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The Representation of a Lattice-Ordered Group As a Group Of THE REPRESENTATION OF A LATTICE- ORDERED GROUP AS A GROUP OF AUTOMORPHISMS H. Jean Springer B. Sc. , London (university College of the West ~ndies), 1962 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in the Department of Mathematics @ H. JEAN SPRINGER 1969 SIMON FRASER UNIVERSITY JULY, 1969 EXAMINING COMMITTEE APPROVAL ,> ,> N. R. Reilly, Senior Supe6isor A. H. Iachlan, Examining Committee Berggren, Exam C5mmit t e"e. T. C. Brown, Examining Committee D. Ryebuyn, Examining Committee (ii) The purpo e of this study is to review some of the developments in.i: the theory of lattice-ordered groups closely related to the Holland representation for lattice ordered groups. In Chapter 0, bask definitions and results required throughout this study are reviewed. Chapter 1 contains a study of regular and prime subgroups of a lattice-ordered group and concludes with the very important Holland repre- sentation theorem. In Chapter 2, the Holland representation is used to derive the very nice result: %very lattice-ordered group can be embedded in a divisible lattice-ordered group. Finally, Chapter 3 contains a study of transitive lattice ordered groups of order preserving permutations on a totally ordered set and also a discussion of a class of simple lattice ordered groups. (iii) TABLE OF CONTENTS Page Introduction v Notation i viii Chapter 0 Fundamental concepts 1 Chapter 1 Regular and prime subgroups of 8 an 1-group Chapter 2 An application of the Holland representat ion Chapter 3 The 1-group of o-permutations on a totally ordered set Bibliography I wish to take this opportunity to express my appreciation to Dr. N. R. Reilly for his advice, suggestions and encouragement during the preparation of my thesis. I would also like to thank Mrs. Joyce Simpson for doing the typing. The support received fro grant No. A-4-044is also Lattices and gro?ap< seem to provide two of the most basic tools in the study of UnLversal Algebra. Also, algebraic systems endowed with a partial or total order are important in many branches of MaKhema%fcs ,, It is therefore not surprising that there should be LncreasEng interest in the study of lattice-ordered groups, Prior to 1941, only lattice-ordered groups which are abelLan or totally ordered had been studied; the most notable contribution madcp to that time was probably due to Hahn in 1907 who developed an embedding for abelian totally ordered groups. This embeddfng was later extended to fnclude abelian lattice-ordered groups by P. F. Conrad, J. Harvey and C. Holland in 1963, In 1941, Garrett Birkhoff published a paper which appeared in the Annals of Ma%heri~atiec( 1942) and in which he fnvest'igated properties of non-abelian lattice-ordered groups. This, no doubt, formed the basis for further investfgation and, since then, many of the problems and conJeetures listed in the conclusion of that paper have been resolved. In the study which follows, some of the more recent developments in the theory of lattice-ordered groups have been reviewed, and an attempt has been made to make this presentation self-contained as far as possfble, However, a basic knowledge of group theory has been assumed. Chapter 0 contains basic results and definitions which are used throwghcxk the study with references befng given whenever proofs are omitted. These results, and a general basic theory of lattice-orderee ~TG~Scan be found in BSrh%offls Sook on "Iattice theory" and fn Fmchts book on "Pa~.tfallyordered algebraic systems " . In Chapter 1, a detailed discussion of the properties of regular and prime subgroups of an, 1-group is presented. Prime subgroups are of particular importance in obtaining representations of lattice-ordered groups. For, if M is a prime subgroup of a lattice-ordered group @, then the set of cosets of M can be endowed with a natural total order. It follows that if M is both prime and normal, then the set of cosets of M is a totally ordered group. Thfs property was utilised by Holland ig his representation theorem which is discussed at the end of this chapter. The Holland representation of a lattice-ordered group is a representation of a lattice-ordered group as a subdirect sum of ILK where P each K is a tra,nsitfve I-subgroup of the lattice-ordered P group of all order-preserving permutations on some totally ordered set. This answered a problem originally posed by Birkhoff in the second edition of his book on latticeI theory, Though this representation throws little light on tiinternal structure of a lattice-ordered group, it is an invaluable tool in the study of the nature and occurrence of lattice ordered groups. An examp1.e of this is given in Chapter '2 when an application of the Holland representation is used to obtain an elegant embedding theorem: re very lattice- ordered group can be embedded in a divisible lattice-ordered group". Finally, Chapter 3 contains a study of the lattice- ordered group of crezr-p~cszrvf~gperm~-Z%tfons oc a tctally ordered set. This chaptw is divided into two sections. Section I contains a study of lattice-ordered groups of order-preservhg permutations oz a totally ordered set, which are transitive on %hat set, while, in Section 2, a class of simple lattice-ordered grcra-ps is discussed. That knowledge of the propertfes of the lattice-ordered group of order-preserving permutations on a totally ordered set yields important informstfon about lattice-ordered groups in general, is clear from the Holland represen$,ation theorem. (viii) x and y are disjoint x and y are incomparable The positive cone of a partially ordered group G The set of all convex 1-subgroups of an 1-group G The set of all 1-ideals of an 1-group G The sct of all right cosets of C The set of all o-preserving permutations on the totally ordered set S The convex 1-subgroup generated by g The convex 1-subgroup generated by M and a The cardinal sum of 1-groups G and H The lexicographic sm of G and H ordered from the left G is 1-isomorphic to H The normalizer of M in G CHATTER 0 For a general theory of lattice-ordered groups, the reader is referred to Birkhoff ("1 and Fuchs (8). Included here are some basic definitions and resuiSs which are used throughout this study. In general, addftfve rotatfon is used unless otherwise mentioned. Definition 0.1 A , G, is a set C such that G is a group; G is a partially ordered set (p.0, set) under a relation *; If a, bcG and a&, then c+asc+b and a+csb+c for every CEG. A p.0, group which is totally ordered is an o-group, Definition 0,2 (a) A lattice-ordered group (1-group) is a p.0. grouF which is also a lattice under the relation s. (b) If G is an 1-group, then a subset H of G is an 1-subga of G if and only if H is a subgroup of @ and H is also a sublattice of @. In a lattice, L, if x, ycL, then denote glb{x,y] by xAy and lub[x,y] by xVy. Deffnftion 0,3 , (a) Let G and H be p.0. groups. Then an o-homomorphism 8 from G into H is an fsotone group homomorphfsm. That is to say 0 is a groAp homomorphism such Chat for any x, ycG if xsy, then x0gy0. (b) 8 is an o-isomorphism If 0 fs a 1-1 o-homomorphism. (c) If G and H are 1-groups, then an 1-homomorphism 8 from G into H is an o-homomorphism such that for any x, yeG, (XVY)~= xevye (5) (xny) 8 = xe~ye ( ii) (d) An 1-isomorphism 0 from an i-group G into an 1-group H is an o-isomorphism from G into H such that (i)and (ii) of (c) hold. When G and H are 1-isomorphic 1-groups L we write G=H. I Remark: Clearly if G is an 1-group and if H fs a subgroup of G such that for each XEH, XVOEH then it follows easily that H is an 1-subgroup of G. Some elementary properties of 1-groups. L(1) In any 1-group G, addition is distributive on meets and joins. That is, ifa, x, y, bcG, then, a+(xvy) = (a+x)~(a+y),(xvy)+b = (x+b)v(y+b) (I) a+(xAy) = (a+x)A(a+y), (xAy)+b = (x+b)A(y+b) (2) L( 2) In an 1-group G, if a, bcG, then aAb = -(-av-b) (3) L( 3) As a result of ~(2)the mapping 8:G+G, defined on an I-group G such lthat xe = a-x-n-5, a, b, XEG is a 1-1 mapping such that for x, ~EGwith xly then x0ry0. Also Ff xemye then this implies xry. Such a mapping fs sometimes termed a dual isomorphism. In any 1-group G, a-(xvy) tb = (8-x+b)~(a-y+b) (4) this follows f~omthe dud of ~(2). L( 4) In any 1-group C, the generalisations of (I), (2) and (4) hold. That is to say a+(Vxo)+b = ~(a+x~+b) (5) a+(Axo)+b = A (acxD+b) (6) a-(Vxo)+b = (a-xoib) and dually (7) I where o ranges over some finite arbitrary index set. Definition 0.4 I If a is an element of an 1-group, G, then a+ = avo and I - - I a = aho; a* is called the positive part of a and a is called I the negative part of a.
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