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Jo.اO? LATTICE-ORDERED GROUPS of ORDER AUTOMORPHISMS OF I /Jo.ÇO? LATTICE-ORDERED GROUPS OF ORDER AUTOMORPHISMS OF PARTIALLY ORDERED SETS Maureen A. Bardwell A Dissertation Submitted to the Graduate College of Bowling Green State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY August 1978 ii ABSTRACT In recent years, groups of order automorphisms of totally ordered sets have been carefully studied. Two of the primary motivations for this study are the facts that these groups are lattice-ordered groups, and that every lattice-ordered group can be embedded in the full group of order automorphisms of some totally ordered set. An interesting property of the lattice-ordered group of order automorphisms of a chain is that positive elements of the group are algebraically disjoint if and only if they have disjoint supports. There are a number of examples of partially ordered sets which are not totally ordered, but whose groups of order automorphisms are lattice-ordered. In this dissertation, we classify the partially ordered sets ft whose groups of order automorphisms are lattice-ordered groups which enjoy the "disjoint support property" mentioned above. We also explicitly describe which lattice-ordered groups arise as the full group of automorphisms of such a partially ordered set. Ill ACKNOWLEDGMENT The author wishes to express her appreciation to Professor W. Charles Holland for his encouragement and assistance in the writing of this dissertation. IV TABLE OF CONTENTS Page CHAPTER 1: Introduction;........................................................... 1 CHAPTER 2: Preliminaries ........................................................... 5 CHAPTER 3: A (ft) with Two Orbits in ft............................... 26 CHAPTER 4: Examples................................ 101 CHAPTER 5: The General Classification................................... 116 BIBLIOGRAPHY 126 CHAPTER I INTRODUCTION In the 1930's several mathematicians considered the group G of order preserving permutations of the real unit interval [0,1]. One of the important properties of this group is the fact that it is a lattice-ordered group (¿-group) under the order induced from [0,1]. That is, we define a partial order of the set G by for f,g e G, f < g if and only if for all x e [0,1], xf <_ xg. With this ordering, each pair of elements of G f,g has a greatest lower bound f A g and a least upper bound f V g. Moreover, if f,g,h,k e G, then h(f A g)k = (hfk) A (hgk) [2]. This group provided the first important example of a non-abelian ¿-group. Now suppose ft is any partially ordered set. We let »(ft) denote the collection of all permutations (automorphisms) f of ft such that both f and f”1 preserve the order relation on ft. Then, »(ft) is a group under function composition, and we may partially order »(ft) as before: f £ g means that for all a e ft, af £ ag. The example presented in the first paragraph does not depend upon the interval [0,1]. In fact, if ft is any chain (totally ordered set), »(ft) is a lattice-ordered group under the induced order ([2], Example 0.8). G. Birkhoff raised the following question: when is an ¿-group isomorphic to the group of all order preserving permutations of some -1- 2 chain [1]? In 1963, W. Charles Holland gave a partial answer to this question by proving that every ¿-group can be embedded in the group of all order preserving permutations of some chain ([4], page 1). His result provided many new examples of non-abelian lattice-ordered groups, and motivated further research in ordered permutation groups as a tool for examining lattice-ordered groups. In 1970, E. Scrimger gave a collection of necessary and sufficient conditions for an abstract ¿-group to be isomorphic to /(ft) for d chain ft [10]. There are many examples of partially ordered sets ft and groups /(ft) where ft is not a chain but /(ft) is an ¿-group under the induced order. A generalization of Birkhoff's original question is the following: when is an ¿-group isomorphic to the group of all order preserving permutations of some partially ordered set? A number of attempts have been made to answer this question, but, because of its generality, prior to this dissertation no real progress has been made towards a solution. We study this question, but we place an additional restriction upon the ¿-groups /(ft). Suppose ft is a partially ordered set, /(ft) is the group of order automorphisms of ft, and /(ft) is an ¿-group under the induced order. Elements f and g of /(ft) are said to be algebraically disjoint if f a g = e (e is the identity of the group /(ft)), and these elements are said to have disjoint supports if {ct c ft | af / a) A {8 e ft | Bg / 8) = 0. When ft is totally ordered, positive elements of /{ft) are algebraically disjoint if and only if they have disjoint supports [6]. This fact has proved to be quite useful in the study of groups of order preserving permutations of 3 chains since it corresponds the algebraic notion of disjointness to the geometric notion of disjoint functions. Keeping this fact in mind, we state the major question answered in this dissertation: characterize the partially ordered sets ft and the groups /(ft) such that /(ft) is an ¿-group under the induced order and /(ft) has the property that positive elements of /(ft) are algebraically disjoint if and only if they have disjoint supports. The method of characterization proceeds as follows. In Chapter 2 we prove that (i) /(ft) is an ¿-group where positive elements are algebraically disjoint if and only if they have disjoint supports, is equivalent to (ii) every orbit of /(ft) in ft is a chain. Condition (ii) provides a complete classification of the ¿-groups /(ft) satisfying our hypotheses which are transitive on ft. We then derive a third equivalent condition which allows us to consider /(ft) as an ¿-subgroup of the direct product of a collection of full groups of order automorphisms of chains. The two equivalences are crucial to all the theorems which follow. We also present in this chapter some background material on the problem, on groups of order automorphisms of totally ordered sets, and on wreath products of permutation groups. In Chapter 3 we proceed by assuming that /(ft) has precisely two orbits in ft. The partially ordered sets ft and the ¿-groups /(ft) which satisfy these hypotheses naturally fall into one of six categories. In the beginning of this chapter we give a general description of each category for reference purposes. We then give a complete and specific classification of each category. In the 4 process of this, classification we generalize the previous definition of a "periodic ¿-group of permutations". An example of this new type of ¿-group is given. In Chapter 4 we present examples of partially ordered sets ft and ¿-groups »(ft)« for each of the six categories, of Chapter 3. These examples illustrate that the classification theorems of Chapter 3 provide a constructive method for giving examples. In Chapter 5 we examine the general case where »(ft) has more than two orbits in ft. Results obtained for this case will be generalizations of those obtained when we assumed »(ft) had precisely two orbits in ft. Once the general case has been discussed, we give a nontrivial example of a po-set ft and an ¿-group »(ft) satisfying our hypotheses such that »(ft) has an infinite number of orbits in ft. We conclude this chapter, and the dissertation with a short list of open questions. CHAPTER II PRELIMINARIES §1. Preliminaries on po-groups. A partially ordered group (po-group) is a group (G,*) together with a partial order <_ of the set G such that f £ g implies hfk £ hgk for all f,g,h,k e G. A po-group G is a lattice-ordered group (¿-group) if and only if every pair of elements f,g in G has a supremum f V g and an infimum f A g. In this dissertation, groups will be written multiplicatively, and the identity element of the group will be denoted by e. A subgroup and a sublattice of an ¿-group G is called an /-subgroup. A subset H of a partially ordered set G is convex if f s H, g e H, x e G, and f £ x £ g imply that x e H. If G is an ¿-group, an /-ideal of G is a normal convex ¿-subgroup of G. Suppose * is a homomorphism from an ¿-group G to an ¿-group H. If * preserves v and a, * is an /-homomorphism. If * is a one-to-one ¿-homomorphism, * is an /-embedding, and * is an /-isomorphism if * is a group isomorphism which preserves both v and A. A convex ¿-subgroup H of G is an ¿-ideal of G if and only if H is the kernel of some ¿-homomorphism * on G; if Hg denotes a typical element of the quotient structure G/H, the mapping Hg g* is an ¿-isomorphism of G/H onto G*. For any nonempty collection of ¿-groups (G^ j a e A), we may form the product /-group -5- 6 y G where the group and lattice operations are defined as follows: aeA a for f,g e y G , fg(a) = f(a)g(a), (f v g)(a) = f(a) V g(a), and aeA a (f A g) (a) = f (a) A g(a) . Elements f and g of an ¿-group G are said to be algebraically disjoint if and only if f A g - e. Let f be an arbitrary element of an ¿-group G. Define - f v e (the positive part of f), f = f 1 V e (the negative part of f), and ]f| = f v f 1 (the + - + - -1 absolute value of f).
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