Varieties of Lattice Ordered Groups / by Mary Elizabeth Huss
VARIETIES OF LATTICE ORDERED GROUPS
Mary Elizabeth Huss
B.Sc., University of Nottingham, 1975 M.Sc., Simon Fraser University, 1981
A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in the Department
Mathematics and Statistics
@ Mary Elizabeth Huss 1984 SIMON FRASER UNIVERSITY
All right reserved. This thesis may not be reproduced in whole or in part, by photocopy or other means, without permission of the author. APPROVAL
Name : Mary Elizabeth Huss
Degree : Doctor of Philosophy (Mathematics)
Title of Thesis: Varieties of lattice ordered groups.
Examining Committee:
Chairman: Dr. A.R. Freedman
Dr. N.R. Reilly Senior Supervisor
- - -- Dr. T.C. Brow.
Dr. S.K. Thomason
Dr. W.C. Holland External Examiner Professor Mathematics and Statistics Department Bowling Green State University
Date approved: May 11, 1984 PART l AL COPY R l GHT L l CENSE
I hereby grant to Simon Fraser University the right to lend my thesis, project or extended essay (the title of which is shown below) to users of the Simon Fraser University Librsry, and to make partial or single copies only for such users or in response to a request from the library of any other university, or other educational institution, en its own behalf or for one of its users. I further agree that permission for multiple copying of this work for scholarly purposes may be granted by me or the Dean of Graduate Studies. It is understood that copying or publication of this work for financial gain shall not be allowed without my written permission.
Title of Thesis/Project/Extended Essay
Author: (signature)
\\ (date) ABSTRACT
For any type of abstract algebra, a variety is an equationally defined class of such algebras. Recently varieties of lattice ordered groups ( L-groups) have been found to be of interest and this thesis continues their study. For any L-group C, C x Z denotes the product of C with the integers Z, ordered lexicographically from the right.
For a variety V of 1-groups let vL = VcUz(G x Z I G E GI. L It has been an open question as to whether or not V = V , for every variety V of € -groups. Examples are given to answer this question negatively, and properties of the varieties vL are developed.
For a variety V, another closely associated variety is the variety VK , obtained by reversing the order of the € - groups in V. It is shown that there are varieties R for which V # V and that the mapping 0: V o VR is both a lattice and semigroup automorphism of the set of varieties
Kopytov and Medvedev, and independently Reilly and
Feil, have shown that there are uncountably many € -group varieties. By considering further uncountable collections of varieties of L-groups, it is shown that the breadth of the lattice of representable e-gsoups has cardinality of the
1 continuum. ACKNOWLEDGEMENT
I would like to thank Dr. N.R. Reilly for his assistance and encouragement during the preparation of the thesis.
I would also like to thank Ms. Kathy Hannon for typing this thesis. I
TABLE OF CONTENTS
Approval Abstract Acknowledgement Table of Contents Introduction 1 Chapter 1. &-Groups and Varieties 5 Chapter 2. Reversing the Order of an e-group 13 1. Basic Observations 2. An Automorphism of L 3. Varieties Invariant under o 4. Varieties moved by Chapter 3. Lex Products by the Integers 2 9 1. The Lex Property 2. Varieties without the lex property 3. Laws for V' Chapter 4. Uncountable Collections of Varieties of 53 &-groups Chapter 5. Further Results 6 1 1. Lex products of varieties 2. Mimicking References 7 3 INTRODUCTION
For any type of abstract algebra,. a variety is an equationally defined class of such algebras. Equivalently,
by Birkhoff [21, a variety is a class of algebras closed under subalgebras, direct products and homomorphic images. The extensive work on varieties of groups, much of which is described by H. Neumann [181, prompted an interest
in the study of varieties of lattice ordered groups. The early work in this area was mainly concerned with specific
varieties. For example, Weinberg [241 studied abelian t- groups and showed that the abelian variety A is the
smallest non-trivial variety of lattice ordered groups. A more comprehensive investigation of varieties of lattice
ordered groups was begun by Martinez, [141, [151, and [161. He described an associative multiplication of 4 -group varieties and determined that the set L of all lattice ordered group varieties forms a lattice ordered semigroup under this multiplication, the partial order being inclusion.
Glass, Holland and McCleary [71 have extended this work. One of their main results shows that the powers of the abelian variety, A , generate the normal valued variety, N , shown by Holland [I01 to be the largest proper variety of lattice ordered groups. In this thesis, the study of varieties of lattice ordered groups is continued. Chapter 1 contains background material and introduces many of the commonly studied varieties. For any lattice ordered g-roup G there are two closely associated 1-groups: G~ , which is obtained from G
by reversing the order, and GW, which is obtained from G reversing the multiplication. GR and GU isomorphic 1-groups, and one can ask whether G and CR 2 GW
the same variety of 1 -groups, and if not, what the relationship between the varieties they generate is. This question is considered in Chapter 2, where it is shown that
there are 1 -groups G for which G and GR .generate
different varieties. If, for any variety V of 4-groups,
(GR]G E V 1, then it is established that the mapping @ vR=
0: v I+ VR is both a lattice and semigroup automorphism of L, the set of all varieties of 1-groups. Further, the class
of varieties which are invariant under 8 , and the class of those which are not, are both shown to have the cardinality of the continuum.
For e-groups G and H, if ti is totally ordered
then the product G x H may be ordered lexicographically by
(g, h) 2e if h>e or h=e and gke. Theproduct with this order is denoted by G ;H and called the lex product of G by H. Even in the simplest case when H is restricted to being the -group of integers, Z, it has been an open question as to whether or not varieties of lattice ordered groups are closed under taking lexicographically ordered products. The problem was first considered by Smith C231 who demonstrated that many varieties (in particular
P, 9 N 9 R r W and Sn(n E N) ) are closed under lexicographically ordered products by the integers. In Chapter 3, it is shown that not all varieties of lattice ordered groups have this property. For any variety V, vL is defined to be the variety generated by
(G f Z / G E V 1 and the properties of these varieties are discussed. It is proved that for any variety of 1-groups, vL is closed under lexicographically ordered products by
2. Moreover, any variety is closed under lex products by Z if and only if it is closed under lex products by all totally ordered abelian groups, so that for every variety LJ,
vL = Vat ( {G :A IG E G, A E A , A totally ordered)). The work of Chapter 3 suggests that the more general situation of the lex product of two varieties be considered. This is done in the first part of Chapter 5. That chapter concludes with a discussion of mimicking, a property introduced by Glass, Holland and McCleary [?I who used it in their study of product varieties. 4 The lattice L of varieties of L -groups was first shown to be uncountable by Kopytov and Medvedev C131 and independently by Feil 151 and Reilly Clgl. Reillyts work demonstrated the existence of an uncountable collection of pairwise incomparable varieties each containing the representable variety R , while Feil constructed an uncountable tower of representable L -group varieties. Thus the height and breadth of L both have cardinality of the continuum. In Chapter 4 further uncountable collections of varieties of -groups are considered and it 'is shown that the lattice of subvarieties of R n ~2 contains a sublattice isomorphic to I x I where I is the unit interval (0, I). CHAPTER 1
4 -Groups and Varieties
A hfkic.e srdered guu~I_ is a group with a lattice structure that is compatible with the group operations. That is, a(xvy)b = (axb) v (ayb) and a(xhy)b =
(axb) A (ayb). An 1-group in which the lattice order is a total order is a totd1.y Wd -. A subgroup of an 1-group which is also a sublattice is called an l -sub-. A subgroup H of an &-group G is said to be conva if whenever h -< g -< k and h, k, E H, then g E H. An LrLdeal is a normal convex 1-subgroup.
An l -h-- (respectively, 1 ---) between two [-groups is a mapping which is both a group and a lattice homomorphism (respectively, group and lattice isomorphism). Further background information and terminology relating to [-groups may be found in [I].
A uriet~of l-groups is the class of all €-groups which satisfy a (possibly infinite) set of equations.
Equivalently, it is a nonempty collection of &-groups closed under [-subgroups, f-homomorphic images and direct products.
An equation used in defining a variety of 1 -groups has
the form w(2) = e, where w(x) is an element of the free e -group F on a countably' infinite set X. By the compatibility of the group and lattice operations, and since the lattice structure of any L-group is distributive (see [I, Chapter 111, all elements of F can be written in the form w(;) = VA Wij(x) = Vfin xijk. Here the index sets I, J, I J I JK.- and K are finite, each xijk E x U x-l el and each w~~(~) is in reduced form as an element of the free group on X. A
titution for. a word w(x) in an [-group G is a mapping (xijk1 -+ G, 'ijk* gijk such that (i) if xijk - .xi, j,k, then gijk - gitjtkt, (ii) if xijk - then gijk - gitjtkt and - (iii) if xijk = e then gijk = e. Then w(g) is the'
A G, and is also referred to as a IJ K gijk of substitution for w(x) in G. Equations written in the above form can become unwieldy and it is therefore usual to use some abbreviations: x v y = Y - -1 is written x -< y,lxl = x v x-I , x+ = x v e, x = Y ve,
[x,yl = x -Iy-1 XY 1 and if it is required that x 2 e then this is simply noted, instead of replacing x by x v e. The .set of all varieties of L -groups is denoted by L. It is clear that the intersection of any collection of varieties is again a variety. Thus by considering L to be partially ordered by inclusio~,it is possible to define lattice operations on L: These definitions make L a complete lattice.
L is also a semigroup, where multiplication of varieties of l?-groups is defined as follows: for U, V E L, an 1-group G belongs to UV if and only if there is an
L -ideal H of G such that HE U and G/H E V . A variety V is said to be nenerat_ed by a family of
-groups {Gil i E I) if V is the smallest variety containing eachGi, i.e. if V =~){uELIU2{~~/i €11 1. The variety generated by {Gi li E I] is written Va,(GiI i E I). Some examples of varieties of L-groups are as follows.
ExamDle U. ltrivial m,E , consists of all one - element t -groups and is defined by the equation x = e.
EhUUd.2 l-2- At the other extreme is the variety L consisting of all 1-groups. L is defined by the equation
e = e.
EUiJIUkld. Tbf:&-elialYXi&Y I A , is defined by the equation [x,yl = e. Weinberg 1241, has shown that A is the
smallest non-trivial variety of 1-groups.
E.-e u. Ihe nDrmal yahed yariety, N, is defined by the law x2y2 -> yx for x, y 2 e. Holland [I01 has shown 8
.. that N is the largest proper' variety of 1-groups.
A connection between the smallest property variety, A, and the largest proper variety, N , was given by Class, w n Holland and McCleary 171 who proved that N = VA . n=l
Eh-uuh.1.5. An 1-group is said to be re~res- if it is a subdirect product of totally ordered groups. The
collection of all representable [-groups is a variety,
denoted by R , with defining law x2 y2 = (x A y)2 . (see, for example, Bigard et a1 [I, Proposition 4.2.9.1)
-u. m-w-u, W , is defined by x22 y0'xy for x ~e.
If r is a totally ordered set, then the group of all order- isomorphisms of r is an C-group, if an order-isomorphism be
called positive when 8g 2 8 for all X c r . Conversely, by a result of Holland [9l, every 1-group G can be considered as a
transitive C-group of order-isomorphisms of some totally ordered set r , in which case G is often denoted (C,r 1. 1f (G, r) and (H, r) are .l -groups then their Hreath
(, r,1 is the C -group (W, s2 1, where
(ordered lexicographically from the right) Up to isomorphism (C,r)Wr(H,h ) depends on G, H and A but not on r , and the wreath product (W, a) is often written
G Wr(H, A 1.
A The -subgroup of (W, Q consisting of those (g, h) such
A that g( A ) f e for only finitely many E: A is called the restrict_ed wreath p?of (G, r ) with (H, ) and is denoted by (G, r)wr(H, A). Further details of the wreath product construction may be found in Holland and McCleary [I11 or in Glass C6, Chapter 51.
Wreath products involving the integers occur frequently in the study of varieties of 1 -groups, see for example,
Examples 1.7 and 1.9. Let (2, 2) be the regular represent- ation of the integers. It is useful to note that multipli- cation in ZWr(Z, 2) is given by, for (F, n), (G, rn) in
ZWr(Z,Z),
(F, n)(G, m) = (F + Gn, n + m) where G"(z) = G(n + z) fo'r all z e Z. The inverse of (F, n) is (-F-", -n).
Example 1.7. For each positive integer n, let Gn be the l-subgroup of ZWr(Z, Z) consisting of all those elements
(F, k) for which F(i) = F(j) whenever i J mod n. Then 10 the Scrimger variety, Sn,'is the variety of 1-groups generated by Gn. Scrimger [221 showed that for each prime p, Sp covers A. Yedvedev [I71 described three further covers of A :
Exarn~leu. Let No be the free nilpotent class two group on two generators a and b, ordered so that a >> b >> [a,bl > e, i.e ambn[a,blk > e if i) m > 0 or ii) m = 0 and n > 0 or iii) m = n = 0 and k > 0. VOc(No) is then a cover of the abelian variety.
L~~IILQ~19, Zwr(Z, 2) with the usual order for wreath products is not totally ordered, or even representable; however there are two similar total orders on Zwr(Z, 2). Let W+ be Zwr(Z, Z) totally ordered by defining (F, k) = e to be positive if either k > 0 or k = 0 and F(r) > 0 where r is the maximum element of the support of F. Let W' be Zwr(Z, Z) totally ordered by defining (F, k) e to be positive if either k > 0 or k = 0 and F(s) > 0 where s is the minimum element of the support of F. V~R(W+) and Votr(W-) are then distinct covers of A .
The lattice of L -group varieties was shown to be uncountable by Kopytov and Medvedev [131. Independently Reilly [I91 and Feil [51 proved the same result, Reilly showing that L has uncountable breadth, Feil constructing an uncountable tower of varieties of l-groups. 11 1.18, Let F be the'free group on a countably infinite set X and let z- k X. For each fully invariant subgroup U of F letqu be the variety of [-groups satisfying the laws z+r\ u"z' u = e for all u & U. Such varieties are called wsi - r- and form the uncountable collection of varieties described by Reilly.
ExamDle 11 Feil [4, 51 constructed two uncountable towers of varieties within R /7 . For p, q integers with 0 -< p/q < llUp,q is the variety defined by the law
together with the laws of R (r2 A Y2 = (x A y)2) and of ([Iz\A~[x,Y] 1, IW~A\[X, ylll = el; u is the variety defined by the law
together with the laws of R and . For 0 < r < 1, r irrational, Ur is defined to be n U and W, is defined p/q>r ~19 to be 0 W pIq>r p/q," Also for t c R, 0 < t (1, e-groups, Gt and Ht are defined in the following way:
Gt is R x Z with multiplication (r, m)(s, n) = t (r + ( Im S, m + n) and order (r, m) _> (0, 0) if multiplication (r, m)(s, n) = (r + (T)t+l m s, m + n) and order (r, m)) (0, 0) if m > 0 or m = 0 and r20.
The properties of the two Feil towers are summarized in the next result.
i) Gt E Ur if andonly if t -< r.
ii) HtE if and only if t r. Wr -<
I
iii) {UrI 0 < r -< 11 is an uncountable tower of varieties with U, E; Us if and only if r -< s.
iv) {wr 10 < r <- 1) is an uncountable tower of
varieties with wr G W, if and only if r -< s. , CHAPTER 2 Reversing The Order of An !-Group
For any 1-group G there are two closely associated
X-groups, G~ which is obtained from G by reversing the order, and GW which is obtained from G by reversing the multiplication. For any 1-group G, G~ and GW are isomorphic 1-groups and it is natural to consider the relationship between the varieties generated by G and by G~ 2 GW . The following work on this question was done in collaboration with N.R. Reilly and appears in [121.
~e'ction1. Basic Observations
Definu 2.1.1. For any !-group (G, -0, let G~ = (G, 5 R denote the 1-group obtained from G by reversing the order; thus a - (GW' -< denote the 2.-group obtained from G by reversing the multiplication; that is, with multiplication * given by a*b = ba. It is easily seen that both G~ and GW are .e -groups. Not For any !-group G we denote the lattice R R R operations in C by v and A . Also, for x E, G I write R Note that for x,y E G, x vR y = x A y and x A y = x v y. Lemma 2.1.3. The mapping + : g ~g - 1 is an l - isomorphism of G R onto GW . PrPof, For g,h E GR , (gh)$= (gh)-l = h- g (h?)(g+)= (g+)*(h~) so that is a group isomorphism. Since (g v hj* = (g vR h)-I = (g A h)- = g-l v h-l = gl'v h+, and similarly (g nR h) = g +lrh t , is also a lattice isomorphism. Corollary 2.1.4. For any L -group G, v~(G~)= v~(G~). Although it will be seen that some properties of G and R G can be quite different, other properties are invariant under order reversal. Lemma- Let G be an 1-group and H S G. H is a sublattice (respectively, subgroup, t -subgroup or e-ideal) of G if and only if HR is a sublattice (respectively, subgroup, t -subgroup or t' -ideal) of GR . Furthermore, if H is an C-ideal of G, then (G/HIR is Y-isomorphic to G~/HR . 15 I flotation 2.1.6. Let F denote the free L-group on a countably infinite set X. For any variety V , of 1 -groups let VR = {CR\ C E V) and, for any word u = x in i jk R I JK F let u = v A( n xijk)-l. IJ K Also for any product yk, let FI'yk denote the K K - - product taken in the reverse order. Thus if 9 yk = Yl -Y, K then n' yk - yn ... y1 . With u as above, K.v write ut = v A n' xi jk. I JK Lemma- For any [ -group G and any u = V A n xijk IJK the following are equivalent. (i) The identity u = e holds in G. (ii) The identity uR = e holds in cR. (iii) The identity u' = e holds in cR. PrOOfa Clearly C satisfies the law V fl = e I J K 'ijk if and only if CR satisfies the law R R f VJ R "ijr = e R R R R -1 -1 But, A V n xijk e n (A (mijk) 1 = e IJK I J K Thus, G satisfies an identity u = e if and only if G~ 16 satisfies uR = e, establishing the equivalence of (i) and (ii). The mapping x ++ x-l of X into F extends to an automorphism g , say, of F. Hence, the identity uR = e holds in G~ if and only if the identity uR$ = e holds and the equivalence of (ii) and (iii) follows. Corollarv2.1.8k FO~any variety of L-groups V, vR is a variety. Moreover the following are equivalent. (i) V has a basis of identities [u e, a E .A]. 2 = (-ii) has a basis of identities [uR = e, a E A]. vR a (iii) = a E A]. vR has a basis of identities [uta e, A question which naturally arises is whether or not it is always the case that V = vR, or equivalently, whether or not it is the case that for all [-groups G the varieties Vnh(G) and v&GR ) = Vm(GW)are the same. Examples will be given in Section 4 to answer this question negatively. Section 2. An Automorphism 'of L. Since it will be shown that there are varieties V for which vR 4 V , it is of interest to consider the properties of the mapping V- V R . Botatipn U Let 8 : L -t L be the mapping defined by V8 = vR and let F E {V E LlvR = V}. . . l'roposdiun zL2LL The mapping 8 is a lattice automorphism of L with the following properties: (i) Q2 is the identity mapping (ii) 8 preserves arbitrary joins and meets (iii) F is a complete sublattice of L (iv) for anyv~L, VvvR~~and VIIV~EF. proof. For any word u E F, it is clear that (uR)R = u and by Corollary 2.1.8, VQ = V for all V E L- Thus (i) holds. It is clear that Us V if and only if U8 G V0 , and since (i) implies that 8 is a bijection, 8 is a complete lattice automorphism and (ii) and (iii) follow. Finally, for any V c L, (V v vR)0 = (vvve)e = vev ve2. ve v v = vv v R R and similarly (V n V )€I= vflvR so that both V v vR R and V f\ V are in F. 18 I The following corollary is an immediate consequence of Proposition 2.2.2 (iv). hlwJalX2.2.3, Let V & L. Then the follwing are equivalent. R (ii) VGV (iii) V 2vR. Recall that the lattice of varieties of 1-groups, L, has a semigroup structure. A variety U is said to be indecomposable if U = U1U2 implies that efther u, or u, is the trivial variety. Glass, Holland, and McCleary [71 have proved the following. . . *. PropoUion 2,2.4_. The set L , of varieties of 1-groups properly contained in the normal valued variety N, forms a' free semigroup on the set of indecomposable varieties. ikQE2- i2'2L.L The mapping 8 is an automorphism of the semigroup structure of L. h2QL Since 8 is bijective it remains to show that 8 is a semigroup homomorphism. Let U, V E L. Then ~ GE ( UV)0 Q GR E UV there exists an -ideal H of GR with H E U and G~/HE V @P there Lxists an l -ideal K of G (K = HR) with K E UQ and G/K (=(G~/H ve cs G E (UG) (VQ) Thus (UV)Q = (UQ) (VG) , . as required. Proposition Let E L* and let 2.2.6. U U = U1...U n where each Ui E L (i = 1, ...,n) is indecomposable. Then UE F if and only if Uic F for all i = 1, ...,n. Proof By Proposition 2.2.5. UQ = (U1.. .U )€I = (UIQ). (U 8) n . . n ' where, since 8 is an automorphism of the semigroup L, each LliQ is indecomposable. By Proposition 2.2.4. the factorization of varieties of L* into indecomposable varieties is unique and therefore U = UQ if and only if U UiO for i 1, n. Thus E F if and only if U E F for i = = ..., u i i = 1, ...,n. Corollary 2.2 -7. The complement F' of F in L is a prime semigroup ideal (i.e. UV E. FC implies U E: FC or V E FC) . In particular F and FC are both subsemigroups of L. Proof_. By their positions in the lattice L it is easily seen that the trivial variety, the variety of all [-groups and the normal valued variety are in F . Thus FC C L* and the result follows from Propositions 2.2.5 and 2.2.6. , Section 3. Varieties Invariant Under 8 . Although it will be shown that not all varieties of l -groups are fixed by 8, many of the commonly studied varieties are invariant under 8 . Examples of such varieties will be given in this section. Forany l -group G, G and G~ have the same underlying group structure and thus satisfy the same group laws. This proves the following. . . humi&Un 2A.LL If V is a variety of [-groups defined by laws involving only the group operations, then R R y y . In particular A -= A. CoroUxy 2.3.2. For all positive integers n, An is in F. FYQQf, This follows from Proposition 2.3.1 and corollary 2.2.7. ProDosU 22LL The normal valued variety N is in F. Proof. N is the largest proper variety of l-groups and 8 is a lattice automorphism of L. Picpppsiiinn 2.3.4, The representable variety R is in f . Proof, Let G be a totally ordered group, then GR is also totally ordered whence GR E R and G = (GRIR E R~. Thus R~ E. R and therefore 'by Corollary 2.2.3 RR = R . . . Proposla 2.7.5. The weakly abelian variety W is in F. Proaf. W is defined by the law y-l(x v e)y -< (x v el2 . In any t -group G, yW1(x v e)y ( (x v el2 for all x, y E G (x /P el2 - Iy-'(x ely]-' -cR [(x hR el2]-' for all x, y E G y-l(x-l vR e)y -cR (x-I vR el2 for all x, y E G y-l(z vR e)y It then follows that G E W if and only if G~.E w and therefore W E F, Recall that No is the free nilpotent class 2 group on two generators a and b ordered so that a >> b >> [a,bl > e, and that Van (No) is a cover of A . ih.U2sition L.3AA (Medvedev C171) If V is a variety of nilpotent [-groups with A V, then Vm(N0) 6 V. Corollary 2.3.7. ItLt.(No) is in F. Proof. Since No is nilpotent V(h.(NO) is a variety of R nilpotent P -groups and hence so is Vn,y(NO) , and by Proposition 2.3.6, vafi(NO) E ~al(~~)~. Then by Corollary 2 2 For each prime p, the Scrimger variety S P' described in Example 1.7, is a cover of A. Each variety P is solvable but not representable. . . n 2.3.8. (Gurchenkov [81). If V E L is a variety of solvable 1-groups which covers A and is not a variety of representable 1-groups, then V = S for some P ' prime p. Proposition 2.3.9. For all primes p, S is in P h2QL Let p be a prime. Since G and G~ have the same group structure ahd f3 is an automorphism of L, S P is a cover of A containing only solvable 1 -groups. Also since 0 is an automorphism of L, RR = R and S $ R, P SpR$~. Therefore by Proposition 2.3.8, S = S for P 9 some prime q. For any n the identity xnyn = ~"x"holds in S n and hence also in Sn . If p f q, then both the identites xPyP = yPxP and xqyq = yqxq hold in S = S - P q ' and it follows that xy = yx holds, contradicting that R S is not abelian. Therefore S E S P P P. . . PropQaALun 2m. Every quasi-representable variety lies in F. Proof_. Recall that a quasi-representable variety, RU, is defined by identities of the form [(u) = e, where l(u) = z* u-lz-u, u is a group word and the variable z does not appear in u. l(u) = e is a law of RU if and only if l(u-') = e is a law of RU . Let C be an l-gro~p and consider any substitution in G. Then Thus G E RU if and only if G~ E RU, so that RU r 5 as required. Since Reilly 11.91 has shown that there are uncountably many quasi-representable varieties, the following corollary is established. Corolliary ?aCF has the cardinality of the continuum. Section 4. Varieties Moved 'BY Q In this section examples will be given of varieties for which vR V Let 0 < t 5 1 and consider G~~,where Gt is the Feill-group. Let x = (0, -11, y = (-1, 0). Then x(y 1 and l[x, yl/ = ( - - 0 A similar calculation gives t+l , 1 I [x, /[x, yllR]lR = ( - - 0). Hence, (c+l) (1 [x, / [x, yl lR1l 9' = ( -L, 0) and (t+l) If 0 < t 1 Lx1 so that L > P -P t+l and - > . Therefore (t+l) (t+l) But by Proposition 1.12, Gt E U Thus G~~ # Up/q. p/q' Hence UpIq R f Up,q. Also for r ER\Q, 0 < r < 1, Gr cur, U R however, as above, GrRg r. Therefore, Ur + Ur for r E R\Q,andthe following result is then established. R Pro~ositionu*ur # Ur for all 0 < r Proaf. Let x,y,z E W. If x and y commute then (I) clearly holds. Otherwise I[x, yll = (F, 0) where if r is the maximum'element of the support of F, F(r) > 0. Let lz / = (G, n) where n 2 0. If n = 0 then z commutes with (F, 0) so that (I) again holds. So assume n > 0. Then where the maximum element of the support of F'" is n + r > r. Hence in this case, (F-",O) > (F, 0) and again (I) holds. Thus (I) holds in w+. However in (w+lR , let x = (0, 11, y = (F, 0) z = (0, -1) where 1 if i = 0 0 otherwise. Then x-ly-'ry = (G, 0) where G(i) = r-1 if i.1 0 otherwise. where i = {-1 if i=O 1 if i = -1 0 otherwise. 2 7 But then (G, 0) < (GI, 0) and (GI, 0) Using the following result, it is possible to identify v~w+)~and Va.t(~-)~. . . 2.4.3, (Medvedev E1711.If VE L is such that A S V and every element of Vis solvable and representable then Van(No) E V , Vak(w+) G V or Vn?.(w') G V . Proof. By Proposition 2.4.2, (w+)~# V~(W+).But since v~&(w+) is a cover of A, so also is V&+L (W + ) R , by Proposition 2.2.2. VLUL (w+) consists of solvable groups and by Proposition 2.3.4, every element of va;~(w+)~ is representable. It now follows from Proposition 2.4.3 + R that Va'4W = Va,+L(NO) Or (w+)~= Vm(W-). However every 1- group in Vfi,t(NO) is nilpotent class 2, but + R + R (W ) is not. Therefore Vm(W ) # Vah(NO) and so + R V&t (W ) = V&'L(W-).Then v~/?(w-)~= v~ The following result, due to Feil [41, is used to produce an entire interval in L which is moved by 8. 2 8 . . Pro~o-n 2.4.5. For an; positive integers, p, q, with 0 < p/q 5 1, (i) Vm(NO ) 9 up/^' (ii) w + upIq and (iii) V~(W+) G UPlq . The proof of the next result was suggested by A.M.W. Glass. Proposition 2.4.6. The intervals [ Vm(W+), U1] and [ Vah(W*), U 9 are disjoint. Proof, Let V be a variety which lies in both intervals. Then vdw-) . Vah (W+) 9 S v 5 U1. which contradicts Proposition 2.4.5. ,CHAPTER 3 Lex Products By The Integers Given any two e-groups G and H the most natural order to consider on their product G x H is the direct product order: (g, h) -> e if and only if g -> e and h -> e. However, in the case where H is totally ordered, there is another order which arises naturally, namely the order: (g, h) ) e if and only if either i) h > e or ii) h = e and g ) e. The product of G and H with this lexicographic order is denoted by G t H. Although by definition a variety of l-groups is closed with respect to taking direct products, it has been an open question as to whether or not a variety of -groups is closed under lexicographic products even in the simplest case when H = Z. Several examples hill be given to answer this question negatively, but first some general theory is developed. Section 1. The Lex Property. DefinW Xu-a For any variety V of 1 -groups let vL = ~cn({G: ZjG E Vl). -3- If V = Vah({GilicI)) then += .e vcn ({Gi x Z I i ~11). 30 Proof, Let G = Vc~t({GI li E I)). It is sufficient to show that G k ZE Vah({Gi $'~li€1)). Since GEVCLY({G~I~EI)) there exist B -< n Aj where each Aj ~EJ is some Gi and 2-epimorphism r : 8 J G. Define a' : B 2 2 -+G 2 Z by (b, z) (ub, z); a' is an l-epimorphism. Consider the (-subgroup H of n (A tx Z) given by iz J j The map B :HL~B % Z given by (a3 ,zIj a J,~) is an l-isomorphism. Thus there exists an g-subgroup H of (A f Z) and an l-epimorphism BQ' : H- GCZ, jtzJ j whence G f Z E V~?({G~2 Z 1 i E I]). ProDo~3s1a3L For any varieties U. V. Vi (i E I), Proof. (i) V Vi = Vm(U Vi). Therefore by the previous ic1 i~1 proposition, (V vif= ~ah({G: Z IC E U $1) = f €1 i~1 L = V Vai({G x ZIG E Vi)) = V Vi. i~I i~1 then G E Vi for all i E I and thus G % Z E V: I, L for all i~ I. SO G': Z E T\Vi , whence (nvi) G igI i~1 vL ~EIn i (iii) Let G E UV ; then there exists an [-ideal H of G such that H E U and G/H E V . Then H (01 is an 4- C €-ideal of G x Z and (G 5 Z)/(H 2 {O)) = G/H x Z. 4 C Further, H x (0) E U and G/H x Z E vL ; therefore 4- L G x Z E UIV~) and (LIV) L~ U(V ) . It is clear from the definition that for each variety V L L of l-groups, V 2 V. Those varieties V for which V = V are said to have the Lex merLy. Since a lexicographically ordered product is a direct product of groups, and the 1 -group of integers, Z, belongs to every non-trivial variety of 1-groups, varieties with the lex property include all non-trivial varieties of i? -groups which may be defined by identites involving only the group operations. Thus, for example, A and Ln (n E N) have the lex property. The following Corollary is due to J. Smith 1231. Coroll- lJAL Let U be a variety of L-groups and V,V.(i E I) be varieties of .!-groups having the lex 1 property, then (i > V Vi and n Vi have the lex property. i& I ~EI '> '> (ii) UV has the lex property. L Proof (i By Proposition 3.1.3 (V v.)~= V > 1 vi ~EI ieI = V Vi since each Vi has the lex property. Also ~EI (ii) Again by Proposition 3.1.3, if V has the L lex property uv (uvlL G u(v3 = UV whence (UV) = UV . CoroJJar~ 3.1.5. For each positive integer n, (~~1'= A". 1 Examples of 4 -group varieties V for which V = V are by no means limited to those defined by group theoretic laws and their products. Pro~osiLian (J. Smith [231) N, R and W have the lex property. Smith proved this result by considering the defining laws of the varieties. For the normal valued variety and the representable variety, the following alternative proofs are also straightforward. (i) By Glass et a1 [?'I, N = V A". Thus by L Proposition 3.1.3 and Corollary 3.1.4, NL = (V An) = V A"L = V A" = N. (ii) R = vat ({GI G is totally ordered)). Therefore by Proposition 3.1.2, R~ = Vivr ({G ;2 1 G is totally t ordered)). However if G is totally ordered, G x Z is also totally ordered and thus R L G R whence R L =R* I ProDosw LJ4L. Quasi-representable varieties have the lex property. P1'00f. Recall that quasi-representable varieties are defined by laws of the form l(u) = e where l(u) = y+~u-ly-u, u is a group word, and the variable y does not appear in u. Thus it i.s sufficient to show that if G satisfies 4- X(U> = e then so does G x Z . Let u + (uG, uZ), C Y + (YG ,yZ) be any substitution for u, y in G x Z . Since u is a group word, u - uG , y + yG is a substitution for u, y in G. There are three cases. since u + UG, y + y~ is a substitution for u, y in G and G satisfies l(u) = e. 3 4 and therefore G f; Z satisfiks the identity e(u) = e as required. Carol- 3.1.8.. There are uncountably many varieties of 1 - groups which have the lex property. Further examples of varieties having the lex property can be obtained using the following result. &Q.Im~i_tion LJdL Let G be an [-group. If th3re exists an element x in the centre of G such that for all g.~G, g < xn for some integer n, . then G; Z E Vm(G). - m Let H = n G/ c G. Then H E Vu(G). For each i=l i=1 ~EGlet g = (g,g,g ,...)ZG EH. Clearly (gig o GI is an -subgroup of H which is l-isomorphic to G. Let - -- -- a = (1x1, Ix12 ,...)ZG E H; then for each go G, ag = ga and a >> g. Thus the [-subgroup of H generated by {;I U ( g E GI is .( -isomorphic to G Z and therefore For any 1 -permutation group (H, , and any integer n > 1, HWr(n)(Z,Z) is defined to be the l-subgroup of the wreath product HWr(Z,Z) consisting of those elements (F, k) for which F(i) = F(j) if i E j(mod n). If H generates a variety V , then ~(n)A denotes the variety generated by HWr(n)(Z,Z). Such varieties were studied by I Reilly and Wroblewski [211. In particular A(n)A is the Scrimger variety S . !hUlh~YLILmA For each variety V and each n > 1, V(~)A has the lex property. PrOOfL By Proposition 3.1.9, it is sufficient to show that \ each HWr(n)(Z,Z) contains a central element x such that for each h E HWr(n) (Z,Z), h < xm for some integer m. If E: Z -+ H is given by E(z) = e for all z E Z, then (E, n) is such an element. Another interesting application of Proposition 3.1.9 is given .in the following Corollary. hXQ~hLY3.AJ.l. For any variety V, vL has the lex property. Proof. For any &-group G, (e, 1) is in the centre of C ;Z, and for any (g, n) E G; Z, (g, n) < (e, n + 1) = (e, 1) whence by Proposition 3.1.9 G; Z; Z r vm(G; Z) and the result follows. A collection 6 of [-permutation groups (G,Q ) is said to mimil: a variety V if 6 G V and whenever (H, A ) E V is a transitive (-permutation group, A EA, wl(E), ...,wn(?) are words and is a substitution in H, then there are (C, R ) &C 36 - ~iJ2and a substitution g in G such that I)< hwj(h) if and only if a ~~(2)< a wj(s) for i,j = l,...,n. This definition is due to Glass et a1 [71, and it should be noted that if mimics V then Vatr ({GI G E G 1)) = V. . prop^^ 3.1.32, (Glass et a1 C71) The regular representation (Z, Z) of the 2-group of integers mimics A. Using this mimicking property of the integers, it is possible to strengthen Corollary 3.1.11. EKQJLQ- 3LlA3A Let O be a coilection of totally ordered groups (viewed as their right regular representations: G = (G, GI). If e mimics Uthen for any totally ordered H E U and any C-group K, Kf; HE Vm({K; GIG E 0)). Proof, Suppose K % H does not satisfy the identity w(x) = V A wij(;) = e. We must show that w(5) = e is not a law IJ of Vcvr ({K i GIG E e 1. Consider two cases. i) If w(x) = e is not a law of U , then for some - - G E e there is a substitution x -. g in G such that C w(g) # e. Consider the substitution x -t 7 in K x G given by gi); = (el w(g)) # el hence yi = (el - w(y) =+ K x G does not satisfy the identity w(x) = e which is therefore not a law of Vad{K :GI G E Cl). ii) Suppose w(E) = el is a law of U . Since K H does not satisfy w(2) = e, there is a substitution - C x -+ y, Xi y = k,hi in K x H such that w(7) $ e. - Now, w(y) = (k, w(h)) for some k E K where x. -+ h is the is a law of U and H E U, w(h> = e and w(y) = (k, e) # for some e # k E K. Since H is totally ordered, e = w(h) = V Awi j(h) = max min vi j(h). For each i E I let IJ IS c i = min wij(h) and let J(i) = (j E JIwij(h) = ci1. Then S max ci = e; let IO = {i E 11 ci = el* In 3: G - K x H, for each i~ I,w) = fi wij(y) = J- J (1) ( A wi j(Z), ci) where x + 2 is the substitution in . ~(i) - given by xi+ ki. Therefore w(y) = V A wij(y) = - 7 - 1J V( A wij(Z), ci) = V( /i wij(k), e) = (V A. wij(i), e) I J(:) I~J(1) I, J(1) Thus, V A wij(L) = k # e. I, J ti', - - Claim: There is a substitution x ,g in G for some g E: G such that b) forall ir I, {j E JIw (9) } = J(i) where - i j di = Q Wij(g). C) {i E Ildi = e} = IO. - f- If the claim holds consider the substitution x +l in K x G given by xi+ li = (kit gi).' ~(i)= V w(k), e) = (k, e) I, ~(i) # e so that K G does not satisfy the identity w(T) = e and therefore w(?) = e is not a law of Vm({K f; G I G E e 1). proof Qf claimr By hypothesis w('ji) = e is a law of U , so every substitution for w(2) in G E e satisfies condition a). H is a totally ordered group in U , so the right regular representation of H is a transitive L-permutation group in U. Since e mimics U and {wi;(ji)) is a finite set of words, there exists G E e and a substitution x g in G such that ( < wij if and only if w i j (g) < witjt(g). Recall, for each i E I c i = minJ wij (b) and J(i) = {j EJI wij(L) = cil. Let di = min wij(E) and Jt(i) = {j E Jlwij(g) = dil. .T- Let jO &J(i)( then for j / J(i) ci = wij (%) < wij(a and by the mimicking property di < w .(g) < wij(g) i J, whence j k Jf(i) and Jt(i) E. J(i). On the other hand - if jl E Jf(i) then for j k Ji,di = wijt(g) < - - w (g) and by'the mimicking property ci < wijt(h) < W..(h) i j 1 J whence j k J(i) and J(i) GJt(i) Thus J(i1 = Jt(i) and condition b) is satisfied. Now for each i E I, let j(i) E J(i) and let If i E IOtthen wi(h) < wi(h) = e so that by the mimicking 0 39 property wi(i) < w.(g) < e,' whence i E 1' and It lo IO. Similarly let it E I,if i E It then wi(i) < wit(i) = e - and by the mimicking property wi(h) < wit (h) < e, whence i c IO and IOG 1'. Thus I = It and condition c) is satisfied, and the claim is proved. -CoroU Let A be an abelian totally ordered group. For any L-group G, Vm(G ;A) = Vm(G % Z). Proof, (Z, Z) mimics A. Section 2. Varieties Without The Lex Property. All the varie-ties considered so far have the lex property. However there are varieties for which I/ L 2 0. =I= The next results give examples of such varieties. P~UDQ- 3.2 The solvable Medvedev varieties, VL~(w+) and VW,do not have the lex property. Proof, i.1 The identity lzl-' /[x, ylllz I -> I[x, yll holds in v~(W+) by Proposition 2.4.2. Consider W+ f Z and let 47 x and y be any two non-commuting elements of W+ x Z, then l[x, yll has the form ((F, O), 0) where if r is the maximum element of the 'support of F, F(r) > 0. Let z be the element , -1, 1. Then Now the maximum element of the support of F' is r - 1 < r and therefore in w+; 2, ((F1, O), 0) < ((F, 01, 0). ii) Van (W-) satisfies the identity 1z1-l[[x, yll/zl( I[x, yll . Consider W-% Z and let z = , -11) and I [x, ylI = ((F, 01, 0) where for s the minimum element of the support of F, F(s) > 0. Then lzl-' l[x, yl/lzl = ((F1,O),O) and sinci the minimum element of the-support of F1 is Proposition 3.2.2. The nilpotent Medvedev variety, Vm(NO), does not have the lex property. Recall No is the free nilpotent class two group on two generators a, b, ordered so that a >> b >> [a, bl > e. 4 1 Glass and Reilly (unpublishbd) have shown that No satisfies the the following identity C However No x Z does not satisfy (*I, for let xl = (ab, 01, -1 1 f xt = (a, 11, x3 = (a b- , 2kN0 x Z. Then e -< xl <_x25x3, [x3, xll =e while [x3, x21 = ([a, bl, 0) = [x2, xll, 4 thus 1x3, x2I"[x2, xll > [x3, x11 whence No x Z L does not satisfy (*) and vm(N0) =J Vah(NO). # Proposition 3.2.3. The Feil varieties Lf ,, O a+ &QQ~. Consider Gr x Z and let x = 1,-1, 1) and y = ((0, 11, 0). Then x 2 y -> e . -1 -1 Cx, YI = x y xy Thus for integers p, q with p 5 q, p/q F 1 < r + 1 and therefore p/(r + 112 < q/(r + 1) whence in Gr +x Z C Therefore Gr x z I! up/q. Thus if r is rational Gr x Z Up. If r is irrational Ur = n ~>p/q>r~19 t. L and clearly Gr x Z d Ur. In either case Ur S Ur and the Feil varieties thus do not have the lex property. Corollary 3.2.4. There are uncountably many varieties which do not have the lex property. Having seen that there are varieties V which do not have the lex property, it is natural to Anyestigate by how I much V' is larger than V . i A Proof, For any function F: Z+ Z let F: Z + Z be A defined by F(z) = F(-z). Then r is the minimum element of the support of F if and only if -r is the maximum element of the support of F. Consider the 1-subgroup K = {((F, z),-z) I F: Z+ Z, z E Z) of W+ Z and let A a : K + W' be defined by a((F, z), -z) = (F, -z). Clearly a is 1-1 and onto. Let ((F, z), -z) e K be positive. If -z > 0, then in W' (F, -2) > e. Otherwise z = 0 and F(r) > 0 where r is the maximum element of the support of F. Then the minimum A A element of the support of F 1s -r and F(-r) = F(r) > 0 and in W-, (a, -z) > e. Therefore for ((F, z), -2) EK, positive, a((F, z), -z) is positive. a is thus an C l -isomorphism of K onto W- whence W- VUA(W+ x Z) = v~~(w+)L. A similar argument shows that W+ E ~W'(w-1~. 4 4 I L Proof. By the above Proposition W'E v~(w+) L Therefore Vat (W-) G V~A(w+lLL . But by Corollary 3.1.11 L Voil(~+)~~ Vm(W + ) whence VCL+L(W-)~ G VCVL(W+)~.Similarly Van (w+lL G Van(W0) and the result follows. Corollary 3.2.7 Vm(W+ L 3 Vnt(W+) v VGL(W-) and hence is not a cover of Vat (w+). 4- Proof. Since. W+ x Z is an o-group, by Martinez [161, W+ f Z E VWL(W+) v Van(W-1 if and only if W+ f Z e v~(w+)L) v~*(w-). ~utVWL (W+I = v~(w-1 < vat(w+), L L Vati (W-1. Therefore Vm(W+ ) c VOL(W+) v Vm(W-) Vm(W+ f whence VNI(W+)~ is not a cover of v~(w+). L In Proposition 3.1.3 (ii) it was seen that ( Vi) c_ i~ I A n vi: The above example shows that even for the i~l intersection of two varieties this containment can be proper: L ( VN~(W+) A VMW-) f = A = A c Va'i(~+ t Z) =f + L L = Vnt(W ) r\vah(w-) . This proves the following: cQuuLxY 3iL.4~ The mapping V u vL is not a lattice homomorphism. Recall that for any variety of 1-groups V , vR is the variety obtained by reversing the order of all the 1-groups 45 in V . Although v and V' are both obtained from V by changing the order of 1 -groups inV , V = V ' is neither a L necessary nor a sufficient condition for V =- V . L ExamDle 3.7.9. Let V = U& then V = V while . . Prop- Li'LlL For each variety V of 1-groups vL R - ,,RLe Proof. It is straightforward to show that the mapping (G 2 z)~+GR x Z given by (g,n)w (g,-n) is an 1-isomorphism. The result then follows since RL R + vLR = VU({(G? z)~/GE G}) while V = VU ({G x Z I G EG}). It has been seen that for a variety V with vL $ V , L V need not be a cover of V . By considering the Feil groups L Gt, 0 < t <- 1 it will be shown that the interval [ V, V I may be very large. Proposition 3.2.12. For the Feil groups Gt, 0 < t -< 1, ~nr(~~)~~= v~(G~)~. Proof. Consider cx : GtR -+ Ct x Z given by (r, n) H r,n, -1. Clearly a is 1-1. a[(r,n)(s,m>l = a (r + ( lns, n + m) t+l Thus is an 1-embedding of GtR into Gt +x Z, whence L GtR E VdGt) and by Corollary 3.1.11, GtR Vak(Gt). L Therefore Vm (GtlR L~ Vat(Gt) . Similarly 0 : Gt + ctR: Z given by (r,n)~-,((-r,-n),n) is an 1-embedding whence Gt, L RL and thus also Gt % Z E VOA (GtIRL and VcdGt) 5 Vm(Gt) L Therefore VL-UL(G~)~~=Vel (Gt). then for 0 < s,t <- 1 Vs c_ Vt if and only if s( t. R since Further for 0 < s < t -< 1, V1 v Vs c+V1 v Vt R the o-group Gt is in V1 v Vt but is in neither V1 nor VsR. Using Proposition 3.2.12, it is easily RL L seen that Vt = Vt. L C_orollarv- [ V , V I I may be uncountable. L Proof Let V = V v v~/~~. hen V = (V1 v VlI2 R)L = 1 1 L V1 v v~/~~~= V1 vV~/~~=V1 . For 1/2 < t 5 1 R L VZVl v vtRS(V1 v Vt ) = V1 = V L . Thus { V v fRl 112 ( t 2 1) is an uncountable collection L of varieties in the interval [V, V 1. Section 3. Laws for vL . In general for a variety of -!-groups defined by a set of generators, not much is known about the laws defining that variety. The exception is the abelian variety, A, which may be defined both in terms of a generator (the .[ -group of integers, Z) and a defining law (x-Iy-1 xy = e). Thus for a variety vL # V, it is not surprising that the defining laws of vL are not known. However the following results L give some information about the laws of V . - For each t -group word w(x) = V A wij(x), and each - IJ substitution x in Z, define a new 1 -group word wi(?) as follows: for each i E I let yi = min w (?I, let - i j J(i) = {j E J Iwij(z) = yi) and let IO = {i E I Iyi = max yil . . Prop- 3.1..1. ~(2)=' e is a law of vL if and, only if - - w(x) = e is satisfied by Z and. for each substitution x + z in Z, wZ(f) = e is a law of V . hXQf. Let C E V and consider a substitution 2 4 E, C X. -t ki = (gig zi) in G x Z. Corresponding to this there 1 - - - are substitutions x -+ ?, x i + zi in Z and x + g, xi + g i in G. Now w(E) = w),w. Therefore, w(c) = e if and only if w(?) = 0 and w7(Z) 5 e. The result then follows. In order to clarify the equation of the following I proposition, I [x, yl I will be written Ex, yj . f- Proof. Let t <- p/q and x, y E Gt x Z. Let x = ((r, n),m). Bx, yD= ((k, O), 0) for some 0 < k ER. Then ii) If nL 1, then (ttllmn = (? > 1. Thus 11 (t+l 1 -1 = 4. - (t:,~nl= " 7'- , t and if follows that n iii) If n<-1, then ('1 1 - t+l t > so that11 - ( t)n/ = (Lf- 1 > LA. -I= 1. tcl t+l - t t > ((0, 01, 0) since t > p/q and thus q/t - p/t2 > 0. Roughly speaking the above result is obtained using the fact that Gt satisfies the law Ex, nx,yBP tx, yJq for x 2 y e if and only if t F p/q. When considering an f element x of Gt x Z, there is no means to ensure that the c. Gt component of x is positive. However, since Gt x Z is totally ordered, either x or x-' will have a positive Gt component. Using this, and noting that for an o-group a 4 b = e if and only if either a = e or b = e, the original law of Gt can be modified to produce a law satisfied by f Gt X Z. L It is clear that for each variety V , V G VA The last result of this section gives a condition under which this containment will be proper. Lemma L3-A Let H be an 1-group. Any element of HWr(Z, Z) ofthe form (K, 0) is a commutator. Proof. Consider (K, 0) E HWr(Z, Z). Define F: Z + H by F(0) = e and for each positive integer n, F(n) = F(n - 1)K(n - 1) and F(-n) = F(-n + 1) K(-n + 1)- 1 Let E: Z -+ H be defined by E(z) = e for all z c Z and let x = (F, 01, y = (E, -1). Then Ix, yl = (F,o)-~(E,-~)-~(F,o)(E,-~)= (L,O) where L: 2 + H is given F(z)-~F(z)K(z)= K(z). Therefore (K, 0) = [x, yl as required. . . Proposi?ium UrnLet' V be a variety of !-groups generated by an &-group G such that the derived subgroup L G1 of G is an 1-subgroup. If Van (G1) 3 V , then V 5 VA. h2QL vA= V~(GWr(Z, 2)) so it suffices to find an e-group word w(xl, ...,x n 1 such that w(x,,...,x n 1 = e is satisfied by G t Z but not by G Wr(Z, Z). Now by Lemma 3.3.3 the commutators of G Wr(Z, Z) are exactly the elements of the form (F, 0) and thus form an !-subgroup isomorphic to TG,and so satisfy all the laws of G. ZEZ Since vm(G1)cv = van (G), there is a law w(xl,. ,x ) = e ? . . n of G1 not satisfied by G. Thus, w(xl,. ..,x ) = e n is not a law of G and w([xl,yll [xn,ynl) = e is not z&Z ,..., a law of G Wr(Z, Z). However w([xl,yl], ...,[xn,ynl) = e is a law of G t Z since for any substitution in Gf Z [xi, yiIH([gi, hi] $0) € Gt x {O) C G1, and w(x,, . . . ,xn) = e is a law of G, Clearly 1 -groups G for which Gt is an l -subgroup include all totally ordered groups. Among the o-groups already considered are examples for which vm(G1) 5 Vm(G). In particular it is easily seen that w+, W- and the Feil !- groups all have derived subgroups which are abelian, while none of these !-groups is abelian itself. However not all o-groups have this property as is shown by the 5 2 following example due to Chehata C31. Chehatats o-group C is defined as the group of order preserving permutations of R whose graphs consist of a finite number of linear pieces and have bounded support. An element is positive if its left- most non-identity piece has slope greater than 1. Chehata showed that C is simple as a group and since it is not abelian, CI = C and thus Vcur(C1) = Vatt(C). CHAPTER 4 Uncountable Collections Of Varieties Of l-groups The lattice L of varieties of e-groups was first shown to be uncountable by Kopytov and Medvedev [I31 and independently by Reilly 1191 and Feil C51. Reilly proved the existence of an uncountable family of pairwise incomparable varieties of e-groups each containing the representable variety R , while Feil constructed two uncountable towers of representable varieties. Thus the breadth and height of L both have cardinality of the continuum. For a variety V of [-groups, the lattice of subvarieties of V will be denoted by L(V). Using Feilfs varieties described in Example 1.11, it will be shown that it is possible to construct a sublattice of L(R A2) isomorphic to I x I, where I is the unit interval (0, .I). The following lemma, due to Martinez C161 is required. Lemma 4.1. If G is an o-group and U, V are varieties of €-groups with G E Uv V, then G E U U V. Proposition 4.2. s = {Ur v Us I 0 < r,s < 1) is a sublattice of Leo A2) that is isomorphic to I x I. Proof. It is first shown that S is a sublattice of L( u~A~).By Proposition 1.12, (Ur v Us) v (U rf v US!) = rn v WS,c S where rn = max{r, rl} and sn = maxis, sf}. By Proposition 1.12, and noking that L(R f3 A~)is distributive, it is seen that ( ur V Ws)n(Ur, v Us,) = ( urnur,)v (urn(",) v v (w~~w~,)= U rn v A v A v WSn = Urn v us. E s where rn = min{r, rtl and sn = rnin{s, st). Thus S is a sublattice of L(R A A~). -+S Now leta: I x I be given by (r, s)a = Ur Clearly a is onto. r, s r,s r 5 rt and s <_st 3 Ur E U rt and Us C- W s ' 3 Ur v Us Curt v 'St, and thus& is order preserving. By Lemma 4.1 and Proposition 1.12, Gt E U if and only if t 5 r and Ht~U rv s r v us if and only if t <- s. If r, s # r,st either r # rt and without loss of generality r < rt and Cr, E Urt v Wst \ Ur v Ws, or s # st and w.1.o.g. s < st and Hs, E urt v us, \ ur v us. Therefore, a is 1-1, and hence S is isomorphic to I x I. I x I contains a subset of 2'0 pairwise incomparable elements. Hence: ihulhn! 4.7. There is an uncountable family of pairwise incomparable varieties within n A~. For 0 < r < 1, let Tr = (Ur v Ws I 0 < s < 11, then Tr is an uncountable tower of varieties within R n A2 qnd for r f s, Tr $ Ts . Thus: '! lhLQuaU 4-4, There 'is an uncountable collection of uncountable towers of varieties within A2. W It should be noted that while S = Iu r v s 10 < r,s < 11 2 is a sublattice of L(R nA ), it is not complete. In fact the tower T = {Ur 10 < r < 11 is not complete, for consider V = 1>m/n>p/q n u m/n v cannot be defined by a single equation whereas U p/q can, thus V C Up/q If r > p/q then there are integers m, n such that r > m/n > p/q and V G U c ur. If =P r < p/q then 'r S Up/q* Therefore V $ T and T is not complete. For each prime p Scrimger [231 constructed an L-group variety S that covers A and is contained in A~,with the P property that if V is any representable variety, V fl Sp = A. Since Sp covers A, Sp n S = A for p, q 9 distinct prime numbers. For each prime p consider the collection of varieties L = {Ur v Ws v S 10 < r,s < 11. P P It is straightforward to verify that L is a sublattice of P 2 L(A ) isomorphic to I x I. Furthermore if p # q, then = since for V E V whereas for all L~ r\ Lq L~'P 0 < 1 1, Spn( Ur vWs v Sq) = ( Spn Up) v (S nW ) < P s v (SPAS) = A. q 4.5L There is a countably infinite collection of 2 sublattices of L(A ) isomorphic to I x I. Recall that A2 is thd variety of all [-groups which have an abelian [-ideal H such that G/H is abelian. Glass, Holland and McCleary 171 have shown that is Let A gp be the variety of those [-groups which as groups -. 2 are in the group variety A . Thus A gp consists of all -groups C which have an abelian normal subgroup H such that the group G/H is abelian, and is defined by the law [[x, yl, [z, wl I = e Since an L-ideal H of G is necessarily a normal subgroup of C, A A2 gP ' but how large is the interval t. Lemma 4.6. For 0 < r, p/q < 1, Qr x Gr AU if and p/q only if r <- p/q. Proof- Consider Gr: Gr and let x = ((1,-11, (1, 0)) and Y = 0,1, 0 0 Then x 2 y 2 e, I [x, yll q = (q/(r + 1,0,0,0 and l[x,Kx, yl/l lP = ((p/(r + I)~,o),(o,o)L Since p/q 5 1 and r > 0, p/q < r + 1 and thus p/(r + 112 < q/(r + 1) so that ([x, l[x, yl t I = r,0 (0, 0))lr ERI and (Gr: Gr)/12 Z x Gr, C thus Gr Gr e A:l if and only if Z x Gr e [Iplq. p/q 5 7 + 4 If r > p/q -then Z x Gr b' UpIq since Z x Gr has an i? i? -subgroup isomorphic to Gr and Gr ' . ~f r < p/q it is straightforward to verify that Z ;Gr satisfies the laws of U p/q For 0 < r < 1, r irrational, Ur = U and thus plq > r P/q n u I= It follows that for A"r = A( p/q > r " AUp/q* ~19> r r > p/q Gr x Gr € AUr \ AUpiq. It is clear that r \A~and the following result is obtained. Proposition 4.7. { AUr nAZgpI0 < r < 1) is an uncountable tower of varieties of L -groups contained in the 2 interval [A2 , A gpl. In a similar way, by considering the o-groups Hr Hr 2 it is possible to show that { AWr A gp/O < r < 1) is a second uncountable tower of varieties in the interval The following Proposition is then easily established (c.f. Proposition 4.2). Proposition 4.8. ((4~ v AW~) A I 0 < r,s < 11 is gP 2 a sublattice of [A, A 2~ isomorphic to I x I. gP lbLQlkau The height and breadth of [A~, gP 1 both have the cardinality of the continuum. Having constructed a sublattice of L isomorphic to I~,it is of interest to consider whether larger powers of the unit interval I can be found as sublattices of L. For 0 < s 5 1, let V, = Vaz.({Ct 10 < t Is)), and let v; = vm({Gt t Z I 0 < t 5 s 1). By Proposition 3.3.2, Gt t Z satisfies * Ex, hih(rx,~ll'(@, [X,YJJ-~Ati*-',[x,YJ]-P) <- e if and only if t i p/q. Therefore V; satisfies (*) if and + only if s p/q and hence Gr Z E -< x V: if and only if r Is. . . Pro~osltlon G, ;...; cr E V: An-l r\ AV' A"-~ n... 1 n 1 S2 n A~"~/An A vsn if and only if ri -< si for i = 1, ...,n. -1 C t. Proof. Let Q.. c = cr x.. .x cr e vi An-1 . ~n-1~S 1 n 1 n Since C € us, A n-1 , there is an [ -ideal H1 of G such L f that H1 E V.;l and G/H1 & A n-1 . Then H12Gr x R; J. / therefore G ;R r yS, and by Proposition 3.3.2, rl 5 sl . r,1 1 G ~~-~1/and thus there is an [-ideal Kn of C such s, that Kn & A and G/Kn e vs. Since Kn L 4n-1 , C 4- C- + Kn & G x...x G x R and G/Kn Z x Gr, rf rn- 2- therefore Z 2 C E and rn ( sn. For' 1 < i < n, 'n v% Ai-l An-i vg and thus there is an 1 -ideal Hi of G 1 such that Hi E ~i-1V I and G/Hi E Also there is an Si . [-ideal K~ of H~ such that K~ ~i-1and Hi/~i E V' Si n-i c + t Since G/Hi E A , Hi2 G x.. .x G x R; since '1 'i i-1 4- + .e Kie~ Kit G x... %& x R. Therefore Hi/Ki 2 t-7I C Z;G xR and ZtG %R 6 whence . ri 'i vsI ri 5 si e t. c = 11let Hi = G x...x G x R and for r1 rL i = 2,...,n let Ki = G tx...x 4 Gr 4x R . Then for t -L i- 2 i G/Hi x G, +x...x + G E An- i A~-l 1 < < n, = z Ki E rn , C i+2 / / and Hi/~i = Z x G ;R E Vs. , whence Hi E Ai-' V "i % *- / and G r Ai-l A n-i. Also HI = Gr x R E I$ vsi ' 1 and G/H1 E an-' so that G E V Kn = G cx...x c G cx RE A"-' S~ rl 'n- 2 4 and G/Kn = Z x Grn E V, whence G E A"-' Vs. Therefore, n...q an-1 '" n Vs n For 0 < re,n < 1 let u(~,*..,r ) be the n variety vrl an-' An-2 r\ ***nlAn-l $< 1 avr2 n Corollary 4.11. { u 0 < rl, (r, , . . ,rn) 1 ..., rn < 11 is a meet subsemilattice of L isomorphic to 1". Proof. Let a : 1" * L by given by (rl,. ..,rn )a = By the preceding proposition, it is clear U(r, ,...,rn) . that 4 is one to one. Also, CHAPTER 5 Further Results Section 1. Lex Products of Varieties. The study in Chapter 3 of the relationship between V and prompts a more general consideration of lexicographic products and varieties of 1-groups. In particular it is possible to introduce the notion of the lex product of two varieties. ... Ref mltlDn 5.1 1 For varieties V and U with UGR f let the Lex af V and U be the variety V x U = The following result is immediate from the definition. 5.1.2. For varieties V and Uwith U G R , Clearly by Corollary 3.1.14, for any variety V, t v r A is the variety vL described earlier. It can easily be seen that some of the properties of vL carry over to the + more general product v x U. The proofs of the following two results are similar to those of Propositions 3.1.2 and 3.1.3 respectively. Proposition 5.1.3. If V = vah({Gi 1 i E 1 ]) and U C then C e- V XU= Vat ( {Gi x HI i E I, H E U, H is totally ordered] ) . PropoW 5.1 .4L For ?ny' varieties of 1 -groups, U,V1,V2, Vi (i E I), Further properties of the lex product of varieties are as follows. . . Prop- 5.1 15L For varieties of l -groups V, U1 , Uz Proof. = Vat({G: H / G E V, HE U1 v U2, H is totally ordered)) C = Va({G x H /G E V, HE UIU U2, H is totally ordered)) = Vat({G% H /GE V , H F Ul, H is totally ordered)) v 4- = Von({G x H Ib E V, HE Up, H is totally ordered)) t = ( vx U1) v (v: U2). & (iii) V x ( U1U2 n R) is generated by the -groups 4- G x H where G E V and H ELI1 U2 is totally ordered. Since H E UlU2 there is an [-ideal K of H such that Kr U1 and H/Kf Up. Also both K and H/K are totally ordered. G 2 K E V ? U1 is an &ideal of G? H and G~H/G?K = H/K E U2, whence G f H E ( v % U1 )U2 and therefore 6- V x ( u lU2 0 R) G ( V? U,) U2. A natural question to ask is for which representable e 6- varieties u does U x U = U . Clearly since UsU x A S U?C U L such varieties must satisfy U = U and therefore not all representable varieties have this property. . . ooo- 5.1.6= AfA= A, R% R=R and w ?W = W. Proaf, That A; A= A is clear. By Proposition 5.1.3 C R x R = vat( {C t H I G, H are totally ordered)), thus R 2 R 4- is generated by totally ordered groups whence R x R c R and therefore R R = R . Let G, H E W, H totally ordered. Recall that w is defined by the law x2 2 xY for x ) e. C e Let e <- x = (gl, h,) E G x H and let y = (g2, h2) c G x H. 2 2 2 g2 2 Then x = (gl , hl and xY = (gl , hl . Since x 2 e and 2 If h12 > hl 2 then x2 > xy* H E W, hl 2 e and hl 2 hl . h Otherwise, h12 = hl 2 and, by Martinez [161, hl = e. In this case, gl 2 e and since G EW, x 2 = (gl 2 , e) 2 (gT2 , e) = xy Therefore x2 2 xy and G f H r W whence W 5 W = (JJ . Further examples are obtained by considering varieties defined by the representable law together with laws involving only the group operations. For such varieties L C U =U = Ux'U. since any lexicographically ordered product G f H with G, H E u satisfies all t\he group laws of and is representable. In Chapter 3, examples were given of representable L varieties U for which U 7 U. Clearly for these varieties U TU 2 uL . The following result yields examples for which this containment is proper. Proof Let GEL(\A be tptally ordered and let H be any abelian [-ideal of G 5 G. Since, G b A "-l , G is not abelian and H is an abelian t - ideal of G. Therefore, (G G)/H 'Z G/H t G An-' since G d An-'. Thus G 2 G k An and so U; U $ R nAn. However since U E R 17 An, C L In particular if A c u c_ R n A2 then U x U g U. =t 65 Such varieties include the ~bdvedevcovers of A, vdNO), Vm (w+) and Vm(W-1 and the Feil varieties Ur (0 < r <- 1). Furthermore these examples show that repeating the lex product may lead to the construction of an infinite tower of varieties. . . Pro~osltloKl 5.1.8. For A c* u c, R n~Ldefine a sequence of varieties by UO = U, and for each positive integer n, Un = e x U. Then UnG R n A n+2 but U $R An+'., 'n-1 r'l and in particular Un+l 3+ Un. Proof, By hypothesis the result holds for n = 0. Suppose c R A but u @ R n A Then, using Propositions 'n-1 - n-1 *. + 5.1.4 and 5.1.5, U, ,= Un_l XU c (RnAn+l)$ UC(R$U)A- -c R~(A :A ) A = n ( an+' A) = R n A"**. Also let Un-1 \ An and let H E U\ A be totally ordered, 47 n+l then it is easily seen that Gfr H , ( u n-l x U) \A The above proposition is in contrast with the 1 . corresponding result for y , Corollary 3.1.11 showing that 1 for any variety of e-gr~~psy, yLL = v. I Section 2. Mimicking. A question yet to be considered is whether the lex product of representable varieties is associative. Let U1,U2 and U3 be representable varieties. Then ( U1 :U2) ;U3 = van ( {G % H ': K IG E U, , HE U2, K E U 3, H ,K are totally ordered) ) + while U,? (u27U3) = Vm({G; LICE U1, L Eu2 x U3, L is totally ordered)). for any o-groups H, K with HE U2 and c. KE u3 it is clear that H x KE U2 x U3 is totally ordered and 6 t. C therefore (U, x U2) x U3 c U, x (Up 4-x 5). One approach in trying to investigate whether or not this containment is proper, is to attempt to narrow down the number of totally ordered groups of a variety U that need 6- to be used in describing a generating set for V x U. In -5 particular it has been shown that for any variety V, V xA= C Vm ({G x ZIG E v)) so that when forming the lex product V?A not all abelian o-groups have to be considered. More generally Proposition 3.1.13 asserts that if G,G)l G E el C mimics a varietyu then vxu = VCUL({G: KI KC V, G el). In this context the mimicking property of by {(G, G) (C C G 1 seems to be stronger than is necessary, the argument of Proposition 3.1.13 requiring only that the behavior of those transitive [-permutation groups in U which are the regular representations of totally ordered groups be mimicked. In the remaining part of this chapter some results on mimicking are developed. This concept was introduced by Glass et al.[71 who used it in their study of product varieties. In particular, they proved that if U= Vm ({Hs 1s c S} 1 and {(Gt, n t)l t E TI mimics V , then UV = {Hs~r(~t,n, ) 1s E S, t E T} ) . They also showed that the regular representation of the integers (Z, Z) mimics A . A convex L-subgroup C of an 1 -group G is Drime if f, g E G and f~ g = e imply f E C or g~ C. For a prime subgroup C, the set of right cosets of C, R(C), is totally ordered by Cf < Cg if 3c 6 C with cf < g. Let R be the set of all right cosets of all prime subgroups of an [-group G. Holland [gl showed that with a suitable ordering of n, (G, n) is an [-permutation group. Although the order on n is not unique, (G, n) is referred to as the Holland pf G. The Holland representation of an [-group is useful when considering mimicking. For example, the following Proposition is due to Reilly and Wroblewski 1211. Pro-Wn 5.2. I, Let G be the relatively free .k' -group of countable rank in a variety V. Let (G,n) be the Holland representation of G. Then (G, Q) mimics V . Other. results are as follows. Proposition 5.2.2. Let V c'R and let C = {(G,flG) G E V is totally ordered, (G, nG) is the Holland representation of GI. Then C mimics V . Proof* Let (H,A ) be a transitive 1 -permutation group, H E V. Then since H is representable, H is totally ordered and (H, flH)~C. Let A , {wp(f)) a finite set of words and {wp(h)) a substitution for these in H. Let C = HA , the stabilizer of A , then C is a prime subgroup of and therefore C fl Also iwp(h) < iwq(%) wq(h)wp(h)-l <+ c < cwq(~)wp(W1- cwP (ib < cwq(i;). Thus C mimics 1 V. . . 5.2 .Xr Let V be a variety of L -groups. Let (Gi li E I) be a collection of 1-groups such that for all i Gi E V, and every finitely generated [-group H in V is a homomorphic image of some Gi. Let (Gi,ni) be the Holland representation of Gi Then = {(Ci, ni)l i E I} mimics V. Proof_. Let (H, A ) be a transitive .t -permutation group, a substitution for these in H. Let % = h,.. . h and let K be the L-subgroup of H generated by {hl,...,hn}. K E Vis finitely generated, thus there is an i I and a homomorphism a :Gi -+ K. Let C = {g E Gi / F(ga ) = h } Then C is a prime subgroup of G and therefore C E i ni. 69 For i j < n let g e Gi 'be such that gja = hj. Then < - j ,wp(h) < hwq(h) " < A ~~(h)w~(h)-~6>h < N(wq(~)wp(g)-l)a) & C < Cwq(g)wp(;)-I W Cw P(2) < cup. Therefore C mimics V. Corollary 5.2.4. Let C = {(F, nF) I (F, nF) is the Holland representation of F, F a free group with some total order). Then C mimics R. lksU2.L Let (G, Q) be a representable transitive C - permutation group. Then G is totally ordered (see Bigard et a1 11, Corollary 4.2.6.1) and thus G is a hornomorphic image of a free group with some total order. Therefore the previous proposition applies. - J- J- 512-5 Let H be a weakly abelian totally ordered group. Then H is the homomorphic image of a free group with a weakly abelian order. Proof, Let H be a weakly abelian totally ordered group. Then as a group H is isomorphic to a quotient F/G where F is a free group. Let F/G be ordered so that it is o-isomorphic to H. Since F is a free group, F can be ordered with a weakly abelian order (see Martinez [141). The restriction of this order to G gives an order on G with the property that vf 'f F, tfe <_ g E. G, f-lgf 2 e. Now defin.e a new order on F by x. -> e if either x E G and x 2 e in the order on G or.. x E 'F\G and xG > G in the order on F/G. Routine verification shows that this gives a weakly abelian order on F in which G is an &ideal and F/G is o-isomorphic to H. Corollary 5.2.6. Let C = {(F, nF] (F, nF) is the Holland representation of F, F a free group with a weakly abelian order). Then C mimics W , the weakly abelian variety. When considering' 1 -permutation groups that mimic a representation of an p -group. In particular, for totally ordered groups the right regular representation (G, G) of an o-group G is a natural one to consider. However not all representable varieties may be mimicked by collections of f -permutation groups which are regular representations of o-groups. In the following discussion the notation (G, G) will always mean the right regular representation of an o-group G. Proposition 5.2.7. Let v E R be mimicked by I(Gs, G,) 1 E S]. Then vc_ W. Proof. Let H be a regular subgroup of G E V . Then H is prime and R(H), the set of right cosets of H, is totally ordered. If H is not normal in G, then 3 e 5 h E H, - g c G such that g 'hg 4 H and ~~-~h~> H. Consider the transitive C-permutation group (G, R(H)) and the words Wl(x,y) = y, w2(x,y) = X-'YXS W3(x,y) = e. Consider the substitution y -+ h, x -+ g in G. Then ~~-~h~> H = Hh, that is, Hw2(g,h) > Hw3(g,h) = Hwl(g,h). Let s E S and x+ a y -+ b be any substitution in Gs such that w2(a,b) > wl(a,b) and W2(8,b) > w3(a,b). Then a-lba > b and a-'ba > e. Since a-'ba > e, b > e, that is wl(a,b) > w3(a,b). However, Hwl(g,h) = Hw3(g,h). This contradicts the hypothesis that {(.Gs, cs)l s E: S) mimics V . Therefore H is normal in G. However the weakly abelian variety w is the largest variety of 1-groups with the property that every regular subgroup is normal 1201. Therefore V W. It must now be shown that weakly abelian varieties can be mimicked by regular representations of totally ordered groups. . . Pro~om5.?.8L Let V c_W. Then C= {(G,G) IGis totally ordered) mimics V. Proof, Let (G, R) be a transitive l -permutation group, C E V. Let u E R . Then (G, d is 1 -isomorphic to (G, R(G )) and since (G, n) is transitive C; a g~n g-'~,i = {el. However V W so every convex [-subgroup of G is normal. In particular Ga is normal and thus Ga = g,G.g-l~ug = {el. Therefore (G, Q) is 1-isomorphic to (G,R({el)) = (G,G), and clearly C mimics V . 7 2 , In the case of the Holland representation it was possible to restrict a mimicking set for W to the collection of free groups with all possible weakly abelian orders. However the final result shows that this is not possible for the regular representation. . . Prop- 5- Let C= {(F, F) I F is a free group with a weakly abelian order). Then C does not mimic w. Proof* Consider No, the free two generator nilpotent class two group, freely generated by {a, b) and ordered such that a >> b >> [a,bl > e. (No, No) is a transitive 1- permutation group with No E W . Let wl(x,y) = [[~,y],y], W2(x,y) = e, w3(x,y) = x A y and wg(x,y) = [x,y]. Consider the substitution x + a, y + b in No. Then e = wl(a,b) = w2(a,b) < wq(a,b) = [a,b] < w3(a,b) = b. Let (F, F) E C and consider any substitution x -t f, y -+ g in F such that e = ~~(f,~)< w4(f,g) < w3(f,g), i.e. such that e < [f, gl < g. Then since F is a weakly abelian, g >> [f, gl. Thus there do not exist hc H and n, m E Z such that g = hn and [f,gl = hm, and then since F is a free group, [[f,gl,gl # e whence [[f,gl,gl > e or [[f,gl,gl < e. Therefore wl(a,b) = w2 (a,b) but either wl(f,g) > w2(f,g) or wl(f,g) < w2(f,g) and C does not mimic 6.1. REFERENCES 1. A. Bigard, K. Keimel, and S. Wolfenstein, Groupes et anneaux reticules, Lecture Notes in Mathematics, 608, Springer- Verlag, 1977. 2. G. Birkhoff, On the structure of abstract algebra, Proc. Cambridge Phil. Soc, 31(1935), 433-454. 3 C.. Chehata, An algebraically simple ordered group, Proc. London Math. Soc., 2(1952), 183-197. 4. T. Feil, Ph.D. Thesis, Bowling Green State University, 1980. 5. T. Feil, An uncountable tower of 4 -group varieties, Algebra Universalis, 14(1981), 129-131. 6. A.M.W. Glass, Ordered permutation groups, London Math. Soc. Lecture Note Series 55, Cambridge University Press, 1981. 7. A.M.W. Glass, W.C. Holland and S.H. McCleary, The structure of l!.-group varieties, Algebra Universalis, 10(1980), 1-20. 8. S .A. .Gurchenkov, Minimal varieties of e -groups, Algebra i Logika, 2(1982), 131-137 (Russian). 9 .W.C. Holland, The lattice ordered group of automorphisms of an ordered set, Mich. Math. J., 10(1963), 399-408. 10. W.C. Holland, The largest proper variety of lattice ordered groups, Proc. Amer. Math. Soc., 57(1976), 25-28. 11. W.C. Holland and S.H. McCleary, wreath products of ordered permutation groups, pacific J. ~ath;, 31 (19691, 703-716. 12. M.E. Huss and N.R. Reilly, On reversing the order of a lattice ordered group, J. Algebra, to appear. 13. V.M. Kopytov and N. Ya. Medvedev, Varieties of lattice ordered groups, Algebra i Logika, 16(1977), 417-423 , .yks (Russian) . 14. J. Martinez, Free products in varieties of lattice ordered groups, Czech. Math. J., 22(1972), 535-553. J. Martinez, Archimedean-like classes of lattice ordered groups, Trans. Amer. Math. Soc., 186(1973), 33-49. J. Martinez, Varieties of lattice ordered groups, Math. Z., 137(1974), 265-2840 N. Ya. Medvedev, The lattice of varieties of lattice ordered groups and Lie algebras, Algebra i Logika, 16(1977), 40-45 (Russian). H. Neumann, Varieties of groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 37, Springer- Verlag, New York, 1967. N.R. Reilly, A subsemilattice of the lattice of varieties of lattice ordered groups, Can. J. Math., 13(1981), 1309-1318. N.R. Reilly, Nilpotent, weakly abelian and Hamiltonian lattice ordered groups, Czech. Math. J., 33(1983), 348-353. N.R. Reilly and R. Wroblewski, Suprema of classes of generalized Scrimger varieties of lattice ordered groups, Math. Z., 1(1981), 293-309. E.B. Scrimger, A large class of small varieties of lattice ordered groups, Proc. Amer. Math. Soc., 51(1975), 301-306. J.E.H. Smith, The lattice of 1-group varieties, Trans. Amer. Math. Soc., 257(1980), 347-357. E.C. Weinberg, Free lattice ordered abelian groups, Math. Ann., 151(1963), 187-199.