VARIETIES OF LATTICE ORDERED GROUPS

Mary Elizabeth Huss

B.Sc., University of Nottingham, 1975

M.Sc., Simon Fraser University, 1981

A THESIS SUBMITTED IN PARTIAL FULFILLMENT

OF THE REQUIREMENTS FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

in the Department

Mathematics and Statistics

@ Mary Elizabeth Huss 1984

SIMON FRASER UNIVERSITY

All right reserved. This thesis may not be reproduced in whole or in part, by photocopy

or other means, without permission of the author.

APPROVAL

Name : Mary Elizabeth Huss

Degree : Doctor of Philosophy (Mathematics)

Title of Thesis: Varieties of lattice ordered groups.

Examining Committee:

Chairman: Dr. A.R. Freedman

Dr. N.R. Reilly Senior Supervisor

- - --

Dr. T.C. Brow.

Dr. S.K. Thomason

Dr. W.C. Holland External Examiner

Professor Mathematics and Statistics Department

Bowling Green State University

Date approved: May 11, 1984

PART l AL COPY R l GHT L l CENSE

I hereby g ran t t o Simon Fraser U n i v e r s i t y the r i g h t t o lend

my thes is , p r o j e c t o r extended essay ( t h e t i t l e o f which i s shown below)

t o users o f t he Simon Fraser U n i v e r s i t y L ib rs ry , and t o make p a r t i a l o r

s i n g l e copies on l y f o r such users o r i n response t o a request from the

l i b r a r y o f any o the r u n i v e r s i t y , o r o ther educat ional i n s t i t u t i o n , en

i t s own behalf o r f o r one o f i t s users. I f u r t h e r agree t h a t permission

f o r m u l t i p l e copying o f t h i s work f o r scho la r l y purposes may be granted

by me o r t he Dean o f Graduate Studies. I t i s understood t h a t copying

o r p u b l i c a t i o n o f t h i s work f o r f i n a n c i a l ga in s h a l l no t be al lowed

w i thout my w r i t t e n permission.

T i t l e o f Thesis/Project/Extended Essay

Author:

(s ignature)

\ \ (da te)

ABSTRACT

For any t y p e o f a b s t r a c t a l g e b r a , a v a r i e t y i s an

e q u a t i o n a l l y d e f i n e d c l a s s o f s u c h a l g e b r a s . R e c e n t l y

v a r i e t i e s o f l a t t i c e o r d e r e d g r o u p s ( L-groups ) have been

found t o b e of i n t e r e s t and t h i s t h e s i s c o n t i n u e s t h e i r

s t u d y .

F o r any L - g r o u p C , C x Z d e n o t e s t h e p r o d u c t o f C

w i t h t h e i n t e g e r s Z , o r d e r e d l e x i c o g r a p h i c a l l y f rom t h e r i g h t .

For a v a r i e t y V o f 1 - g r o u p s l e t v L = V c U z ( G x Z I G E G I . L

I t h a s been an open q u e s t i o n a s t o w h e t h e r o r n o t V = V , f o r e v e r y v a r i e t y V o f € - g r o u p s . Examples a r e g i v e n t o

answer t h i s q u e s t i o n n e g a t i v e l y , and p r o p e r t i e s o f t h e

v a r i e t i e s vL a r e d e v e l o p e d .

Fo r a v a r i e t y V , a n o t h e r c l o s e l y a s s o c i a t e d v a r i e t y i s

t h e v a r i e t y V K , o b t a i n e d by r e v e r s i n g t h e o r d e r of t h e

€ - g r o u p s i n V . I t i s shown t h a t t h e r e a r e v a r i e t i e s R f o r which V # V and t h a t t h e mapping 0 : V o VR i s b o t h

a l a t t i c e and semig roup au tomorphism o f t h e s e t o f v a r i e t i e s

Kopytov and Medvedev, and i n d e p e n d e n t l y R e i l l y and

F e i l , h a v e shown t h a t t h e r e a r e u n c o u n t a b l y many € -group

v a r i e t i e s . By c o n s i d e r i n g f u r t h e r u n c o u n t a b l e c o l l e c t i o n s o f

v a r i e t i e s o f L-groups , i t i s shown t h a t t h e b r e a d t h o f t h e

l a t t i c e o f r e p r e s e n t a b l e e - g s o u p s h a s c a r d i n a l i t y o f t h e 1

con t inuum.

ACKNOWLEDGEMENT

I w o u l d l i k e t o t h a n k Dr. N . R . R e i l l y f o r h i s

a s s i s t a n c e a n d e n c o u r a g e m e n t d u r i n g t h e p r e p a r a t i o n o f t h e

t h e s i s .

I would a l s o l i k e t o t h a n k Ms. K a t h y Hannon f o r t y p i n g

t h i s t h e s i s .

A p p r o v a l

A b s t r a c t

Acknowledgement

T a b l e o f C o n t e n t s

I

TABLE OF CONTENTS

I n t r o d u c t i o n 1

C h a p t e r 1 . &-Groups and V a r i e t i e s 5

C h a p t e r 2 . R e v e r s i n g t h e O r d e r o f a n e - g r o u p 13

1 . B a s i c O b s e r v a t i o n s 2. An Automorph i sm o f L 3. V a r i e t i e s I n v a r i a n t u n d e r o 4 . V a r i e t i e s moved by

C h a p t e r 3. Lex P r o d u c t s by t h e I n t e g e r s 2 9

1 . The Lex P r o p e r t y 2. V a r i e t i e s w i t h o u t t h e l e x p r o p e r t y 3. Laws f o r V'

C h a p t e r 4 . U n c o u n t a b l e C o l l e c t i o n s o f V a r i e t i e s o f 53 & - g r o u p s

C h a p t e r 5. F u r t h e r R e s u l t s 6 1

1 . Lex p r o d u c t s o f v a r i e t i e s 2 . M i m i c k i n g

R e f e r e n c e s 7 3

INTRODUCTION

For any t y p e o f a b s t r a c t a l g e b r a , . a v a r i e t y i s an

e q u a t i o n a l l y d e f i n e d c l a s s o f s u c h a l g e b r a s . E q u i v a l e n t l y ,

by B i r k h o f f [21 , a v a r i e t y i s a c l a s s o f a l g e b r a s c l o s e d

u n d e r s u b a l g e b r a s , d i r e c t p r o d u c t s and homomorphic

images . The e x t e n s i v e work on v a r i e t i e s o f g r o u p s , much o f

which i s d e s c r i b e d by H . Neumann [ 1 8 1 , prompted an i n t e r e s t

i n t h e s t u d y o f v a r i e t i e s o f l a t t i c e o r d e r e d g r o u p s . The

e a r l y work i n t h i s a r e a was m a i n l y c o n c e r n e d w i t h s p e c i f i c

v a r i e t i e s . F o r e x a m p l e , Weinberg [ 2 4 1 s t u d i e d a b e l i a n

t - g r o u p s and showed t h a t t h e a b e l i a n v a r i e t y A i s t h e

s m a l l e s t n o n - t r i v i a l v a r i e t y o f l a t t i c e o r d e r e d g r o u p s . A

more c o m p r e h e n s i v e i n v e s t i g a t i o n o f v a r i e t i e s o f l a t t i c e

o r d e r e d g r o u p s was begun by M a r t i n e z , [ 1 4 1 , [ 1 5 1 , and [161 .

He d e s c r i b e d an a s s o c i a t i v e m u l t i p l i c a t i o n o f 4 -g roup

v a r i e t i e s and d e t e r m i n e d t h a t t h e s e t L o f a l l l a t t i c e

o r d e r e d g r o u p v a r i e t i e s fo rms a l a t t i c e o r d e r e d semig roup

unde r t h i s m u l t i p l i c a t i o n , t h e p a r t i a l o r d e r b e i n g i n c l u s i o n .

G l a s s , H o l l a n d and McCleary [ 7 1 have e x t e n d e d t h i s work. One

o f t h e i r main r e s u l t s shows t h a t t h e powers o f t h e a b e l i a n

v a r i e t y , A , g e n e r a t e t h e no rma l v a l u e d v a r i e t y , N , shown

by H o l l a n d [ I 0 1 t o b e t h e l a r g e s t p r o p e r v a r i e t y o f l a t t i c e

o r d e r e d g r o u p s .

I n t h i s t h e s i s , t h e s t u d y o f v a r i e t i e s o f l a t t i c e

o r d e r e d g r o u p s i s c o n t i n u e d .

C h a p t e r 1 c o n t a i n s b a c k g r o u n d m a t e r i a l and i n t r o d u c e s

many o f t h e commonly s t u d i e d v a r i e t i e s .

F o r a n y l a t t i c e o r d e r e d g - roup G t h e r e a r e t w o c l o s e l y

a s s o c i a t e d 1 - g r o u p s : G~ , w h i c h i s o b t a i n e d f r o m G

by r e v e r s i n g t h e o r d e r , and G W , w h i c h i s o b t a i n e d f r o m G

r e v e r s i n g t h e m u l t i p l i c a t i o n . G R a n d GU i s o m o r p h i c

1 - g r o u p s , a n d o n e c a n a s k w h e t h e r G a n d C R 2 GW

t h e same v a r i e t y o f 1 - g r o u p s , a n d i f n o t , w h a t t h e

r e l a t i o n s h i p b e t w e e n t h e v a r i e t i e s t h e y g e n e r a t e i s . T h i s

q u e s t i o n i s c o n s i d e r e d i n C h a p t e r 2 , w h e r e i t i s shown t h a t

t h e r e a r e 1 - g r o u p s G f o r w h i c h G a n d G R . g e n e r a t e

d i f f e r e n t v a r i e t i e s . I f , f o r a n y v a r i e t y V o f 4 - g r o u p s ,

@ v R = ( G R ] G E V 1 , t h e n i t i s e s t a b l i s h e d t h a t t h e m a p p i n g

0 : v I+ V R i s b o t h a l a t t i c e a n d s e m i g r o u p a u t o m o r p h i s m o f

L , t h e s e t o f a l l v a r i e t i e s o f 1 - g r o u p s . F u r t h e r , t h e c l a s s

o f v a r i e t i e s w h i c h a r e i n v a r i a n t u n d e r 8 , a n d t h e c l a s s o f

t h o s e w h i c h a r e n o t , a r e b o t h shown t o h a v e t h e c a r d i n a l i t y

o f t h e c o n t i n u u m .

F o r e - g r o u p s G a n d H , i f ti i s t o t a l l y o r d e r e d

t h e n t h e p r o d u c t G x H may b e o r d e r e d l e x i c o g r a p h i c a l l y by

(g, h ) 2 e i f h > e o r h = e a n d g k e . T h e p r o d u c t

w i t h t h i s o r d e r i s d e n o t e d by G ; H a n d c a l l e d t h e l e x p r o d u c t

o f G by H . Even i n t h e s i m p l e s t c a s e when H i s r e s t r i c t e d t o

b e i n g t h e - g r o u p o f i n t e g e r s , Z , i t h a s b e e n a n o p e n

q u e s t i o n a s t o w h e t h e r o r n o t v a r i e t i e s o f l a t t i c e o r d e r e d

g r o u p s a r e c l o s e d u n d e r t a k i n g l e x i c o g r a p h i c a l l y o r d e r e d

p r o d u c t s . The p r o b l e m was f i r s t c o n s i d e r e d by S m i t h C231

who d e m o n s t r a t e d t h a t many v a r i e t i e s ( i n p a r t i c u l a r

P, 9 N 9 R r W a n d S n ( n E N) ) a r e c l o s e d u n d e r

l e x i c o g r a p h i c a l l y o r d e r e d p r o d u c t s by t h e i n t e g e r s .

I n C h a p t e r 3 , i t is shown t h a t n o t a l l v a r i e t i e s

o f l a t t i c e o r d e r e d g r o u p s h a v e t h i s p r o p e r t y . F o r a n y

v a r i e t y V, vL i s d e f i n e d t o b e t h e v a r i e t y g e n e r a t e d by

(G f Z / G E V 1 a n d t h e p r o p e r t i e s o f t h e s e v a r i e t i e s a r e

d i s c u s s e d . I t i s p r o v e d t h a t f o r a n y v a r i e t y o f 1 - g r o u p s ,

vL i s c l o s e d u n d e r l e x i c o g r a p h i c a l l y o r d e r e d p r o d u c t s by

2 . M o r e o v e r , a n y v a r i e t y i s c l o s e d u n d e r l e x p r o d u c t s by Z

i f a n d o n l y i f i t i s c l o s e d u n d e r l e x p r o d u c t s by a l l t o t a l l y

o r d e r e d a b e l i a n g r o u p s , s o t h a t f o r e v e r y v a r i e t y LJ,

vL = V a t ( { G : A I G E G, A E A , A t o t a l l y o r d e r e d ) ) .

The work o f C h a p t e r 3 s u g g e s t s t h a t t h e more g e n e r a l

s i t u a t i o n o f t h e l ex p r o d u c t o f two v a r i e t i e s b e c o n s i d e r e d .

T h i s i s d o n e i n t h e f i r s t p a r t o f C h a p t e r 5. T h a t c h a p t e r

c o n c l u d e s w i t h a d i s c u s s i o n o f m i m i c k i n g , a p r o p e r t y

i n t r o d u c e d by G l a s s , H o l l a n d and M c C l e a r y [ ? I who u s e d i t i n

t h e i r s t u d y o f p r o d u c t v a r i e t i e s .

4

The l a t t i c e L o f v a r i e t i e s o f L -g roups was f i r s t

shown t o b e u n c o u n t a b l e by Kopytov and Medvedev C131 and

i n d e p e n d e n t l y by F e i l 151 and R e i l l y Clgl. R e i l l y t s work

d e m o n s t r a t e d t h e e x i s t e n c e o f an u n c o u n t a b l e c o l l e c t i o n o f

p a i r w i s e i n c o m p a r a b l e v a r i e t i e s e a c h c o n t a i n i n g t h e

r e p r e s e n t a b l e v a r i e t y R , w h i l e F e i l c o n s t r u c t e d an

u n c o u n t a b l e t o w e r o f r e p r e s e n t a b l e L -g roup v a r i e t i e s . Thus

t h e h e i g h t and b r e a d t h o f L b o t h have c a r d i n a l i t y o f t h e

con t inuum. I n C h a p t e r 4 f u r t h e r u n c o u n t a b l e c o l l e c t i o n s o f

v a r i e t i e s o f -g roups a r e c o n s i d e r e d and i t 'is shown t h a t

t h e l a t t i c e o f s u b v a r i e t i e s o f R n ~2 c o n t a i n s a s u b l a t t i c e

i s o m o r p h i c t o I x I where I is t h e u n i t i n t e r v a l (0, I ) .

CHAPTER 1

4 -Groups a n d V a r i e t i e s

A hfkic.e srdered g u u ~ I_ i s a g r o u p w i t h a

l a t t i c e s t r u c t u r e t h a t i s c o m p a t i b l e w i t h t h e g r o u p

o p e r a t i o n s . T h a t i s , a ( x v y ) b = ( a x b ) v ( a y b ) a n d a ( x h y ) b =

( a x b ) A ( a y b ) . An 1 - g r o u p i n w h i c h t h e l a t t i c e o r d e r

i s a t o t a l o r d e r i s a t o t d 1 . y W d -. A s u b g r o u p o f a n 1 - g r o u p w h i c h i s a l s o a s u b l a t t i c e i s

c a l l e d a n l -sub-. A s u b g r o u p H o f a n & - g r o u p G

i s s a i d t o b e c o n v a i f w h e n e v e r h < g < k a n d h , k , E H , - -

t h e n g E H . An LrLdeal i s a n o r m a l c o n v e x 1 - s u b g r o u p .

An l -h-- ( r e s p e c t i v e l y , 1 ---) b e t w e e n

two [ - g r o u p s i s a m a p p i n g w h i c h i s b o t h a g r o u p a n d a l a t t i c e

homomorphism ( r e s p e c t i v e l y , g r o u p a n d l a t t i c e i s o m o r p h i s m ) .

F u r t h e r b a c k g r o u n d i n f o r m a t i o n a n d t e r m i n o l o g y r e l a t i n g

t o [ - g r o u p s may b e f o u n d i n [ I ] .

A u r i e t ~ o f l - g r o u p s i s t h e c l a s s o f a l l € - g r o u p s

w h i c h s a t i s f y a ( p o s s i b l y i n f i n i t e ) s e t o f e q u a t i o n s .

E q u i v a l e n t l y , i t i s a nonempty c o l l e c t i o n o f & - g r o u p s c l o s e d

u n d e r [ - s u b g r o u p s , f - h o m o m o r p h i c i m a g e s a n d d i r e c t

p r o d u c t s .

An e q u a t i o n u s e d i n d e f i n i n g a v a r i e t y o f 1 - g r o u p s h a s

t h e fo rm w ( 2 ) = e , w h e r e w ( x ) i s a n e l e m e n t o f t h e f r e e

e - g r o u p F on a c o u n t a b l y ' i n f i n i t e s e t X . By t h e

c o m p a t i b i l i t y o f t h e g r o u p a n d l a t t i c e o p e r a t i o n s , a n d s i n c e

t h e l a t t i c e s t r u c t u r e o f a n y L - g r o u p i s d i s t r i b u t i v e ( s e e

[ I , C h a p t e r 1 1 1 , a l l e l e m e n t s o f F c a n b e w r i t t e n i n t h e f o r m

w(;) = V A Wij (x ) = V f i n x i j k . H e r e t h e i n d e x s e t s I , J , I J I J K . -

and K a r e f i n i t e , e a c h x i j k E x U x - l el a n d e a c h w ~ ~ ( ~ )

i s i n r e d u c e d f o r m a s a n e l e m e n t o f t h e f r e e g r o u p on X . A

t i t u t i o n f o r . a word w ( x ) i n a n [ - g r o u p G i s a

m a p p i n g ( x i j k 1 -+ G, ' i jk* g i j k s u c h t h a t - ( i ) i f x i j k - . x i , j , k , t h e n g i j k - - g i t j t k t ,

- ( i i ) i f x i j k - then g i j k - - g i t j t k t a n d -

( i i i ) i f x i j k = e t h e n g i j k = e . Then w ( g ) is t h e '

A g i jk o f G, a n d i s a l s o r e f e r r e d t o a s a I J K

s u b s t i t u t i o n f o r w ( x ) i n G .

E q u a t i o n s w r i t t e n i n t h e a b o v e f o r m c a n become u n w i e l d y

a n d i t i s t h e r e f o r e u s u a l t o u s e some a b b r e v i a t i o n s : x v y = Y

- -1 i s w r i t t e n x < y , l x l = x v x-I , x + = x v e , x = Y v e , -

[ x , y l = x - I y - 1 X Y 1 a n d i f i t i s r e q u i r e d t h a t x 2 e t h e n

t h i s i s s i m p l y n o t e d , i n s t e a d o f r e p l a c i n g x by x v e .

The .set o f a l l v a r i e t i e s o f L - g r o u p s i s d e n o t e d by L .

I t i s c l e a r t h a t t h e i n t e r s e c t i o n o f a n y c o l l e c t i o n o f

v a r i e t i e s i s a g a i n a v a r i e t y . T h u s by c o n s i d e r i n g L t o b e

p a r t i a l l y o r d e r e d by i n c l u s i o ~ , i t i s p o s s i b l e t o d e f i n e

l a t t i c e o p e r a t i o n s on L:

T h e s e d e f i n i t i o n s make L a c o m p l e t e l a t t i c e .

L i s a l s o a s e m i g r o u p , w h e r e m u l t i p l i c a t i o n o f

v a r i e t i e s o f l ? - g r o u p s i s d e f i n e d a s f o l l o w s : f o r U , V E L ,

a n 1 - g r o u p G b e l o n g s t o U V i f a n d o n l y i f t h e r e i s a n

L - i d e a l H o f G s u c h t h a t H E U a n d G / H E V . A v a r i e t y V i s s a i d t o b e nenerat_ed by a f a m i l y o f

- g r o u p s {Gil i E I ) i f V i s t h e s m a l l e s t v a r i e t y c o n t a i n i n g

e a c h G i , i . e . i f V = ~ ) { u E L I U 2 { ~ ~ / i € 1 1 1. The

v a r i e t y g e n e r a t e d by {Gi li E I] i s w r i t t e n Va,(GiI i E I).

Some e x a m p l e s o f v a r i e t i e s o f L - g r o u p s a r e a s f o l l o w s .

ExamDle U. ltrivial m, E , c o n s i s t s o f a l l o n e - e l e m e n t t - g r o u p s a n d i s d e f i n e d by t h e e q u a t i o n x = e .

EhUUd.2 l-2- A t t h e o t h e r extreme i s t h e v a r i e t y

L c o n s i s t i n g o f a l l 1 - g r o u p s . L i s d e f i n e d by t h e e q u a t i o n

e = e .

E U i J I U k l d . Tbf:&-el ialYXi&Y I A , i s d e f i n e d by t h e

e q u a t i o n [ x , y l = e . W e i n b e r g 1 2 4 1 , h a s shown t h a t A i s t h e

s m a l l e s t n o n - t r i v i a l v a r i e t y o f 1 - g r o u p s .

E.-e u. Ihe nDrmal yahed y a r i e t y , N, i s d e f i n e d by

t h e l a w x2y2 - > yx f o r x , y 2 e . H o l l a n d [ I 0 1 h a s shown

8

.. t h a t N i s t h e l a r g e s t p r o p e r ' v a r i e t y o f 1 - g r o u p s .

A c o n n e c t i o n b e t w e e n t h e s m a l l e s t p r o p e r t y v a r i e t y , A ,

a n d t h e l a r g e s t p r o p e r v a r i e t y , N , was g i v e n by C l a s s , w n

H o l l a n d a n d M c C l e a r y 171 who p r o v e d t h a t N = V A . n=l

Eh-uuh.1.5. An 1 - g r o u p i s s a i d t o b e r e~ re s - i f i t

i s a s u b d i r e c t p r o d u c t o f t o t a l l y o r d e r e d g r o u p s . The

c o l l e c t i o n o f a l l r e p r e s e n t a b l e [ - g r o u p s i s a v a r i e t y ,

d e n o t e d by R , w i t h d e f i n i n g l a w x 2 y 2 = ( x A y ) 2 . ( s e e , f o r e x a m p l e , B i g a r d e t a 1 [ I , P r o p o s i t i o n 4.2.9.1)

-u. m - w - u , W , i s d e f i n e d by

x 2 2 y0'xy f o r x ~ e .

If r i s a t o t a l l y o r d e r e d s e t , t h e n t h e g r o u p o f a l l o r d e r -

i s o m o r p h i s m s o f r i s a n C - g r o u p , i f a n o r d e r - i s o m o r p h i s m b e

c a l l e d p o s i t i v e when 8 g 2 8 f o r a l l X c r . C o n v e r s e l y , by a

r e s u l t o f H o l l a n d [ 9 l , e v e r y 1 - g r o u p G c a n b e c o n s i d e r e d a s a

t r a n s i t i v e C - g r o u p o f o r d e r - i s o m o r p h i s m s o f some t o t a l l y

o r d e r e d s e t r , i n w h i c h c a s e G i s o f t e n d e n o t e d ( C , r 1. 1 f

(G, r ) a n d (H, r ) a r e .l - g r o u p s t h e n t h e i r H r e a t h

( , r , 1 i s t h e C - g r o u p (W, s2 1, w h e r e

( o r d e r e d l e x i c o g r a p h i c a l l y f r o m t h e r i g h t )

Up t o i s o m o r p h i s m ( C , r ) W r ( H , h ) d e p e n d s on G , H and A b u t n o t

on r , a n d t h e w r e a t h p r o d u c t (W, a ) i s o f t e n w r i t t e n

G Wr(H, A 1. A

The - s u b g r o u p o f (W, Q c o n s i s t i n g o f t h o s e ( g , h ) s u c h A

t h a t g ( A ) f e f o r o n l y f i n i t e l y many E: A i s c a l l e d t h e

restrict_ed wreath p ? o f (G, r ) w i t h (H, ) a n d i s

d e n o t e d by (G, r ) w r ( H , A ) . F u r t h e r d e t a i l s o f t h e w r e a t h

p r o d u c t c o n s t r u c t i o n may b e f o u n d i n H o l l a n d a n d McClea ry [ I 1 1

o r i n G l a s s C6, C h a p t e r 51.

W r e a t h p r o d u c t s i n v o l v i n g t h e i n t e g e r s o c c u r f r e q u e n t l y

i n t h e s t u d y o f v a r i e t i e s o f 1 - g r o u p s , see f o r e x a m p l e ,

E x a m p l e s 1.7 a n d 1 .9 . L e t ( 2 , 2 ) b e t h e r e g u l a r r e p r e s e n t -

a t i o n o f t h e i n t e g e r s . I t i s u s e f u l t o n o t e t h a t m u l t i p l i -

c a t i o n i n ZWr(Z, 2 ) i s g i v e n b y , f o r ( F , n ) , (G, rn) i n

ZWr(Z ,Z) ,

( F , n ) ( G , m ) = ( F + G n , n + m )

w h e r e G " ( z ) = G(n + z ) f o ' r a l l z e Z . The i n v e r s e o f

( F , n ) i s (-F-", - n ) .

Example 1 .7 . F o r e a c h p o s i t i v e i n t e g e r n , l e t G n b e

t h e l - s u b g r o u p o f ZWr(Z, Z) c o n s i s t i n g o f a l l t h o s e e l e m e n t s

( F , k) f o r w h i c h F ( i ) = F ( j ) w h e n e v e r i J mod n . Then

10

t h e S c r i m g e r v a r i e t y , S n , ' i s t h e v a r i e t y o f 1 - g r o u p s

g e n e r a t e d by G n . S c r i m g e r [ 2 2 1 showed t h a t f o r e a c h p r i m e

p , Sp c o v e r s A .

Yedvedev [ I 7 1 d e s c r i b e d t h r e e f u r t h e r c o v e r s o f A :

E x a r n ~ l e u. L e t N o b e t h e f r e e n i l p o t e n t c l a s s t w o g r o u p

on two g e n e r a t o r s a a n d b , o r d e r e d s o t h a t a >> b >> [ a , b l

> e , i . e a m b n [ a , b l k > e i f i ) m > 0 o r i i ) m = 0

a n d n > 0 o r i i i) m = n = 0 a n d k > 0 . V O c ( N o ) i s t h e n

a c o v e r o f t h e a b e l i a n v a r i e t y .

L ~ ~ I I L Q ~ 19, Z w r ( Z , 2 ) w i t h t h e u s u a l o r d e r f o r w r e a t h

p r o d u c t s i s n o t t o t a l l y o r d e r e d , o r e v e n r e p r e s e n t a b l e ;

h o w e v e r t h e r e a r e t w o s i m i l a r t o t a l o r d e r s on Z w r ( Z , 2 ) . L e t

W+ b e Z w r ( Z , Z ) t o t a l l y o r d e r e d by d e f i n i n g ( F , k) = e t o

b e p o s i t i v e i f e i t h e r k > 0 o r k = 0 a n d F ( r ) > 0 w h e r e

r i s t h e maximum e l e m e n t o f t h e s u p p o r t o f F . L e t W' b e

Z w r ( Z , Z ) t o t a l l y o r d e r e d by d e f i n i n g ( F , k ) e t o b e

p o s i t i v e i f e i t h e r k > 0 o r k = 0 a n d F ( s ) > 0 w h e r e s

i s t h e minimum e l e m e n t o f t h e s u p p o r t o f F . V ~ R ( W + ) a n d

V o t r ( W - ) a r e t h e n d i s t i n c t c o v e r s o f A .

The l a t t i c e o f L - g r o u p v a r i e t i e s was shown t o b e

u n c o u n t a b l e by Kopytov a n d Medvedev [ 1 3 1 . I n d e p e n d e n t l y

R e i l l y [ I 9 1 a n d F e i l [ 5 1 p r o v e d t h e same r e s u l t , R e i l l y

s h o w i n g t h a t L h a s u n c o u n t a b l e b r e a d t h , F e i l

c o n s t r u c t i n g a n u n c o u n t a b l e t o w e r o f v a r i e t i e s o f l - g r o u p s .

11

1 . 1 8 , L e t F b e t h e ' f r e e g r o u p on a c o u n t a b l y

i n f i n i t e s e t X a n d l e t z- k X . F o r e a c h f u l l y i n v a r i a n t

s u b g r o u p U o f F le tqu b e t h e v a r i e t y o f [ - g r o u p s s a t i s f y i n g

t h e l a w s z + r \ u"z' u = e f o r a l l u & U . S u c h v a r i e t i e s

a r e c a l l e d wsi - r- a n d fo rm t h e u n c o u n t a b l e

c o l l e c t i o n o f v a r i e t i e s d e s c r i b e d by R e i l l y .

ExamDle 1 1 F e i l [ 4 , 5 1 c o n s t r u c t e d two u n c o u n t a b l e

t o w e r s o f v a r i e t i e s w i t h i n R /7 . F o r p , q i n t e g e r s w i t h

0 - < p / q < l l U p , q is t h e v a r i e t y d e f i n e d by t h e l a w

t o g e t h e r w i t h t h e l a w s o f R (r2 A Y 2 = ( x A y ) 2 ) a n d of

( [ I z \ A ~ [ x , Y ] 1 , I W ~ A \ [ X , y l l l = e l ; u i s t h e v a r i e t y

d e f i n e d by t h e l a w

t o g e t h e r w i t h t h e l a w s o f R and . F o r 0 < r < 1, r

i r r a t i o n a l , U r i s d e f i n e d t o b e n U p / q > r ~ 1 9

a n d W, i s d e f i n e d

t o b e 0 W p I q > r p/q,"

A l s o f o r t c R , 0 < t ( 1 , e - g r o u p s , G t and H t

a r e d e f i n e d i n t h e f o l l o w i n g way:

G t i s R x Z w i t h m u l t i p l i c a t i o n ( r , m ) ( s , n) = t ( r + ( I m S , m + n ) a n d o r d e r ( r , m ) _> ( 0 , 0 ) i f

t+l m multiplication (r, m)(s, n) = (r + ( T ) s, m + n) and

order (r, m)) (0, 0) if m > 0 or m = 0 and r 2 0.

The properties of the two Feil towers are summarized in

the next result.

i) Gt E Ur if andonly if t < r. -

ii) H t E Wr if and only if t < r. - I

iii) {UrI 0 < r < 11 is an uncountable tower of -

varieties with U, E; Us if and only if r < s. -

iv) {wr 10 < r < 1) is an uncountable tower of -

varieties with wr G W, if and only if r < s. -

,

CHAPTER 2

R e v e r s i n g The O r d e r o f An !-Group

F o r any 1 - g r o u p G t h e r e a r e two c l o s e l y a s s o c i a t e d

X-groups , G~ which i s o b t a i n e d f rom G by r e v e r s i n g t h e

o r d e r , and GW which i s o b t a i n e d f rom G by r e v e r s i n g t h e

m u l t i p l i c a t i o n . F o r any 1 -g roup G , G~ and GW a r e

i s o m o r p h i c 1 - g r o u p s and i t is n a t u r a l t o c o n s i d e r t h e

r e l a t i o n s h i p be tween t h e v a r i e t i e s g e n e r a t e d by G and by

G~ 2 GW . The f o l l o w i n g work on t h i s q u e s t i o n was done i n

c o l l a b o r a t i o n w i t h N . R . R e i l l y and a p p e a r s i n [121.

~ e ' c t i o n 1 . B a s i c O b s e r v a t i o n s

D e f i n u 2.1.1. F o r any !-group ( G , 0 , l e t G~ = - R ( G , 5 d e n o t e t h e 1 - g r o u p o b t a i n e d f rom G by r e v e r s i n g

t h e o r d e r ; t h u s a - < R b i f and o n l y i f b - < a . L e t GW =

( G < d e n o t e t h e 2.-group o b t a i n e d f rom G by r e v e r s i n g W' -

t h e m u l t i p l i c a t i o n ; t h a t i s , w i t h m u l t i p l i c a t i o n * g i v e n by

a*b = b a . I t i s e a s i l y s e e n t h a t b o t h G~ and GW a r e

.e -g roups .

N o t F o r any !-group G we d e n o t e t h e l a t t i c e R R R o p e r a t i o n s i n C by v and A . A l s o , f o r x E, G

I

wri te

R N o t e t h a t f o r x , y E G , x v R y = x A y a n d x A y = x v y .

Lemma 2.1.3. - 1 The m a p p i n g + : g ~ g i s a n

R l - i s o m o r p h i s m o f G o n t o GW .

PrPof, F o r g , h E G R , ( g h ) $ = ( g h ) - l = h- g ( h ? ) ( g + ) =

( g + ) * ( h ~ ) s o t h a t i s a g r o u p i s o m o r p h i s m . S i n c e ( g v h j *

= ( g v R h ) - I = ( g A h ) - = g-l v h - l = g l ' v h + ,

a n d s i m i l a r l y ( g nR h ) = g + l r h t , i s a l s o a l a t t i c e

i s o m o r p h i s m .

C o r o l l a r y 2 . 1 . 4 . F o r a n y L - g r o u p G , v ~ ( G ~ ) = v ~ ( G ~ ) .

A l t h o u g h i t w i l l b e s e e n t h a t some p r o p e r t i e s o f G a n d

R G c a n b e q u i t e d i f f e r e n t , o t h e r p r o p e r t i e s a r e i n v a r i a n t

u n d e r o r d e r r e v e r s a l .

Lemma- L e t G b e a n 1 - g r o u p a n d H S G . H i s a

s u b l a t t i c e ( r e s p e c t i v e l y , s u b g r o u p , t - s u b g r o u p o r e - i d e a l )

o f G i f a n d o n l y i f H R i s a s u b l a t t i c e ( r e s p e c t i v e l y ,

s u b g r o u p , t - s u b g r o u p o r t' - i d e a l ) o f G R . F u r t h e r m o r e , i f

H i s a n C - i d e a l o f G, t h e n ( G / H I R i s Y - i s o m o r p h i c t o

G ~ / H R .

15 I

f l o t a t i o n 2.1.6. L e t F d e n o t e t h e f r e e L - g r o u p on a

c o u n t a b l y i n f i n i t e s e t X . F o r a n y v a r i e t y V , o f 1 - g r o u p s

l e t V R = { C R \ C E V ) a n d , f o r a n y word u = x i j k i n

R I J K F l e t u = v A ( n x i j k ) - l .

I J K

A l s o f o r a n y p r o d u c t y k , l e t FI' y k d e n o t e t h e K K - -

p r o d u c t t a k e n i n t h e r e v e r s e o r d e r . T h u s i f 9 yk = Y l -Y, K - t h e n n ' yk - y n ...

.v

y 1 . W i t h u a s a b o v e , K

wri te u t = v A n ' x i j k . I J K

Lemma- F o r a n y [ - g r o u p G a n d a n y u = V A n x i jk I J K

t h e f o l l o w i n g a r e e q u i v a l e n t .

( i ) The i d e n t i t y u = e h o l d s i n G .

( i i ) The i d e n t i t y uR = e h o l d s i n cR.

( i i i ) T h e i d e n t i t y u ' = e h o l d s i n cR.

PrOOfa C l e a r l y C s a t i s f i e s t h e l a w V fl I J K ' i j k

= e

i f a n d o n l y i f CR s a t i s f i e s t h e l a w R R f VJ R " i j r = e

R R R R -1 -1 But, A V n x i jk e n ( A ( m i j k ) 1 = e

I J K I J K

T h u s , G s a t i s f i e s a n i d e n t i t y u = e i f a n d o n l y i f G~

16

s a t i s f i e s u R = e , e s t a b l i s h i n g t h e e q u i v a l e n c e o f ( i ) a n d

( i i ) .

The m a p p i n g x ++ x - l o f X i n t o F e x t e n d s t o a n

a u t o m o r p h i s m g , s a y , o f F. H e n c e , t h e i d e n t i t y u R = e

h o l d s i n G~ i f a n d o n l y i f t h e i d e n t i t y uR$ = e h o l d s

a n d t h e e q u i v a l e n c e o f ( i i ) a n d ( i i i ) f o l l o w s .

Corollarv2.1.8k F O ~ a n y v a r i e t y o f L - g r o u p s V , v R

i s a v a r i e t y . M o r e o v e r t h e f o l l o w i n g a r e e q u i v a l e n t .

( i ) V h a s a b a s i s o f i d e n t i t i e s [ u = e , 2

a E . A ] .

( - i i ) v R h a s a b a s i s o f i d e n t i t i e s [ u R = e , a E A ] . a

( i i i ) v R h a s a b a s i s o f i d e n t i t i e s [ u t = e , a

a E A ] .

A q u e s t i o n w h i c h n a t u r a l l y a r i s e s i s w h e t h e r o r n o t i t

i s a l w a y s t h e c a s e t h a t V = v R , o r e q u i v a l e n t l y , w h e t h e r

o r n o t i t i s t h e c a s e t h a t f o r a l l [ - g r o u p s G t h e

v a r i e t i e s Vnh (G) and v & G R ) = Vm(GW) a r e t h e s a m e .

E x a m p l e s w i l l b e g i v e n i n S e c t i o n 4 t o a n s w e r t h i s q u e s t i o n

n e g a t i v e l y .

S e c t i o n 2 . An Automorph i sm 'o f L .

S i n c e i t w i l l b e shown t h a t t h e r e a r e v a r i e t i e s V f o r

w h i c h v R 4 V , i t is o f i n t e r e s t t o c o n s i d e r t h e

R p r o p e r t i e s o f t h e m a p p i n g V - V .

B o t a t i p n U L e t 8 : L -t L b e t h e m a p p i n g d e f i n e d by

V8 = vR a n d l e t F E { V E L l v R = V } .

l ' r o p o s d i u n . .

zL2LL The m a p p i n g 8 i s a l a t t i c e

a u t o m o r p h i s m o f L w i t h t h e f o l l o w i n g p r o p e r t i e s :

( i ) Q 2 i s t h e i d e n t i t y m a p p i n g

( i i ) 8 p r e s e r v e s a r b i t r a r y j o i n s a n d meets

( i i i ) F i s a c o m p l e t e s u b l a t t i c e o f L

( i v ) f o r a n y v ~ L , V v v R ~ ~ a n d V I I V ~ E F .

p r o o f . F o r a n y word u E F , i t i s c l e a r t h a t ( u R ) R = u

a n d by C o r o l l a r y 2 .1 .8 , VQ = V f o r a l l V E L - T h u s

( i ) h o l d s . I t i s c l e a r t h a t Us V i f a n d o n l y i f U8 G V0 ,

a n d s i n c e ( i ) i m p l i e s t h a t 8 i s a b i j e c t i o n , 8 i s a c o m p l e t e

l a t t i c e a u t o m o r p h i s m and ( i i ) and ( i i i ) f o l l o w .

F i n a l l y , f o r a n y V c L,

( V v v R ) 0 = ( v v v e ) e = v e v v e 2 . ve v v = v v v R

R a n d s i m i l a r l y ( V n V )€I = v f l v R s o t h a t b o t h V v v R R

a n d V f\ V a r e i n F .

18

I

The f o l l o w i n g c o r o l l a r y i s an immed ia t e c o n s e q u e n c e o f

P r o p o s i t i o n 2.2 .2 ( i v ) .

hlwJalX2.2.3, L e t V & L . Then t h e f o l l w i n g a r e

e q u i v a l e n t .

( i i ) R V G V

( i i i ) V 2 v R .

R e c a l l t h a t t h e l a t t i c e o f v a r i e t i e s o f 1 - g r o u p s , L ,

h a s a s emig roup s t r u c t u r e . A v a r i e t y U i s s a i d t o b e

indecomposab le i f U = U 1 U 2 i m p l i e s t h a t e f t h e r u , o r u,

i s t h e t r i v i a l v a r i e t y . G l a s s , H o l l a n d , and McCleary [ 7 1 have

p roved t h e f o l l o w i n g .

P r o p o U i o n . .

2,2.4_. * .

The s e t L , o f v a r i e t i e s o f 1 - g r o u p s

p r o p e r l y c o n t a i n e d i n t h e no rma l v a l u e d v a r i e t y N , f o r m s a '

f r e e s emig roup on t h e s e t o f i ndecomposab le v a r i e t i e s .

ikQE2- i2'2L.L The mapping 8 i s an au tomorphism o f

t h e s emig roup s t r u c t u r e o f L .

h 2 Q L S i n c e 8 i s b i j e c t i v e i t r e m a i n s t o show t h a t

8 i s a s emig roup homomorphism. L e t U , V E L . Then ~

G E ( UV)0 Q G R E UV

t h e r e e x i s t s an - i d e a l H o f G R

w i t h H E U and G ~ / H E V

@P there Lxists an l -ideal K of

G (K = HR) with K E UQ and

G/K (=(G~/H ve

cs G E (UG) (VQ)

Thus (UV)Q = (UQ) ( V G ) , . as r equ i r ed .

Proposition 2.2.6. Let U E L* and let U = U1...U n

where each U i E L (i = 1 , ..., n) is indecomposable. Then

U E F if and only if Uic F for all i = 1, ..., n.

Proof By Proposition 2.2.5. UQ = (U1.. .U )€I = (UIQ). . . (U 8) n n

' where, since 8 is an automorphism of the semigroup L,

each LliQ is indecomposable. By Proposition 2.2.4. the

factorization of varieties of L* into indecomposable

varieties is unique and therefore U = UQ if and only if

U = UiO for i = 1, ..., n. Thus u E F if and only if U E F for i i

i = 1, ..., n.

Corollary 2.2 - 7 . The complement F' of F in L is a

prime semigroup ideal (i.e. UV E. F C implies U E: FC or

V E FC) . In particular F and F C are both subsemigroups

of L.

Proof_. By their positions in the lattice L it is easily

seen that the trivial variety, the variety of all [-groups

and the normal valued variety are in F . Thus FC C L*

and the result follows from Propositions 2.2.5 and 2.2.6.

, S e c t i o n 3 . V a r i e t i e s I n v a r i a n t Under 8 .

Al though i t w i l l b e shown t h a t n o t a l l v a r i e t i e s o f

l -g roups a r e f i x e d by 8 , many o f t h e commonly

s t u d i e d v a r i e t i e s a r e i n v a r i a n t u n d e r 8 . Examples o f

s u c h v a r i e t i e s w i l l b e g i v e n i n t h i s s e c t i o n .

F o r a n y l -group G , G and G~ have t h e same u n d e r l y i n g

g r o u p s t r u c t u r e and t h u s s a t i s f y t h e same g r o u p l a w s . T h i s

p r o v e s t h e f o l l o w i n g .

. . humi&Un 2 A . L L I f V i s a v a r i e t y o f [ -g roups

d e f i n e d by l a w s i n v o l v i n g o n l y t h e g r o u p o p e r a t i o n s , t h e n

R R y y . I n p a r t i c u l a r A - = A.

C o r o U x y 2.3.2. F o r a l l p o s i t i v e i n t e g e r s n , An i s i n F.

FYQQf, T h i s f o l l o w s from P r o p o s i t i o n 2 . 3 . 1 and c o r o l l a r y

2.2.7.

P r o D o s U 2 2 L L The no rma l v a l u e d v a r i e t y N i s i n F.

Proof. N i s t h e l a r g e s t p r o p e r v a r i e t y o f l - g r o u p s

and 8 i s a l a t t i c e au tomorphism o f L .

P i c p p p s i i i n n 2.3.4, The r e p r e s e n t a b l e v a r i e t y R i s i n f .

Proof, L e t G b e a t o t a l l y o r d e r e d g r o u p , t h e n G R i s

a l s o t o t a l l y o r d e r e d whence G R E R and G = (GRIR E R ~ .

Thus R~ E. R and t h e r e f o r e 'by C o r o l l a r y 2 .2 .3 R R = R . P r o p o s l a

. . 2.7.5. The weakly a b e l i a n v a r i e t y W i s i n F.

Proaf. W i s d e f i n e d by t h e l a w y - l ( x v e ) y - < ( x v e l 2 . I n any t -g roup G ,

yW1(x v e ) y ( ( x v e l 2 f o r a l l x , y E G

( x /P e l 2 - < R y - l ( x R e ) y f o r a l l x , y E G

I y - ' ( x e l y ] - ' c R [ ( x hR e l 2 ] - ' f o r a l l x , y E G -

y - l ( x - l v R e ) y c R (x- I v R - e l 2 f o r a l l x , y E G

y - l ( z vR e ) y < R - ( z v e l 2 f o r a l l z , y E G .

I t t h e n f o l l o w s t h a t G E W i f and o n l y i f G ~ . E w and

t h e r e f o r e W E F,

R e c a l l t h a t N o i s t h e f r e e n i l p o t e n t c l a s s 2 g r o u p on

two g e n e r a t o r s a and b o r d e r e d s o t h a t a >> b >> [ a , b l > e ,

and t h a t Van (No) i s a c o v e r o f A .

ih.U2sition L.3AA (Medvedev C171) I f V i s a v a r i e t y o f

n i l p o t e n t [ -groups w i t h A V , t h e n Vm(N0) 6 V .

C o r o l l a r y 2 .3 .7 . ItLt.(No) i s i n F .

P r o o f . S i n c e N o i s n i l p o t e n t V(h.(NO) i s a v a r i e t y o f

R n i l p o t e n t P - g r o u p s and h e n c e s o i s Vn,y(NO) , and by

P r o p o s i t i o n 2 . 3 . 6 , vafi(NO) E ~ a l ( ~ ~ ) ~ . Then by C o r o l l a r y

2 2

F o r e a c h p r i m e p , t h e S c r i m g e r v a r i e t y S P '

d e s c r i b e d i n Example 1 . 7 , i s a c o v e r o f A . E a c h v a r i e t y

i s s o l v a b l e b u t n o t r e p r e s e n t a b l e . P

. . n 2.3.8. ( G u r c h e n k o v [ 8 1 ) . If V E L i s a

v a r i e t y o f s o l v a b l e 1 - g r o u p s w h i c h c o v e r s A a n d i s n o t a

v a r i e t y o f r e p r e s e n t a b l e 1 - g r o u p s , t h e n V = S f o r some P '

p r i m e p .

P r o p o s i t i o n 2.3.9. F o r a l l p r i m e s p , S i s i n P

h 2 Q L L e t p b e a p r i m e . S i n c e G a n d G~ h a v e t h e

same g r o u p s t r u c t u r e a h d f3 i s a n a u t o m o r p h i s m o f L , S P

i s a c o v e r o f A c o n t a i n i n g o n l y s o l v a b l e 1 - g r o u p s . A l s o

s i n c e 0 i s a n a u t o m o r p h i s m o f L , R R = R a n d S $ R, P

S p R $ ~ . T h e r e f o r e by P r o p o s i t i o n 2.3.8, S = S f o r P 9

some p r i m e q . F o r a n y n t h e i d e n t i t y x n y n = ~ " x " h o l d s i n

S and h e n c e a l s o i n Sn . n If p f q , t h e n b o t h t h e

i d e n t i t e s xPyP = yPxP a n d xqyq = y q x q h o l d i n S = S - P q '

a n d i t f o l l o w s t h a t xy = yx h o l d s , c o n t r a d i c t i n g t h a t

R S is n o t a b e l i a n . T h e r e f o r e S E S . P P P

PropQaALun . . 2 m . E v e r y q u a s i - r e p r e s e n t a b l e v a r i e t y l i e s

i n F.

Proof_. R e c a l l t h a t a q u a s i - r e p r e s e n t a b l e v a r i e t y , R U , i s

d e f i n e d by i d e n t i t i e s o f t h e f o r m [ ( u ) = e , w h e r e

l(u) = z* u-lz-u, u is a group word and the variable

z does not appear in u. l(u) = e is a law of RU if and

only if l(u-') = e is a law of RU . Let C be an l - g r o ~ p

and consider any substitution in G. Then

Thus G E RU if and only if G~ E RU, so that RU r 5

as required.

Since Reilly 11.91 has shown that there are uncountably

many quasi-representable varieties, the following corollary

is established.

Corolliary ?aC F has the cardinality of the continuum.

Section 4. Varieties Moved 'BY Q

In this section examples will be given of varieties

for which vR V

Let 0 < t 5 1 and consider G ~ ~ , where Gt is the

Feill-group. Let x = (0, -11 , y = (-1, 0). Then x(y < e

- R so that e <R y 5 x. Also,

1 [x, y l = (0, -11- (-1, o r h o , -IN-1, 0)

1 and l[x, yl/ = ( - - t+l , 0 A similar calculation gives

1 I [x, /[x, yllR]lR = ( - - 0). Hence, ( c + l )

( 1 [x, / [x, yl lR1l 9' = ( -L , 0) and ( t+l)

1 If 0 < t <p/q < I , then q/p - > 1 L x so that

L > P -P t+l and - > . Therefore

( t+l) ( t+l)

T h u s G~~ # U p / q . B u t by P r o p o s i t i o n 1 . 1 2 , G t E U p / q ' Hence UpIq R f Up,q. A l s o f o r r E R \ Q , 0 < r < 1 , G r c u r ,

U R h o w e v e r , a s a b o v e , GrRg r . T h e r e f o r e , U r + U r f o r

r E R \ Q , a n d t h e f o l l o w i n g r e s u l t i s t h e n e s t a b l i s h e d .

R P r o ~ o s i t i o n u * u r # Ur f o r a l l 0 < r < I , s o t h a t

h a s t h e c a r d i n a l i t y o f t h e c o n t i n u u m .

Medvedev [ I 7 1 d e s c r i b e d t h r e e c o v e r s o f t h e a b e l i a n

v a r i e t y . One o f t h e s e , v a t ( N O ) h a s b e e n shown t o b e

i n v a r i a n t u n d e r 8 . The n e x t r e s u l t s h o w s t h a t t h e d t h e r t w o

Medvedev v a r i e t i e s v ~ ( w + ) a n d vat(W') a r e moved

by 8.

Proaf. L e t x , y , z E W . I f x and y commute t h e n ( I )

c l e a r l y h o l d s . O t h e r w i s e I[x, y l l = ( F , 0) w h e r e i f r i s

t h e m a x i m u m ' e l e m e n t o f t h e s u p p o r t o f F , F ( r ) > 0. L e t

l z / = ( G , n ) w h e r e n 2 0. I f n = 0 t h e n z commutes w i t h

( F , 0 ) s o t h a t ( I ) a g a i n h o l d s . S o a s s u m e n > 0. Then

where the maximum element of the support of F'" is

n + r > r. Hence in this case, (F-",O) > (F, 0) and again (I) holds. Thus (I) holds in w+. However in (w+lR , let x = (0, 11, y = (F, 0) z = (0, -1) where

1 if i = 0

0 otherwise.

Then x-ly-'ry = (G, 0) where

G(i) = r-1 if i.1

0 otherwise.

where i = {-1 if i = O

1 if i = -1

0 otherwise.

2 7

But t h e n ( G , 0 ) < ( G I , 0 ) and ( G I , 0 ) < R ( G , 0 ) . Thus

(I) i s n o t v a l i d i n ( w + l R . The p r o o f o f ( i i ) is s i m i l a r t o t h a t o f ( i ) .

Us ing t h e f o l l o w i n g r e s u l t , i t i s p o s s i b l e t o i d e n t i f y

v ~ w + ) ~ and V a . t ( ~ - ) ~ .

. . 2 .4 .3 , (Medvedev E1711.I f V E L i s s u c h t h a t

A S V and e v e r y e l e m e n t o f Vis s o l v a b l e and r e p r e s e n t a b l e

t h e n Van(No) E V , Vak(w+) G V o r Vn?.(w') G V .

P r o o f . By P r o p o s i t i o n 2 .4 .2 , ( w + ) ~ # V ~ ( W + ) . But

+ R s i n c e v ~ & ( w + ) i s a c o v e r o f A , s o a l s o i s V&+L ( W ) , by

P r o p o s i t i o n 2.2.2. VLUL ( w + ) c o n s i s t s o f s o l v a b l e g r o u p s

and by P r o p o s i t i o n 2 .3 .4 , e v e r y e l e m e n t o f va;~ ( w + ) ~

i s r e p r e s e n t a b l e . I t now f o l l o w s f rom P r o p o s i t i o n 2 .4 .3

+ R t h a t V a ' 4 W = V a , + L ( N O ) O r ( w + ) ~ = Vm(W-). However

e v e r y 1 - g r o u p i n V f i , t ( N O ) i s n i l p o t e n t c l a s s 2 , b u t + R + R ( W ) i s n o t . T h e r e f o r e Vm(W ) # V a h ( N O ) and s o

+ R V & t ( W ) = V&'L(W-). Then v ~ / ? ( w - ) ~ = v ~ < w + ) ~ ~ ; V U ( W + )

by P r o p o s i t i o n 2.2.2.

The f o l l o w i n g r e s u l t , d u e t o F e i l [ 4 1 , i s u s e d t o

p r o d u c e an e n t i r e i n t e r v a l i n L which i s moved by 8 .

2 8

Pro~o-n . . 2.4.5. F o r an; p o s i t i v e i n t e g e r s , p , q , w i t h

0 < p /q 5 1, ( i ) Vm(NO ) 9 up/^' ( i i ) w + u p I q a n d ( i i i ) V ~ ( W + ) G UPlq .

The p r o o f o f t h e n e x t r e s u l t was s u g g e s t e d by A.M.W.

G l a s s .

P r o p o s i t i o n 2.4.6. The i n t e r v a l s [ Vm(W+), U1] a n d

[ V a h ( W * ) , U 9 a r e d i s j o i n t .

Proof, L e t V b e a v a r i e t y w h i c h l i e s i n b o t h i n t e r v a l s .

Then

v d w - ) . V a h (W+) 9 S v 5 U1.

w h i c h c o n t r a d i c t s P r o p o s i t i o n 2.4.5.

, C H A P T E R 3

Lex P r o d u c t s By The I n t e g e r s

G i v e n a n y two e - g r o u p s G a n d H t h e m o s t n a t u r a l

o r d e r t o c o n s i d e r on t h e i r p r o d u c t G x H i s t h e d i r e c t

p r o d u c t o r d e r : ( g , h ) > e i f a n d o n l y i f g > e a n d - -

h > e . However , i n t h e c a s e w h e r e H i s t o t a l l y o r d e r e d , - t h e r e i s a n o t h e r o r d e r w h i c h a r i s e s n a t u r a l l y , n a m e l y t h e

o r d e r : ( g , h ) ) e i f a n d o n l y i f e i t h e r

i) h > e o r i i ) h = e a n d g ) e . The p r o d u c t o f G

a n d H w i t h t h i s l e x i c o g r a p h i c o r d e r i s d e n o t e d by G t H.

A l t h o u g h by d e f i n i t i o n a v a r i e t y o f l - g r o u p s i s c l o s e d w i t h

r e s p e c t t o t a k i n g d i r e c t p r o d u c t s , i t h a s b e e n a n o p e n

q u e s t i o n a s t o w h e t h e r o r n o t a v a r i e t y o f - g r o u p s i s

c l o s e d u n d e r l e x i c o g r a p h i c p r o d u c t s e v e n i n t h e s i m p l e s t c a s e

when H = Z . S e v e r a l e x a m p l e s h i l l b e g i v e n t o a n s w e r

t h i s q u e s t i o n n e g a t i v e l y , b u t f i r s t some g e n e r a l t h e o r y i s

d e v e l o p e d .

S e c t i o n 1. The Lex P r o p e r t y .

D e f i n W Xu-a F o r a n y v a r i e t y V o f 1 - g r o u p s l e t

vL = ~ c n ( { G : Z j G E V l ) .

-3- If V = V a h ( { G i l i c I ) ) t h e n + =

.e vcn ( { G i x Z I i ~ 1 1 ) .

30

P r o o f , L e t G = V c ~ t ( {GI li E I ) ) . I t i s s u f f i c i e n t t o

show t h a t G k Z E Vah({Gi $ ' ~ l i € 1 ) ) . S i n c e

G E V C L Y ( { G ~ I ~ E I ) ) t h e r e e x i s t B < n A j w h e r e e a c h A j - ~ E J

is some G i and 2 -ep imorph i sm r : 8 J G . D e f i n e a ' :

B 2 2 - + G 2 Z by ( b , z) ( u b , z ) ; a' i s a n l - e p i m o r p h i s m .

t C o n s i d e r t h e ( - s u b g r o u p H o f n ( A x Z) g i v e n by i z J j

The map B : H L ~ B % Z g i v e n by ( a , z I j 3 a J , ~ )

i s a n l - i s o m o r p h i s m . T h u s t h e r e e x i s t s a n g - s u b g r o u p H o f

(A f Z) a n d a n l - e p i m o r p h i s m BQ' j : H - G C Z ,

jtzJ whence G f Z E V ~ ? ( { G ~ 2 Z 1 i E I]).

P r o D o ~ 3s1a3L F o r a n y v a r i e t i e s U. V. V i ( i E I ) ,

Proof.

( i ) V Vi = V m ( U V i ) . T h e r e f o r e by t h e p r e v i o u s i c 1 i ~ 1 p r o p o s i t i o n , (V v i f = ~ a h ( { G : Z I C E U $1) =

f €1 L

i ~ 1 = V Vai({G x Z I G E V i ) ) = V V i . i~ I i ~ 1

t h e n G E V i f o r a l l i E I a n d t h u s G % Z E V:

I, L f o r a l l i~ I . SO G': Z E T \ V i , whence ( n v i ) G

n v L i g I i ~ 1

~ E I i

( i i i ) L e t G E U V ; t h e n t h e r e e x i s t s a n [ - i d e a l H o f G

s u c h t h a t H E U a n d G / H E V . Then H (01 i s a n 4- C

€ - i d e a l o f G x Z and (G 5 Z ) / ( H 2 {O)) = G / H x Z . 4 C

F u r t h e r , H x (0) E U a n d G / H x Z E v L ; t h e r e f o r e 4- L G x Z E U I V ~ ) a n d (LIV) L~ U ( V ) .

I t i s c l e a r f r o m t h e d e f i n i t i o n t h a t f o r e a c h v a r i e t y V

L L o f l - g r o u p s , V 2 V. T h o s e v a r i e t i e s V f o r w h i c h V = V

a r e s a i d t o h a v e t h e Lex merLy. S i n c e a

l e x i c o g r a p h i c a l l y o r d e r e d p r o d u c t i s a d i r e c t p r o d u c t o f

g r o u p s , and t h e 1 - g r o u p o f i n t e g e r s , Z , b e l o n g s t o e v e r y

n o n - t r i v i a l v a r i e t y o f 1 - g r o u p s , v a r i e t i e s w i t h t h e l e x

p r o p e r t y i n c l u d e a l l n o n - t r i v i a l v a r i e t i e s o f i? - g r o u p s

w h i c h may b e d e f i n e d by i d e n t i t e s i n v o l v i n g o n l y t h e g r o u p

o p e r a t i o n s . T h u s , f o r e x a m p l e , A a n d L n ( n E N) h a v e

t h e l e x p r o p e r t y . The f o l l o w i n g C o r o l l a r y i s d u e t o

J . S m i t h 1231.

Coroll- lJAL L e t U b e a v a r i e t y o f L - g r o u p s and

V , V . ( i E I) b e v a r i e t i e s o f .!-groups h a v i n g t h e lex 1

p r o p e r t y , t h e n

(i > V Vi a n d n Vi h a v e t h e l ex p r o p e r t y . i& I ~ E I

'> ( i i ) U V h a s t h e l ex p r o p e r t y .

Proof ( i > By P r o p o s i t i o n 3 . 1 . 3 ( V v . ) ~ = V vi L 1

~ E I i e I = V Vi s i n c e e a c h Vi h a s t h e l e x p r o p e r t y . A l s o

~ E I

( i i ) A g a i n by P r o p o s i t i o n 3 . 1 . 3 , if V h a s t h e

L lex p r o p e r t y uv ( u v l L G u(v3 = U V w h e n c e ( U V ) = U V .

C o r o J J a r ~ 3 . 1 . 5 . F o r e a c h p o s i t i v e i n t e g e r n , ( ~ ~ 1 ' = A".

1 E x a m p l e s o f 4 - g r o u p v a r i e t i e s V f o r w h i c h V = V

a r e by n o m e a n s l i m i t e d t o t h o s e d e f i n e d b y g r o u p t h e o r e t i c

l a w s a n d t h e i r p r o d u c t s .

P r o ~ o s i L i a n ( J . S m i t h [ 2 3 1 ) N, R a n d W h a v e t h e

l e x p r o p e r t y .

S m i t h p r o v e d t h i s r e s u l t by c o n s i d e r i n g t h e d e f i n i n g

l a w s o f t h e v a r i e t i e s . F o r t h e n o r m a l v a l u e d v a r i e t y a n d t h e

r e p r e s e n t a b l e v a r i e t y , t h e f o l l o w i n g a l t e r n a t i v e p r o o f s a r e

a l s o s t r a i g h t f o r w a r d .

( i ) By G l a s s e t a 1 [ ? ' I , N = V A". T h u s by L

P r o p o s i t i o n 3 . 1 . 3 a n d C o r o l l a r y 3 . 1 . 4 , N L = ( V A n ) =

V A"L = V A" = N .

( i i ) R = vat ({GI G i s t o t a l l y o r d e r e d ) ) . T h e r e f o r e by

P r o p o s i t i o n 3 . 1 . 2 , R~ = V i v r ( { G ; 2 1 G i s t o t a l l y t

o r d e r e d ) ) . However i f G i s t o t a l l y o r d e r e d , G x Z

L i s a l s o t o t a l l y o r d e r e d a n d t h u s R G R whence

L R = R *

I

P r o D o s w LJ4L. Quasi-representable varieties have the

lex property.

P1'00f. Recall that quasi-representable varieties are

defined by laws of the form l(u) = e where l(u) = y+~u-ly-u,

u is a group word, and the variable y does not appear

in u. Thus it i.s sufficient to show that if G satisfies 4-

X(U> = e then so does G x Z . Let u + (uG, uZ),

C Y + (YG ,yZ) be any substitution for u, y in G x Z .

Since u is a group word, u - uG , y + yG is a substitution

for u, y in G. There are three cases.

since u + UG, y + y~ is a substitution for

u, y in G and G satisfies l(u) = e.

3 4

and t h e r e f o r e G f; Z s a t i s f i k s t h e i d e n t i t y e ( u ) = e

a s r e q u i r e d .

Carol- 3.1.8.. T h e r e a r e u n c o u n t a b l y many v a r i e t i e s o f

1 - g r o u p s w h i c h h a v e t h e l e x p r o p e r t y .

F u r t h e r e x a m p l e s o f v a r i e t i e s h a v i n g t h e l e x p r o p e r t y

c a n b e o b t a i n e d u s i n g t h e f o l l o w i n g r e s u l t .

&Q.Im~i_tion LJdL L e t G b e a n [ - g r o u p . If t h 3 r e

e x i s t s a n e l e m e n t x i n t h e c e n t r e o f G s u c h t h a t f o r a l l

g . ~ G , g < x n f o r some i n t e g e r n , . t h e n G ; Z E V m ( G ) .

m - L e t H = n G / c G . Then H E V u ( G ) . F o r e a c h

i=l i= 1 ~ E G l e t g = ( g , g , g ,... ) Z G E H . C l e a r l y (gig o G I i s

a n - s u b g r o u p o f H w h i c h i s l - i s o m o r p h i c t o G . L e t - - - - -

a = ( 1 x 1 , I x 1 2 ,... ) Z G E H ; t h e n f o r e a c h g o G , a g = g a

a n d a >> g. T h u s t h e [ - s u b g r o u p o f H g e n e r a t e d by

{;I U ( g E GI is .( - i s o m o r p h i c t o G Z a n d t h e r e f o r e

F o r a n y 1 - p e r m u t a t i o n g r o u p ( H , , and a n y i n t e g e r

n > 1 , H W r ( n ) ( Z , Z ) i s d e f i n e d t o b e t h e l - s u b g r o u p o f

t h e w r e a t h p r o d u c t HWr(Z,Z) c o n s i s t i n g o f t h o s e e l e m e n t s

( F , k) f o r w h i c h F ( i ) = F ( j ) if i E j (mod n ) . I f H

g e n e r a t e s a v a r i e t y V , t h e n ~ ( n ) A d e n o t e s t h e v a r i e t y

g e n e r a t e d by H W r ( n ) ( Z , Z ) . S u c h v a r i e t i e s were s t u d i e d by

I

Reilly and Wroblewski [211. In particular A(n)A is the

Scrimger variety S .

! h U l h ~ Y LILmA For each variety V and each n > 1, V(~)A has the lex property.

PrOOfL By Proposition 3.1.9, it is sufficient to show that \

each HWr(n)(Z,Z) contains a central element x such

that for each h E HWr(n) (Z,Z), h < xm for some integer m.

If E: Z -+ H is given by E(z) = e for all z E Z, then

(E, n) is such an element.

Another interesting application of Proposition 3.1.9 is

given .in the following Corollary.

h X Q ~ h L Y 3.AJ.l. For any variety V, vL has the lex property.

Proof. For any &-group G, (e, 1) is in the centre of

C ; Z, and for any (g, n) E G; Z, (g, n) < (e, n + 1 ) =

(e, 1) whence by Proposition 3.1.9 G ; Z; Z r vm(G; Z)

and the result follows.

A collection 6 of [-permutation groups (G,Q ) is said

to mimil: a variety V if 6 G V and whenever (H, A ) E V is a

transitive (-permutation group, A E A , wl(E), ..., wn(?) are

words and is a substitution in H, then there are (C, R ) &C

36 -

~ i J 2 a n d a s u b s t i t u t i o n g i n G s u c h t h a t I ) < h w j ( h )

if a n d o n l y if a ~ ~ ( 2 ) < a wj(s) f o r i , j = l , . . . , n .

T h i s d e f i n i t i o n i s d u e t o G l a s s e t a 1 [71 , a n d i t s h o u l d b e

n o t e d t h a t i f mimics V t h e n V a t r ( { G I G E G 1 ) ) = V .

prop^^ . 3 . 1 . 3 2 , ( G l a s s e t a 1 C71) The r e g u l a r

r e p r e s e n t a t i o n (Z, Z) o f t h e 2 - g r o u p o f i n t e g e r s mimics A.

U s i n g t h i s m i m i c k i n g p r o p e r t y o f t h e i n t e g e r s , i t is

p o s s i b l e t o s t r e n g t h e n C o r o l l a r y 3 . 1 . 1 1 .

EKQJLQ- 3LlA3A L e t O b e a c o i l e c t i o n o f t o t a l l y

o r d e r e d g r o u p s ( v i e w e d a s t h e i r r i g h t r e g u l a r r e p r e s e n t a t i o n s :

G = ( G , G I ) . If e mimics U t h e n f o r a n y t o t a l l y o r d e r e d

H E U and a n y C - g r o u p K , K f ; H E Vm({K; G I G E 0)).

Proof, S u p p o s e K % H d o e s n o t s a t i s f y t h e i d e n t i t y w ( x )

= V A w i j ( ; ) = e . We m u s t show t h a t w ( 5 ) = e is n o t a l a w I J

o f Vcvr ( { K i G I G E e 1. C o n s i d e r t w o c a s e s .

i ) I f w ( x ) = e i s n o t a l a w o f U , t h e n f o r some - -

G E e t h e r e i s a s u b s t i t u t i o n x -. g i n G s u c h t h a t C w(g) # e . C o n s i d e r t h e s u b s t i t u t i o n x -t 7 i n K x G

g i v e n b y y i = ( e l gi); w ( y ) = ( e l w(g)) # e l h e n c e - =+

K x G d o e s n o t s a t i s f y t h e i d e n t i t y w ( x ) = e w h i c h i s

t h e r e f o r e n o t a l a w o f Vad{K : G I G E Cl).

i i ) S u p p o s e w ( E ) = el i s a l a w o f U . S i n c e K H

d o e s n o t s a t i s f y w ( 2 ) = e , t h e r e i s a s u b s t i t u t i o n - C

x -+ y, X i y = k , h i i n K x H s u c h t h a t w(7) $ e . -

Now, w ( y ) = ( k , w(h ) ) f o r some k E K w h e r e x. -+ h is t h e

is a law o f U a n d H E U, w(h> = e a n d w ( y ) = ( k , e ) #

f o r some e # k E K . S i n c e H i s t o t a l l y o r d e r e d , e = w ( h ) =

V Awi j ( h ) = max min v i j (h ) . F o r e a c h i E I l e t I J I S c i = min wij(h) a n d l e t J ( i ) = ( j E J Iw i j (h ) = c i 1 . Then

S

max c i = e ; l e t IO = {i E 11 c i = e l * I n 3: G -

K x H , f o r e a c h i~ I , w ) = fi w i j ( y ) = J J (1) -

( A w i j ( Z ) , c i ) w h e r e x + 2 i s t h e s u b s t i t u t i o n i n . ~ ( i ) -

g i v e n by x i + k i . T h e r e f o r e w ( y ) = V A w i j ( y ) = - 7 - 1 J

V ( A w i j ( Z ) , c i ) = V( /i w i j ( k ) , e ) = ( V A . w i j ( i ) , e ) I J(:) I ~ J (1) I, J(1) T h u s , V A w i j ( L ) = k # e .

I, J ti', - -

Claim: T h e r e i s a s u b s t i t u t i o n x , g i n G f o r some

g E: G s u c h t h a t

b ) f o r a l l i r I , { j E JIw ( 9 ) } = J ( i ) w h e r e i j -

d i = Q W i j ( g ) .

C ) { i E I l d i = e } = IO.

- f-

If t h e c l a i m h o l d s c o n s i d e r t h e s u b s t i t u t i o n x + l i n K x G

g i v e n by x i + l i = ( k i t g i ) . ' ~ ( i ) = V w ( k ) , e ) = ( k , e ) I, ~ ( i )

# e s o t h a t K G d o e s n o t s a t i s f y t h e i d e n t i t y w(T) = e

a n d t h e r e f o r e w(?) = e i s n o t a l a w o f V m ({K f; G I G E e 1 ) .

p r o o f Qf claimr By h y p o t h e s i s w('ji) = e i s a l a w o f U , s o

e v e r y s u b s t i t u t i o n f o r w ( 2 ) i n G E e s a t i s f i e s c o n d i t i o n a ) .

H i s a t o t a l l y o r d e r e d g r o u p i n U , s o t h e r i g h t

r e g u l a r r e p r e s e n t a t i o n o f H i s a t r a n s i t i v e L - p e r m u t a t i o n

g r o u p i n U. S i n c e e mimics U a n d {wi ; ( j i ) ) i s a

f i n i t e s e t o f w o r d s , t h e r e e x i s t s G E e a n d a s u b s t i t u t i o n

x g i n G s u c h t h a t ( < w i j i f a n d o n l y

if w i j (g ) < w i t j t ( g ) . R e c a l l , f o r e a c h i E I

c = min w i j i J (b) a n d J ( i ) = { j E J I w i j ( L ) = c i l .

L e t d i = min w i j ( E ) a n d J t ( i ) = { j E J l w i j ( g ) = d i l . .T -

L e t jO & J ( i ) ( t h e n f o r j / J ( i ) c i = w i j (%) < w i j ( a

a n d by t h e m i m i c k i n g p r o p e r t y d i < w . ( g ) < w i j ( g ) i J,

whence j k J f ( i ) a n d J t ( i ) E. J ( i ) . On t h e o t h e r h a n d -

if j l E J f ( i ) t h e n f o r j k J i , d i = w i j t ( g ) < - -

w ( g ) a n d b y ' t h e m i m i c k i n g p r o p e r t y c i < w i j t ( h ) < W . . ( h ) i j 1 J

whence j k J ( i ) a n d J ( i ) G J t ( i ) T h u s J(i1 = J t ( i )

a n d c o n d i t i o n b ) i s s a t i s f i e d .

Now f o r e a c h i E I , l e t j ( i ) E J ( i ) a n d l e t

I f i E I O t t h e n w i ( h ) < w i ( h ) = e s o t h a t by t h e m i m i c k i n g 0

39

p r o p e r t y w i ( i ) < w.(g) < e,' whence i E 1' a n d I t IO. lo

S i m i l a r l y l e t i t E I , if i E I t t h e n w i ( i ) < w i t ( i ) = e -

a n d by t h e m i m i c k i n g p r o p e r t y w i ( h ) < w i t (h) < e , w h e n c e

i c IO and IO G 1'. T h u s I = I t a n d c o n d i t i o n c ) i s

s a t i s f i e d , a n d t h e c l a i m i s p r o v e d .

-CoroU L e t A b e a n a b e l i a n t o t a l l y o r d e r e d

g r o u p . F o r a n y L - g r o u p G , Vm(G ; A ) = V m ( G % Z ) .

Proof, ( Z , Z ) mimics A .

S e c t i o n 2 . V a r i e t i e s W i t h o u t The Lex P r o p e r t y .

A l l t h e v a r i e - t i e s c o n s i d e r e d s o f a r h a v e t h e l e x

L p r o p e r t y . However t h e r e a r e v a r i e t i e s f o r w h i c h I/ 2 0. =I=

The n e x t r e s u l t s g i v e e x a m p l e s o f s u c h v a r i e t i e s .

P~UDQ- 3 . 2 The s o l v a b l e Medvedev v a r i e t i e s ,

V L ~ ( w + ) a n d V W , d o n o t h a v e t h e l e x p r o p e r t y .

Proof,

i.1 The i d e n t i t y l z l - ' / [ x , y l l l z I - > I [ x , y l l h o l d s i n

v ~ ( W + ) by P r o p o s i t i o n 2 . 4 . 2 . C o n s i d e r W + f Z a n d l e t 47

x a n d y b e a n y two non-commuting e l e m e n t s o f W+ x Z ,

t h e n l [ x , y l l h a s t h e f o r m ( ( F , O), 0 ) w h e r e i f r i s

the maximum element of the 'support of F, F(r) > 0. Let z

be the element , - 1 , 1 . Then

Now the maximum element of the support of F' is r - 1 < r and therefore in w+; 2 , ((F1, O), 0) < ((F, 01, 0).

ii) Van (W-) satisfies the identity 1z1-l [[x, yll/zl (

I[x, yll . Consider W - % Z and let z = , - 1 1) and

I [x, ylI = ((F, 01, 0) where for s the minimum element of

the support of F, F(s) > 0. Then lzl-' l[x, yl/ lzl = ((F1,O),O)

and sinci the minimum element of the- support of F1 is

Proposition 3.2.2. The nilpotent Medvedev variety, Vm(NO),

does not have the lex property.

Recall No is the free nilpotent class two group on

two generators a, b, ordered so that a >> b >> [a, bl > e.

4 1

G l a s s and R e i l l y ( u n p u b l i s h b d ) h a v e shown t h a t N o s a t i s f i e s

t h e t h e f o l l o w i n g i d e n t i t y

C However No x Z d o e s n o t s a t i s f y ( * I , f o r l e t x l = ( a b , 0 1 ,

-1 1 f x t = ( a , 1 1 , x 3 = ( a b- , 2 k N 0 x Z . Then e - < x l < _ x 2 5 x 3 ,

[ x 3 , x l l = e w h i l e [ x 3 , x21 = ( [ a , b l , 0 ) = [ x 2 , x l l , 4

t h u s 1 x 3 , x 2 I " [ x 2 , x l l > [ x 3 , x11 whence No x Z L

d o e s n o t s a t i s f y ( * ) and vm(N0) =J V a h ( N O ) . #

P r o p o s i t i o n 3.2.3. The F e i l v a r i e t i e s Lf ,, O < r < 1 , d o n o t

have t h e l e x p r o p e r t y .

a+ &QQ~. C o n s i d e r Gr x Z and l e t x = 1 , - 1 , 1 ) and

y = ( ( 0 , 1 1 , 0 ) . Then x 2 y - > e . -1 -1 C x , Y I = x y x y

Thus f o r i n t e g e r s p , q w i t h p 5 q , p / q F 1 < r + 1 and

+ t h e r e f o r e p / ( r + 112 < q / ( r + 1) whence i n G r x Z

C T h e r e f o r e G r x z I! u p / q . T h u s i f r is r a t i o n a l

G r x Z Up. I f r i s i r r a t i o n a l Ur = n ~ > p / q > r ~ 1 9

t. L and c l e a r l y G r x Z d U r . I n e i t h e r c a s e U r S U r a n d t h e

F e i l v a r i e t i e s t h u s d o n o t h a v e t h e l e x p r o p e r t y .

C o r o l l a r y 3.2.4. T h e r e a r e u n c o u n t a b l y many v a r i e t i e s w h i c h

d o n o t h a v e t h e lex p r o p e r t y .

Hav ing s e e n t h a t t h e r e a r e v a r i e t i e s V w h i c h d o n o t

h a v e t h e l e x p r o p e r t y , i t i s n a t u r a l t o A n y e s t i g a t e by how I

much V' i s l a r g e r t h a n V . i

A

P r o o f , F o r a n y f u n c t i o n F: Z + Z l e t F: Z + Z b e A

d e f i n e d by F ( z ) = F ( - z ) . Then r i s t h e minimum e l e m e n t o f

t h e s u p p o r t o f F i f a n d o n l y i f -r i s t h e maximum e l e m e n t

o f t h e s u p p o r t o f F . C o n s i d e r t h e 1 - s u b g r o u p

K = { ( ( F , z ) , - z ) I F: Z + Z , z E Z) o f W + Z a n d l e t A

a : K + W' b e d e f i n e d by a ( ( F , z ) , - z ) = ( F , - z ) .

C l e a r l y a i s 1-1 a n d o n t o .

L e t ( ( F , z ) , - z ) e K b e p o s i t i v e . If -z > 0, t h e n i n W'

( F , - 2 ) > e . O t h e r w i s e z = 0 a n d F ( r ) > 0 w h e r e r i s

t h e maximum e l e m e n t o f t h e s u p p o r t o f F . Then t h e minimum A A

e l e m e n t o f t h e s u p p o r t o f F 1s -r a n d F ( - r ) = F ( r ) > 0

and i n W - , (a, - z ) > e . T h e r e f o r e f o r ( ( F , z ) , - 2 ) E K ,

p o s i t i v e , a ( ( F , z ) , - z ) i s p o s i t i v e . a i s t h u s a n C

l - i s o m o r p h i s m o f K o n t o W- whence W- V U A ( W + x Z )

= v ~ ~ ( w + ) L . A s i m i l a r a r g u m e n t s h o w s t h a t W+ E ~ W ' ( w - 1 ~ .

4 4

I

L P r o o f . By t h e above P r o p o s i t i o n W ' E v ~ ( w + )

L T h e r e f o r e V a t ( W - ) G V ~ A ( w + l L L . But by C o r o l l a r y 3.1.11

+ L V o i l ( ~ + ) ~ ~ V m ( W ) whence V C L + L ( W - ) ~ G VCVL(W+)~ . S i m i l a r l y

V a n (w+ lL G V a n ( W 0 ) and t h e r e s u l t f o l l o w s .

C o r o l l a r y 3.2.7 + L V m ( W 3 V n t ( W + ) v V G L ( W - ) and

h e n c e i s n o t a c o v e r o f V a t (w+) .

4- Proof. S i n c e . W+ x Z i s an o -g roup , by M a r t i n e z [ 1 6 1 ,

W+ f Z E VWL(W+) v V a n ( W - 1 i f and o n l y i f W+ f Z

e v ~ ( w + ) L) v ~ * ( w - ) . ~ u t VWL (W+I = v ~ ( w - 1 < v a t ( w + ) , + L + L V a t i ( W - 1 . T h e r e f o r e V m ( W ) c VOL(W+) v V m ( W - ) V m ( W

f whence V N I ( W + ) ~ i s n o t a c o v e r o f v ~ ( w + ) .

L I n P r o p o s i t i o n 3.1 .3 ( i i ) i t was s e e n t h a t ( V i ) c_

A

i~ I n vi: The above example shows t h a t even f o r t h e i ~ l

i n t e r s e c t i o n of two v a r i e t i e s t h i s c o n t a i n m e n t c a n b e p r o p e r :

L ( V N ~ ( W + ) A VMW-) f = A = A c V a ' i ( ~ + t Z )

=f + L L = V n t ( W ) r\vah(w-) . T h i s p r o v e s t h e f o l l o w i n g :

c Q u u L x Y 3 i L . 4 ~ The mapping V u vL i s n o t a l a t t i c e

homomorphism.

R e c a l l t h a t f o r any v a r i e t y o f 1 - g r o u p s V , v R i s t h e

v a r i e t y o b t a i n e d by r e v e r s i n g t h e o r d e r o f a l l t h e 1 - g r o u p s

45

i n V . A l t h o u g h v and V' a r e b o t h o b t a i n e d f rom V by

c h a n g i n g t h e o r d e r o f 1 - g r o u p s i n V , V = V ' i s n e i t h e r a L n e c e s s a r y n o r a s u f f i c i e n t c o n d i t i o n f o r V =- V .

L ExamDle 3.7.9. L e t V = U& t h e n V = V w h i l e

Prop- . . Li'LlL F o r e a c h v a r i e t y V o f 1 - g r o u p s

v L R - - ,,RLe

Proof. I t i s s t r a i g h t f o r w a r d t o show t h a t t h e mapping

R ( G 2 z ) ~ + G x Z g i v e n by ( g , n ) w (g,-n) i s an

1 - i s o m o r p h i s m . The r e s u l t t h e n f o l l o w s s i n c e R L R + vLR = V U ( { ( G ? z ) ~ / G E G } ) w h i l e V = VU ( { G x Z I G E G } ) .

I t h a s been s e e n t h a t f o r a v a r i e t y V w i t h vL $ V , L

V need n o t b e a c o v e r o f V . By c o n s i d e r i n g t h e F e i l g r o u p s

L G t , 0 < t < - 1 i t w i l l b e shown t h a t t h e i n t e r v a l [ V , V I

may b e v e r y l a r g e .

P r o p o s i t i o n 3.2.12. F o r t h e F e i l g r o u p s G t , 0 < t < 1 , - ~ n r ( ~ ~ ) ~ ~ = v ~ ( G ~ ) ~ .

P r o o f . C o n s i d e r cx : G t R -+ C t x Z g i v e n by ( r , n ) H

r , n , - 1 . C l e a r l y a i s 1-1.

a [ ( r , n ) ( s , m > l = a ( r + ( l n s , n + m ) t + l

+ T h u s i s a n 1 - e m b e d d i n g o f G t R i n t o G t x Z , w h e n c e L

G t R E V d G t ) a n d by C o r o l l a r y 3 . 1 . 1 1 , G t R V a k ( G t ) .

L T h e r e f o r e Vm ( G t l R L~ V a t ( G t ) . S i m i l a r l y 0 : G t + ctR: Z

g i v e n by ( r , n ) ~ - , ( ( - r , - n ) , n ) i s a n 1 - e m b e d d i n g w h e n c e G t ,

a n d t h u s a l s o G t % Z E VOA ( G t I R L a n d L RL V c d G t ) 5 V m ( G t )

L T h e r e f o r e V L - U L ( G ~ ) ~ ~ = Vel ( G t ) .

t h e n f o r 0 < s , t < - 1 Vs c_ Vt i f a n d o n l y i f s ( t . R F u r t h e r f o r 0 < s < t - < 1 , V 1 v V s c + V 1 v V t s i n c e

t h e o - g r o u p G t R i s i n V1 v Vt b u t i s i n n e i t h e r

V 1 n o r VsR. U s i n g P r o p o s i t i o n 3 . 2 . 1 2 , i t i s e a s i l y

R L L s e e n t h a t V t = V t .

C_orollarv- L

[ V , V I I may b e u n c o u n t a b l e .

Proof L e t V = V v L v ~ / ~ ~ . hen V = ( V 1 v V l I 2 R ) L = 1 1 L

V 1 v v ~ / ~ ~ ~ = V 1 v V ~ / ~ ~ = V 1 . F o r 1 / 2 < t 5 1 R L V Z V l v v t R S ( V 1 v Vt ) = V 1 = V L . Thus

{ V v f R l 112 ( t 2 1 ) i s an u n c o u n t a b l e c o l l e c t i o n L o f v a r i e t i e s i n t h e i n t e r v a l [ V , V 1.

S e c t i o n 3 . Laws f o r vL .

I n g e n e r a l f o r a v a r i e t y o f -! - g r o u p s d e f i n e d by a s e t

o f g e n e r a t o r s , n o t much i s known a b o u t t h e l a w s d e f i n i n g t h a t

v a r i e t y . The e x c e p t i o n i s t h e a b e l i a n v a r i e t y , A , which

may b e d e f i n e d b o t h i n t e r m s o f a g e n e r a t o r ( t h e .[ -group o f

i n t e g e r s , Z) and a d e f i n i n g l a w ( x - Iy-1 xy = e ) . Thus f o r a

v a r i e t y vL # V , i t i s n o t s u r p r i s i n g t h a t t h e d e f i n i n g

l a w s o f vL a r e n o t known. However t h e f o l l o w i n g r e s u l t s L g i v e some i n f o r m a t i o n a b o u t t h e l a w s o f V .

- F o r e a c h t -group word w ( x ) = V A w i j ( x ) , and e a c h

- I J s u b s t i t u t i o n x i n Z , d e f i n e a new 1 -g roup word w i ( ? )

a s f o l l o w s : f o r e a c h i E I l e t y i = min w ( ? I , l e t i j -

J ( i ) = {j E J I w i j ( z ) = y i ) and l e t IO = {i E I Iyi = max y i l

Prop- . . 3.1..1. ~ ( 2 ) = ' e is a law of vL if and, only if

- - w(x) = e is satisfied by Z and. for each substitution x + z

in Z, wZ(f) = e is a law of V . hXQf. Let C E V and consider a substitution 2 4 E,

C X. -t 1 ki = ( g i g zi) in G x Z. Corresponding to this there

- - - are substitutions x -+ ?, x + zi in Z and x + g, xi + i g i

in G. Now w(E) = w ) , w . Therefore, w(c) = e if

and only if w(?) = 0 and w 7 ( Z ) 5 e. The result then

follows.

In order to clarify the equation of the following I

proposition, I [x, yl I will be written Ex, y j .

f- Proof. Let t < - p/q and x , y E Gt x Z. Let x = ((r, n),m).

Bx, y D = ((k, O), 0) for some 0 < k E R . Then

i i ) If n L 1, t h e n ( t t l l m n = ( ? > 1 .

" 7'- 1 , T h u s 11 - ( t : , ~ n l = ( t + l - 1 t

= 4.

a n d i f f o l l o w s t h a t

n i i i ) If n < - 1 , t h e n ( ' 1 > 1 - t+l t

s o t h a t 1 1 - ( t ) n / = ( L f - 1 > LA. t c l - I =

t + l - t 1. t

> ((0, 0 1 , 0 )

s i n c e t > p / q a n d t h u s q / t - p / t 2 > 0 .

Roughly s p e a k i n g t h e above r e s u l t i s o b t a i n e d u s i n g t h e

f a c t t h a t G t s a t i s f i e s t h e law E x , nx,yBP t x , yJq

f o r x 2 y e i f and o n l y i f t F p /q . When c o n s i d e r i n g an f

e l e m e n t x of G t x Z , t h e r e i s no means t o e n s u r e t h a t t h e c.

G t component o f x i s p o s i t i v e . However, s i n c e G t x Z i s

t o t a l l y o r d e r e d , e i t h e r x o r x-' w i l l h ave a p o s i t i v e G t

component . Us ing t h i s , and n o t i n g t h a t f o r an o-group

a 4 b = e i f and o n l y i f e i t h e r a = e o r b = e , t h e o r i g i n a l

law o f G t c a n b e m o d i f i e d t o p r o d u c e a l aw s a t i s f i e d by f

G t X Z .

L I t i s c l e a r t h a t f o r e a c h v a r i e t y V , V G V A

The l a s t r e s u l t o f t h i s s e c t i o n g i v e s a c o n d i t i o n u n d e r

which t h i s c o n t a i n m e n t w i l l b e p r o p e r .

Lemma L3-A L e t H b e an 1 - g r o u p . Any e l e m e n t of

H W r ( Z , Z) o f t h e form ( K , 0 ) i s a commuta to r .

P r o o f . C o n s i d e r ( K , 0 ) E H W r ( Z , Z ) . D e f i n e F: Z + H

by F ( 0 ) = e and f o r e a c h p o s i t i v e i n t e g e r n , F ( n ) = - 1 F ( n - 1 ) K ( n - 1 ) and F ( -n ) = F(-n + 1 ) K(-n + 1 )

L e t E : Z -+ H be d e f i n e d by E ( z ) = e f o r a l l z c Z and l e t

x = ( F , 0 1 , y = (E, -1 ) . Then I x , y l =

(F,o)-~(E,-~)-~(F,o)(E,-~) = ( L , O ) where L: 2 + H i s g i v e n

F ( z ) - ~ F ( z ) K ( z ) = K ( z ) . T h e r e f o r e ( K , 0 ) = [ x , y l a s r e q u i r e d .

P r o p o s i ? i u m . . Urn L e t ' V be a v a r i e t y o f ! - g r o u p s

g e n e r a t e d by a n & - g r o u p G s u c h t h a t t h e d e r i v e d s u b g r o u p

L G 1 o f G i s a n 1 - s u b g r o u p . If Van ( G 1 ) 3 V , t h e n V 5 VA.

h 2 Q L v A = V ~ ( G W r ( Z , 2 ) ) s o i t s u f f i c e s t o f i n d an

e -g roup word w ( x l , ..., x 1 s u c h t h a t w ( x , , . . . , x 1 = e n n i s s a t i s f i e d by G t Z b u t n o t by G W r ( Z , Z ) . Now by

Lemma 3 .3 .3 t h e c o m m u t a t o r s o f G Wr(Z, Z ) a r e e x a c t l y t h e

e l e m e n t s o f t h e form (F, 0) and t h u s form a n !-subgroup

i s o m o r p h i c t o TG, and s o s a t i s f y a l l t h e l a w s o f G . ZEZ

S i n c e vm(G1)c v = van ( G ) , t h e r e i s a l a w w ( x l , . . . , x ) = e ? n

o f G 1 n o t s a t i s f i e d by G . T h u s , w ( x l , . . . , x ) = e n i s n o t a l a w o f G and w ( [ x l , y l l ,..., [ x n , y n l ) = e is n o t

z &Z a l a w o f G W r ( Z , Z ) . However w ( [ x l , y l ] , ..., [ x n , y n l ) = e i s

a l a w o f G t Z s i n c e f o r any s u b s t i t u t i o n i n G f Z

[ x i , y i I H ( [ g i , h i ] $0) € G t x {O) C G 1 , and w ( x , , . . . , x n ) = e

i s a l a w o f G ,

C l e a r l y 1 - g r o u p s G f o r which G t i s a n l - s u b g r o u p

i n c l u d e a l l t o t a l l y o r d e r e d g r o u p s . Among t h e o - g r o u p s

a l r e a d y c o n s i d e r e d a r e examples f o r which v m ( G 1 ) 5 Vm(G).

I n p a r t i c u l a r i t i s e a s i l y s e e n t h a t w+, W- and t h e F e i l

! - g r o u p s a l l h a v e d e r i v e d s u b g r o u p s which a r e a b e l i a n ,

w h i l e none o f t h e s e !-groups i s a b e l i a n i t s e l f . However

n o t a l l o - g r o u p s have t h i s p r o p e r t y a s i s shown by t h e

5 2

following example due to Chehata C31. Chehatats o-group C

is defined as the group of order preserving permutations of

R whose graphs consist of a finite number of linear pieces

and have bounded support. An element is positive if its left-

most non-identity piece has slope greater than 1. Chehata

showed that C is simple as a group and since it is not

abelian, C I = C and thus Vcur(C1) = Vatt (C) .

CHAPTER 4

Uncountable Collections Of Varieties Of l-groups

The lattice L of varieties of e-groups was first

shown to be uncountable by Kopytov and Medvedev [I31 and

independently by Reilly 1191 and Feil C51. Reilly proved the

existence of an uncountable family of pairwise incomparable

varieties of e-groups each containing the representable

variety R , while Feil constructed two uncountable towers

of representable varieties. Thus the breadth and height of

L both have cardinality of the continuum.

For a variety V of [-groups, the lattice of subvarieties

of V will be denoted by L(V). Using Feilfs varieties

described in Example 1.11, it will be shown that it is

possible to construct a sublattice of L(R A2) isomorphic

to I x I, where I is the unit interval (0, . I ) .

The following lemma, due to Martinez C161 is required.

Lemma 4.1. If G is an o-group and U, V are varieties

of €-groups with G E U v V , then G E U U V .

Proposition 4.2. s = {Ur v Us I 0 < r,s < 1) is a

sublattice of Leo A 2 ) that is isomorphic to I x I.

Proof. It is first shown that S is a sublattice of

L( u ~ A ~ ) . By Proposition 1.12, (Ur v Us) v (U v US!) = r f

rn v WS,c S where rn = max{r, rl} and sn = maxis, sf}.

By P r o p o s i t i o n 1 . 1 2 , a n d n o k i n g t h a t L ( R f3 A ~ ) i s

d i s t r i b u t i v e , i t i s s e e n t h a t

( u r V Ws)n(Ur, v U s , ) = ( u r n u r , ) v (urn(",) v

v ( w ~ ~ w ~ , ) = U r n v A v A v WSn = U r n v us. E s w h e r e r n = m i n { r , r t l a n d sn = r n i n { s , s t ) . T h u s S i s a

s u b l a t t i c e o f L(R A A ~ ) .

Now l e t a : I x I -+S b e g i v e n by ( r , s ) a = U r C l e a r l y a i s o n t o . r , s r , s r 5 r t a n d s < _ s t

3 Ur E U a n d Us C- W r t s ' 3 U r v Us C u r t v ' S t , a n d t h u s &

i s o r d e r p r e s e r v i n g . By Lemma 4 .1 a n d P r o p o s i t i o n 1 . 1 2 ,

G t E U i f a n d o n l y i f t 5 r a n d H t ~ U r v s r v u s i f

a n d o n l y i f t < s . If r , s # r , s t e i t h e r r # r t -

a n d w i t h o u t l o s s o f g e n e r a l i t y r < r t a n d C r , E U r t v W s t \

U r v W s , o r s # s t a n d w.1 .o .g . s < s t a n d H s , E

u r t v u s , \ u r v u s . T h e r e f o r e , a i s 1 - 1 , a n d h e n c e S i s

i s o m o r p h i c t o I x I .

I x I c o n t a i n s a s u b s e t o f 2'0 p a i r w i s e i n c o m p a r a b l e

e l e m e n t s . Hence :

ihulhn! 4.7. T h e r e i s a n u n c o u n t a b l e f a m i l y o f p a i r w i s e

i n c o m p a r a b l e v a r i e t i e s w i t h i n n A ~ .

F o r 0 < r < 1 , l e t Tr = ( U r v Ws I 0 < s < 1 1 ,

t h e n Tr i s a n u n c o u n t a b l e t o w e r o f v a r i e t i e s w i t h i n R n A2

q n d f o r r f s , Tr $ Ts . T h u s : '!

lhLQuaU 4 - 4 , T h e r e ' i s a n u n c o u n t a b l e c o l l e c t i o n o f

u n c o u n t a b l e t o w e r s o f v a r i e t i e s w i t h i n A2.

I t s h o u l d b e n o t e d t h a t w h i l e S = I u v W 1 0 < r , s < 1 1 r s 2 i s a s u b l a t t i c e o f L(R nA ) , i t i s n o t c o m p l e t e . I n f a c t

t h e t o w e r T = { U r 1 0 < r < 11 i s n o t c o m p l e t e , f o r

c o n s i d e r V = n u 1 >m/n>p/q m/n v c a n n o t b e d e f i n e d by a

s i n g l e e q u a t i o n w h e r e a s U p /q c a n , t h u s V C U p / q

I f r > p / q t h e n t h e r e a r e i n t e g e r s m , n s u c h t h a t

r > m/n > p /q a n d V G U c u r . =P If r < p / q t h e n

'r S U p / q * T h e r e f o r e V $ T a n d T i s n o t c o m p l e t e .

F o r e a c h p r i m e p S c r i m g e r [ 2 3 1 c o n s t r u c t e d a n L-group

v a r i e t y S t h a t c o v e r s A a n d i s c o n t a i n e d i n A ~ , w i t h t h e P

p r o p e r t y t h a t i f V i s a n y r e p r e s e n t a b l e v a r i e t y ,

V f l Sp = A . S i n c e S p c o v e r s A , S p n S = A f o r p , q 9

d i s t i n c t p r i m e n u m b e r s . F o r e a c h p r i m e p c o n s i d e r t h e

c o l l e c t i o n o f v a r i e t i e s L = { U r v Ws v S 1 0 < r , s < 1 1 . P P

I t i s s t r a i g h t f o r w a r d t o v e r i f y t h a t L i s a s u b l a t t i c e o f P

2 L(A ) i s o m o r p h i c t o I x I . F u r t h e r m o r e i f p # q , t h e n

L~ r\ Lq = s i n c e f o r V E V w h e r e a s f o r a l l L ~ ' P

0 < 1 < 1 , S p n ( U r v W s v Sq) = ( S p n U p ) v ( S nW ) P s

v ( S P A S ) = A . q

4 . 5 L T h e r e i s a c o u n t a b l y i n f i n i t e c o l l e c t i o n o f

2 s u b l a t t i c e s o f L(A ) i s o m o r p h i c t o I x I .

R e c a l l t h a t A 2 i s t h d v a r i e t y o f a l l [ - g r o u p s w h i c h

h a v e a n a b e l i a n [ - i d e a l H s u c h t h a t G / H i s a b e l i a n .

G l a s s , H o l l a n d a n d M c C l e a r y 171 h a v e shown t h a t i s

L e t A g p b e t h e v a r i e t y o f t h o s e [ - g r o u p s w h i c h a s g r o u p s -.

2 a r e i n t h e g r o u p v a r i e t y A . T h u s A gp c o n s i s t s o f a l l

- g r o u p s C w h i c h h a v e a n a b e l i a n n o r m a l s u b g r o u p H s u c h

t h a t t h e g r o u p G / H i s a b e l i a n , a n d i s d e f i n e d by t h e l a w

[ [ x , y l , [z, w l I = e

S i n c e a n L - i d e a l H o f G i s n e c e s s a r i l y a n o r m a l

s u b g r o u p o f C , A A2 gP ' b u t how l a r g e i s t h e i n t e r v a l

Lemma 4 .6 . t. F o r 0 < r , p / q < 1 , Qr x G r AU i f a n d p / q

o n l y i f r < p / q . -

Proof- C o n s i d e r G r : G r a n d l e t x = ( ( 1 , - 1 1 , ( 1 , 0 ) )

a n d Y = 0 , 1 , 0 0 Then x 2 y 2 e , I [ x , y l l q =

( q / ( r + 1 , 0 , 0 , 0 a n d l [ x , K x , y l / l l P = ( ( p / ( r + I ) ~ , o ) , ( o , o ) L

S i n c e p / q 5 1 a n d r > 0 , p / q < r + 1 a n d t h u s

p / ( r + 1 1 2 < q / ( r + 1 ) s o t h a t ( [ x , l [ x , y l < I [ x , y l P +

whence G r x G r h u p I q . C The o n l y n o n - t r i v i a l a b e l i a n [ - i d e a l o f G r x G r i s

t I = r , 0 ( 0 , 0 ) ) l r E R I a n d (Gr: G r ) / 1 2 Z x G r , C

t h u s G r G r e A:l p / q

i f a n d o n l y i f Z x G r e [ I p l q .

5 7 +

I f r > p/q - t h e n Z x G r b' UpIq 4

s i n c e Z x G r h a s an

i? - subgroup i s o m o r p h i c t o G r and Gr ' . ~ f r < p/q

i t i s s t r a i g h t f o r w a r d t o v e r i f y t h a t Z ; G r s a t i s f i e s t h e

l a w s o f U p /q

F o r 0 < r < 1 , r i r r a t i o n a l , U r = U and t h u s p l q > r P/q

n u I = A"r = A ( p / q > r " A U p / q * I t f o l l o w s t h a t f o r ~ 1 9 > r

r > p/q G r x G r € AUr \ A U p i q . I t i s c l e a r t h a t

r \ A ~ and t h e f o l l o w i n g r e s u l t i s o b t a i n e d .

P r o p o s i t i o n 4.7 . { AUr nAZgpI 0 < r < 1) is an

u n c o u n t a b l e t o w e r o f v a r i e t i e s o f L - g r o u p s c o n t a i n e d i n t h e

2 2 i n t e r v a l [ A , A g p l .

I n a s i m i l a r way, by c o n s i d e r i n g t h e o -g roups Hr Hr 2 i t i s p o s s i b l e t o show t h a t { A W r A g p / O < r < 1 ) i s a

s econd u n c o u n t a b l e t o w e r o f v a r i e t i e s i n t h e i n t e r v a l

The f o l l o w i n g P r o p o s i t i o n i s t h e n e a s i l y e s t a b l i s h e d ( c . f .

P r o p o s i t i o n 4 . 2 ) .

P r o p o s i t i o n 4.8. ( ( 4 ~ v A W ~ ) A gP

I 0 < r , s < 11 i s

2 a s u b l a t t i c e o f [ A , A 2~ i s o m o r p h i c t o I x I . gP

lbLQlkau The h e i g h t and b r e a d t h o f [ A ~ , 1 gP b o t h have t h e c a r d i n a l i t y o f t h e c o n t i n u u m .

Having c o n s t r u c t e d a s u b l a t t i c e o f L i s o m o r p h i c t o

I ~ , i t i s o f i n t e r e s t t o c o n s i d e r w h e t h e r l a r g e r powers

of t h e u n i t i n t e r v a l I c a n b e found a s s u b l a t t i c e s o f L .

Fo r 0 < s 5 1 , l e t V, = Vaz.({Ct 1 0 < t I s ) ) , and l e t

v; = vm({Gt t Z I 0 < t 5 s 1 ) . By P r o p o s i t i o n 3 .3 .2 ,

G t t Z s a t i s f i e s

* E x , h i h ( r x , ~ l l '(@, [ X , Y J J - ~ A ti*-', [ x , Y J ] - P ) < - e

if and o n l y i f t i p / q . T h e r e f o r e V; s a t i s f i e s ( * ) i f and +

o n l y i f s < p/q and h e n c e G r x Z E V: i f and o n l y i f r I s . -

P r o ~ o s l t l o n . .

G, ;...; cr E V: An- l r\ A V ' A " - ~ n... 1 n 1 S2

i f and o n l y i f r i < si f o r i = 1 , ..., n. n A ~ " ~ / A n A vsn - -1

P r o o f . C t. L e t c = cr x. . . x cr e v i An-1 Q.. . ~ n - 1 ~ 1 n 1

S

n-1 n S i n c e C € us, A , t h e r e i s an [ - i d e a l H 1 of G s u c h

L

t h a t H 1 E V.;l n-1 f

and G / H 1 & A . Then H 1 2 G r x R ; J. /

t h e r e f o r e G ; R r yS, and by P r o p o s i t i o n 3 . 3 . 2 , r l 5 s l . r, 1

G ~ ~ - ~ 1 / 1

and t h u s t h e r e i s an [ - i d e a l K n o f C s u c h s,

t h a t K n & A and G / K n e vs. S i n c e K n L 4 n-1 , C 4- C- + K n & G x . . . x G x R and G / K n Z x G r ,

rf rn- 2 - t h e r e f o r e Z 2 C E

v % and r n ( sn . For ' 1 < i < n ,

'n A i - l A n - i

vg and t h u s t h e r e i s an 1 - i d e a l H i o f G 1

s u c h t h a t H i E ~ i - 1 V I Si

and G / H i E . A l s o t h e r e i s an

[ - i d e a l K~ o f H~ s u c h t h a t K~ ~ i - 1 and H i / ~ i E V'

n- i c + t Si

S i n c e G / H i E A , H i 2 G x . . . x G x R ; s i n c e '1 'i

i - 1 4- + .e K i e ~ K i t G x . . . %& x R . T h e r e f o r e H i / K i 2

t-7 C

I

Z ; G x R a n d Z t G % R 6 vsI whence r i 5 s i . ri 'i

e t. c = 1 1 l e t H i = G x . . . x G x R a n d f o r r1 rL

t 4 4 i = 2 , . . . , n l e t K i = G x . . . x G r x R . Then f o r t -L i- 2

1 < i < n , G / H i = z x G, + x . . . x + G E An- i A ~ - l rn

, K i E

C i+2 / / a n d H i / ~ i = Z x G ; R E Vs. , whence H i E Ai-' V "i %

*- / and G r Ai-l ' A n - i . A l s o H I = G r x R E I$ a n d vsi 1

G / H 1 E an- ' s o t h a t G E V K n = G c x . . . x c G c x R E A"-'

4 S~ rl 'n- 2

a n d G / K n = Z x Grn E V, whence G E A"-' Vs. T h e r e f o r e ,

an-1 '" n n...q Vs n

F o r 0 < r e , < 1 l e t u ( ~ , * . . , r n ) b e t h e n

v a r i e t y vrl an- ' avr2 An-2 r\ * * * n l A n - l $< 1 n

C o r o l l a r y 4 . 1 1 . { u 0 < r l , ..., (r , , . . , r n ) 1 rn < 11 i s a

meet s u b s e m i l a t t i c e o f L i s o m o r p h i c t o 1".

P r o o f . L e t a : 1" * L by g i v e n by ( r l , . . . , r n ) a =

U(r, ,..., r n ) . By t h e p r e c e d i n g p r o p o s i t i o n , i t i s c l e a r

t h a t 4 i s o n e t o o n e . A l s o ,

CHAPTER 5

F u r t h e r R e s u l t s

S e c t i o n 1 . L e x P r o d u c t s o f V a r i e t i e s .

The s t u d y i n C h a p t e r 3 o f t h e r e l a t i o n s h i p be tween V and

p rompt s a more g e n e r a l c o n s i d e r a t i o n o f l e x i c o g r a p h i c

p r o d u c t s and v a r i e t i e s o f 1 - g r o u p s . I n p a r t i c u l a r i t i s

p o s s i b l e t o i n t r o d u c e t h e n o t i o n o f t h e l e x p r o d u c t o f two

v a r i e t i e s .

Ref mltlDn . . . 5 . 1 1 F o r v a r i e t i e s V and U w i t h U GR f

l e t t h e Lex af V and U b e t h e v a r i e t y V x U =

The f o l l o w i n g r e s u l t i s immed ia t e f rom t h e d e f i n i t i o n .

5.1.2. F o r v a r i e t i e s V and U w i t h U G R ,

C l e a r l y by C o r o l l a r y 3 .1 .14 , f o r any v a r i e t y V ,

t v r A i s t h e v a r i e t y v L d e s c r i b e d e a r l i e r . I t c a n e a s i l y b e

s e e n t h a t some o f t h e p r o p e r t i e s o f vL c a r r y o v e r t o t h e +

more g e n e r a l p r o d u c t v x U . The p r o o f s o f t h e f o l l o w i n g

two r e s u l t s a r e s i m i l a r t o t h o s e o f P r o p o s i t i o n s 3.1.2 and

3 .1 .3 r e s p e c t i v e l y .

P r o p o s i t i o n 5 .1 .3 . If V = v a h ( { G i 1 i E 1 ] ) and U C t h e n C e-

V X U = Vat ( {Gi x H I i E I , H E U , H i s t o t a l l y o r d e r e d ] ) .

P r o p o W 5.1 . 4 L F o r ? n y ' v a r i e t i e s o f 1 - g r o u p s ,

U , V 1 , V 2 , Vi ( i E I ) ,

F u r t h e r p r o p e r t i e s o f t h e l e x p r o d u c t o f v a r i e t i e s a r e a s

f o l l o w s .

Prop- . . 5 . 1 1 5 L F o r v a r i e t i e s o f l - g r o u p s V , U1 , Uz

Proof.

= V a t ( { G : H / G E V , H E U 1 v U2,

H i s t o t a l l y o r d e r e d ) ) C = V a ( { G x H / G E V , H E U I U U 2 ,

H i s t o t a l l y o r d e r e d ) )

= V a t ( { G % H / G E V , H F U l ,

H i s t o t a l l y o r d e r e d ) ) v

4- = Von({G x H I b E V , H E U p ,

H i s t o t a l l y o r d e r e d ) ) t

= ( v x U 1 ) v ( v : U 2 ) . &

( i i i ) V x ( U1U2 n R ) i s g e n e r a t e d by t h e - g r o u p s 4-

G x H where G E V and H ELI1 U2 i s t o t a l l y o r d e r e d .

S i n c e H E U l U 2 t h e r e i s an [ - i d e a l K o f H s u c h

t h a t Kr U1 and H / K f U p . A l s o b o t h K and H / K

a r e t o t a l l y o r d e r e d . G 2 K E V ? U1 i s an

& i d e a l o f G? H and G ~ H / G ? K = H / K E U 2 ,

whence G f H E ( v % U1 ) U 2 and t h e r e f o r e 6-

V x ( u l U 2 0 R ) G ( V? U , ) U2.

A n a t u r a l q u e s t i o n t o a s k i s f o r which r e p r e s e n t a b l e e 6- v a r i e t i e s u d o e s U x U = U . C l e a r l y s i n c e U s U x A S U?C U

L s u c h v a r i e t i e s mus t s a t i s f y U = U and t h e r e f o r e n o t a l l

r e p r e s e n t a b l e v a r i e t i e s have t h i s p r o p e r t y .

. . ooo- 5 . 1 . 6 = A f A = A , R % R = R and w ? W = W .

P r o a f , T h a t A; A = A i s c l e a r . By P r o p o s i t i o n 5 . 1 . 3 C

R x R = vat( {C t H I G , H a r e t o t a l l y o r d e r e d ) ) , t h u s R 2 R 4-

i s g e n e r a t e d by t o t a l l y o r d e r e d g r o u p s whence R x R c R and

t h e r e f o r e R R = R . L e t G , H E W , H t o t a l l y o r d e r e d . R e c a l l

t h a t w i s d e f i n e d by t h e law x 2 2 x Y f o r x ) e . C e L e t e < - x = ( g l , h , ) E G x H and l e t y = ( g 2 , h 2 ) c G x H .

2 2 2 g2 Then x = ( g l , h l and xY = ( g l 2 , h l . S i n c e x 2 e and

2 If h12 > h l 2 t h e n x2 > x y * H E W , h l 2 e and h l 2 h l . h O t h e r w i s e , h12 = h l 2 a n d , by M a r t i n e z [ 1 6 1 , h l = e . I n t h i s

2 2 c a s e , g l 2 e and s i n c e G E W , x = ( g l , e ) 2 ( g T 2 , e ) =

xy T h e r e f o r e x2 2 xy and G f H r W whence W 5 W = (JJ . F u r t h e r examples a r e o b t a i n e d by c o n s i d e r i n g v a r i e t i e s

d e f i n e d by t h e r e p r e s e n t a b l e law t o g e t h e r w i t h l a w s

i n v o l v i n g o n l y t h e g r o u p o p e r a t i o n s . F o r s u c h v a r i e t i e s L C

U = U = U x ' U . s i n c e any l e x i c o g r a p h i c a l l y o r d e r e d p r o d u c t

G f H w i t h G , H E u s a t i s f i e s a l l t \he g r o u p l a w s o f

and i s r e p r e s e n t a b l e .

I n C h a p t e r 3 , examples were g i v e n o f r e p r e s e n t a b l e L v a r i e t i e s U f o r which U 7 U . C l e a r l y f o r t h e s e v a r i e t i e s

U TU 2 uL . The f o l l o w i n g r e s u l t y i e l d s e x a m p l e s f o r which

t h i s c o n t a i n m e n t i s p r o p e r .

Proof L e t G E L ( \ A b e t p t a l l y o r d e r e d and l e t H be

any a b e l i a n [ - i d e a l o f G 5 G . S i n c e , G b A "-l , G i s n o t

a b e l i a n and H i s an a b e l i a n t - i d e a l o f G . T h e r e f o r e ,

( G G ) / H 'Z G / H t G A n - ' s i n c e G d A n - ' . Thus G 2 G k A n

and s o U; U $ R nAn. However s i n c e U E R 17 A n ,

C L I n p a r t i c u l a r i f A c u c_ R n A2 t h e n U x U g U .

=t

65

Such v a r i e t i e s i n c l u d e t h e ~ b d v e d e v c o v e r s o f A , v d N O ) ,

Vm (w+) and Vm(W-1 and t h e F e i l v a r i e t i e s U r ( 0 < r < - 1 ) .

F u r t h e r m o r e t h e s e examples show t h a t r e p e a t i n g t h e l e x

p r o d u c t may l e a d t o t h e c o n s t r u c t i o n o f an i n f i n i t e t o w e r

o f v a r i e t i e s .

P r o ~ o s l t l o K l . . 5.1.8 . F o r A c u c, R n ~ L d e f i n e a s e q u e n c e o f *

v a r i e t i e s by U O = U , and f o r e a c h p o s i t i v e i n t e g e r n , U n = e

'n-1 x U . Then U n G R n A n+2 b u t U $ R An+'., r'l

and i n p a r t i c u l a r U n + l 3 + U n .

P roo f , By h y p o t h e s i s t h e r e s u l t h o l d s f o r n = 0 . Suppose

c R A b u t u @ R n A *. Then , u s i n g P r o p o s i t i o n s 'n-1 - n-1 + 5.1.4 and 5.1 .5 , U, ,= U n _ l X U c ( R n A n + l ) $ U C ( R $ U ) A -

c - R ~ ( A : A ) A = n ( a n + ' A ) = R n A"**. A l s o l e t

Un-1 \ A n and l e t H E U \ A b e t o t a l l y o r d e r e d ,

t h e n i t i s e a s i l y s e e n t h a t G f r H , ( u n - l 47

x U ) \ A n + l

The above p r o p o s i t i o n i s i n c o n t r a s t w i t h t h e

1 . c o r r e s p o n d i n g r e s u l t f o r y , C o r o l l a r y 3 .1 .11 showing t h a t

1 f o r any v a r i e t y o f e - g r ~ ~ p s y , y L L = v .

I

S e c t i o n 2 . Mimicking.

A q u e s t i o n y e t t o b e c o n s i d e r e d i s w h e t h e r t h e l e x

p r o d u c t o f r e p r e s e n t a b l e v a r i e t i e s i s a s s o c i a t i v e . L e t U 1 , U 2

and U 3 b e r e p r e s e n t a b l e v a r i e t i e s . Then ( U1 : U2) ; U 3 =

van ( { G % H ': K I G E U, , H E U 2 , K E U 3 , H ,K a r e t o t a l l y o r d e r e d ) ) +

w h i l e U , ? ( u 2 7 U 3 ) = Vm({G; L I C E U 1 , L E u 2 x U 3 , L i s

t o t a l l y o r d e r e d ) ) . f o r any o -g roups H , K w i t h H E U2 and c.

K E u3 i t i s c l e a r t h a t H x K E U 2 x U 3 i s t o t a l l y o r d e r e d and 6 t. C 4- t h e r e f o r e ( U , x U 2 ) x U 3 c U, x ( U p x 5).

One a p p r o a c h i n t r y i n g t o i n v e s t i g a t e w h e t h e r o r n o t

t h i s c o n t a i n m e n t i s p r o p e r , i s t o a t t e m p t t o n a r r o w down t h e

number o f t o t a l l y o r d e r e d g r o u p s o f a v a r i e t y U t h a t n e e d 6-

t o b e u s e d i n d e s c r i b i n g a g e n e r a t i n g s e t f o r V x U . I n -5

p a r t i c u l a r i t h a s been shown t h a t f o r any v a r i e t y V , V x A = C

Vm ( { G x Z I G E v ) ) s o t h a t when f o r m i n g t h e l e x p r o d u c t V?A

n o t a l l a b e l i a n o -g roups have t o b e c o n s i d e r e d . More

g e n e r a l l y P r o p o s i t i o n 3.1 .13 a s s e r t s t h a t i f G , G ) l G E e l C

mimics a v a r i e t y u t h e n v x u = VCUL({G: K I K C V , G el).

I n t h i s c o n t e x t t h e mimicking p r o p e r t y o f by { ( G , G ) ( C C G 1

seems t o b e s t r o n g e r t h a n i s n e c e s s a r y , t h e a rgumen t o f

P r o p o s i t i o n 3.1 .13 r e q u i r i n g o n l y t h a t t h e b e h a v i o r o f t h o s e

t r a n s i t i v e [ - p e r m u t a t i o n g r o u p s i n U which a r e t h e r e g u l a r

r e p r e s e n t a t i o n s o f t o t a l l y o r d e r e d g r o u p s b e mimicked .

I n t h e r e m a i n i n g p a r t o f t h i s c h a p t e r some r e s u l t s on

m i m i c k i n g a r e d e v e l o p e d . T h i s c o n c e p t was i n t r o d u c e d by

G l a s s e t a l . [ 7 1 who u s e d i t i n t h e i r s t u d y o f p r o d u c t

v a r i e t i e s . I n p a r t i c u l a r , t h e y p r o v e d t h a t i f U =

Vm ( { H s 1s c S } 1 a n d { ( G t , n t ) l t E TI mimics V , t h e n U V =

{ H s ~ r ( ~ t , n, ) 1s E S , t E T} ) . They a l s o showed t h a t

t h e r e g u l a r r e p r e s e n t a t i o n o f t h e i n t e g e r s (Z, Z ) mimics A . A c o n v e x L - s u b g r o u p C o f a n 1 - g r o u p G i s Drime i f

f , g E G a n d f~ g = e i m p l y f E C o r g ~ C . F o r a p r i m e

s u b g r o u p C , t h e s e t o f r i g h t c o s e t s o f C , R ( C ) , i s t o t a l l y

o r d e r e d by C f < Cg i f 3 c 6 C w i t h c f < g . L e t R b e t h e

s e t o f a l l r i g h t c o s e t s o f a l l p r i m e s u b g r o u p s o f a n

[ - g r o u p G . H o l l a n d [ g l showed t h a t w i t h a s u i t a b l e

o r d e r i n g o f n, ( G , n) i s a n [ - p e r m u t a t i o n g r o u p .

A l t h o u g h t h e o r d e r on n i s n o t u n i q u e , ( G , n ) i s r e f e r r e d

t o a s t h e Holland pf G. T h e H o l l a n d

r e p r e s e n t a t i o n o f a n [ -group i s u s e f u l when c o n s i d e r i n g

m i m i c k i n g . F o r e x a m p l e , t h e f o l l o w i n g P r o p o s i t i o n i s d u e t o

R e i l l y a n d W r o b l e w s k i 1 2 1 1 .

P r o - W n 5 . 2 . I, L e t G b e t h e r e l a t i v e l y f r e e .k' - g r o u p

o f c o u n t a b l e r a n k i n a v a r i e t y V . L e t (G,n) b e t h e

H o l l a n d r e p r e s e n t a t i o n o f G . Then ( G , Q ) mimics V .

O t h e r . r e s u l t s a r e a s f o l l o w s .

P r o p o s i t i o n 5.2.2. L e t V c'R and l e t C = {(G, f lG) G E V i s

t o t a l l y o r d e r e d , ( G , n G ) i s t h e H o l l a n d r e p r e s e n t a t i o n o f

G I . Then C mimics V .

Proof* L e t ( H , A ) b e a t r a n s i t i v e 1 - p e r m u t a t i o n g r o u p ,

H E V . Then s i n c e H i s r e p r e s e n t a b l e , H i s t o t a l l y

o r d e r e d a n d ( H , f l H ) ~ C . L e t A , { w p ( f ) ) a f i n i t e s e t o f

w o r d s a n d { w p ( h ) ) a s u b s t i t u t i o n f o r t h e s e i n H . L e t

C = H A , t h e s t a b i l i z e r o f A , t h e n C i s a p r i m e s u b g r o u p o f

a n d t h e r e f o r e C fl A l s o i w p ( h ) < iwq(%)

w q ( h ) w p ( h ) - l <+ c < c w q ( ~ ) w p ( W 1 - cw (ib < c w q ( i ; ) . P T h u s C mimics 1 V .

. . 5 . 2 . X r L e t V b e a v a r i e t y o f L - g r o u p s . L e t

(Gi li E I ) b e a c o l l e c t i o n o f 1 - g r o u p s s u c h t h a t f o r a l l i

G i E V , a n d e v e r y f i n i t e l y g e n e r a t e d [ - g r o u p H i n V i s a

homomorphic i m a g e o f some G i . L e t ( G i , n i ) b e t h e H o l l a n d

r e p r e s e n t a t i o n o f G i Then = { ( C i , ni) l i E I } mimics V.

Proof_. L e t (H , A ) b e a t r a n s i t i v e .t - p e r m u t a t i o n g r o u p ,

a s u b s t i t u t i o n f o r t h e s e i n H . L e t % = h , . . . h and

l e t K b e t h e L - s u b g r o u p o f H g e n e r a t e d by { h l , . . . , h n } .

K E Vis f i n i t e l y g e n e r a t e d , t h u s t h e r e i s a n i I

a n d a homomorphism a : G i -+ K . L e t C = {g E G i / F ( g a ) = h }

Then C is a p r i m e s u b g r o u p o f G a n d t h e r e f o r e C E ni . i

69

F o r i < j < - n l e t g e G i ' b e s u c h t h a t g j a j

= h j . Then

,wp(h) < hwq(h) " < A ~ ~ ( h ) w ~ ( h ) - ~ 6 > h < N ( w q ( ~ ) w p ( g ) - l ) a )

& C < Cwq(g)wp(;)-I W C w P (2) < c u p .

T h e r e f o r e C mimics V .

C o r o l l a r y 5 . 2 . 4 . L e t C = {(F, n F ) I ( F , nF) i s t h e H o l l a n d

r e p r e s e n t a t i o n o f F , F a f r e e g r o u p w i t h some t o t a l o r d e r ) .

Then C mimics R.

l k s U 2 . L L e t (G, Q ) b e a r e p r e s e n t a b l e t r a n s i t i v e

C - p e r m u t a t i o n g r o u p . Then G i s t o t a l l y o r d e r e d ( s e e B i g a r d

e t a 1 11, C o r o l l a r y 4 .2 .6 .1 ) and t h u s G i s a hornomorphic

image o f a f r e e g r o u p w i t h some t o t a l o r d e r . T h e r e f o r e t h e

p r e v i o u s p r o p o s i t i o n a p p l i e s .

- J- 5 1 2 - 5 L e t H b e a weakly a b e l i a n t o t a l l y o r d e r e d

g r o u p . Then H i s t h e homomorphic image o f a f r e e g r o u p

w i t h a weakly a b e l i a n o r d e r .

P r o o f , L e t H b e a weakly a b e l i a n t o t a l l y o r d e r e d g r o u p .

Then a s a g r o u p H i s i s o m o r p h i c t o a q u o t i e n t F/G where

F i s a f r e e g r o u p . L e t F/G b e o r d e r e d s o t h a t i t i s

o - i s o m o r p h i c t o H . S i n c e F i s a f r e e g r o u p , F c a n b e

o r d e r e d w i t h a weakly a b e l i a n o r d e r ( s e e M a r t i n e z [ 1 4 1 ) . The

r e s t r i c t i o n o f t h i s o r d e r t o G g i v e s an o r d e r on G w i t h

t h e p r o p e r t y t h a t v f 'f F , tfe <_ g E. G , f - l g f 2 e. Now d e f i n . e

a new o r d e r on F by x. - > e i f e i t h e r x E G and x 2 e

i n t h e o r d e r on G o r . . x E 'F\G a n d xG > G i n t h e o r d e r on

F/G. R o u t i n e v e r i f i c a t i o n s h o w s t h a t t h i s g i v e s a w e a k l y

a b e l i a n o r d e r on F i n w h i c h G i s a n & i d e a l a n d F/G i s

o - i s o m o r p h i c t o H .

C o r o l l a r y 5.2.6. L e t C = {(F, nF] (F, nF) i s t h e H o l l a n d

r e p r e s e n t a t i o n o f F, F a f r e e g r o u p w i t h a w e a k l y a b e l i a n

o r d e r ) . Then C mimics W , t h e w e a k l y a b e l i a n v a r i e t y .

When c o n s i d e r i n g ' 1 - p e r m u t a t i o n g r o u p s t h a t m i m i c a

r e p r e s e n t a t i o n o f a n p - g r o u p . I n p a r t i c u l a r , f o r t o t a l l y

o r d e r e d g r o u p s t h e r i g h t r e g u l a r r e p r e s e n t a t i o n (G, G) o f a n

o - g r o u p G i s a n a t u r a l o n e t o c o n s i d e r . However n o t a l l

r e p r e s e n t a b l e v a r i e t i e s may b e m i m i c k e d by c o l l e c t i o n s o f

f - p e r m u t a t i o n g r o u p s w h i c h a r e r e g u l a r r e p r e s e n t a t i o n s o f

o - g r o u p s . I n t h e f o l l o w i n g d i s c u s s i o n t h e n o t a t i o n (G, G)

w i l l a l w a y s mean t h e r i g h t r e g u l a r r e p r e s e n t a t i o n o f a n

o - g r o u p G.

P r o p o s i t i o n 5 .2 .7 . L e t v E R b e m i m i c k e d by I(Gs, G,) 1 E S].

Then v c _ W .

Proof. L e t H b e a r e g u l a r s u b g r o u p o f G E V . Then H

i s p r i m e a n d R ( H ) , t h e s e t o f r i g h t c o s e t s o f H , i s

t o t a l l y o r d e r e d . If H i s n o t n o r m a l i n G, t h e n 3 e 5 h E H , -

g c G s u c h t h a t g 'hg 4 H a n d ~ ~ - ~ h ~ > H . C o n s i d e r

t h e t r a n s i t i v e C - p e r m u t a t i o n g r o u p (G, R ( H ) ) a n d t h e w o r d s

W l ( x , y ) = y , w 2 ( x , y ) = X - ' Y X S W3(x,y) = e. C o n s i d e r t h e

s u b s t i t u t i o n y -+ h , x -+ g i n G . Then ~ ~ - ~ h ~ > H = Hh,

t h a t i s , Hw2(g,h) > Hw3(g,h) = H w l ( g , h ) . L e t s E S and x + a

y -+ b b e any s u b s t i t u t i o n i n Gs s u c h t h a t w 2 ( a , b ) >

w l ( a , b ) and W2(8,b) > w 3 ( a , b ) . Then a - l b a > b and

a-'ba > e . S i n c e a- 'ba > e , b > e , t h a t i s w l ( a , b ) >

w 3 ( a , b ) . However, Hwl (g ,h ) = Hw3(g ,h) . T h i s c o n t r a d i c t s t h e

h y p o t h e s i s t h a t {(.Gs, cs)l s E: S) mimics V . T h e r e f o r e

H i s no rma l i n G . However t h e weakly a b e l i a n v a r i e t y w i s

t h e l a r g e s t v a r i e t y o f 1 -g roups w i t h t h e p r o p e r t y t h a t e v e r y

r e g u l a r s u b g r o u p i s norma l 1201. T h e r e f o r e V W .

I t must now b e shown t h a t weakly a b e l i a n v a r i e t i e s c a n

be mimicked by r e g u l a r r e p r e s e n t a t i o n s o f t o t a l l y o r d e r e d

g r o u p s .

P r o ~ o m . . 5 .? .8L L e t V c _ W . Then C = {(G,G) I G i s

t o t a l l y o r d e r e d ) mimics V .

Proof, L e t (G, R ) b e a t r a n s i t i v e l - p e r m u t a t i o n g r o u p ,

C E V . L e t u E R . Then ( G , d i s 1 - i s o m o r p h i c t o

( G , R ( G ) ) and s i n c e ( G , n ) i s t r a n s i t i v e a g~ n C; g - ' ~ , i = { e l .

However V W s o e v e r y convex [ - s u b g r o u p o f G i s n o r m a l .

g - l ~ u g = { e l . I n p a r t i c u l a r Ga i s no rma l and t h u s G a = g , G .

T h e r e f o r e ( G , Q ) i s 1 - i s o m o r p h i c t o ( G , R ( { e l ) ) = (G,G) ,

and c l e a r l y C mimics V .

7 2 ,

I n t h e c a s e o f t h e H o l l a n d r e p r e s e n t a t i o n i t was

p o s s i b l e t o r e s t r i c t a mimicking s e t f o r W t o t h e c o l l e c t i o n

o f f r e e g r o u p s w i t h a l l p o s s i b l e weakly a b e l i a n o r d e r s .

However t h e f i n a l r e s u l t shows t h a t t h i s i s n o t p o s s i b l e f o r

t h e r e g u l a r r e p r e s e n t a t i o n .

Prop- . . 5- L e t C = { ( F , F ) I F i s a f r ee g r o u p w i t h

a weakly a b e l i a n o r d e r ) . Then C d o e s n o t m i m i c w.

P r o o f * C o n s i d e r N o , t h e f r e e two g e n e r a t o r n i l p o t e n t c l a s s

two g r o u p , f r e e l y g e n e r a t e d by { a , b ) and o r d e r e d s u c h t h a t

a >> b >> [ a , b l > e . ( N o , N o ) i s a t r a n s i t i v e

1 - p e r m u t a t i o n g r o u p w i t h N o E W . Let w l ( x , y ) = [ [ ~ , y ] , y ] ,

W 2 ( x , y ) = e , w 3 ( x , y ) = x A y and w g ( x , y ) = [ x , y ] .

C o n s i d e r t h e s u b s t i t u t i o n x + a , y + b i n N o . Then

e = w l ( a , b ) = w 2 ( a , b ) < w q ( a , b ) = [ a , b ] < w 3 ( a , b ) = b . Let

( F , F ) E C and c o n s i d e r any s u b s t i t u t i o n x -t f , y -+ g

i n F s u c h t h a t e = ~ ~ ( f , ~ ) < w 4 ( f , g ) < w 3 ( f , g ) ,

i . e . s u c h t h a t e < [ f , g l < g . Then s i n c e F i s a weakly

a b e l i a n , g >> [ f , g l . Thus t h e r e d o n o t e x i s t h c H and

n , m E Z s u c h t h a t g = hn and [ f , g l = h m , and t h e n s i n c e

F i s a f r e e g r o u p , [ [ f , g l , g l # e whence [ [ f , g l , g l > e o r

[ [ f , g l , g l < e . T h e r e f o r e w l ( a , b ) = w 2 ( a , b ) b u t e i t h e r

w l ( f , g ) > w 2 ( f , g ) o r w l ( f , g ) < w 2 ( f , g ) and C d o e s n o t

mimic 6.1.

REFERENCES

1 . A . B i g a r d , K . Keimel , and S . W o l f e n s t e i n , Groupes e t anneaux r e t i c u l e s , L e c t u r e Notes i n Mathemat ic s , 6 0 8 , S p r i n g e r - V e r l a g , 1977.

2 . G. B i r k h o f f , On t h e s t r u c t u r e o f a b s t r a c t a l g e b r a , P r o c . Cambridge P h i l . Soc , 3 1 ( 1 9 3 5 ) , 433-454.

3 C . . C h e h a t a , An a l g e b r a i c a l l y s i m p l e o r d e r e d g r o u p , P roc . London Math. S o c . , 2 ( 1 9 5 2 ) , 183-197.

4 . T . F e i l , Ph.D. T h e s i s , Bowling Green S t a t e U n i v e r s i t y , 1980.

5 . T . F e i l , An u n c o u n t a b l e tower o f 4 -group v a r i e t i e s , Algebra U n i v e r s a l i s , 1 4 ( 1 9 8 1 ) , 129-131.

6 . A . M . W . G l a s s , Ordered p e r m u t a t i o n g r o u p s , London Math. Soc. L e c t u r e Note S e r i e s 5 5 , Cambridge U n i v e r s i t y P r e s s , 1981.

7 . A . M . W . G l a s s , W.C. Hol land and S.H. McCleary, The s t r u c t u r e o f l!.-group v a r i e t i e s , A lgebra U n i v e r s a l i s , 1 0 ( 1 9 8 0 ) , 1-20.

8 . S . A . .Gurchenkov, Minimal v a r i e t i e s o f e - g r o u p s , Algebra i Log ika , 2 ( 1 9 8 2 ) , 131-137 ( R u s s i a n ) .

9 . W . C . H o l l a n d , The l a t t i c e o r d e r e d g roup o f automorphisms o f an o r d e r e d s e t , Mich. Math. J . , 1 0 ( 1 9 6 3 ) , 399-408.

10. W . C . H o l l a n d , The l a r g e s t p r o p e r v a r i e t y o f l a t t i c e o r d e r e d g r o u p s , P roc . Amer. Math. S o c . , 5 7 ( 1 9 7 6 ) , 25-28.

11. W . C . Hol land and S.H. McCleary, w r e a t h p r o d u c t s of o r d e r e d p e r m u t a t i o n g r o u p s , p a c i f i c J . ~ a t h ; , 31 (19691, 703-716.

12. M.E. H u s s and N.R. R e i l l y , On r e v e r s i n g t h e o r d e r of a l a t t i c e o r d e r e d g r o u p , J . A l g e b r a , t o a p p e a r .

13. V . M . Kopytov and N. Ya. Medvedev, V a r i e t i e s o f l a t t i c e o r d e r e d g r o u p s , Algebra i Log ika , 1 6 ( 1 9 7 7 ) , 417-423 , .y ks

( R u s s i a n ) . 14. J . M a r t i n e z , F r e e p r o d u c t s i n v a r i e t i e s o f l a t t i c e

o r d e r e d g r o u p s , Czech. Math. J . , 2 2 ( 1 9 7 2 ) , 535-553.

J . M a r t i n e z , Archimedean- l ike c l a s s e s o f l a t t i c e o r d e r e d g r o u p s , T r a n s . Amer. Math. Soc. , 1 8 6 ( 1 9 7 3 ) , 33-49.

J . M a r t i n e z , V a r i e t i e s of l a t t i c e o r d e r e d g r o u p s , Math. Z . , 1 3 7 ( 1 9 7 4 ) , 265-2840

N . Ya. Medvedev, The l a t t i c e o f v a r i e t i e s of l a t t i c e o r d e r e d g r o u p s and L i e a l g e b r a s , A lgebra i L o g i k a , 1 6 ( 1 9 7 7 ) , 40-45 ( R u s s i a n ) .

H . Neumann, V a r i e t i e s o f g r o u p s , E r g e b n i s s e d e r Mathematik und i h r e r G r e n z g e b i e t e , Vol. 3 7 , S p r i n g e r - V e r l a g , New York, 1967.

N . R . R e i l l y , A s u b s e m i l a t t i c e o f t h e l a t t i c e o f v a r i e t i e s o f l a t t i c e o r d e r e d g r o u p s , Can. J . Math., 1 3 ( 1 9 8 1 ) , 1309-1318.

N . R . R e i l l y , N i l p o t e n t , weakly a b e l i a n and H a m i l t o n i a n l a t t i c e o r d e r e d g r o u p s , Czech. Math. J . , 3 3 ( 1 9 8 3 ) , 348-353.

N . R . R e i l l y and R . Wroblewski , Suprema o f c l a s s e s o f g e n e r a l i z e d Sc r imger v a r i e t i e s of l a t t i c e o r d e r e d g r o u p s , Math. Z . , 1 ( 1 9 8 1 ) , 293-309.

E . B . S c r i m g e r , A l a r g e c l a s s o f s m a l l v a r i e t i e s o f l a t t i c e o r d e r e d g r o u p s , P r o c . Amer. Math. Soc . , 5 1 ( 1 9 7 5 ) , 301-306.

J.E.H. S m i t h , The l a t t i c e o f 1-group v a r i e t i e s , T r a n s . Amer. Math. S o c . , 2 5 7 ( 1 9 8 0 ) , 347-357.

E . C . Weinberg, F r e e l a t t i c e o r d e r e d a b e l i a n g r o u p s , Math. Ann., 1 5 1 ( 1 9 6 3 ) , 187-199.