orbit general recursion generates (as an Jl-group). global component global part ($-) greatest lower bound . . . group ring

6 6 232 .217 118 120 . vi 74

H hole 22 Holland's Theorem 240 homogeneous (chain) 6 homogeneous (congruence) . . 84 homogeneous, m29 homogeneous, # -

187

homogeneous, rigidly . . . . 98 I image initial character

vi 23

inner automorphism insoluble word problem . . insular integral interval interval, closed interval, open interval support property. invariant set, Gisotropy subgroup

71 .231 131 87 vii vii vii . 32 6 11

K kernel (equivalence relation) kernel (homomorphism). . . .

9 4

L ^-characteristic subgroup. .173 ^-embedding 7,8 5,-group 7 I -homomorphism 7,8 jl-ideal 65

&-isomorphism £-permutation group il-simple ^-subgroup lateral completion laterally complete lattice lattice, distributive. lattice isomorphism lattice-ordered group. lazy subgroup least upper bound left (f of g) lies between linearly ordered set . local completion . . . local component local part (£-) locally primitive long line long orbit

7,8 5 129 7 250 222 vi . . . vi vii ... 7 4 vi 43 52 . . . vi .141,173 118 120 141 69 18

N natural block 83 natural congruence 84 natural partial congruence . 85 negative orbit 18 negative parity 25 normal valued 207 normaliser 212

xlv 0 o-embedding o-group o-homomorphism o-isomorphism O-group one bump open interval orbit, Gorbit, long orbit, negative orbit, positive orbit Wreath product . . . orbital ordered permutation group. ordermorphic ordermorphism outer automorphism overlapping

R 7,8 7 7,8 7,8 116 25 vii 6 18 18 18 .196 60 . 5 vii vii 71 206

P p.o. group 6 p.o. set vi parity 25 parity, negative 25 parity, positive 25 parity, of orbital 60 partial congruence 85 partially ordered group. . . 6 partially ordered set. . . . vi pathological 30 period 93 periodic 93 plenary 212 pointwise order 2 pointwise suprema 151 positive 7 positive orbit 18 positive parity 25 positive, strictly 7 presentable function . . . .236 presentation of Ji-group. . .231 prime subgroup . . .143,148,2/40 primitive 14,86 primitive chain 14 primitive component. . . .20,87 primitive components, set of 78,87 primitive, locally 141 primitive segments 141 primitive, weakly 141

range of function vi recursive function 232 recursive set 232 regular cardinal 188 representation 4 representation, right regular 10 representing subgroup. . . .143,148,243 restricted wreath product. .119 restriction of function. . . vi right cosets, set of . . . .vii right regular representation 10 right totally ordered group.244 rigidly homogeneous 98 S sentence separation property set, aset, a- of degree K set, n -

viii 15 187 199 187

singleton congruence . . . . 10 splitting subset, a- . . . .199 stabiliser subgroup 11 standard embedding 128 static 14 strictly between vii strictly positive 7 subalgebra 6 support 24 support property . . . .141,153 support property, interval . 32 supporting interval 26 T totally ordered group. totally ordered set transitive transitive, doubly transitive, mtransitive, uniquely . transitively derived . transversal trivial congruence trivially ordered set.

...

7 vi 6 15 28 . . . 98 . . . 94 125 10 . . . vi

xlvi U unbounded 25 uniquely transitive 98 universal congruence . . . . 11 V value 85,207 value of (a,3) 77,86 value of g 207 Wweakly doubly transitive . .109 weakly primitive 141 word problem, insoluble. . .231 wreath product . . . . . . .119 wreath product, restricted .119 Wreath product . . .117,118,123 Wreath product, orbit. . . .196

xlvii INDEX OF SYMBOLS

(The characters are listed in order of appearance.) A ^ B

0 a)

vi vi vi vi vi ' vi vi

0)

Vi

|A|

vi

a < 3 a v 3

vi vi

a A 3

vi

[a,3]

vii

(a, 3)

vii

fl*

vii

A § B A\B Af f|C

,: i € 1}

vii

e

vii

G *= 6 . . . "

viii

G E H

viii

A x B

ix,13

/((n)

2

£ n V £T

2

Ag 2 ker(e)

4

aG.

A' . a-configuration

10 11 11 11 11 16 17

component

18

vii vii

7L

%2

vii vii

G . a

vii vii

19

vii

22

vii H o G

vii

ci(A)

23

H xi G

vii

cf(A)

23

c

23

R(C)

vii,240

xlviii c „

23

I(G)

c

*v ** supp(g) Conv^(A) supp(g) m-transitive m-homogeneous B(Q) [x,y]

2

c

2

•± R[G] K < KT fc(G,fi)

|g| L(n) R(n) bump(g) bump(g,h) g1fg2 R(g) L(g) B(g) Adj(g 1 ,g 2 ) f « g C Q (X) C(X) C(g) B(f,g,h) B(g) L(a,b,f,g) L'(a,b,f,g) £(a,b,f,g) <X> ILf IL Fx(g) Aut(G)

36 40 40 42 43 43 44 44 44 44 46 47 47 47 47 53 53 53 54 54 60 69 69 69 71

3 3 24 25 25 28 29 30 32

71

fc(fl) K

0f K ,5 )

71 74 76,86 76,85 76 76

Val(o,3) 6 v

77,86 77

(GK,fiK)

78,87

F(A)

dom(K) G+

82 85 97

Y p (G)

100

F5F(Ga) Config(n) Config(oo)

106 110 HO

117 {gT> Wr{(GK,ftK): K € k}. . .117,123 G Wr H 118 (G,fi) Wr (H,T) (G,Q) wr (H,T)

118 119

L(C)

120

(G,ft) Wr (H,T) Wr (L,B) . .121 Qf

122

supp(a)

122

0

122

a

122

fe(o,B)

122

W1

122

K

122

=

^K aK g K K,a

122

123 123

xlix [H,H] Q° G~

129 HI,172 148

C D(G)

153 157

I

157 160

J A£ £ C* Fx(C) G(Fx(C))
G.C.H Tb T*

160 165 165 165 168 187 187 187 188 188 194,196 201

Kg*) F*(g*) A(a,g) X X

203 203 206 208 208

A % V • \J

208 208 208

Vn+1

208

N(H) G <_ H

212 221 231 234 237

UJ

x*y xty