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VARIETIES OF LATTICE ORDERED GROUPS
by
R o g e r W r o b l e w s k i
A THESIS SUBMITTED I N PARTIAL FULFILLMENT OF
THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE
i n t he D e p a r t m e n t
of
M a t h e m a t i c s
0 R o g e r W r o b l e w s k i 1979
SIMON FRASER UNIVERSITY
A p r i l 1979
A l l r i gh t s reserved. T h i s thesis may not be reproduced i n w h o l e or i n part, by photocopy
or other m e a n s , w i t h o u t p e r m i s s i o n of the author.
APPROVAL
Name : Roger Wrob lewski
Degree : Master of Science
Title of Thesis: ~ a h e t i e s of Lattice Ordered Groups
Examining Committee:
Chairman: Dr. S . K . Thomason
N . R . Reilly Senior Supervisor
A. Das External Examiner
Date Approved: April 2 6 1 1979
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ABSTRACT
In approaching the study of 1-groups, there are two w e l l known
methods. F i r s t , the study of the i n t e rna l s t ruc tu re ( t he subgroups,
the elements, e tc . ) of 1-groups. By t h i s method the answers are given
t o questions such as: "When does the t r i ang le of inequali ty hold i n a
1-group?"
The second method is concerned with an external view of 1-groups,
i . e . the study of col lect ions of 1-groups which have some common a t t r ibu tes .
In Chapter I , the study of 1-groups begins with a consideration of
the i n t e rna l s t ruc ture of 1-groups, and then (v ia Birkhoff's Theorem) pro-
gresses t o a specif icat ion of the col lect ions (va r i e t i e s ) of 1-groups and
the common a t t r i bu t e s (laws) . In Chapter I1 the mathematical too ls (Holland's Representation Theorem,
wreath products of ordered permutation groups) u t i l i z e d i n t he study of
va r i e t i e s of 1-groups are presented and examined.
Several examples of va r i e t i e s of 1-groups a r e presented i n Chapter 111.
The containment relat ionships between the v a r i e t i e s a r e a l so examined.
Final ly , i n Chapter I V a mathematical s t ruc tu re is defined on the
collection of a l l va r i e t i e s of 1-groups. The s t ruc ture consists of a
binary operation cal led v a r i e t a l mul t ipl icat ion, and an order re la t ionship -
induced by the containment re la t ionship presented i n Chapter 111. Then
the connection between these two re la t ionships is examined i n the conclusion
of Chapter I V .
(iii)
ACKNCMLEDGEMENTS
1 w o u l d l i k e to thank D r . N.R. R e i l l y for h i s supervision. A l s o ,
t h a n k s t o my dearest C.C. .
TABLE OF CONTENTS
Page
i T i t l e page
Approval
Abstract
Ackncw ledgments
Table of Contents
I. L-groups and Universal Algebra
1. Lat t i ce Ordered Groups
i) Riesz Decomposition Theorem
ii) Prime and Regular Subgroups
iii) L-homomorphisms
2. Birkhof f ' s Theorem
i) Varie t ies
ii) Laws of a Variety
iii) Birkhof f' s Theorem
11. Ordered Permutation Groups
1. Lat t i ce Ordered Permutation Groups
i) 0-k t r ans i t i ve
ii) 0-primitive
2. Holland's Representation Theorem
3 . Wreath Products of Ordered Permutation Groups
i) Properties of the Wreath Product
ii) An Example
iii) An hnbedding Theorem
i i
iii
i v
v
1
1
4
8
12
14
17
22
2 4
26
2 6
27
2 7
28
32
3 7
3 8
39
4. Wreath Products of the Real Numbers
i) 0-primitive Components
ii) Wreath Product of 0-primitive Components
iii) Holder ' s Theorem
iv) An Application of Holder's Theorem and Wreath Products
v) Orbits
v i ) Paired Orbits
v i i ) 0-2 transitive subgroups
111. Examples of Varieties of Lattice Ordered Groups
1. Examples
i) Abelian 1-groups
ii) Representable 1-groups
iii) Normal Valued 1-groups
iv) Scrimger Varieties
2. Containment Relationships
i) The smallest proper variety of 1-groups
ii) Varieties of 1-groups which cover A
iii) The largest proper variety of 1-groups
I V . The Lattice L of Varieties of Lattice Ordered Groups
1. Torsion Classes
i) Torsion Radical
ii) The Torsion Class
iii) Every 1-group Variety is a Torsion Class
Page
41
41
4 2
2. Varietal Products
i) Multiplication is Associative
ii) The Idempotent
3. The Generation of Varieties
i) Definition of Mimics
ii) The Main Theorem
iii) (Z,Z) Mimics
4. The Factorization of 1-group Varieties
Appendix
Bibliography
Page
81
83
86
89
90
91
94
97
99
10 1
v i i
CHAPTER 1
L-groups and Universal Algebra
This chapter i s devoted t o the discussion of some of the basic
proper t ies of l a t t i c e ordered groups (1-groups), which a re important
requirements fo r the presentation of examples of v a r i e t i e s of 1-groups
in Chapter 3. (For a f u l l e r treatment of the proper t ies of 1-groups
see Conrad [21, and Fuchs [31.) Then, i n the second section of t h i s
chapter the reader i s introduced t o the background material f o r B i r k h o f f ' s
Theorem and t o the proof of t h i s theorem.
Section 1. Lat t i ce Ordered Groups
A group ( G I + ) endowed with a p a r t i a l order 5 is a p a r t i a l l y
ordered grouE i f f o r a l l a ,b ,x ,y C G
a 5 b implies t h a t x + a + y 5 x + b + y .
I f the p a r t i a l order 5 i s a l a t t i c e order, then G is cal led
a l a t t i c e ordered group (1-group). I f the p a r t i a l order is a t o t a l order
then G i s ca l led a t o t a l l y ordered group (0-group). I f , f o r a r b C G ,
an 1-group, one denotes by a V b (a A b) the l e a s t upper bound o r join
(greates t lower bound o r meet) of a and b then, fo r a,b,x,y 6 G ,
With t h i s introduction, we now exhibi t some of the r i ch s t ructure of
1-groups.
Proposi t ion 1.1.1. A p a r t i a l l y ordered group G i s an 1-group, i f
for a L 1 g 6 G , g V 0 € G .
Proof: I f G is an 1-group, then it i s evident from the d e f i n i t i o n
of G, t h a t g V 0 C G f o r a l l g € G.
Conversely, i f g v 0 € G f o r a l l g € G, then it is s t r a i g h t
forward t o show t h a t
[(a-b) V 0] + b = a v b
and,
-[-a + [(a-b) v 011 = a A b .
This second equa l i ty follows from the i d e n t i t y
- ( a V b ) = (-a) A (-b) . ( S e e A p p e n d i x )
Lemma 1 .l. 2. For an 1-group G, and a p o s i t i v e i n t e g e r n , if
n g 1 O then g 1 0 .
Proof: By an induction argument it may be shown t h a t
n ( a A 0) = na A ( n - l ) a A ... A a A 0 . (See A p p e n d i x )
Now i f na L 0 then na A 0 = 0, and hence,
Therefore
O r
propos i t i on 1.1.3. For an 1-group G, and a p o s i t i v e i n t e g e r n ,
n ( a A 0) = (na) A 0
n ( a V 0 ) = (na) V O .
Proof: One should f i r s t n o t e t h a t f o r k and m non-negative
i n t e g e r s , where 0 5 k 5 m ; and a € G
m[ (m-k) a V -kal = . . . V [k (m-k) a + (m-k) (-ka) 1 V . . . 2 0 .
By lemma 1.1.2 we then have
o r
( m a ) V 0 1_ ka ( O 5 k 5 m ) . Therefore
n ( a V 0 ) = n a V ( n - l ) a V ... V a V 0 = na V 0
and,
Corol la ry 1.1.4. Fox an 1-group G I and a , b € G ; i f a + b = b + a - -
n ( a A b) = na A n b .
Proof: Notice f i r s t t h a t
n ( a V b) = n i [ (a-b) V 01 + b} ,
and s i n c e a and b commute, t h i s imp l i e s t h a t
[ (a-b) V 01 + b = b + [(a-b) V 0] .
Hence, n ( a V b ) = na V nb, and n ( a A b) = na A nb.
Propos i t ion 1.1.5. i s a n 1-group, t h e c a r d i n a l i t y o f
least countable , ( G # 0 ),
Propokit ion 1.1.6. For an 1-group G a , b , c C GI if a V c = b V c
and a A c = b A c then a = b . -
Proof: Since a V c = a' (a c ) + c , we have
Consequently, G is a d i s t r i b u t i v e l a t t i c e s i n c e it does n o t con ta in a
Propos i t ion 1.1.7. Riesz .!&composition). For an 1-group G, i f - - 0 < x 5 d + ... + dn , where 0 5 di (i=l, ..., n ) , then t h e r e e x i s t
1
c l l*-rc such t h a t O 5 x = c + c + ... + c
1 2 where 0 5 c. di
n n 1 - 1
for i = l , 2 1 . . . I n - I
Proof: (By Induc t ion ) . If 0 x dl + d2 , l e t c l = x A dl L 0 - and c = -c + x ? 0. Then,
2 1
and,
Now assume t h a t the statement o f t h e proposi t ion is t r u e f o r a l l
p o s i t i v e i n t e g e r s k such t h a t 0 i k 5 n. Then f o r
there exist c c . I C 19 such t h a t , n-1
where 0 5 c . 5 di (1 5 i 5 n-1) and 0 5 g 5 dn + dn+l
. By another 1
app l i ca t ion o f the induction hypothesis, there e x i s t cn and c n + l
such t h a t 0 5 g = Cn + C n + ~ dn + dn+l where 0 5 c 5 d and n n
O ' 'n+l ' dn+l '
Now we consider the subgroups of an 1-group G. These play an
important r o l e i n v a r i e t i e s of 1-groups, and represen ta t ions of 1-groups.
A subgroup S o f an 1-group G is an 1 - s u b g r o u ~ of G i f S is
a s u b l a t t i c e of G. An 1-subgroup S df an 1-group G i s c a l l e d a
convex 1-subgroup i f 0 5 g 5 c 6 S and g C G imply g 6 S .
A normal convex 1-subgroup S of an 1-group G is c a l l e d an 1-ideal
of G. An 1-ideal p lays the same r o l e i n t h e theory of 1-groups a s a
normal subgroup plays i n the theory of groups.
If one denotes by G(a), ( a f G, a 2 O ) , the smallest convex
1-subgroup of G containing a , then G(a) A G(b) = G(a A b) . The
inclusion one way is t r i v i a l . The o ther inclusion, G(a) A G (b) c G(a A b) , - may be seen by noticing t h a t f o r non-negative elements a ,b ,c € G
a A (b+c) 5 ( a A b) + ( a A c ) . Amore general r e s u l t which w i l l be
used i n l a t e r pa r t s of the t he s i s is: f o r M a convex 1-subgroup of
G, and two non-negative elements a and b i n G, G(M,a) A G(M,b) =
G(M,a A b) where G(M,x) denotes the smallest convex 1-subgroup of
G containing M and the non-negative element x.
For C a convex 1-subgroup of G , the r i g h t cosets of C may
be ordered a s follows: C + x 5 C + y i f and only i f there ex i s t s an
element b i n C such t h a t b + x 5 y. I t can be shown t h a t 5 i s
a p a r t i a l order on the r i g h t cosets of C i n G, which we s h a l l c a l l
the induced order.
The absolute value of an element a E G denoted by la1 is equal
t o a V -a .
Theorem 1.1.8. For an 1-subgroup C of G, t he following are equivalent. -
1) C is a convex 1-subgroup of G.
2) The s e t of r i g h t cosets of C fo rma d i s t r i bu t ive Lat t ice ,
under the induced order with
and, -
Proof: (1 -* 2 ) . Since x V y 5 x,y ; l e t b = 0 i n t h e
d e f i n i t i o n of t h e p a r t i a l order on the cose tsof C. Then,
Now i f C + d l (C+x) , (C+y), then the re e x i s t a , b € C such
t h a t a + d 5 x and b + d 2 y. Let s be an upper bound i n C f o r
both a and b. Then s + d 2 x,y , and hence s + d Z x V y. Con-
sequently, C + d ? C + (xi V y) . Therefore, the join ( 0 x 1 V (C+y)
e x i s t s and is equal t o C + ( x V y) . Similar ly , (C+x) A (C+y) = C + ( x A y ) ,
To see t h a t the r i g h t cose t s of C form a d i s t r i b u t i v e l a t t i c e ,
we use the above r e s u l t s , and the f a c t t h a t G i s a d i s t r i b u t i v e l a t t i c e .
= [ (C+x) V (C+z) 1 A [ (C+y) V (C+z) 1
The dual i s s i m i l a r l y proved.
( 2 + 3 ) . For g E G and s E C i f Igl 5 Is1 then,
-Is] 5 g 5 Is1 (141 2 -41 9)
Hence,
C = C + (-lsl) 5 c + g r c + Is1 = C I
and then g € C.
( 3 -+ 1). I f 0 5 g 5 s where g C G and s C C, then Ig1 5 1st
and by the hypothesis, t h i s implies g C C. Likewise f o r s C C,
0 5 I S V 0 1 5 Is 1 so t h a t s V 0 C C , and consequently C i s an
1-subgroup of G.
Corollary 1.1.9. I f M i s an 1-ideal of GI then the r i gh t cosets of - - M form an 1-group.
A convex 1-subgroup M of an 1-group G' is cal led regular, i f M
i s maximal with respect t o not containing an element g f 0 of G. I n
t h i s
zero
case, M i s a l so cal led a value of g. By Zorn's Lemma each non-
element of G has a value.
A convex 1-subgroup P of G i s cal led a prime subgroup i f f o r
convex 1-subgroups of G M 2 A B implies t h a t e i t h e r M -> A -
Theorem 1.1.10. For an 1-group G and a convex 1-subgroup M of G I - - the following a r e equivalent.
1) M i s regular.
2) M C M* = II {C 1 M 2 C , C convex 1 -subgroup of G} . #
3) M i s meet i r reducible i n the col lect ion of convex 1-subgroups
of G. - If M is normal, then each of t he above is equivalent t o - 4) G/M is an O-group, with a convex 1-subgroup t h a t covers the
i den t i t y M - i n G/M.
Proof: (1 -+ 2) . Suppose t h a t M is a value of g € G. Then fo r -
c 2 M, g € C. Hence, g 6 M * \ M and M* is c a l l e d t h e cover o f M.
(2 -+ 3) . Since A and n agree f o r convex 1-subgroups,
M 2 M* = A {C 1 C 2 M I C convex 1-subgroup o f G) . Therefore, M i s
m e e t i r r e d u c i b l e i n t h e c o l l e c t i o n o f convex 1-subgroups o f G.
( 3 -+ 1). For M a convex 1-subgroup which i s n o t r egu la r ,
M = n IK I K 3 M , K i s r egu la r ) , s i n c e f o r each nonzero element g i n #
G wi th g n o t i n M , t h e r e e x i s t s a va lue K o f g w i th K 3 M. #
Consequently, i f M i s meet i r r e d u c i b l e , then M i s regular .
For t h e remaining
(2 -+ 4 ) . Assume
p a r t o f t h e proof o f t h e theorem assume M i s normal.
by way o f c o n t r a d i c t i o n t h a t G/M is n o t an
0-group. Then t h e r e e x i s t elements M + x and M + y s t r i c t l y g r e a t e r
than M i n G/M such t h a t (M+x) A (M+y) = M. Consequently, i n t h e
group G/M
But t h i s c o n t r a d i c t s M c M* . #
( 4 -t 2 ) . Let (4) hold. It is then straightforward t o ver i fy t h a t .
there is a one t o one correspondence which preserves containment between
the convex 1-subgroups of G/M t h a t contain and the convex 1-subgroups
of G t h a t contain M. Thus, since t he convex 1-subgroups of G/M
which contain M form a t o t a l l y ordered s e t under inclusiontthe cor-
responding convex 1-subgroups of G which contain M a l so form a
t o t a l l y ordered s e t under inclusion. Therefore, by the assumption
tha t there exists a convex 1-subgroup of G/M which covers M , we
conclude t h a t there is a convex 1-subgroup of G t ha t covers M.
Theorem 1.1.11. For an 1-group G and a convex 1-subgroup M of - G
the following a r e equivalent.
1) M i s prime.
2) If A a& I3 are convex 1-subgroups of G M 5 M c B then M c A n B.
# - #
3) I f a , b C G W and a ,b 2 0 then a A b E GW. - - 4) The l a t t i c e of r igh t cosets of M is t o t a l l y ordered.
I f M is normal, each of the above is equivalent t o - 5) G/M is an 0-group.
Proof: (1 o 2) . This is simply a restatement of the def ini t ion
( 2 -+ 3) . I f a , b € G W and a ,b > 0 then,
Consequently,
(3 + 4 ) . Consider M + a, M + b with a,b E G and l e t
- b - b ' ( a A b ) .
- Then, a A I;- = 0 and by hypothesis
- (3) either b C M or a f M.
M + b S M + [b+ ( a A b ) ] = M + ( a A b ) S M + a .
- On the other hand, i f a C M then,
M + a = M + [a+ ( a h b ) ] = M + ( a A b ) 5 M + b .
(4 -t 1). Suppose by way of contradiction that hypothesis (4) holds,
and (1) does not. Then there exis t elements a f A \ M, b f B \ M where
a rb > 0 such that
Hence,
M + a > M and M + b > M .
However,
(M+a) A (M+b) = M + (a A b) = M ,
which contradicts the assumption that the right cosets of M are total ly
ordered.
( 5 o 4 ) . T h i s is clear when M is assumed normal since only then
i s G/M a group.
corollary 1.1.12. - If M is regular, then M is prime.
Proof: For a and b two strictly positive elements of G \ M
Hence, a A b j? M and therefore, M is prime.
This section closes with several properties of 1-homomorphisms
which we will not prove. The proofs of these properties are similar
to their counterparts in the theory of groups.
A map f : G -t H between two 1-groups G and H is called an
1-homomorphism (1-isomorphism) , if f is a group homomorphism (iso-
morphism) and f(h A g) = f(h) A f (g), or equivalently, f (h V 0) =
f (h) V 0 . The kernel of an 1-homomorphism, h : A -t B denoted by
ker h is {a E A I h(a) = 0) .
Theorem 1.1.13. Given A,B,C,D 1-groups and f,g,h 1-homomorphisms
such that f : A -t B, g : B -+ C, h : A -+ DI and the ker h ismapped
into ker g by f, then thereexists a unique 1-homomorphism k such - - that koh(a) = gof (a) - for all a C A. Moreover, k is an 1-isomor-
-1 phism if and only if ker h = h (ker g) = {a 6 A f(a) 6 ker 53)
Proof: This is a standard theorem of group theory, and we need
only to show that k is an 1-homomorphism.
For d C D
= (go•’ (a)) V 0 = k(d) V 0 .
Theorem 1.1.14. For an 1-subgroup C - of G the following a r e equivalent.
1) C is an 1-ideal. - - 2) C i s the kernel of an 1-homomorphism.
and - are - and - then - B/A is an 1-ideal of G/A - and G/B is 1-isomorphic to (G/A) / (B/A) .
Theorem 1.1.16. Suppose G is an 1-subgroup of an 1-group H, C
i s a convex 1-subgroup of H , - and C is normal i n the 1-group generated
G U C. Then C n G i s an 1-ideal of G I C + G i s an 1-subgroup of - and G/(G n C ) is 1-i&omorphic to (C+G)/C. H f -
I f { ~ ( i ) ( i E I) is a col lect ion of 1-groups, then n ~ ( i ) (ZG(i .1)
w i l l denote the cardinal product (cardinal sum) of t he G ( i ) ' s . This i s
the d i r e c t product (d i r ec t sum) of the G ( i ) ' s a s groups with the order
defined componentwise, i .e . ,
f o r g,h C n G ( i ) , g ,< h i f and only i f g ( i ) 6 h ( i ) f o r a l l i E I .
4. If G i s an 1-group and H i s an 0-group, then G x H w i l l denote
the lexicographical product of G and H. This is the d i r e c t product of
G and H a s group, with the order defined by:
o r h = h and gl 5 g2 . 1 2
i C An 1-group G i s a subdirect product of
i i f G is 1-isomorphic t o an 1-subgroup of f l G ( i ) such t h a t G'S image
e x i s t 1-ideals N ( i ) (i € I) in G such that ntJ(j.1 = ( 0 ) (where 0
is the group identity) and G / N ( i ) is 1-isomorphic to G ( i ) .
An 1-group G is subdirectly irreducible, i f whenever G is a
subdirect product of l-groups { ~ ( i ) li € I}, there exis ts j 6 I such
that G and ~ ( j ) are 1-isomorphic. This corresponds to G having
a smallest nontrivial 1-ideal.
In th i s section we develop the necessary ideas and notation to
prove Birkhoff's Theorem. The approach to Birkhoff's Theorem is by way
of Universal Algebra. There are two reasons for th i s approach. F i r s t ,
Universal Algebra in i ts abstractness allows the reader to apply i t s
results toawider class of algebras (rather than 1-groups alone).
Second, the old adage "simplicity in abstractness" could not be more
aptly applied in algebra as in th is case. For more deta i ls concerning
Universal Algebra see Cohn 1 1 1 .
An operator domain is a se t 52 with a mapping a : $2 + N where
the elements of 52 are called operators; and i f w € fi then a(w)
i s called the ari ty of w, If a (w) = n then we say w is n-ary, and
write
Consequently, 00
5 2 = U R(n) . n=O
Let A be a s e t and $2 an operator domain. Then anabstract
algebra denoted by [A,QI is a family of maps
~ h u s , with each U C R(n) there is associated an n-ary operation on
A. The usual notation for th i s is, for a 1 , a n € A and w € Q(n),
a l , a 1 € A, i.e. we identify w with i t s associated n-ary n
operation.
A subalgebra [ ~ , f i ] of [A,Q] is an abstract algebra such that
B C A and for each w 6 Q(n) i f -
then the restr ict ion of w to B~ agrees with w, and maps B" to B.
W e say that [A,Q] and [B,Q'] are 1 when
!d = 52' and i f th is is the case, we c a l l a map J, : A -+ B a homomorphism
from [A,Q] into I B , ~ ] i f and only i f for a l l w 6 fl and aiC A i f
A homomorphism IJJ is called an isomorphism, i f J, is also one t o one
and onto.
If [A,Q] and [B,Rl are similar abstract algebras, then one may
I form a new algebra (called the direct product of [A,QI and [B,Rl
i denoted by [ A X B , ~ ] ) by defining for a l l w € $2, and [ai.bil d A x B,
1 if w C R ( n ) , then
The d i r e c t product may be genera l ized t o an a r b i t r a r y c o l l e c t i o n
{ [A i I 1 i € I of s imi la r a lgebras i n t h e n a t u r a l way, and t h e
r e s u l t i n g o b j e c t w i l l be denoted by ll [ ~ ( i ) ,Q].
For an a b s t r a c t a lgebra [A,fi], an e q u i ~ a l e n c e ~ r e l a t i a n c on
A i s s a i d t o be a congruence r e l a t i o n on [A,Q] i f r f o r a l l w C fi ,
whenever w C $2. (n) , and [ai .bil 6 c. W e can make the s e t A/c of
c-equivalence c l a s s e s i n t o an a b s t r a c t a lgebra [A,ll]/c s imi la r t o
I A , ~ ~ I by defining. f o r a l l w C Q and ai C A,
from [A,RI t o [~,$2.1 t h e
i f O € Q ( n ) . It can be shown t h a t t h e opera t ion is w e l l defined.
s i m i l a r a lgebras and J, a homomorphism
kernel of J, (ker $1 is defined a s
Consequently, t h e ke rne l of a homomorhpism J, is a congruence r e l a t i o n
on [~,!d] and i f I/J : [A,Q] -t [B,Q], then [A,Q]/ker $ i s isomorphic
to a subalgebra of [B,Q] . This subalgebra i s denoted by $J [A$] . The congruence r e l a t i o n s { c ( i ) 1 i C I) i n an a lgebra [AI$2.1
form a complete l a t t i c e where
i f and
I 1 Vc(i) = n(cl 1 c1 3 Uc(i), c' i s a congruence relation on [A,Q]). - I
r We may extend the results from the previous paragraph to a family of
congruence relations on [A,a], an abstract algebra. For a family
I {c (i) I i € 1) of congruence relations on [A,Ql, [A,Ql / n c (i)
i s isomorphic to a subalgebra of n ( [A,al/c(i)) . I i 1 With th is background we may now turn our attention to the properties i i of varieties that are useful i n Chapter I V .
l 1 1 A class C of similar algebras is called abstract, i f [A,Q] € C ! i whenever there exists a [B,Q] € C such that [A,Q] i s isomorphic to
[B,Ql . A nonempty abstract class C of algebras is called a varieey,
i f the following hold:
1) If EA,QI € C and [B,Ql is a subalgebra of [A,Q] then
2) If [A,Ql € C and c i s a congruence relation on [A,n]
then [A,Q]/c € C.
3) If [A(i) ,!dl € c (i € I) then n[A(i) ,a] € C.
If C is a nonempty class of similar algebras, we say S i s a
C-free se t of generators of [A,QI -- i f ,
1) S c A and the smallest subalgebra of [A,Q] containing - * S is [A,QI i t se l f .
2) For every algebra [B,Ql € C and every mapping $ : S -+ B,
t h e r e e x i s t s a homomorphism $ from [A,Ql to fB,Gl which
when r e s t r i c t e d t o S agrees wi th I).
~f [A,R] i s a l s o an element of C, [A,Q] is called a f r e e C -algebra.
W e w i l l now const ruct a word algebra W [ X , ~ ] over a set X with
c a r d i n a l i t y g r e a t e r than zero. A convention which is t a c i t l y assumed
throughout t h e construction i s :
i f G(n) = then ~ ( i ) ~ x Q(n) = 0 .
With a word algebra W[X,G], it w i l l be proved t h a t each v a r i e t y V
has a f r e e V algebra a s a member. And t h i s r e s u l t i s of g r e a t importance
i n t h e proof of B i rkof f ' s Theorem.
For a set X with c a r d i n a l i t y g r e a t e r than zero l e t A(0) = X
and def ine A ( i ) recurs ively a s follows: f o r i 1 1 l e t
w
A(i+ l ) = ~ ( i ) u ( U ~ ( i ) ~ x R ( n ) ) . n=O
w
Note t h a t t h e elements of U ~ ( i - 1 " Q(n) a r e of t h e form n=O
a 2 a L O where o C Q(n) and ak C A ( i ) (1 5 k C n), and n
t h a t A(0) c A ( 1 ) 5 ... . L e t -
A = U A ( i ) .
By defining f o r a l l bl,b 2,. . . ,b C A and fib) n
[ A I Q I becomes an a b s t r a c t algebra, s ince i f bl ,b2, . . . , b C A and n
W F Q ( n ) , then bl,b2, ..., bn C A(k) f o r some f ixed k and
This algebra is denoted by w[x,Q].
Lemma 1.2.1. - If X is a s e t of cardinality greater than zero and
C is a class of abstract algebras with WIX,Q]C C , then X is a - - set of C-free generators of W[X,Q].
Proof: For [B,Q] € C and 8 a map from X t o B it w i l l be - shown that 6 may be extended to a homomorphism $ from w[x,QI to
[B,Ql which when restricted to X agrees with 0.
Let $(O) = 6 and define 4 ( 1 : A -+ B recursively as
follows: for i Z 1
$ ( i ) ( a ) i f a.6 A ( i ) $ (i+l) (a) =
w[$(i) (al) , . . . $ ( i ) (an) I i f 00
n a = ((al , . - - ,a )w) C U A ( i ) x ~ ( n ) . n
n=O
Consequently, $ 1 , when restricted to A ( i ) , is $ ( i ) . And i f
$ denotes the union of the @ (i) 0 1 , . . , then for each i
such that a € A ( i )
The map i s an extension of 6, and $ is a homomorphism from
W[x,QI to [B,Q] since, i f al,a2, ..., a C A, then al,a2, ..., a C A ( k ) n n
for some fixed k, and for w € Q(n),
~ i n a l l . y , by the construction of W[X,Ql it is c l e a r t h a t t h e smal les t
subalgebra of W[X,Ql containing X i s W[x,Q].
Theorem 1.2.2. - Let C be a v a r i e t y of algebras, which contains an
algebra of c a r d i n a l i t y g r e a t e r than zero. Then f o r any cardinal m
g r e a t e r than zero, the re e x i s t s a f r e e C-algebra wi th C-free generat ing
s e t of c a r d i n a l i t y m.
Proof: Consider W[X,Ql where t h e c a r d i n a l i t y of X i s m. Let
c = nc (i) where ( c (i) I i C I) is the c o l l e c t i o n of a l l congruence
r e l a t i o n s on w[x,Ql such t h a t W[X,Q]/c(i) € C. The algebra W[X,Ql/c
i s isomorphic t o a subalgebra of JIW[X,Q]/c(i). Since C i s a va r ie ty ,
w[x,Q]/c € C . The s e t X c = {xc I x 6 X ) genera tes w[X,~] /C . To
see t h i s l e t x denote t h e algebra generated by X c . W e f i r s t show
t h a t x 3 - W[X,Q]/c .
The algebra X, conta ins A(0)c. I f X conta ins A ( i ) c , then
f o r alc, ..., a c € A ( i ) c and w € Q(n) n
Hence A(i4-l)c c X , and s o Ac c X. Therefore, w [X,al/c 5 X - Con- - - versely, it i s c l e a r t h a t X c W[x,QI/c. -
To show t h a t t h e set X c has c a r d i n a l i t y m, it s u f f i c e s to show
t h a t i f x,y C x and x # y then xc # yc. Let [B,Q] be an element
of C such t h a t the re e x i s t a ,b 6 B with a # b. ~ e f i n e a map 8
from X t o B such t h a t
8 ( x ) = a and 8(y) = b .
~ x t e n d 8 t o a homomorphism JI from w[x,QI to [B,Q]. Then, i f
c (i) is t h e kernel o f t h i s
c c c ( i ) and therefore xc - To s e e t h a t X c is a
and 0 ' a map from Xc t o
homomorphism, w [x,Q]/c (i) € C. Consequently
# yc.
C-free set of genera tors , le t [BIRl € C
B. Define a map $' from X t o B by,
Since X is a C-free set of genera tors f o r W[X,Q], $ may be
extended t o a homomorphism $ from W [X,Q] to [B,Q]. Hence
w [x,R] /ker @ i s isomorphic t o a subalgebra o f [B,Ql , and c o n s e q u e n t l ~ ,
~ [ ~ , R ] / k e r @ € C. Therefore, c c - ker 4. F i n a l l y , de f ine a map 8
from W[X,R]/c i n t o [B,Q] by ,
where rl i s the na tu ra l homomorphism from W[X,QI t o W[X,~] /C.
The map 0 i s a homomorphism which extends 8'.
Corollary 1.2.3. For a v a r i e t y and - with - there
exists a homomorphism from a s u i t a b l y chosen f r e e C- a lgebra onto
Proof: For [B,R] C C , chose a f r e e C-algebra, with C-free - set of genera tors with c a r d i n a l i t y equal t o t h e c a r d i n a l i t y of B.
Then the bijection between the C-free s e t of generators and B may
be extended to a homomorphism onto [B ,Q].
By a or identity in w[x,QI we mean an ordered pair
(olIW2) 6 W[x,Ql x W[X,Q]. Sometimes th i s ordered pai r i s written as
u1 = w We say t h i s law holds i n an algebra [A.Ql, i f under every 2 '
homomorphism 9 from w[X,QI to [A,fi'], $ ( y ) = $(u2). O r , in other
words, the ordered pair (u1,w2) i s in the kernel of every homomorphism
from W [x, R l to [A,QI . When a law holds in every element i n a class
C of abstract algebras, we say that the law holds in C, or tha t
C I s a t i s f i e s the law.
Theorem 1.2.4. If there exists a s e t of laws C such that C is the
class of abstract algebras which sa t i s fy the laws of C, then C is a
variety .
Proof: Let [B,R] be a subalgebra of [A,Q] 6 C . If @ i s a
homomorphism from w [ X , Q I to [B,Q], then $ is a homomorphism from
W [ X , R ] to [A,Q], since [B,Q] is a subalgebra of [A,R]. Consequently,
for any law u1 = w in C, $(ul) = $(u2). Hence, [B.Q] sa t i s f i e s 2
the laws of C, and therefore, C i s closed under the formation of
subalgebras . Let { [A(i) , R ] I i € I) be a collection of algebras from C.
4 i s a homomorphism between W [X,Q] and [ A ( i ) Q l , and W1 = W 2
one i € 1 the law w i l l not hold in [A(i) ,a] under the composit
If
is
leas t
ion
of $ and the projection map from the product onto [A (i) ,n] . Con-
sequently, t h e hypothesis [A(i) ,Ql € C f o r a l l i C I is
contradic ted , and therefore one concludes t h a t TI [A (i) $1 € C.
F ina l ly , f o r E A , ~ ] C C and c a congruence r e l a t i o n on [A,Q] ,
l e t be a homomorphism from w[X,QI to [A,Q]/c. Let $ be a
homomorphism from W[X,QI t o [A,R] defined by $ x = x , where
rl i s the n a t u r a l map between [A,Q] and [A,Ql/c. Then f o r any law
Therefore, [A,Q]/c € C. And thus C is a v a r i e t y .
Theorem 1.2.5. - I f C i s a v a r i e t y and C is t h e set of a l l laws which
hold i n C, - then C', t h e c o l l e c t i o n of a l l a b s t r a c t algebras s a t i s f y i n %
the laws i n C, - is C.
Proof. Clear ly C ' 3 C. Conversely, i f [A,R] € C ' , then the re - - e x i s t s a word a lgebra w[x,R] such t h a t W[X,Q] is l a r g e enough to
wri te a l l t h e laws o f C, and [A,Q] is a homomorphic image of WEX,Ql
by t h e homomorphism . Since C is a v a r i e t y , it has a f r e e C - a lgebra
F, on X , and the ident i ty 'map
i : X - ) X
may be extended t o a su r j ec t ion I,
I f ul,u2 f W[x,Q] and I(%) = I(uZ) then wl = w i n F 2
and'consequently s ince every algebra i n C i s a homomorphic image of
a free C-algebra, wl = o is i n Z. Therefore u = o is s a t i s f i e d 2 1 2
by a l l elements of C' and i n p a r t i c u l a r [A,Q].
This implies t h a t the law U1 = w is i n t h e kernel of $ and, 2
therefore , the re e x i s t s an epimorphism $* ,
g u t w[x,Q]/ker I is isomorphic t o f. Thus [A,Q] i s a homomorphic
image of an element of C and s o [A,G] € C. Therefore, C' c C. -
Summing up Theorems 1.2.4 and 1.2.5, w e have Birkhoff ' s Theorem.
Theorem 1 .2 - 6 . C i s a v a r i e t y i f and only i f t h e r e e x i s t s a set of
laws C such t h a t C is the c l a s s of a lgebras s a t i s f y i n g the laws of C. -
We end t h i s chapter with severa l no tes on v a r i e t i e s , and 1-group
v a r i e t i e s .
Every v a r i e t y is generated by i ts subd i rec t ly i r r educ ib le members.
Consequently, it is s u f f i c i e n t , i n many of t h e theorems concerning var-
i e t i e s , t o show t h a t these theorems hold f o r t h e subd i rec t ly i r r educ ib le
members of a va r i e ty .
For every v a r i e t y V , t h e f r e e word a lgebra w[X,Ql suppl ies an
alphabet f o r the laws of V . The laws of V may then be wr i t t en using
a s e t X which i s countably i n f i n i t e . Hence, s ince the c o l l e c t i o n of
a l l 1-groups forms a v a r i e t y w e have:
Theorem 1.2.7. For t h e f r e e group F on a countably i n f i n i t e set X,
each and F the f r e e 1-group over the t r i v i a l l y ordered f r e e group F,
l a w of a v a r i e t y of 1-groups has the form
where t h e index sets a r e f i n i t e , e is t h e i d e n t i t y of the f r e e 1-group
and x € X U {el U I x € XI . Moreover, o (x) is i n reduced i j k i j
form a s an element o f F.
m e form o ( x ) w i l l be c a l l e d a - word, and we denote by (A? (x) k
i j
the product II xijm . I f J, : F -+ H is an 1-homomorphism from t h e m = l
f r e e 1-group F t o an 1-group H defined by $(xijk) = hijk ,
( X i j k € X U {el U {x-I I x € XI), then
= X - 1) Xijk i ' j ' k '
implies hijk -
- -1 -1 2) - implies h i , j , k , = (hijk)
3) xijk = e implies hijk = e .
In t h i s case , we w r i t e
and say t h a t is a s u b s t i t u t i o n which w i l l be w r i t t e n simply a s x -+ h.
An 1-group v a r i e t y V is generated by a c o l l e c t i o n
{G(s) I s € S, G ( s ) € V) of 1-groups, i f V is t h e smallest v a r i e t y
containing each element of { ~ ( s ) I s € S, G ( s ) € V). Equivalently,
V is generated by { ~ ( s ) I s € S, G ( s ) € V), i f f o r every word U ( X )
fo r which the re is a s u b s t i t u t i o n x -+ h such t h a t ~ ( h ) # e i n H V
there e x i s t s a s u b s t i t u t i o n x -+ g such t h a t w(g) # e i n some G ( s ) .
I n t h i s ins tance V is denoted by 1-var { ~ ( s ) I s € s).
Ordered Permutation Groups
This chapter introduces ordered permutation groups. In particular
the discussion focuses on two fundamehtal tools utilized in examining
1-groups and varieties of 1-groups. These are: Holland's Representation
Theorem and ordered wreath products of ordered permutation groups. For
an extensive treatement of ordered permutation groups see Glass [ 4 1 .
Section 1. Lattice Ordered Permutation Groups
An ordered permutation 9rouE (G,S2) is a permutation group (GIm)
(where 1 denotes the identity) acting on a totally ordered set !J where
1) for all a,f3 C !J, a < B if and only if ag < Bg for all g € G
The group G is then a partially ordered group with respect to the
partial order given by:
1) for g,h € G I g 5 h if and only if ag 5 ah for all a € !J .
If this partial order on G is a lattice order so that G is
an 1-group, then (G,!J) is called a lattice ordered permutation group
(1-permutation group). In this case,for all a € !J and g,h C GI
a(g A h) = ag A ah
and .
For any totally ordered set 52, we shall denote by P(S2) the
group of all order preserving permutations of 52. Then P(Q) is an
1-permutation group.
For a positive integer k, an ordered permutation group (G,Q)
is called 0-k transitive, if whenever
and
there exists g 6 G such that aig = Bi (1 5 i 5 k). If (G.52)
is 0-k transitive, then we say G acts 0-k transitively on . Also, if k is one, then we will refer to (G,Q) as being transitive
rather than 0-1 transitive. If (G,SZ), is an 1-permutation group which
is 0-2 tranhitive, then for any positive integer n greater than two,
(G,Q) is 0-n transitive. Moreover, if one also assumes that the
cardinality of Q is greater than two, then (G,Q) is transitive.
However, the 1-permutation group (Z,Z), which is the right regular
representation of the integers, is transitive, but not 0-n transitive
for n a positive integer greater than one. Thus not all transtitive
ordered permuation groups are 0-2 transitive.
For an ordered permutation group (G,Q) and an equivalence relation
R on R, R is called a convex congruence of (G,Q), if each equivalence
class is a convex subset of R and for a and 8 in 9, C~R@ implies
agRBg for all g € G. If (G,Q) is a transitive ordered permutation
group with no proper convex congruence, it is said to be 0-primitive.
section 2. Holland's Representation Theorem
Holland's Representation Theorem i n the theory of 1-groups is
an analogue t o Cayley's theorem i n group theory and it is a very useful
tool fo r studying 1-groups. For more d e t a i l s see Holland [ 9 I .
Lemma 2.2.1. For an 1-permutation group (GIQ) and a € n, - s e t Ga = {g ( G ag = a] is a prime subgroup of G . -
Proof: Let g,h € G \ G where g and h a r e nonnegative. a
Then, ag > a and ah > a. Hence, a ( g A h) = ag A cch > a. Thus,
g A h J! Ga. By Theorem 1.1.11, and the observation t h a t Ga is a convex
1-subgroup of G I we conclude t h a t Ga is prime. The prime subgroup
Ga of G is ca l led the s t a b i l i z e r of a .
Lemma 2.2.2. - Let C be a prime subgroup of an 1-group G. Then - the map
defined by
(C+x) (gq) = C + x + g f o r a l l x € G ,
is an 1-homomorphism of G i n t o P (G/c) where P (G/c) is the 1-group - - of a l l order preserving permutations of the s e t G/C of r i g h t cosets
of C under t he na tura l order. Moreover, G$ a- G/C.
*Proof: From Lemm 1.1 .ll we know t h a t the s e t of r i g h t cosets of C i n
G is t o t a l l y ordered. Clearly g$ i s a permutation of G/c, f o r a l l
For elements x and y of G, suppose C + x 5 C + y.
Then,there exists an element s in C such that, s + x 5 y . Hence,
and therefore, g$ f P (G/C) . The map $ is clearly a group homomorphism. In order to shaw that
it is an 1-homomorphism, let x C G. Then
(c+x) (gJ, v 1) = max{c+x+g, C+X)
= (C+x+g) v (C+X)
Hence is an 1-homomorphism.
In order to see that G$ acts transitively on G/C, let C + X
and C + y be two elements of G/C and let h = -x + y. Then
Lemma 2.2.3. If G is an 1-group I then G can be 1-embedded - in a cardinal product of transitive 1-pennutation groups {(I( ,n ) 1 4 G I *
9 9
9 f d .
Proof: For each g C GI g # 0, there exists a regular subgroup - of G having g as a value. The regular subgroup H
9 9
is a prime subgroup by Corollary 1.2.12. Let !G! = 6/Hg . 9
BY Lemma 2.2.2 each map
$g : G -+ P(G/Hg)
is an 1-homomorphism. where Gqg acts transitively on 6/Hg . If we denote by K the 1-permutation group , then (Kg. ng) is a
9 4
transitive 1-permutation group.
consider the cardinal product of the 1-permutation groups
{(K~, "1. The map 9
+ : G + n R OZgEG
defined by
(hqJIg = (h)$ (for all h € G and 0 # g C G) 9
is a 1-homomorphism. Moreover, since
(the last equality. because for each g # 0 in GI H is a value of g), 9
the map + is injective, and therefore. an 1-embedding.
Holland's Theorem 2.2.4. - - If G is an 1 - u p , G - is 1-isomorphic
to an 1-permutation group. - proof: BY Lemma 2.2.3 thereexists an 1-embedding
q J : G + II P ( R ~ L O#g CG
well order G - {o) by ( , and let I
1 f
r I R = u n I O#gEG
! 1 I
be the lexicographic union of the (0 # g f G). The order on R g
is; for a,@ E R, a < B if and only if
i) a E P , 6 Enh and g (h 55
or ii) a.B C Rg and a < B in Sl . g
With this order, R is a totally ordered set. We may 1-embed the
cardinal product Il P (.fJ in P(Q) via the map O#gfG
g
defined by
A representing subgrou~ C of an 1-group G is a prime 1-subgroup
of G which contains no 1-ideals of G other than (0).
Corollary 2.2.5. Let G be an 1-yroup. For some totally ordered
set RI (G,Q) is a transitive 1-permutation group, if and only if G - contains a representing subgroup.
Proof: If (GIG) is transitive and a f RI then by Lemma 2.2.1,
Ga is a prime 1-subgroup; and transitivity yields
- 1 Howeverr g G g is the l a rges t l - ideal of G contained i n
gCG " he ref ore Ga i s a representing subgroup.
Conversely, i f C is a representing subgroup of G I then t he
s e t G/C of r i gh t cosets of C a r e t o t a l l y ordered. The map
defined by
(C+x) (g$) = C + x + g ( fo r a l l x € G)
is an in jec t ive 1-homomorphism such tha t , GJ, ac t s t r ans i t i ve ly on G/C.
Section 3. Wreath Products of Ordered Permutation Groups
The wreath product of ordered permutation groups provides a method
for studying the algebraic proper t ies of the l a t t i c e of v a r i e t i e s of
1-groups, i n much the same way a s wreath products of groups a id s i n
determining proper t ies of the algebraic s t ruc ture of the l a t t i c e of
var ie t ies of groups. Because the construction of the generalized
wreath product of ordered permutation groups is complicated, it w i l l
be presented by a s e r i e s of lemmas. For
properties of wreath products of ordered
and McCleary [61.
more d e t a i l s concerning the
permutation groups see Holland
Throughout the constructian w e w i l l assume t h a t
{(G fiy) ( y € r, I' i s a t o t a l l y ordered s e t ) Y'
is a col lect ion of ordered permutation groups. Let A = P Q , . Y
Choose a reference point i n h, say 0, and f o r each h € A le t
and
R = { r € A 1 supp r is inversely w e l l ordered) .
Lemma 2.3.1. R can be t o t a l l y ordered.
Proof: Let r s = { C 1 r y # s yj . For r # s.
4 # I'(r,s) 5 supp r U supp s,
and since supp r U supp s i s inversely well ordered, l'(r,s) is a l so
inversely well ordered. Let a be the maximal element i n r(r ,s) .
Define an order on R by,
r < s i f and only i f r < s . a a
Since 5101 is t o t a l l y ordered, the order t h a t has been defined on R
i s likewise a t o t a l order.
Lemma2.3.2. - For y C r and S R , define r z Y s . i f and
only i f r = s fo r a l l a > y . Similarly, define r f s i f and only a a Y
if r * = - s for a l l a ? y. - Then E' - and a r e convex equivalence u a Y re la t ions on R.
Proof: It is clear that 3 and E are equivalence relations. Y
To Show that the equivalence classes of E~ are convex, let s,r C R
be in the same equivalence class, then r =Y s. For t C R such that
r < t < s , let f3 and 6 denote the maximal elements of r(r,t)
and r(t,s) respectively. Then for E = max(y,$,6), assume E > y.
We then have r = s E '
and either E
r < tE and tE 5 sE , E
or
r 5 tE and tE < sE . E
Hence, E > y contradicts the fact that r = s . Thus, E = y and I E E
Y 1 6,6 . Consequently, for all a > Y, r = ta = s 0 1 '
and theref ore 01
r GY t 5's.
- Similarly, = can be shown to be a convex equivalence relation.
Y
Lemma 2.3.3. If -
W' = { g C P(R) I for all r C ; r Z' 5 , if and only if
-Y rg = sg, and r E s if and only if rg 1 sg) , Y
then (W8,R) is a permutation group. Moreover, for each Y C - - = and 5' are convex congruences on R. Y -
Proof: This is a summary of the previous work.
We may characterize each g in W' as a matrix, since each g
will induce a permutation gy,r(~ C r, r f R) on Qy defined in the
following way. If a f R the re e x i s t s s f R such t h a t s E~ r Y
and s = a. Now def ine Y
The map g i s w e l l defined, s ince t - Y s and t = a k.ply Y I r Y
t by S. Hence, t g 5 sg f o r every g € W'. In o t h e r words, Y
( t g ) y = (sg) Y
To show t h a t g is one t o one, l e t a,B C f2 and Y r r Y
Let S r t € R such t h a t
Then
Consequently, sg E t g and so s E t. Hence, Y Y
a = s = t = f 3 . Y Y
The map g (y € l', r € R) i s onto, s ince f o r a € $2 choose s f R Y 1 r Y
such t h a t
s zY r g and s.. = a . Y
There e x i s t s t C R such t h a t t g = s, and hence t g 3 r g so t h a t
t rY r. Also
So we see that each g € W' is characterized by the matrix
%. r y € T , r CR} of its components. The
reader may notice that from the definition of g ~ I r
we have r -Y s
implies that g = y,r
Lemma 2.3.4. Let g € W'. Then g - is an order preservine y,r - permutation on Q for each y € and r € R.
Y -
Proof: Suppose g C . Let a, $ f $I with a < fi. Choose Y
r,s,t C R suchthat r 3 s cYt and s =a, t = $ . Then s < t, Y Y
and consequently sg < tg. But since s E~ t, we also have sg zY tg.
Therefore,
- Wy,, = (59) < (tg) - Y Y $gY,r
as required.
Recalling that {(G ,Q ) Y C l'1 is a collection of ordered Y Y
permutation groups, we have the following corollary.
Corollary 2.3.5. For W = ig € W' 1 g € G for all y € r , Y I ~ Y
r € R). (W.R) is an ordered permutation qroup. If each (G~.$I~)
is an 1-permutation qroup, then (W,R) is an 1-permutation group.
Proof: For convenience
identity of W. Then for g
let ~r~ = cY and let 1 denote the group
C W g V 1 = h where
The ordered permutation
of {(G~,G+) I Y C P I .
group (W,R) is called the Wreath product
We now list several properties of the Wreath product which will be
useful in the later chapters.
5) If (W,R) and (V,S) are two wreath products of the collection
((G R ) 1 y € I' a totally ordered set) of transitive 1-permutation Y' Y
groups, then (W,R) and (V,S) are also transitive. Moreover, if 0
and O* are the reference points used in the constructions of (WIR)
and (V, S) respectively, then (W ,R) and (V, S ) are 1-isomorphic .
6 ) The restricted wreath product or small wreath product of the
Collection { (GSy) I y C r a totally ordered set) of ordered permu-
tation groups is {g C (W.R) I gyIr = 1 except on finitely many zY Y
classes). The wreath product of G , 1 y C a totally Y Y
ordered set) will be denoted by Wr( (G ,Q ) 1. The small wreath product Y 5
will be denoted by (Gy. fiY) 1 .
38.
For a transitive 1-permutation group (G,Q), and a convex congruence
c of G I each g € G will map a C-class A to itself or to
another C-class B. Since the C-classes are convex, there is a natural
order on them, which forms a total order on Q/c. The lazy subgroup of
6 with respect to the convex congruence C is L(C) = {g € G A = ACJ
for all C-classes A) , The pair (G/L(C) , Q/C) is a transitive
1-permutation group. For a C-class A, let G(A) = Cg € G Ag = A)
and let L(A) denote the set {g C G(A) 1 ag = a for all a € A).
Then (G(A)/L(A),A) is an 1-permutation group.
Let us apply the previous results of wreath products to a wreath
product of (G (A) /L (A) ,A) and (G/L (C) ,fi/~) . A permutation g in
(G (A) /L (A) ,A) wr (G/L (C) ,Q/c) will be written as an ordered pair ( G ,GI - For an element a = (a,b) 6 (A X fi/C),
where is a map from Q/c to G(A) /L(A) and Gb is the image of
- b in G(A)/L(A). Also, g is a permutation in G/L(C).
The product gh of two elements g = ($,GI and h = ( f i Ih ) of
the wreath product is determined by,
Hence, for all a t A x R/c
Finally, for an element
1 of the wreath product, g V
g of the wreath product and the identity
1 = h, where for all a € A x Q/c.
An 0-isomorphism from a totally ordered set X to a totally
ordered set Y is a map f from X to Y such that f is one to
one and such that a < b in X if and only if f(a) < f(b) in Y.
One says that an 1-permutation group (G,Q) may be 1-embedded in an
1-permutation group H , if there exists an 0-isomorphism @ from
$2 to r, and a monomorphism $ from G into H, such that for
a € 52 and g E GI
The following procedure is a method by which a transitive l-permuta-
tion group (G,Q) may be 1-embedded in a wreath product. More precisely,
it will be shown that (G,Q) may be 1-embedded in a wreath product of
the ordered permutation groups (G/L (C) ,Q/c) and (G(A) /L (A) ,A) .
Theorem 2.3.6. - If ( ~ ~ 5 2 ) is a transitive 1-permutation group,
C a convex congruence of (G,Q) with A - a C-class, then (GIG) can
Proof: For each C-class B, choose a permutation kg€ G with - ke = e, if B = A, SO that A I'g = B. This is possible since
(G,R) is transitive.
The map $I : fi -+ A x Q/c, defined by
is an O-isomorphism.
The map II, : G -+ W, defined by
is a monomorphism, where g is the image of g in G/L(C) under tHe
cannonical map, and Bg acting on A is given by
GB = kBg(k )-I (restricted to A) f G(A)/L(A).
Bg
To see that the homomorphism $ is one to one let g f G be
- mapped to , f W, where g is the image of g in G/L(C) under
-1 the canonical map, and 6 = kBg(k 1 restricted to A, and so B Bg
$ f G(A)/L(A). Then, if for all (a,B) f A x Q/L(C), (a,B) ($,;I = (a,B)
then
(agBIBg) = (a,B) for all (a,B) f A x Q/L(c) .
Since g fixes every C-class, g € L(C) . Consequently, -1
$B = $g(k )-I = kBg(kB) . since aeB = a for all a € A and Bg
-1 B 6 R/L(c), we have akBg(kB) = a or ag = a for all a € A and
B C fi/L(C). Thus g must be the identity of G I and so $ is one to
one.
Section 4. Wreath Products of the Real Nu;mbers
For a transitive 1-permutation group (G,Q) for which every value
M in G is normal in its cover M*, it will be shown that (G,Q) is
1-embedded in a wreath product of subgroups of the real numbers. Also,
if G does not have the above property, then it will be shown that G
contains an 1-subgroup which is 0-2 transitive on some totally ordered
set. The results in this section may be found in Holland and McCleary
t 6 1 .
For a transitive 1-permutation group (G,Q),
a pair of convex congruences on such that C 5
a K-class. The set of C-classes contained in the
by CIA, G(A) = (9 G Ag = A), and L(A) = (9
a C A). In this context, the 1-permutation group
let C and K be
Kt and let A be
K-class A is denoted
t G a g = a forall
(G(A)/L(A), A/(C~A) is
called the (C,K) component of (~$2). It is 0-primitive if there does not
exist a convex congruence R on 'd such that C 5 R ;Z K. In this case
the (C,K) component of (G,Q) will be called an 0-primitive component.
Lemma 2.4.1. - If (GIG) is a transitive 1-permutation group, the
set of all convex congruences of G is a totally ordered set under
inclusion.
i Consequently, if we denote by (CirC ) the pairs of convex congruences
i on !il for which the (CirC 1 component of (G ,Q) is 0-primitive,
i then C . c is totally ordered. We will write i = (CilC and
1
{(ci,ci) 1 = I. Thus I is a totally ordered set. and we shall write
i (Gi,Qi) as the (CirC ) component of (G,Q).
Lemma 2.4.2. If Ii is a group, there exists a set (T(KI 1 K is a subgroup of HI such that T(K) is a set consisting of exactly
one element from each of the right cosets of K in H, T(K) n K is - the identity of H, and if G and K are subgroups of H with -
G 5 Kr then T(K) c - T(G).
Proof: Well order the elements of H so that the identity is the
smallest element in the well ordering. Let K be a subgroup of H
and h E H. Choose for the unique element of T(K) in Kh the smallest
element in the well ordering of Kh. Then clearly T(K) fl K is the
identity of H. Also if G and K are subgroups of H with G 5 H,
then T (HI c - T (GI . The function T is called a transversal function.
Theorem 2.4.3. Let (G,R) be a transitive 1-permutation group - with 0-primitive components { (Gi .nil i € I). There exists an l-embed- - ding of (G,!d) into w~{(G~,R~) i C 11 = (W,R).
i i Proof: Fix a C R, and for each i = (C ,ci) C I let Qi =(aoc )/ci. 0
The 0-primitive components of (G.Q) are G i i - Let T
be a transversal function for the subgroups of G. FOX each a C !d,
i let g(a,i) be the unique element of T(G(~~c~) ) such that aog(a,i)c a.
(Uniqueness is a consequence of the fact that the right cosets are
disjoint.) Also g(%,i) is the identity in G.
Pick 0 C hi by 0 = 4 where
into
with
by
Then
The map 4 also maps fi in a one-to-one, order preserving manner
the subset R of 5 fli consisting of those points whose supports
respect to 0 = ao4 are inversely well ordered.
Let r € R, i f I and g € G. Define
( (ad $1 if a$Cir for some a € R
(r) otherwise
$ is a monomorphism from G into W. Therefore (G~S~) is
~6lder ' s Theorem 2.4.4. For an 0-group G the following are
equivalent.
1) For a, b € GI 0 c a < b implies that b < na for some positive -
integer n.
2) G is a subgroup of the real numbers.
3) G has no proper convex subgroups.
Theorem 2.4.5. If a transitive 1-permutation group (G,R) has
the property that each value M G is normal in its cover M*,
then ( ~ ~ $ 2 ) is 1-embeddable in the wreath product of subgroups of - the real numbers (permuting themselves in the right regular representa-
tion). - Proof: It will be shown that each 0-primitive component of
(G,R) is 1-isomorphic to a subgroup of the real numbers, permuting
it self in the right regular representation.
Fix 0 € and suppose C is a convex congruence of (G,Q).
Let H = {g € G (0C)g = OC) then H is a convex 1-subgroup of G.
Conversely, each convex 1-subgroup H c G (for some a E R) of - a
G defines a convex congruence C on (G,R) by sCf if and only if
s € {th 1 h € HI.
Thus each (CIK)-component of G which is 0-primitive determines
a pair (A,B) of convex 1-subgroups of G such that A S 3, and there does not exist an 1-subgroup of G strictly between A and B.
Let s C OK \ OC. Then since G is transitive, there exists g € G
such that Og = s. Consequently, g € B \ A. Therefore A is a value
of g, and B is a cover of A. Moreover, A is normal in B and
hence B/A is an 0-group with no non-trivial convex 1-subgroups.
Therefore, B/A is 1-embeddable in a copy of the real numbers.
We now proceed to show that for an 1-group G I if G contains a
value M which is not normal in its cover M*, then G contains an
1-subgroup which is 0-2 transitive on some totally ordered set.
For an 1-permutation group (G,R), a € , and an 1-subgroup H
of G we call aH = {ah ) h € H) an orbit of H containing a.
It may be easily shown that for orbits CYH and BH either aH = BH
or QH n BH = $.
For an 1-permutation group (G,Q) let the Dedekind completion of
R be denoted by fi. Then, for each g € G and &, C fi \ C? we define
For a totally ordered set X we say that Y c X is dense in X - if whenever a,b f X and a < b then there is a y € Y such that
a < y < b .
Lemma 2.4.6. Let - ( G , S Z ) be a transitive 1-permutation group.
Then for a € R the orbits of the s'tabilizer Ga are convex.
Proof: For B € fi and R1, 62 C BG,, there exists g C Gg
such that cllg = 6 If 61 9 O 5 62 then by transitivity there 2 -
exists f C G such that cS1f = a. Let h = (f V l)A(g v 1).
Then, 6 h = a, and since 1 Ga is convex, 1 5 h 5 g V 1 implies
that h € G a-
Lemma 2.4.7. - Let (G,Q) be an 0-primitive 1-permutation group.
Then, for € fi either GG is dense in or fi = Q and R may - - be taken to be the integers.
Proof: If an element B € Q has an immediate successor, then by
transitivity, every element of R has an immediate successor. Con-
sequently there would be an element of fi with an immediate predecessor.
Again by transiti~ity~this implies that every element of 52 would have
an immediate predecessor.
Define an equivalence relation C on 52 by
aCB if and only if there are only a finite number of elements
in fi between a and 8.
Then it is easily shown that C is a convex congruence on fi. Since,
by assumption, fi has an element with an immediate successor, the
classes of C are not all singletons. Thus, since (G,Q) is O-primi-
tive, C has only one class i.e. R. Therefore, 62 may be taken as
the integers and so i?, = fi.
If no element of R has an immediate successor then by the above
argument it follows that R is dense in itself. Since G is transitive
on if a € R, then GG = Q. Thus, EG is dense in fi.
On the other hand, if a € fi \ Qr then define an equivalence
relation C on R by
aCB if and only if there is no g € G such that Gg lies
strictly between a and 6.
Then, it is straight forward to show that C is a convex congruence
(G,R) is O-primitive, which implies that the C-classes are all single-
tons. In other words GG is dense in Q.
For a transitive l-permutation group (G,R) and each orbit A
of Gat we define the reflection of A in a as A' = ccK where
Lemma 2.4.8. - If (G,Q) is a transitive l-permutation group and
then A' is an orbit of Ga and ( A ' ) ' = A. A is an orbit of Ga,
Proof: Clearly A'Ga 3 - A'. Conversely, since KG c K we have a -
47.
We w i l l now show tha t given 6 € A then f o r each (3 C A' t h e r e
e x i s t s g C Ga such t h a t $g = 0. From t h i s f a c t and t h e above argu-
ment we may conclude t h a t $G = A'. a
Since f3,a € A1 the re e x i s t h,k € K such t h a t 6 = ah and
0 = ak. Thus ah-1 and ak-' a r e elements of A Consequently,
t h e r e e x i s t s f € Ga such t h a t ah-'•’ = ak-l. Let g = h-lfk-l.
Then g € Ga and f3g = 0.
Fina l ly , l e t u s note t h a t from t h e d e f i n i t i o n of A' w e have
ag C A ' i f a n d o n l y i f ctg-I € A.
Thus, ag € ( A 1 ) ' i f and only i f ag € A. Therefore, ( A ' ) ' = A.
For t h e following d e f i n i t i o n s it is assumed t h a t (G,n) is a
t r a n s i t i v e 1-permutation group.
Since A ' is an o r b i t of G wherever A is an o r b i t , we w i l l a
c a l l A ' a pa i red o r b i t of A. Also, s ince t h e o r b i t s of G a
p a r t i t i o n R i n t o convex c lasses , t h e o r b i t s of Ga may be t o t a l l y
ordered i n t h e n a t u r a l way i.e. A < A2 when t h e r e e x i s t s 61 € A l , 1
ti2 € A z such t h a t 6 < 62. W e w i l l c a l l an o r b i t A of G 1 a
p o s i t i v e i f {a} < A and negative i f {a) > A.
W e note t h a t f o r o r b i t s Al and A2 of Ga, Ax < bz i f and
only i f A; > A; . Moreover, t h e map between t h e o r b i t s of Ga and
t h e paired o r b i t s , denoted by A -* A ' , is a b i j ec t ion .
An element f3 C is c a l l e d a f ixed po in t of Ga i f f3Ga = {@I.
I f f3 is not a f ixed point then BGol i s c a l l e d a long o r b i t . If
the paired o r b i t f 3 of t h e f ixed po in t $ of Ga is a f ixed
point of Ga then $ is ca l led a strongly fixed point of Gas I f
every fixed point of is a strong fixed point , then, G is ca l led
balanced.
Lemma 2.4.9. - I f (G,Q) is an 0-primitive 1-permutation group
and B i s a f ixed point of - Ga - then f3 is a s t rongly fixed point
i . e . G is balanced. -
Proof: Suppose t h a t (6)' is not a f ixed point of - Then fo r each Ga
Define a r e l a t i on C on a s follows
aC$ i f and only i f a , $ E {@}'g f o r some g € G.
The r e l a t i on C is an equivalence re la t ion , s ince f o r c lasses
Since ( $ 1 ' is convex, each c l a s s of C is convex, and hence C
is d convex congruence on Q which contains the non-trivial c l a s s
($1'. This contradicts the f a c t t h a t (GIG) i s 0-primitive.
Let X be a t o t a l l y ordered set. We say t h a t Y c - X is cof inal
i n X i f f o r a l l x € X there e x i s t y E Y such t h a t y 2 x.
Lemma 2.4.10. - If (G,Q) is 0-primitive and G # (1) f o r some - a
then - has a posi t ive long o r b i t and - is the only
point of between Al h i .
Proof : Since Ga # 11) for some a € $2, Ga has a long orbit
A. Since (G,Q) is balanced,we assume A is negative and so not
cof inal in Q. Let = sup{$ € A3 € a. Choose g € G so that
- 1 ag < A. Define i+ = {$ C fi I Ggk 5 6 B Cgh, k:h € ~ ~ 3 . To see
that Al is a positive orbit of Ga, we first show that there are
no points of Q. between a and Al. The reader may then convince
himself that A is the first positive long orbit. 1
-1 - Since Q is not the integers, and ag < a, there
exists f3 C Q such that a < $ < Gg. Also &G is denee
in so there exists a positive element h < g in 6, Such that
-1 a < b 5 6 . Consequently, ag <ah-' < ,, and so ah-' € A.
Choose a non negative f C Ga such that ag-I < 6' < ag-'f < a. Then. a < a g-lfh and (9-lfh A 1) € Ga. Moreover, since G c G-
a - a
we have &g(g-lfh A 1) = ah A &J < f.3. Thus, ~(g-'fh A 1) 5 $ < 6g
and therefore, 6 € dl.
Lenuna 2.4.11. A transitive 1-permutation group (G,O)
0-2 transitive if and only if whenever a < f.3 < a there exists a
non negative g € Ga such that ag = a and B9 = 0. -
Proof: If (~$1 is 0-2 transitive then there exists g € G
such that ag = a and Bg = a . Let h = g V 1, then h is the
required order preserving permutation.
Conversely, since (~~$2) is transitiverif a < a;! and 1 61 < 62
then there exists f € G such that a f = $ and without loss of 1 1.
t he re e x i s t s a non negative h C G , such t h a t a f h = alf and f3 1 1
a2fh = BZ . Thus a f h = and a2fh = B2 a s required. 1
Theorem 2.4.12. - Let (G,Q) be an 0-primit ive 1-permutation group
with - Ga # { l } f o r some a C a. Then. G conta ins an 1-subgroup which
a c t s 0-2 t r a n s i t i v e l y on some t o t a l l y ordered set. -
Proof. Let A be t h e f i r s t long p o s i t i v e o r b i t of . We Ga
w i l l show t h a t Ga a c t s 0-2 t r a n s i t i v e l y on A by using lemma 2.4.10.
Let f31,f32.f33 € A such t h a t a < B1 < B2 < B3. Since A is an o r b i t
of Ga t h e r e e x i s t s a non-negative h C Ga such t h a t B2h = B3 . NOW
B1< Blh < B2h = B3 , SO the re fo re Blh < A .
We w i l l now show t h a t t h e r e e x i s t s a non-positive g € G such B3
t h a t Blhg = B1. It is then easy t o demonstrate t h a t (hg V 1) C Ga . Bl(hg V 1) = f3 , . and B 2 ( h g V 1 ) = B 3 -
Let f C G such t h a t a f = B3 s ince a < BL < B2 < B 3 , we have
< B f-' < filhf-l < = a. NOW. by d e f i n i t i o n , a f C A i f 1
and only i f a•’" C A ' ( the l a s t negative o r b i t of G,). By lemma 2.4.9. ,
~ ~ f - ' . Blhf'l C A ' . Thus G does not f i x any po in t 0 such t h a t a filf-l 4 0 5 f3 h f - l . NOW s ince f - l ~ ~ f = G . it i s c l e a r t h a t B1
1 3 and Blh a r e i n t h e l a s t negative o r b i t A'f of G . Thus t h e r e s u l t
f33 fol lows,
Theorem 2.4.13. I f G is an 1-group which has a value M not - normal i n i t s cover M*, then every v a r i e t y conta in ing G must con-
t a i n an 1-group which is 0-2 t r a n s i t i v e on some t o t a l l y ordered set.
Proof: Let M be a value i n G which f u l f i l l s t h e condi t ions
of the theorem. Then fI - g + M + g i s an 1 - idea l of M*- The 1-group gCM*
H = M*/( fl -g+M+g) i s 1-isomorphic t o a p r imi t ive 1-subgroup of order g€M*
preserving permutations a c t i n g on t h e t o t a l l y ordered s e t M*/M of r i g h t
c o s e t s of M i n M*. Since M is no t normal i n M*, t h e s t a b i l i z e r
HM 2 ( b g + ~ + g ) . By Theorem 2.4.12, H con ta ins an 1-subgroup which acts g CM*
0-2 t r a n s i t i v e l y on some t o t a l l y ordered set as required .
We end t h i s chapter by introducing some no ta t ion which w i l l be used
i n Chapters
s ign i fy the
denotes the
of the r e a l
I11 and I V . Let Z denote t h e i n t e g e r s , than (Z ,Z) w i l l
r i g h t r egu la r representa t ion o f t h e in tege r s . Likewise, i f R
r e a l numbers then (R,R) i s t h e r i g h t r egu la r representa t ion
numbers. The wreath product of n copies o f ( Z , Z ) [ (R, R) 1
w i l l be denoted simply by wrnz [W~"RI where n i s any f i n i t e cardinal .
+ 00 or r = z (z-) w e write wr{ ( G ~ , Q ~ ) I y c r l = w r { ( G ,Q ) I (w~-{(G ,Q ) 1 ) . Y Y Y Y
The na tu ra l 1-ideal of ( H , r ) W r (G,Q) [ ( ~ , r ) w r (G,Q) I i s t h e c a r d i n a l
ci product !J H~ ( ca rd ina l sum C H ) of copies of H , where
a€R aCQ H~ = {(f i re) I fig = e f o r a l l f3 # a).
CHAPTER III
Examples of Varieties of Lattice Ordered Groups
In this chapter several examples of varieties of 1-groups will
be presented, as well as their positions in the lattice of varieties
of 1-groups. The notation used for naming varieties of 1-groups is
the same notation used in the paper by Glass, Holland and McCleary I5 I .
The material in this chapter may be found in Holland [7], Martinez [lo],
Fuchs [3], Smith [14], Weinberg 1151, Wolfenstein [14] and Scrimger [131.
Section 1. The Examples
Example 3.1.1. The variety E consists of thosel-groups with
one element. It is clear that the equation defining this variety
is: For each G C E,
x = y for all x,y € G.
Moreover, any two elements of E are 1-isomorphic and
up to isomorphism, this variety contains one element.
Example 3.1.2. On the other hand, the variety L consisting
of all 1-groups has as its defining equation the following: For each
G c L ,
x = x for all x C G.
Example 3.1.3. A variety of 1-groups with some interesting
properties, which will be discussed later, is the variety A con-
sisting of all abelian 1-groups. The equation defining this variety
is: For each G € A,
x + y = y + x for all x,y f G.
Example 3.1.4. A lattice ordered group G is called represent-
able if and only if G is 1-embeddable in a cardinal product of
totally ordered groups. The collection of all representable 1-groups
is a variety, which is denoted by R. The proof of this claim is
obtained from Lemma 3.1.5, Lemma 3.1.6 and Theorem 3.1.7.
Lemma 3.1.5. The class C of all 1-groups G such that
a A b = 0 implies that a A(-y+b+y) = 0 is equationally definable.
The equation which defines this class is (x V 0) A [-y + [(-x) V 01 + yl = 0.
Proof. For a,b,y B let x = a - b. Then, a A b = 0 if
and only if a = x V 0 and, b is equal to (-XI V 0.
Consequently, if a A b = 0 implies that a A (-y+b+y) = 0,
then by replacing a and b by x V 0 and (-x) V 0 respectively
we have
,Conversely, if (x V 0) A (-y + [ (-x) V 01 + y) = 0 for all x
and y € GI then upon setting x = a - b, we have* a A b = 0 implies
t h a t a = x V 0 and b = (-XI V 0. Hence,
a A (-y+b+y) = 0 since (x v 0 ) A (-y + [(-XI v 01 + y j = 0
Lemma 3.1.6. For G an 1-group and f o r any X a subset of G , - t he s e t
X I = {g 6 G I Igl A la1
is an 1-ideal i f and only i f
f o r a l l a ,b ,y € G.
= 0 f o r a l l a 6 X)
a A b = 0 i m p l i e s a
Proof: I f X' is an 1-ideal f o r each
1-ideal. Also, i f a A b = 0, then a r b L
X C G - 0, and
then {a)' is an
consequently
0 = a A b = la1 A lbl . Hence, b € {a) ' . Consequently, f o r any y € G,
-y+b+y E {a) ' s ince {a)' is normal. Therefore, since a ,b 2 0,
o = 1 a1 A I-y+b+y 1 = a A (-y+b+y).
Conversely, it is eas i ly shown t h a t X' i s a convex 1-subgroup
of G. To show t h a t X' i s normal i n G l e t b C X' and y € G.
Then 1 a1 A I bl = 0 f o r a l l a € X, which implies t h a t
l a [ -y + lbl + y = 0. B U ~ , [ a ] A I-y+b+yl = la1 A -y+lbl+ y so
thercfore -y+b+y €
Theorem 3 -1.7.
a A b = 0 implies
Proof: I f G *' -
product of 0-groups
X ' , and X' is normal i n G.
An - 1-group G is representable, i f and only i f
a A (-y+b+y) = 0 f o r a l l a ,b ,y € G.
is representable it is 1-embeddable i n a cardinal
{ ~ ( i ) I i C I). I f a A b = 0 i n each G ( i ) ,
then a(i) = 0 or b(i) = 0. Hence, a(i) A (-y(i) + b ( i ) + y(i)) = 0
in each G(i). Consequently, a A b = 0 in G implies a A (-y+b+y) = 0
in G .
Conversely, the class C of all 1-groups {~(i) / i € 11 such that
a A b = 0 implies a A (-y+b+y) = 0, for all a, b and y members
of G, is an equationally definable class by Lemma 3.1.5. Hence, any
1-group G in this class may be 1-embedded in a subdirect product of
subdirectly irreducible 1-groups from C.
Suppose one of these subdirectly irreducible 1-groups is not an
0-group. Call it H. Then there exist strictly positive elements
Let a,b 6 H such that a A b = 0.
B = { g C H I lg
and c = {k E 11 I Ik
Then, by Lemma 3.1.6, B and
and b € C, and B f l C = (0).
H is subdirectly irreducible.
Summing up, the class C
C are 1-ideals of H. However, a € B
This contradicts the assumption that
Therefore, G is representable.
consisting of all 1-groups satisfying
[X A (-y-x+y) 1 V 0 = 0 for all x and y, is the variety R of all
representable 1-groups.
If M is normal in M*, then M is called a normal value.
We shall now show that the class N of all 1-groups G such
that each value M in G is a normal value is a variety. Any
element G of N is called a normal valued 1-group. The defining
equation may be determined by examining the following theorem, which
may be found in Wolfenstein j1Sl.
Theorem 3.1.9. For G an 1-group and r the collection of all - - values in GI the following are equivalent.
2) For all positive elements a,b € G, a + b 5 2b + 2a.
3) For all pairs of convex 1-subgroups A and B of G;
A + B = B + A .
Proof: (1 -t 2.) If a or b is zero, then evidently
2b + 2a L a + b.
Hence, assume a,b > 0 and M is a value of a + b. Then, since
M is normal valued, M*/M is 1-isomorphic to an 1-subgroup of the
real numbers by ~6lder's Theorem. Consequently, M*/M is an abelian
1-group, and M + (2b+2a) = M + 2 (a+b) > M + (a+b) . The strict inequality follows from the assumption that a + b P M.
Suppose now, by way of contradiction, that 2b + 2a # a + b.
Then (-2b+a+b-2a) V 0 > 0. Hence, for N a value of (-2b+a+b-2a) V 0
N + [ (-2b+a+b-2b) V 01 > N. Consequently, N + (a+b) 2 N + (2b+2a).
Thus, for each value M of a + b which contains
M + (2b+2a) which is a contraction.
(2 -t 3) For a € A and b € B, la1 + Ibl 5
Consequently.
Hence,
0 5 21b( + a + b + 21aI 541bl + 41al.
By the Reiz Decomposition Theorem, there exist a' f A and b' € B
such that, 0 5 a' 5 41a1, 0 5 b' 5 4lbl and
21bl + (a+b) + 21al = b' + a'.
This implies,
a + b = (-21b1+b1) + (a1-2)al)
C B + A .
Similarly, it may be shown that for c € A and d € B
d + c C A + B .
Therefore, A + B = B + A.
(3 -+ 1) Suppose, by way on contradition, there exists a value M
in G and a positive element y € M* such that -y+M+y # M. Then,
0 < -y + m + y C M*\M for some positive m € M. Since M* = G(M,-y+m+y)
and G(M,-y+m+y) + M = M + G (MI-y+m+y) , we claim,
M* = {g C G I I CJ~ 5 m* + n* (-y+m+y) , for some m* € M, n* € 2) .
To see this, let g E G ( M I -y+m+y) . Then
' Igl 5 ml + (-y+m+y) + m2 + ... + m + (-y+m+y) n
= m* + (-y+m+y) + . . . + (-y+m+y), .
f o r some m ,m* 'C M and where ( - ~ + m + y ) ~ € G(-y+m+y). Hence t h e r e i
e x i s t s 1 a non-negative in tege r such t h a t i
Consequently,
Let
Upcn l e t t i n g nn = n*, t h e c l a i m has been proved. 1
Returning t o t h e proof of t h e theorem, 0 c y € M*\MI hence,
f o r some m C M and an i n t ege r n. Thus, s ince y j€ M,
However, M + y 5 M + (-y+mn+y) implies t h a t t h e r e e x i s t s an
a C M, such t h a t ,
a + y 5 - y + n m + y
o r ,
-y + M + y = M I and M i s normal.
From statement (3) i n the theorem, the defining equation f o r N
can be determined. For any G 6 #, and a ,b C 6 ,
The reader may a l s o note t h a t t h i s equation is equivalent t o
(a v 0) + (b V 0) = [ ( a V O ) + (b v O)] A [n(b V 0) + n ( a V 011
fo r any integer n greater than two-
Example 3.1.10. For each posi t ive integer n, a var ie ty S(n)
of 1-groups w i l l be constructed. In recent l i t e r a t u r e r t h i s var ie ty
is cal led the Scrimger variety. It was f i r s t introduced by Martinez
1101; Smith 1141 and Scrimger [131 extended t h i s work.
Lemma 3.1.11. Let the s e t of in tegers be denoted by Z. Then,
Proof: The binary operation on Z W r Z 3s defined a s follows:
For (F, k) and ( G , R ) i n Z W r Z
( ~ , k ) + (G, R) = ( F + G ~ . k+R) ,
k where G (2) = G(k+z) f o r a l l z € 2.
Section 2. Containment Relat ionships
60.
-k The inverse of (F,k) is (-F k and t h e i d e n t i t y element of
- Z W r Z i s (0,0) where O(z) = z f o r a l l z € 2.
With t h i s information, t h e reader may convince himself t h a t G(n)
is a subgroup of Z W r 2.
The l a s t i t e m t o check, is t h a t G(n) is a s u b l a t t i c e of Z W r Z .
For (F, k) C G (n) , t h e least upper bound of (F,k) and (5,0) i n
z w r Z is : ( ~ , k ) v (0,0) = ( H , h ) where f o r (a,b) C Z x Z
(a , b) i f k < O
(a ,b) (H,h) = Wb(a) , b+k) i f k > O
b - b (F (a) V 0 (a) ,b) i f k = 0 .
But, (H ,h) is again i n G (n) . Therefore, G(n) is an 1-subgroup of
Z W r Z.
The v a r i e t y generated by G(n) is t h e Scrimger v a r i e t y S(n) .
Example 3.1.12. Let n be a p o s i t i v e in teger . Then L(n) denotes
t h e v a r i e t y of a l l 1-groups G, such t h a t ,
nx + ny = ny + nx f o r a l l x,y C G.
In t h i s sec t ion t h e containment r e l a t i o n s h i p s between t h e v a r i e t i e s
E , A, R , N, S ( n ) , L(n) and L w i l l be exhibited. A var ie ty V is
contained i n a v a r i e t y U i f each element of V is a l s o an element of U.
The var ie ty E is t h e smallest var ie ty of l-groups, since it is
contained i n every other var ie ty of l-groups
The var ie ty L is the la rges t var ie ty of l-groups since it
contains every other var ie ty of l-groups.
Theorem 3.2.1. I f G is an element of A, then G is an element -
of every var ie ty V - of l-groups other than E.
Weinberg [151 constructed the f r e e abelian l-group, and character-
ized it a s a subdirect sum of a family consis t ing of copies of t he
integers.
The f r ee abelian l-group, with a f r e e generators denoted by
is described a s follows:
For a, l e t Ja denote the f r ee abelian group of rank a,
and l e t r denote the family of to ta lorders on Ja (a f r e e group
o r f r ee abelian group can always be t o t a l l y ordered). Then Aa
is a sub la t t i ce of the cardinal product 'a of the family of O-groups
{ L J,, TI , T € I-1. The sub la t t i ce Aa is generated by the diagonal of
the cardinal product. Thus every abelian l-group i s a subdirect sum
of a family of copies of t he ordered group of integers.
Proof of Theorem 3.2.1: L e t V be any var ie ty of l-groups other
than E , and l e t G be an element of V . Then G contains a copy
of the integers Z. Therefore V contains, as an element, the O-group
(Z,+) . . Moreover, t h i s implies t h a t V contains a subdirect sum of a
family of O-groups (Z,+). Consequently, V contains the f r e e abelian
l-groups; and since each G € A i s the l-homomorphic image of some
su i tab le chosen f r e e abelian l-group, A C - V.
Theorem 3.2.1 a l s o implies t h a t t h e v a r i e t y A covers t h e v a r i e t y
E, i. e. t h e r e are no v a r i e t i e s s t r i c t l y contained between A and E.
The next set of proposit ions w i l l show t h a t t h e v a r i e t y S(n)
is contained i n t h e v a r i e t y L(n) ; and S(n) covers A, i f n is
a prime number.
To show S (n) is contained i n f. (n) , it s u f f i c e s t o show t h a t
G(n) s a t i s f i e s t h e def in ing l a w , nx + ny = ny + nx of f . (n) , s ince
S (n) is generated by t h e 1-group G (n) .
Theorem 3.2.2. For (F,k) -
n(F,k) + n(G,h) = n(G,h) + n(F,k
k Proof: n(F,k) = (F+F+..
Hence, n(F,k) + n(G,h)
k (n-1) k+Gnkffink+h+ = (P+F +... 4 F . . . ffi , nk+nh) ,
Corollary 3.2.3. S (n) c - L (n) .
In Smith [14], t h i s r e s u l t has been extended t o S(n) c L(n) f o r f
every composite in teger n.
63.
Corollary 3.2.4. - For n and m p o s i t i v e i n t e g e r s t h e following
statements a r e equivalent .
1) L(n) c - h(m)
3) n d iv ides m
4) S(n) 5 L(m) .
I n showing t h a t S ( n ) covers A i f n is a prime number,
seve ra l o t h e r r e s u l t s w i l l be derived. Namely, n e i t h e r S (n ) nor
L(n) (n > 1) is contained i n R. A l s o , i f m and n are r e l a t i v e l y
Lemma 3.2-5. If C is a convex 1-subgroup of G i n L (n) , -
then -nx + C + nx = C f o r a l l x € G. -
Proof: Let c be a p o s i t i v e element of C then
€ -nx + C + nx .
Hence, c f -nx + C + nx, ( s ince a conjugate o f a convex 1-subgroup
i s another convex 1-subgroup). Therefore, C c - -nx + C + nx. Simi lar ly
it can be shown t h a t -nx + C + nx 5 C.
Lemma 3.2.6. - I f C i s a convex 1-subgroup of G C h(n) and
x C G, then t h e number of d i s t i n c t conjugates, of t h e form - ix + C + i x
(i € Z) , is a d i v i s o r of n.
Proof: For i and k in tegers , w r i t e
i = p n + r and k = q n + s
where 0 I r, s < n and p,q f 2.
I f r = s, t h e n b y Lemma 3.2.5
= -kx + C + kx .
Consequently, i n proving t h e lemma we restrict our a t t e n t i o n t o t h e
in tegers k such t h a t 0 5 k k n.
If - ix + C + i x = C, then i d iv ides n. I n order t o see t h i s ,
suppose -kx + C + kx = C where k > 0 is t h e smallest p o s i t i v e
in teger with t h i s property. I f k does not d iv ide n, then t h e
g r e a t e s t common d i v i s o r b of k and n may be wr i t t en as kp + nq,
f o r p and q some in tegers . Then,
But t h i s c o n t r a d i c t s t h e assumption t h a t k w a s t h e smallest such
in teger ; and therefore , k d iv ides n.
Consequently, i f the re a r e m conjugates of t h e form
- i x + C + i x = C , then m d iv ides n.
Proof: For G € R n L(n), G is a product of totally ordered
groups G and for a,b C G, na + nb = nb + na. Hence, in each
of the totally ordered groups G nc + nd = nd + nc for c and
d elements in Gi . Now if c and d do not commute in Gi , then without loss of
generality, assume that c,d 2 0 and c + d < d + c. This implies
that n(-d+c+d) = -d + nc + d < nc. But,
a contradiction. Hence, c and d commute, and consequently each
of the totally ordered groups is abelian. Therefore G € A.
Corollary 3.2.8. I? P L (n) and L (n) R for n > 1. - -
Corollary 3.2.9. R $ - S(n) and S(n) $ R for n > 1. - -
Proof: The variety S(n) also satisfies the law nx + ny = ny + nx.
then -
This
Lemma 3.2.10. - If m - and n are relatively prime positive integers,
L(m) ll L(n) = A.
Proof: Let G f L(m) fl L(n) be a subdirectly irreducible 1-group.
is equivalent to G having a smallest non-trivial 1-ideal or to G
having a representing subgroup. Thus G acts transitively on some
totally order set Q .
t Choose integers r and s such that n u + sn = I. Then,
= a [mrh + mng + nsh]
= a[h + mngl
i
The permutation mng fixes ah, and consequently g fixes
ah. Hence, since G is transitive, g must be the identity. There-
fore, if an element k of G fixes an element of 52, then k is
the identity; and G must be an 0-group. This implies G € L(n) n R = A.
But, since G is an arbitrary element in the generating set of
L (m) n L (n) , we conclude that L(m) fl L (n) = A.
Corollary 3.2.11. - If m and n are relatively prime, then
S(m) fl S(n) = A.
Lemma 3.2.12. If C is a representing subgroup of G 6 L(n), - G j? A, then there exists a strictly positive element x G such
that -x + C + x # C.
Proof: Suppose, for all x where O c x € G, that - x + C + x = C .
Then C is an 1-ideal of G as well as a representing subgroup. Con-
sequently, C = (01, and therefore, G is an 0-group. Thus G € R
and G € L(n), which by Lemma 3.2.8 implies G € A , a contradiction.
Lemma 3.2.13. - If C is a representing subgroup of G 6 i (n)
which has n distinct conjugates of the form -ix + C + ix = C - for
some x € G, then G contains an 1-subgroup which is 1-isomorphic - -
Proof: An element a will be constructed so that a € G, and
a and x correspond under an 1-isomorphism to the generating elements
(5,l) and (B,O) of G(n) where
0 if z j! 0 (mod n) B(z) =
1 if z E 0 (mod n) -
For 0 5 i I n-1 define C(i) and D(i) by
Let be the totally ordered set of right cosets of C in G.
For 0 5 i I 1 C + ix has C(i) as its stabilizer
subgroup. The set D ( 1 ) consists of all permutations
which fix each C + jx for j # i.
Since by hypothesis C (0) # C (i) (1 C i 5 n-1) there exist
h(i) €C(O)\C(i) (1Ci5n-1) suchthat O < h(i). Whence,
0 < g(0) = h(i) + . . . + h(n-1) € C(0) \ U C(i) . Define d(0) by i#O
Since ng(0) fixes only C, -ix + ng(0) + ix fixes only C + ix
1 5 i 5 1 Thus, d(0) moves C only, and d(0) < x, since
x moves C.
Define d ( i ) by
which by convexity implies that e ( i ) € n D(k) = n C(k). k=O k=O
Now l e t
Then, the following hold:
1) -nx + a ( i ) + n(x) = a ( i ) ,
2 ) O < a ( O ) < x ,
3) 0 < a ( i ) < x,
4) a ( i ) € D ( i ) \ U D(j ) , j # i
5) a ( i ) A a ( j ) = O ( i # j )
6 ) a ( i ) + a ( j ) = a ( j ) + a ( i ) .
The map f : G (n) + G defined by
is a group of homomorphism, since for (F,k) and (G,h) G(n)
k k = [F+G (0) l a ( 0 ) + . . . + [Fffi (n-1) ] a h - 1 ) + (k+h)x
k = F ( O ) ~ ( O ) + ... + F(n-l)a(n-1) - kx + kx + G ( O ) a ( O ) +
k k (x) - k (x) + . . . + k (x) - k lx) + G (n-1) a (n-1) + (k+h) x
k = F(O)a(O) + ... + F(n-l)a(n-1) + k(x) + G a (k ) + ... +
k G (k+n-l)a(n-l+k) + h(x)
= f ( F I k ) + f(GIh) -
The homomorphism f is a l s o an 1-homomorphism. This w i l l be
shown by f i r s t l e t t i n g k = 0, and then k # 0 . For ( F , O ) C G(n) , l e t T = (i I F ( i ) 1 01, then s ince
a ( j ) A a ( i ) = 0 f o r i # j ,
W e now show t h a t f o r (F,k) C G(n) with k # 0,
i ( ~ , k ) v ~ ( O , O ) = f [ ( ~ , k ) v (0,0)1.
To do so, it w i l l be shown t h a t ma(0) < x f o r m € 2. This is
t r u e f o r m C 1. Assume •’or 2 5 i < m t h a t i a ( 0 ) < x. Since,
it s u f f i c e s t o show t h a t one of t h e terms i n ma(0) is less than x,
riamcly t h e t e r m
d ( 0 ) - d(n-1) + [O V m(A (d(0)-d( j ) ) )J+ d(O) - d ( l ) . NOW,
d(0) - d(n-1) + [0 V (m-2) A (d (0 ) -d ( j ) ) ] + d(0) - d(1) < x
i f and only i f ,
But, d(n-1) - d(0) + x + d(1) - d(0) = x. Therefore, ma(0) < x,
f o r a l l m € Z. Moreover, ma(j) < x, (m € 2) ( j = O , l , . . . ,n-1).
With t h i s r e s u l t we have,
F(O)a(O) + .. . + F(n-l)a(n-1) + mx < 0 i f m < 0 ,
and
F(O)a(O) + ... + F(n-l)a(n-1) + m x > 0 i f m > 0 .
Thus, f [(F,k) V (0,0)1 = f (F ,k) V f (5 ,0 ) .
F inal ly , f i s one-to- one, s ince f ( ~ , k ) = 0 i f and only i f
F = and k = 0.
Theorem 3-2.14. If n is a prime number then S(n) covers A.
Proof: I f G is a subdirec t ly i r r e d u c i b l e member of S (n) , but
not an element of A, then G contains a respresent ing subgroup C ,
which i s not an 1-ideal of G. Thus, t h e r e e x i s t s a s t r i c t l y p o s i t i v e
element x i n G such t h a t -x + C + x # C. By Lemma 3.2.6, s ince
n is prime, C must have n d i s t i n c t conjugates of t h e form
-ix'+ C + ix. ThusI G contains an 1-subgroup 1-isomorphic t o G(n).
Consequently, i f V is any v a r i e t y contained i n S(n) which has a
71.
nonabelian member then V contains S (n) . Thus , S (n) covers A.
The variety hJ consisting of normal valued 1-groups will be shown
to be the largest non-trivial variety. This result was established
in Holland [ 7 I .
Theorem 3.2.15, If H is a nontrivial 0-2 transitive 1-group - of order preserving permutations on a totally ordered set S, if F
is the free 1-group on a countable set XI and w € F is not the -
identity element of F then H does not satisfy the law w = e. -
Proof: By Hollands' Representation Theorem, it may be assumed
that F is an 1-permutation group on some totally ordered set T.
Since w € F is not the identity element, there exists an element
t in T such that tw # t. Let w = v A ~ I x , where I and J I JK ijk
are finite sets, and K = 1 2 . . . n . For each (i,j) € I x 3, define
and for 1 5 k 5 n, define
For each element x of X occurring in w, and each pair (i,j) let
'i j ( x ) = { k E K I x = x ijk 1 and
-1 Nij(x) = {k C K I x = x } .
ijk
1f k C Pij (x) I then
t(i, j,k-l)x = t(i,j,k);
and if k E N . . (x) , then 17
The set T' = <t(i,j,k) I (i,j) E I x J, 0 5 k 5 n) is a finite
subset of T. Choose and label any subset {s (i, j ,k) I (i, j E I x J, 05k5n)
of S in one-to-one correspondence with T'; so that the correspondence
t ( i k ) s ( i j k preserves the order on T'. Since, multiplication
by x provides a one-to-one order preserving correspondence such that
t i j k - 1 t ( i j k ) (kEP..(x)) 1 3
$ t(i,j,k) + t(i,j,k-1) (k € N. .(XI) ; 1 3
F g it follows that the correspondence k !
s i j k - 1 +- s i j k (k E P.. (XI) 1 3
and
s i l k + s i j k - 1 (k C Nij(x))
must also be one-to-one and order preserving.
Because H is 0-2 transitive on S, it is also 0-n transitive;
and thus there exist h(x) E H such that,
s(i,j,k-l)h(x) = x(i,j,k) for k E P . (XI , i 3
and
s(i,j,k)h(x) = s(i,j,k-1) for k € N. .(x). 13
Since t = t(i,j,O) for each i , , denote by s the elements
si,j,O for each (i,j). Then for the substitution x -+h(x), we
have for each (i,j) € I x 3,
Because tw # t,
t # t vAIIxijk= VA t IIx = vn t(i,j,n) . I J K IJ K
ijk IJ
Recalling that the set s , k is in one-to-one order preserving
correspondence with T',
s VAII h(xijk) = VA x II h(x 1 I JK IJ K
ijk
= VA s(i,j,n) I J
# s.
Consequently, H does not satisfy w = e.
Corollary 3.2.16. - If G is an 1-group which satisfies a law not
satisified by every 1-group, then G is normal valued.
Proof: Let G be an 1-group which is not normal valued and let
w = e be a law not satisfied by every 1-group. Then by Theorem 2.4.13,
G contains an 1-subgroup which acts 0-2 transitively on some totally
ordered set. Therefore, w = e is not satisifed by any varitety
containing G .
Corollary 3.2.17. N is the largest proper variety o f 1-groups.
CHAPTER IV
The La t t i ce L of Varieties of Latticeordered Groups
In t h i s chapter we w i l l discuss the proper t ies of the col lect ion
L of a l l v a r i e t i e s of 1-groups. I t w i l l be shown t h a t L i s a complete
l a t t i c e ; and t h a t a multiplication on the elements of L can be defined
so t h a t L becomes a l a t t i c e ordered semigroup.
Some of the topics concerning the l a t t i c e ordered semigroup L
which w i l l be discussed i n t h i s chapter are:
1) the idempotents of L;
2 ) the generation of var ie t ies ;
3) the factor izat ion of var ie t ies ;
The material i n t h i s chapter may be found i n Glass, ~ o l l a n d and
McCleary [ 5 1, Martinez 1111, 2121, and Smith [141.
Section 1. Torsion Classes
A torsion c l a s s of 1-groups i s a col lect ion of 1-groups which is
closed with respect t o homomorphic images, convex 1-subgroups, and joins
of convex 1-subgroups i n the c lass . This idea of tors ion c l a s s was
f i r s t introduced and studied by Martinez U21.
A n important consequence derived from t h i s approach is the T-torsion
rad ica l of G denoted by T(G) , which is defined as follows: fo r
an 1-group G and a torsion c l a s s T, T(G) is the join of a l l convex
1-subgroups of G belonging t o T. It should be c l ea r t o the reader
t h a t T(G) i s the l a rges t convex 1-subgroup of G belonging t o TI
and hence, it is an 1-ideal of G.
Several basic proper t ies of the T-torsion rad ica l of an 1-group a r e
found i n the following proposition which is due t o Martinez [121.
Proposition 4.1.1. - Let T be a tors ion c l a s s and G be an
1-group.
1) If A i s a convex 1-subgroup of G, then T ( A ) = A n T(G). - 2) If f : G -+ H i s an 1-homomorphism of G onto H I then
f [T(G) 1 5 - T(H) - 3) T(T(G) = T(G) i .e . , T(G) is closed.
4) If { A ~ I i 6 1) i s a col lect ion of convex 1-subgroups of G,
then T(V Ail = V T ( A ~ ) and a l so T ( A A ~ ) = A T ( A ~ ) . -
Proof of (1). The convex 1-subgroup A 0 T(G) of T(G) is i n TI
and since A fI T (GI i s a l so a convex 1-subgroup of A, A T(G) 5 T (A) . Conversely, T(A) is a convex 1-subgroup of A, and hence, it is
a convex I-subgroup of G. Since T(A) is i n T , T(A) c - A T(G) .
Proof of (2) . Since T i s closed under 1-homomorphic images,
f (T(G)) i s a convex 1-subgroup of H which i s contained i n T. Thus,
f (T(G)) 5 T(H).
The reader should note t ha t i f h : G -+ G i s an 1-automorphism
of G , then f (T(G)) 5 T(G), i n other words T(G) i s a f u l l y invar iant
1-ideal of G.
Proof of ( 3 ) . By ( 1 1 , and the f a c t t h a t T(G) is a convex 1-subgroup
of G r we have T(T(G)) = T(G) fl T(G) = T(G).
Proof of (4) . T(v Ai) 1 T ( A ~ ) fo r a l l i € I, thus,
T ( V A ~ ) 3 - v T ( A ~ ) .
For ( A ~ I i C I) a col lect ion of convex 1-subgroups of G, V Ai
i s a l so a convex 1-subgroup of G. Moreover, T(Ai) 5 T(G) and thus
v T(Ai) c - T(G). Consequently by (11 ,
Since T ( A A . ) C T is a convex 1-subgroup of each Ai , w e have 1
Conversely, A T(Ai) i s a convex 1-subgroup of each T(Ai). hence,
A T ( A ~ ) C T. Also, A T ( A ~ ) i s a convex 1-subgroup of each Ai , so
We now proceed t o show t h a t t he 1-group var ie ty N is a tors ion
c lass . With t h i s proposition we then reproduce a r e s u l t by Holland [8 1
which s t a t e s t h a t every 1-group var ie ty i s a tors ion c lass .
Proposition 4.1.2. The 1-group var ie ty N i s a torsion c lass .
Proof: Since N i s a var ie ty , it is closed with respect to taking
1-subgroups, 1-homomorphic images, and cardinal products of normal valued
1-groups. Thus, t o prove N is a tors ion c l a s s it suf f ices t o show t h a t
N is closed with respec t t o taking jo ins of normal valued convex l-sub-
groups from an l-group G. To t h i s end, l e t { A ~ I i C 11 be a col-
l e c t i o n of normal valued convex l-subgroups o f some l-group G.
We w i l l show t h a t f o r 0 5 x,y C V Ai , t h e inequa l i ty
x + y 5 ny + nx is s a t i s f i e d f o r some p o s i t i v e in teger n.
F i r s t , however, w e w i l l concentrate on t h e spec ia l case of
- O Z a C A and O 5 b C A . Let a = a - ( a A b ) and 6 = b - ( a A b ) .
i j
Then, a A b = 0, a + b = b + a, and a A b, a and b a r e a l l non-
negative . Thus,
a + b = i i + ( a ~ b ) + b
5 2(a A b) + 2; + b (h, aAb C Ail
= 2 ( a ~ b ) + h + b + a
= 2 ( a ~ b ) + b + a + a
5 3b + 3a.
Ut i l i z ing t h i s r e s u l t , we may now show t h a t f o r 0 5 a , b C V Ai ,
a + b 5 nb + na. Since 0 5 a , b V A i t by t h e Reiz Decomposition
Theorem, a and b a r e f i n i t e sums (Eai and b , respect ively)
of non-negative elements from U Ai. Thus,
fo r some p o s i t i v e in teger n. Although t h i s in teger n depends on
a and b it is c l e a r t h a t condit ion (3) of Theorem.3.1.9 i s s a t i s f i e d
by v Ai and therefore t h a t v A i i s normal valued.
; An 1-group G is a lex-extension of a prime subgroup C of G
r: i f O < a € G and a A b = O f o r s o m e O < b € G I then a € C . 6
Lemma 4.1.3. - Let G be a subd i rec t ly i r r e d u c i b l e normal valued
u, 1-group generated by g ,g ,-..,
1 2 gn. Then f o r some 1 5 k I n, G = G(gk).
I I
ig Proof: Let C be a value o f some element i n t h e minimum 1- ideal
of G. Then {glIg2.. . . , g n ~ C.
Since the convex 1-subgroups o f GI which contain a prime subgroup
of G I a r e t o t a l l y ordered under inc lus ion, t h e co l l ec t ion
e h {K 3 - C I K i s a value of some gi} is t o t a l l y ordered. Thus, each b
gi has only one value which conta ins C. Moreover, s ince t h e number 1 of genera tors o f G is f i n i t e , t h e set has a l a r g e s t element which
w i l l be denoted by M, and we w i l l say t h a t M i s a value of g . m
Since any convex 1-group o f G conta in ing M must conta in a l l
of t h e genera tors of G I t h e cover o f M i s a l l of G. By t h e assump-
t i o n t h a t G i s a normal valued 1-group, and by ~ G l d e r ' s Theorem, w e
conclude t h a t G/M is 1-isomorphic t o t h e real numbers.
W e w i l l now show t h a t G is a lex-extension o f M. Let
0 < a , b € G and l e t a A b = 0. Since fl - g + C + g is an g€G
1-ideal of G which does no t conta in t h e minimum 1-ideal of G, t h e
1-ideal A - g + C + g i s the i d e n t i t y element of G. Thus, t h e r e 9 CG
e x i s t s an element g € G such t h a t b P - g + c + g. However, s ince
-g 4- C + g i s a prime subgroup of G and a A b = 0, we conclude
t h a t a C -q + C + q -g + M + g = M. ~ h u s , G is a lex-extension
Finally, gm f M, and G/M i s an archimedean 0-group, and hence
G = G(gm).
Theorem 4.14. Every v a r i e t y V of 1-groups is a tors ion c lass . -
Proof: Let G be an 1-group, and l e t
of convex 1-subgroups of G such t h a t A. € 1
I f V i s the var ie ty consist ing of a l l
V Ai f V .
Thus, l e t V - N and l e t o ( x l....,xn)
{ni I i C 13 be a co l lec t ion
V ( i € I ) .
1-groups then c l ea r ly
= e be a law of V
not s a t i s f i ed by every 1-group. Then Ai € N (i C I). Thus
v f N .
We wish t o show t h a t w(xl, ... ,x ) = e holds i n V Ai . n
For hlrh2,..- .h C V Ai , each h i s a f i n i t e sum X a i j of n i
j elements a C U Ai . Let H be the 1-subgroup of V Ai generated
i j
by {a. . I . Then, H i s a member of N. 1 3
Let H be a subdirectly i r reducible fac tor of H. Then, H is
a l so a member of N . Moreover, i f h -+ h denotes the natural map
- - of H onto H I then . . generates H. Hence, by Lemma 4.1.3,
1 3
we conclude t h a t fi = fi(a. . ) for some i and some j. 1 3 -
The preimage a of a i s an element of $ f o r some k C I. i j i j -
Thus, q( ll H = H. From the assumption t h a t Ak s a t i s f i e s
w(x lI. . .x 1 = e we conclude t h a t 4 ll H = H s a t i s f i e s w(xl,. . . ,x 1 = e. n n
Therefore, since H is a subdirect product of subdirectly
i r reducible fac tors . it s a t i s f i e s w(xl... . , x 1 = e. Thus n
w(hl,...,h = 0 and so v s a t i s f i e s w(xl, ... ,x = e a s required. n n
Corollary 4.1.5. For a n 1-group G and a v a r i e t y of 1-groups U,
t he re e x i s t s a unique 1-ideal U(G) of G such t h a t U(G) € U
and U(G) conta ins every convex 1-subgroup of G which i s a member -
of U . -
The reader i s now aware of t h e f a c t t h a t U(G) is t h e U-torsion
r a d i c a l of G. Consequently, Proposit ion 4.1.1 i s appl icable t o
v a r i e t i e s of 1-groups.
With t h i s background we now proceed t o Section 2, where w e consider
t h e v a r i e t a l product of 1-group v a r i e t i e s .
Section 2 . Var ie ta l products
Since t h e i n t e r s e c t i o n of a c l a s s of 1-group v a r i e t i e s is again
a va r ie ty ; it is na tu ra l t o define a p a r t i a l o rde r , on t h e class of a l l
1-group v a r i e t i e s L, by using s e t containment. For a l l U and V i n
L def ine
V 5 U i f and only i f V C U . -
The p a r t i a l order 5 on L becomes a l a t t i c e order , i f one de f ines
f o r V . EL (i f I ) , 1
A v i = n v , iC1 i f I i
and
Since L conta ins both a l a r g e s t element and a smallest element, t h e
d e f i n i t i o n s of V and A on L s u f f i c e t o make L a complete l a t t i c e .
One may a l s o def ine a mul t ip l i ca t ion on L as follows: f o r
U and V members of L, G € UV i f and on ly i f G conta ins an
1-ideal H such t h a t H € 1% and G/H € V. O r , G € UV i f and only
Proposit ion 4.2.1. I f U and V are m e m b e r s of L, then - - UV i s an element of L.
Proof: I t w i l l be shown t h a t UV s a t i s f i e s t h e d e f i n i t i o n o f a
va r ie ty .
Let G € UV, then G conta ins an 1-ideal H such t h a t H f 1%
and G/H € V . For an 1-subgroup K of G, H A K i s an 1-ideal o f
K, and an 1-subgroup of H. Also K/ (H A K) i s 1-isomorphic t o
HK/H, and HK/H is an 1-subgroup of G/H. Consequently, H A K C U
and K/(H A K) C V , s ince U and V a r e v a r i e t i e s . Thus K € UV.
Next it w i l l be shown t h a t t h e ca rd ina l product of elements from
UV is i n UV.
Let { ~ ( i ) 1 i 6 11 be elements of UV. Then f o r each i € I,
the re e x i s t s an 1- ideal H ( i ) o f G ( i ) , such t h a t , H ( i ) C U and
G H C v. Consequently, f Z H ( i ) € U and G H C . Whence,
I I H ( i ) 6 U and G v. Thus, n G ( i ) € UV.
Final ly , it i s shown t h a t t h e 1-homomorphic image of an element
G i n UV i s i n UV.
Let G € UV, then G conta ins an 1-ideal H such t h a t H € U
and G/H C V. Let f be an 1-homomorphism o f G onto G ' , then
~ / k c r ( f ) is 1-isomorphic t o G' , the 1-homomorphic image of H, f (1.1) ,
i n 6' is an 1-ideal of G' . T h e quotient group G S / f ( H ) is an
1-homomrphic image of G/H by T h e o r e m 1.1.13, T h u s , f (HI C U
and G ' / f (HI € V. T h e r e f o r e G' € UV.
Proposition 4 . 2 . 2 . L e t U, V and 111 be e l e m e n t s of L, then - - U(Vw) = (UV)W. I n other w o r d s , multiplication is associative.
Proof: L e t G € U ( w ) , then there exists an 1-ideal H of G
such t h a t H € U and G/H 6 w. Since G/H € w, G/H contains an
1-ideal K/H such t h a t K/H C V and (G/H)/(K/H) w h i c h is 1-isomorphic
to G/K € W.
NOW, H € U and K/H C V i m p l i e s t h a t K € (UV). B u t , s ince
G/K C W ; it m u s t be t h a t G € (UV)W. T h e r e f o r e U ( w ) 5 (UV)W.
C o n v e r s e l y , l e t G € (UV)W and l e t A be the UV-torsion radical
of G. T h e n G/A € W . L e t B be the U-torsion radical of A,
then A/B E b'.
Since torsion radicals are f u l l y invariant, B is an 1-ideal of
G. T h u s , B € U , A/B € V and since (G/B)/(A/B) is 1-isomorphic
to G/A C W , w e also have (G/B)/(A/B) € W. T h e r e f o r e , G f U ( v t d ) .
Proposition 4 . 2 . 3 . For e l e m e n t s U , V and W of L , i f - - - U 5 b' then UW 5 Vw and WU 5 UV. -
Proof: A s s u m e U 5 V and G € UW. T h e n , there ex i s t s an 1-ideal
H of G such t h a t H € U and G/H f W. Since U 5 V , H € b'
a n d * thus , G C w. T h e r e f o r e , (.@ C - Vw.
S i m i l a r l y , it m a y be s h o w n t h a t WU 5 WV.
By proposition 4.2.1, 4.2.2, and 4.2.3, L is a l a t t i c e , and a
p a r t i a l l y ordered semigroup. The semigroup (I,,*) has an iden t i ty ,
namely E, s ince EV = V and VE = V f o r a l l V € L,
For V and element of L, vn(n € N ) w i l l denote the product
of V with i t s e l f n t i m e s . It is ca l led the n th power of V .
I f v2 = V , then V is ca l led an idempotent of (Lie). It w i l l be
shown t h a t the only idempotents of (Lie) a r e E, N and L.
Proof: Since (H,A) w r ( ~ , B ) is an 1-subgroup of (H,A) W r (G,B)
l-var{ ( H , A ) w r (G,B) ) 5 1-var{ ( H , A ) w ~ ( G , B ) ).
Conversely, suppose w(x) = e i s a law which f a i l s i n (H,A)Wr(G,B),
and l e t x -+ f be a subst i tu t ion i n (H,A)Wr (G,B) f o r which
f for some a = (a ,b) C A x B.
- For ( f i i j Ih . . ) = mijk ( ( i . j ) c I x J ) , a(fiij.h..) + a fo r
K 1 I
a f i n i t e subset of I x J. Therefore, afi i j ( b
# a f o r a f i n i t e
subset of I x J. For a l l c € B \ ( { ~ I U {bh . . ) ) , le t fi i jc
be t he 1 3
iden t i ty . Then, a s required, w(x) = e f a i l s i n ( H , A ) w r ( G , B ) , and
1-var( ( H , B ) W ~ ( G , A ) ) 5 1-var(HIB)wr(G,A)).
Lemma 4.2.5. (H,B) be an 1-permutation group i n t he 1-group
var ie ty U, and l e t (G,A) be a t r a n s i t i v e 1-permutation group i n
the 1-group v a r i e t y V . Then, W = (H,B)Wr(G,A) 6 UV. - -
Proof: The na tu ra l 1-ideal TI H~ of W i s an element of a € a
U and W/ II H' i s 1-isomorphic t o G which i s a member of V. '€A
Therefore, W C UV.
Corollary 4.2.6. For any p o s i t i v e i n t e g e r n, w r n Z C A" n
and W r R € A".
Lemma 4.2.7. For an 1-group G, an 1- ideal H of G I and a - - - value M i n G; i f M does not conta in H I then M n H i s a value - - i n H. -
Proof: W e w i l l show t h a t M n H i s a va lue i n H by showing
t h a t M* n H covers M n H I where M* i s t h e cover of M i n G.
F i r s t , i f H n (M*\ M) = 4 then M 3 H which c o n t r a d i c t s one - of our assumptions. Thus H l l M* 3 H n M.
#
If C i s a proper convex 1-subgroup of H, then
{x 6 G ( 1x1 A Ihl C C f o r a l l h C H) i s a convex 1-subgroup of
G I which we w i l l denote by C ' .
Clear ly , C c - C ' n H. Conversely, i f C ' fl H $ C, then choose - 0 < x C (C ' n H I \ C. Since x 6 C ' r and x C H we have
1x1 A It1 € C f o r a l l t C H implies t h a t x = 1x1 A 1x1 C C I
which i s a cont radic t ion . Consequently, C = C ' n H.
Moreover~if B and C a r e convex 1-subgroups of H such t h a t
B C then C ' n H = C 7 B = B ' fl H. Therefore C ' 7 B'.
Applying the previous two paragraphs t o t h e lemma e s t a b l i s h e s t h a t
H n M is a value of H and M* ll H i s i t s cover.
Theorem 4.2.8. The l-CJr0up v a r i e t y N i s an idempotent i n - & , * I
Proof: It s u f f i c e s t o show t h a t f o r G C k 2 , each regu la r subgroup
M of G is normal valued.
2 Let G C N , then G conta ins an 1- ideal H such t h a t H € N
and G/H C N .
Let M be a r egu la r subgroup of G. I f M 3 H, then M/H - i s a r egu la r subgroup of G/H. Thus, M/H i s normal i n i t s cover
( M / H I X = M*/H (where M* i s t h e cover of M i n G ) . Therefore, M
i s normal i n M*.
If , however, M 2 H , then by Lemma 4.1.2 M Il H i s a r egu la r
subgroup of H. Hence, M fI H is normal i n i t s cover ( M I7 HI* = M* n H.
Observe t h a t by Theorem 1.1.16
(M*nH) / ( M ~ H ) i s 1-isomorphic t o M + (M* I7 H) /MI
and
Therefore, M i s 'normal i n M*.
Theorem 4.2.9. N = An. n=l
Proof: Since A i s contained i n N and N i s an idempotent,
by an induction argument w e ob ta in t h e conclusion t h a t A" c N f o r - a l 1 , p o s i t i v e in tege r s .
CQnversely, l e t w ( x ) = V A n xijk = e be a law which i s not sa t - I JK
i s f i e d by some subd i rec t ly i r r educ ib le element G i n N. Since G
i s an element of N , G may be taken to be an 1-subgroup of t h e wreath
product o f copies of t h e r e a l number, (by Theorem 2.4.5). say
~ ( r I r C = A Hence, t h e r e e x i s t s a s u b s t i t u t i o n X -+ 9
and an element A i n A. such t h a t , A # Aw(g) = VAM gijk . IJ K
W e w i l l show t h a t w(x) = e i s no t s a t i s f i e d by ~ r ( R ( 6 ) 1 6 C I
where A i s a f i n i t e subset of r. k
L e t B = {A} U {Awij(g) I i C I, j C J. k C K}. Then. B i s
a f i n i t e subset of A; and f o r each p a i r of d i s t i n c t po in t s (a ,b)
-r i n B , t h e r e e x i s t s a unique r C , such t h a t , a = b. Let A
be t h i s f i n i t e subset o f p o i n t s of r.
Let A = ~ ( 6 ) and l e t a -+ a denote t h e p ro jec t ion of A
- 6 CA onto A.
For each g C {gijk/i C I, j C J, k C K) t he re e x i s t s
- = (gl -1 C w r { ~ ( 6 ) / 6 C A ) defined by
. a
i f a € B and t h e r e e x i s t s b € B such t h a t
ag = b i n A
otherwise.
- The permutation - i s well defined s ince i f a and C C B
'I. , a - -6 - -
such t h a t a = c i n A I then by t h e way i n which w e choose the po in t s
- 6 of A , a = c i n A. Consequently, - -
'1.a gl.c'
We wish t o show t h a t t h e map x i j k 'ijk
i s a subs t i tu t ion .
To do so, it s u f f i c e s t o show t h a t i f x i j k and x
i .3 'k ' are
inverses , then g i jk and
i ' j are inverses .
I•’ xijk and x are inverses , then gi jk and g i ' y k S i ' j k '
- a r e inverses . Hence, i f ) - = (gijkl6 y g ~
(gi jk 6 ,a then t h e r e e x i s t s
- an element b C B such t h a t b E~ agijk i n A. Consequently,
-6 - (agijk)gigjgk1 = a. and so "i jk'6 ,a; - - (giS j c k m )
i t j ' k ' 69%
i jk a s required.
We w i l l now show t h a t w(x) = e f a i l s i n ~ r { ~ ( 6 ) / 6 C A ) .
k-1 ~ e t g C { g i j k I i C 1 , ~ C J } , a n d l e t a = A w (g) C B
i j
then, for 6 C
- --- Hence, f o r each p a i r ( i , j ) C I x J. Angijk = AIIgi,j,kl .
K K Since a -t a i s one-to-one and an o rde r preserving correspondence
from B t o 6 ;
Thus, w(x) = e f a i l s i n ~ r { ~ ( 6 ) / 6 < A ) .
Consequently, by Corollary 4.1.7, w(x) = e f a i l s i n An, where
03
n i s t h e c a r d i n a l i t y of A; and so w(x) = e f a i l s i n V A". n=l
Corollary 4.2.10. - I f V i s any proper v a r i e t y of 1-groups, then
t h e powers o f V generate. , N .
Upon not ing t h a t both L and E a r e idempotents, w e have t h e
following r e s u l t :
Theorem 4.2.11. The only idempotents of (L,-) are E, hf
and f,.
Section 3. The Generation of Varie t ies
In the f i r s t section of t h i s chapter it was shown t h a t i f (HIA)
and ( GI B) a r e subdirectly i r reducible elements of the v a r i e t i e s
U and V respectively, then ( H , A ) ~ ~ ( G , B ) i s an element of UV.
In t h i s section we w i l l extend t h i s r e s u l t t o determine how par t icu la r
elements from U and V generate UV.
Lemma 4.3.1. Let (GIA) be a t r a n s i t i v e 1-permutation group,
and G € UV. The o r b i t s of U ( G ) determine a convex congruence C -
on ( G I A ) , and the lazy subgroup f o r C is U ( G ) i t s e l f . Let -
(G/U ! G) , A/C ) den0 t e the induced t r a n s i t i v e 1 -permutation group; 81
C1 and l e t ( U ( G ) ,6) denote the t r ans i t i ve 1-permutation group obtained
,., by r e s t r i c t i n q U ( G ) to the C-class a . Then, there is an 1-embedding
4 of ( G r A , onto ( W r B ) = (u(Gla.h) (G/U(G).A/C).
Proof: In l i g h t of Theorem 2.3.6, it su f f i ce s t o show t h a t the
lazy subgroup of C, I!. (C) , i s U ( G ) . To demonstrate t h i s point, we
w i l l need t o introduce some notation f i r s t .
Let A denote the Dedekind conpletion of A. Let F ~ ( U ( G ) =
- - {a € ii I ag = a f o r a l l g € U(G)}, and H = (g C G I ag = a fo r a l l
a 6 F X ( U ( G ) ) 1 . Finally, l e t Conv (aH) = {b € A I c 5 b 5 d f o r some A
We would like to establish that Conv (aH) 5 COW~(~U(G) ) for A
all a € H. To this end, suppose that Cow (W) P cowA(aU(G)), and A -
without loss of generality let 6 = sup(convA(a (G) ) E Conv (a) c i. A -
Then, there exists an h E H such that a 5 i; I ah. However,
6 C FX(U(G)) and h € HI so that ah 5 6h = 6 < ah. Thus,
~ o n v ~ ( a ~ ) c - convA(aU(G)) for all a E A.
Now we will show that H 5 U(G). To this end let a € A and
1 5 g F H. Then, from the previous paragraph there exist 1 5 fa € U(G)
-1 such that ag 5 af . Thus, (g A fa)g E Ga. Also, 1 5 g A f 5 fa E U(G), a a
and hence by convexity, g A fa € U(G).
Moreover, since g A f 5 g for all a € A, V (g A fa) 5 g. a
a €A Conversely, for each b € A, b[ V (g A fa)] 1 b(g A fb) = bh. Thus
a €A g = V ( g A fa) € U(G) since U(G) is closed.
a F A The reader may now convince himself that H = f(C), and consequently
Definition: A family {(Gi,Ri) I i < TI of 1-permutation groups
is said to mimic a variety V if and only if the following two conditions - are satisfied:
(1) Gi € 0, for all i E I;
(2) for any transitive I-permutation group (H,A) with H E V ,
for any A € A, any finite set of words {w (x) and any P
substitution x -t h in (H,A) there exist elements
i E I, a € Ri and a substitution x + g in Gi such
that Aw (h) < Xw (h) if and only if P 9
aw (q) < awq(gl - P
,
91.
Theorem 4.3.2. ~f U = 1-var ( (us1rs) I s E SI and { (GtIQt) I t E
mimics V . then 1-var { (uS,rS) w r (Gt1Qt) I s E S, t E T) = UV.
proof: By lemma4.2.5, 1-var (us.rSl w (G R 1 1 i s t' t
contained i n U V .
Conversely, suppose w(x) = e is a law which f a i l s i n a subdirectly 1
i r reducible member F of V. Then, the 1-group F has a representing I subgroup, and so F may be viewed a s a t r a n s i t i v e 1-permutation group
F A Applying lemma 4.3.1, we embed (FJ) i n (H,A), where (H ,A)
i s
(u(F) ' , = ( F/U(F) . 1 ) and we indentify t he two s e t s denoted by A. Thus, w(x) = e f a i l s i n
( H , A ) . I f w(x) = e f a i l s i n V then w ( x ) = e f a i l s i n some ( G ~ . R ~ ) ,
since { ( G ~ , Q ~ ) } mimics V. Thus. w(x) - e f a i l s i n any (U s ,Ts) w r ( G t l n t ) .
Therefore, throughout the remainder of the proof we w i l l assume t h a t
w(x) = e holds i n V , and i n par t i cu la r F/U(F), but f a i l s i n (H,A). k . . .
For convenience we w i l l m i t e w (h) = hijlhij2 i j h i j k ' 'L
i f x + h i s a subst i tu t ion. Consequently, since there e x i s t s a ( A , A) €
- such t h a t A # Aw(h) = AvNl (hijk,hijk), we w i l l a l so wri te
k Consider the s e t of words { w (x) } U ( e } U w. . (x) . i j 11
Since ( G ~ , Q ~ ) mimics V . there e x i s t s ( G , R ) E { ( G t , R t ) 1 and
E [
a s u b s t i t u t i o n x "g E G such t h a t f o r some or E Q i
I
'L Xw (h) < ?w (h) i f and o n l y i f aw (g) < orw ( g ) ,
P 4 P 9
f o r a l l w (x) and w (x) i n o u r c o l l e c t i o n o f words. Consequently t h i s P 9
f o r c e s t h e fol lowing:
'L 'L (1) I w i j (h) < Xwmn(h) if and o n l y if aw (9) < awm(9) i j
(2) Tw. .(h) < i f and o n l y i f a w (g) < a 13 i j
( 3 ) ?w. . (h) > 1 i f and o n l y i f aw (9) > a 1 7 i j
Q J ~ 'L s s (4) Xw. . (h) = Xw,,(h) i f and o n l y i f awk (g) = aw,,(g) .
1 7 i j
We would now l i k e t o show t h a t t h e r e e x i s t s U E U, such
t h a t t h e word w(x) = e f a i l s i n t h e wreath product ( U r n wr ( ~ $ 1 . 'L 'L 'L 'L
To t h i s end, l e t I = { i I Ahwij(h) = A ) and Ji = { 1 1 Awij(h) = A}
f o r a l l i E I. Then call ?$ x i j k = w' (x) . It is e a s i l y shown
1
that Xw1(h) = X w ( h ) .
W e w i l l now c o n s t r u c t a new word which w e w i l l t hen show f a i l s
i n u (F)B . &t wW(y) ~ 3 " Yijk , where y r e p l a c e s x i jk i jk 1
i n W' (x) as fol lows:
(a) yijk = i f and o n l y i f x = x , and 'mns i j k mns
xw(k - % (s-1) i j
l) (h) = Xw mn (h)
(b) Yi jk -Ymns i f and o n l y i f x i j k t+x mns , and
(c) yijk = e i f and o n l y i f x = e i jk
Now w" ( y ) = e f a i l s i n U ( I?)' , s i n c e t h e s u b s t i t u t i o n g iven
'L (k - by Yijktt \ j k ( hwij (h) ) t a k e s i n v e r s e s t o - i n v e r s e s . More
p r e c i s e l y i f ' i jk and 'mns
a r e i nve r se s , t hen hi jk ( lw:; - l) (h) )
'L (s - 1) and h ( A w (h) ) a r e i nve r se s . Thus, s i n c e mns mn
'L (k - 'L h # hw(h) = Aw8(h) = V A ( Ahijk( hwij l ) (h) , hw (h) i j
'L (k - we have V A ( Ahijk( hwij l) (h) # e , and s o w" (y) # e .
Thus, t h e r e e x i s t s a s u b s t i t u t i o n y ++ u i n some U such t h a t
e # w W ( u ) . combining t h e s e r e s u l t s , w e w i l l d e f i n e a s u b s t i t u t i o n
i n ( U , T ) wr (G,Q) by xijk +-+ ( b ijk,9ijk) = m where bijk:Q -+ U i jk
i s de f ined by
(i) whenever x ++ x and w (x) i s involved i n w' ( X I , i jk mns mn
S -1 t h e n b ( a w (g) = umnS I i j k mn
(ii) whenever x = x and w (x) i s involved i n w' (x ) , i j k mns mn
then b ( S - l) ( g ) ) , u i j k ( awmn mns t
(iii) a l l the o t h e r components of b (@) = e. i jk
Summing up, i n ( U
m) = y 3 n bijk i
~ h u s , w(x) = e f a i l s i n (u ,P ) w r (G,R) as r equ i r ed .
c o r o l l a r y 4 . 3 . 3 . Let 1-var (us,T ) 1 = U { ( G ~ , Q ~ ) } S
be t h e c o l l e c t i o n of a l l t r a n s i t i v e 1-permutation groups i n V ,
NOW t h a t we know how a product of v a r i e t i e s i s generated, it
f s of i n t e r e s t t o study when a subset of a v a r i e t y mimics t h e v a r i e t y .
Also, it i s advantageous t o study some examples. The next theorem
gives an important example of an 1-group which mimics a v a r i e t y .
Theorem 4.3.4. The regular r ep resen ta t ion ( Z , Z ) of t h e 1-group
of i n t e g e r s mimics A .
Proof: Let ( H , A ) be a t r a n s i t i v e abe l i an 1-permutation
group. Since ( H , A ) i s t r a n s i t i v e and abe l i an , H is an o-group,
and so ( H , A ) i s t h e r i g h t regular r ep resen ta t ion (HIH) . Since H i s an abe l i an o-group, H may be embedded i n a Hahn
group v ( ~ , R ) . I n t h i s case t h e Hahn group i s t h e lexicographic
product R6 of copies of t h e ordered r e a l numbers R, where r i s a 6
t o t a l l y ordered s e t .
Let { w (x) 1 be a f i n i t e s e t of words. For each word w (x) P P
l e t x -s denote a s u b s t i t u t i o n i n R6. B y an arguement s imi la r
t o t h a t i n theorem 4.2.9 we may now assume t h a t t h e index s e t
i s f i n i t e , say 1, 2, . . . , m 1
Let D be t h e set of a l l p o s i t i v e d i f f e r e n c e s w (s) - w ( s ) . P 9
A t yp ica l element of D w i l l be d = ( d , d 2 , . . . ,dm) . Let
l / c = min { I di I I di # 0 1 . Let kl = 1 , and f o r 2 5 i C m ,
l e t k > max ( d k c + d2k2c + ' ' ' i 1 1 + di - 1ki - lc I We w i l l now show t h a t the re e x i s t s an 1 - homomorphism
O: V(I',R)-+ R , where R denotes t h e r e a l numbers. Also the
reader should note t h a t the o - permutation groups involved i n t h i s
discussion a r e r i g h t regular representa t ions , and thus we iqnore
t h e sets t h a t t h e groups a c t upon.
Define 0 : V(I ' ,R) + R by e(rlr . . . , r ) = T: r . k . c . n 1 1
Clear ly , O i s a group homomorphism. I f w (s) = (plI . - - I P ) r P m
w (s) = (qlI - . q and w (s) > w (s ) , then (pl-ql,. . . q P q
(pl - qlr - - 'm
non - zero e n t r y from
order) then p - q > j
( p j - q . 1 k . c 3 3
qm) > 0 . Thus i f p - q j i s the f i r s t j
t he r i g h t ( t h e order i s t h e lexicographic
0. Therefore,
k > d . c ( d k c + . . . + d j - l j - l 3 1 1
c 1
> I d k c + . . . + d 1 1
k j - 1 j - 1 ' I
where di = pi - 'i - Thus, O preserves t h e o rde r o f the elements
from { w p ( s ) ).
Now we would l i k e t o show t h a t f o r a f i n i t e s e t o f words
( w (x) ) and a s u b s t i t u t i o n x -+ r € R, t h e r e i s a s u b s t i t u t i o n P
x + z E z ( t h e in tege r s ) such t h a t w (r) < w (r) i f and only P q
i f w ( z ) < w (2 ) . Let x -+ r be a s u b s t i t u t i o n i n R. P 9
Since t h e number of words i s f i n i t e , t h e number o f images under
the s u b s t i t u t i o n -+ i s a l s o f i n i t e , say a l , . . . pn . The
subgroup S o f R generated by these elements is a f r e e abel ian
group.
A s before, l e t D be t h e set of p o s i t i v e d i f fe rences
w (r) - w (r) . Each d E D has t h e form d = C f i (d)a i > 0 P 9
with each f (dl an in tege r . Thus, (al, . . . ,an) i s a
so lu t ion to the system of i n e q u a l i t i e s C f i (d) zi > 0. By
cont inui ty , t h i s system must have a r a t i o n a l s o l u t i o n and thus an
in tege r so lu t ion . That is, C f . (dl n . > 0. Let @ : S + Z 1 1
be the homomorphism defined by Q, ( C m , a . ) = C m.n.. Then, 1 1 1 1
Theorem 4.3.5 For each p o s i t i v e i n t e g e r n,
n 00 -a, 1-var ( W r Z) = A", - and 1-var ( W r Z) = 1-var ( W r Z) = M .
Proof: The theorem has been proved f o r n = 1. For a
n p o s i t i v e i n t e g e r n > 1, wrnZ = ( W r Z) WrZ. Thus, by Theorem 4.3.2
n and by induction, 1-var ( W r 2) = A".
Now s ince any G E !d is 1-isomorphic t o a wreath product
of the r e a l numbers R by Theorem 2.4.5 w e have 1-var ( w ~ ~ z ) 5 El.
Conversely, s ince !d = "An and 1-var (wrnz) = A n i s
w a subset o f 1-var ( W r Z) f o r a l l p o s i t i v e i n t e g e r s n, w e have
N 5 I-var ( W r Z) . 4u
Similar ly , it can be shown t h a t 1-var ( W r Z) = N.
Corollary 4.3.6. The v a r i e t i e s N and f. are t h e only
1-group v a r i e t i e s closed under taking wreath products.
Section 4. The Factor iza t ion of 1-group Var ie t i e s .
I n t h i s sec t ion we w i l l g ive a p a r t i a l converse t o prop-
o s i t i o n 4.2.3.
Lemma 4.4.1. Let H be an 1-group and (G,Q) a t r a n s t i v e - - 1-permutation group. Then every 1-idea1 N of (H,H) w r (G,Q)
i s r e l a t e d by inc lus ion t o C H ~ . Moreover, i f N 5 C H~ then - -
Proof: Let N be an 1-ideal o f the wreath product, and
suppose t h a t N E H ~ . Then the re e x i s t s (h,g) E N such t h a t
g > e ( t h e i d e n t i t y o f G I . Hence, t h e r e e x i s t s a E Q such t h a t
a i
ag > a. B y convexity, H & - N. Let B E S2 , then t h e r e e x i s t s f E G
k - 1 t such t h a t a f = B I and by normality ( e , f ) (h,g) ( e , f ) E N.
- 1 B Moreover, Bf gf > 6 and so H c- N f o r a l l E R a s required.
Lemma 4.4.2. - Let U i , Vi (i = 1. 2) 1-group v a r i e t i e s
with UIVl \ U2V2. If U1 $ U 2 - then AV1 _t V 2 ; and i f V1 V 2
then LflA c= U q .
Proof: Let (G ,R) be a t r a n s i t i v e 1-pennutation group with
G E V1. Let H be an 1-group i n U1\ u2. Then (H.H) w r (G,Q) i s an
element i n UIV1 5 U2V2.
Since H 4 U 2 , U 2 ( (H ,H) w r ( G I G ) ) = F & 1 H ~ . Thus,
a F c 1 H , so t h a t ( ( H , H ) w r (G,Q) ) / F E V2 . Hence,
(Z,Z) w r (G ,Q) (.. ( ( H , H ) w r (G,Q) / F c V2, and so
V 2 -= . 1-var { (z,z) w r (G,Q) I G E V 1 = A V by c o r o l l a r y 4 . 3 . 3 . 1 1 '
i li a I L
I Corol la ry 4 . 4 . 3 . Let U , V I W - be 1-group v a r i e t i e s and
Proof: Since by c o r o l l a r y 4.3.6. , N and L a r e t h e
on ly 1-group v a r i e t i e s c losed under t ak ing wreath products ,
AV 1 V and VA v whenever V is n e i t h e r !.! nor L.
In the f i r s t chapter of t h i s thes i s some of the r e su l t s concerning
the proper t ies of 1-groups and 1-subgroups were presented without proof.
We expound those ideas n w .
Theorem 1. - I f G -- is a 1-group x,y ,a,b, 6 G then -
Proof: We w i l l show t h a t
Since a V b 3 a and b , we have x + ( a V b) + y is grea te r than
or equal t o x + a + y and x + b + y. Thus
Assume t h a t fo r some z 6 G ,
z ? ? ( x + a + y ) V ( x + b + y ) .
Then, z ? x + a + y and x + b + y , a n d s o - x + z - y ' a and b .
Consequently, -x + z - y 2 a V b , and therefore z 2 x + ( a V b) + y . We have shown t h a t x + (a V b) + y i s the least upper bound of
x + a + y and x + b + y .
The proof of t he dual is similar.
Theorem 2. I f G i s a 1-group and a,b E G , then - -
- ( a V b ) = (-a) A (-b) and - (a A b) = (-a) v (-b) .
Proof: We w i l l show t h a t -(a V b) = (-a) A (-b) . Since
a,b 5 a V b , we have -a,-b 2 - ( a V b) . Thus, (-a) A (-b) 1 -(a v b) . I f z E G and z L (-a) A (-b) then -2 > a and b . Thus,
- z ? a V b , o r 2 5 - ( a V b ) . We have sham t h a t - ( a v b) is the g r ea t e s t lower bound of
-a and -b .
Theorem 3. -- For a 1-group G , a E G , and a pos i t ive in teger n , --
Proof: (By Induction) The r e s u l t holds f o r n=l. Assume t h a t
the r e s u l t is t r u e f o r k-1. Then,
k ( a A 0 ) = ( a A 0) + (k - l ) ( a A 0)
= ( a + (k - 1) (a A 0 ) ) A (0 + (k - 1) ( a A 0 ) )
= (ka A (k - l ) a A . . . A a) A ( ( k - l ) ( a A 0))
= k a A ( k - 1 ) a A . . . A a A O .
The reader may a l s o observe t h a t the r e s u l t holds f o r t h e dual.
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