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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.
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