Varieties of lattice ordered groups / by Mary Elizabeth varieties of lattice ordered groups ( L-groups)

  • View
    0

  • Download
    0

Embed Size (px)

Text of Varieties of lattice ordered groups / by Mary Elizabeth varieties of lattice ordered groups (...

  • 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

Recommended

View more >