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). Then fAf = e, f = f (f ) , and this
is the unique representation of f as the difference of algebraically disjoint elements ([2], page 0.20).
For a more thorough discourse on the fundamentals of lattice- ordered groups, the reader should consult Birkhoff [1] or Conrad [2].
§2. Groups of order automorphisms of po-sets.
Let ft be a partially ordered set (po-set). A permutation f of ft is said to be order preserving on ft if for each pair x,y e ft such that x <_ y we have xf <_ yf. If f is a permutation of ft such that f and f 1 preserve the order relation on ft, we say f is an order automorphism of ft. Let /(ft) denote the collection of all order automorphisms of ft. Then /(ft) is a po-group with function composition as the group operation and ordering given by: f <_ g if and only if xf <_ xg for all x e ft. This ordering is the ordering induced on /(ft) by ft. 7
If ft is any chain (totally ordered set), »(ft) is a lattice-
ordered group under the induced order ([2], Example 0.8). For let
f,g e »(ft) and x e ft. Then x(f v g) = max{xf,xg) and
x(f A g) = min(xf,xg).
Let ft be any po-set and suppose f e »(ft). Define the support
of f, denoted by supp f, to be (x e ft | xf / x). When ft is
totally ordered, the ¿-group »-(ft) has the following interesting
property.
PROPOSITION 2.1 [6]. Suppose f,g e »-(ft) and f,g ¿_e. Then f a g — e if and only if supp f A supp g = 0.
Proof : Suppose supp f A supp g / 0. Choose Xg e supp f A supp g
Then Xgf / Xg and Xgg / Xg. Since ft is totally ordered, either
Xgf < Xgg or Xgg Without loss of generality, assume
x f £ xog' 7)1611 x0(£ A g') = min{xofyXog} = xof / x0’ i-e-’ f a g / e.
Thus f A g = e implies supp f A supp g = 0.
The converse is clear. //
We have thus shown that when ft is a chain, positive elements of the ¿-group »(ft) are algebraically disjoint if and only if they have disjoint supports. This property is interesting in that it relates the algebraic notion of disjointness to the geometric notion of disjoint functions. For convenience, we make the following definition. Suppose ft is a po-set, »(ft) is an ¿-group under the induced order, and G is an ¿-subgroup of »(ft). Then G has the disjoint support property if for each pair of elements f,g of G such that f,g > e, f A g = e if and only if supp f A supp g - 0. 8
Throughout this dissertation, we will make use of the following
notation. Suppose ft is a set partially ordered by £. If it is
not clear from context we will denote the partial order on ft by <^.
If E is any subset of the po-set ft, then <( denotes the partial “I Z ordering on ft restricted to the elements of Z. For chains S and
T, the lexicographic product of S by T, denoted by S x T, is
the collection of ordered pairs (s,t) e S x T totally ordered by
the following rule: for (s^tp , (s2,t2) e S x T,
(s1,t1) (ii) t = t„ and s < s . The ordered set of integers will be X Z X O z denoted by Z; Q denotes the rational chain; I denotes the irrational chain; and R denotes the ordered set of real numbers. For any set S, |s| denotes the cardinality of the set S. If {S | A e A} is a collection of disjoint sets, (¿J S will denote A A A the disjoint union of the sets S A We now state some standard results concerning Dedekind completions of chains, and we show how an order preserving permutation of a chain may be extended to an order preserving permutation of the Dedekind completion of the chain. Let S be a totally ordered set. S is a discrete chain if each element of S has an immediate predecessor and an immediate successor. S is dense if for each s^,s2 e $ with s^ < s? there exists an s$ e S such that s^ < s^ < s2- If for each s^,s2 e S there exists f e /(S) such that s^f = s^, then S is discrete or dense. If S and T are totally ordered sets and T i=S, T is dense in S if for each s^,s2 e S with sj there exists t e T such that Sg < t < s2- In an analogous way to constructing tt from Q, for each chain S we may construct its Dedekind completion S by using Dedekind cuts. (The Dedekind completion of a chain will not have endpoints.) S inherits a natural total order from S and for each pair Sg,s2 e G such that Sg < s2 we may find an s e S such that Sg <_ s 5. s2- Also, each s e S\S has the property that s = sup{sa e S | sa If S is a chain and S is its Dedekind completion, there is a natural way to ¿-embed the ¿-group /(S) into /(S) . THEOREM 2.2 [7]. There is a one-to-one ¿-homomorphism - : A(S) ACS). Proof: Let g e X(S). Define g e /(S) by for each s e S, sg = sup{sg | s e S and s <_ s) = inffs'g | s' e S and s <_ s'). The proof that this is an ¿-embedding is routine. // For the remainder of this dissertation, if f e /(S) , f will denote the extension of f to S described in the above theorem. For a thorough discussion of the properties of groups of order automorphisms of totally ordered sets, the reader is referred to Glass [4], 10 §3. Statement of problem and some basic results. 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 (see Example 2.3 and Chapter 4). Attempts have been made by S. H. McCleary, A. M. W. Glass, and other active researchers in ¿-groups to completely characterize the partially ordered sets ft and the groups /(ft) for which /(ft) is a lattice-ordered group under the induced order. This problem appears to be too general, and, as a result, all attempts at a classification have not been fruitful. We will study this problem, but we place an additional restriction upon the groups /(ft). The concept of algebraic disjointness has proved to be an invaluable tool in the study of the structure of ¿-groups. We present the following result as an example of the use of this notion. If A is a subset of an ¿-group G, the polar of A is {x e G | ]x| A ¡a| = e for all a e A). A class of ¿-groups which has been widely studied is the class of representable /-groups. (An ¿-group G is representable if and only if G is an ¿-subgroup of a cartesian product of totally ordered groups.) Sik showed that an ¿-group G is representable if and only if each polar is normal ([2], page 2.1). For further results along these lines, the reader is referred to Conrad [2], In Proposition 2.1, we saw that for ft a chain, /(ft) is an ¿-group with the disjoint support property. Since algebraic disjoint ness is such a useful tool in the study of ¿-groups and since the two notions of disjointness correspond in the important example where 11 is a chain, we further assume that the groups A(^) we will consider have the property that for f,g e A-(ft) where f,g >_ e, fAg=e if and only if supp f D supp g = 0. We now state the major question answered in this dissertation: Classify the partially ordered sets ft and the groups A-(ft) such that A(ft) is a lattice-ordered group having the disjoint support property. The following example due to A. M. W. Glass demonstrates that there are po-sets ft such that A(ft) is an ¿-group under the induced order which does not have the disjoint support property. EXAMPLE 2.3. For arbitrary chains S and T, the cardinal produei-af- S and—T, denoted by SET, is the collection of all .. ordered pairs (s,t) e S x T ordered by the following rule: for (s1,t1), (s2,t2) e S B T, (s^tp s <„ s. and t £ t . For our example, let ft = R 0 Z. Then »(ft) X o Z X 1 z is the cartesian product of the ¿-groups »(HL) and A(Z), i.e., 4(ft) = »(HL) x _4{Z). Let (+m,+n) denote the element of A (HL) x AW which adds the real number m to the first coordinate of each element of RS Z and the integer n to the second coordinate of each element of R S Z. Then (+l,+0) A (+0,+l) - (+0,+0), but supp(+l,+0) = RE 1 = supp(+0,+l). // We need three definitions before stating our first theorem. Suppose G is a group of permutations of a set S. (S is not necessarily partially ordered.) Fix s^ e S. Then s@G - (s^g I g e G) is an orbit of G in the set S. (sG | s e S) forms a partition of 12 the set S. G is said to be transitive on S if G has only one orbit in the set S. If S is partially ordered and /(S) is transitive, S is homogeneous. The following theorem is crucial in our classification. THEOREM 2.4. Suppose ft is a po-set and A (Cl) is the group of order automorphisms of Cl. Then, the following three conditions are equivalent: (i) A (Cl) is an ¿-group having the disjoint support property. (ii) Every orbit of A(Cl) in Cl is a chain. (iii) A (Cl) is an ¿-group and for each pair f,g e A(Cl) and each a e Cl, a(f v g) = max{af,ag} and a(f A g) = min{af,ag}. Proof: (i) (ii) . Let 0 be an orbit of /(ft) in ft and let a,8 £ 0, a / 8. To show 0 is a chain, we need to show a £ 8 or 8 £ a- Since 0 is an orbit of /(ft) in ft, there exists f e /(ft) such that af = 8- Since /(ft) is an ¿-group, f = f+(f ) 1 where f+ a f = e. By assumption, positive elements of /(ft) which are algebraically disjoint have disjoint supports. Therefore, supp f+ A supp f” = 0. This implies that supp f+ A supp(f ) 1 = 0. Thus af = af+(f) 1 = 8 implies that either af+ =8 or a(f ) 1 = 8- If af+ = 8, then a £ 8- If a(f ) 1 = 8, then 8 <_ a. Therefore a f 0 or 0 <_ a. (ii) => (iii). Assume every orbit of /(ft) in ft is a chain. Since /(ft) is partially ordered, to show /(ft) is an ¿-group it suffices to show that for each pair f,g e /(ft), f v g exists. Let 13 f, g e/(ft). Define a function h on ft as follows: Let a e ft; then ah = max{af,ag). h is well defined since there exists a chain 0 , an orbit of /(ft) in ft, such that a,af,ag e 0&. We verify h e /(ft). To check h is order preserving, let a,B e ft -with a £ 6. We show -¿»h £Bh. - If ah =-af- and_-Bh - ff, - or if ah = ag and Bh = Bg, then it is clear that ah £ Bh. Next suppose ah = af and Bh = Bg. a £ B and f e /(ft) so af £ Bf. Bh = max{Bf,Bg) = Bg. Therefore Bf £ Bg. Therefore ah = af £ Bf £ Bg = Bh. Similarly, we show ah £ Bh if ah = ag and Bh = Bf. Thus h preserves order on ft. To show h is one-to-one, use the argument in the previous paragraph and a similar one which shows that if a is unrelated to B then ah is unrelated to Bh. Next we check that h is onto. Let to c ft. We must show that there exists deft such that <5h = m. There exists Bh”1 < ah”1 then B < a, a contradiction. Thus Bh 1 is unrelated to ah’1. But if y is unrelated to 6, yh is unrelated to <$h. 14 This implies 8 is unrelated to a, again a contradiction. Hence ah 1 < 8h 1 and h 1 preserves order on ft. It now follows h e /(ft). It is clear that h 2 f, h £ g, and that any k e /(ft) which has the property that k £f,g also has the property that k > h. Hence h = f v g. Similarly, for any a e ft, a(f A g) = min(af,ag). Thus /-(ft) is an ¿-group where for each pair f,g e /(ft) and each a e ft, a(f V g) = max(af,ag) and a(f A g) - min{af,ag). (iii) =s> (i) . To show /(ft) satisfies (i) , it suffices to show that /(ft) has the disjoint support property. Let f,g e /(ft) with f,g 1 e. Suppose supp f A supp g £ 0. Choose a e supp f A supp g. Since f,g £ e, af > a and ag > a. Therefore a(f a g) = min{af,ag) > a. Thus f A g £ e. Next suppose supp f A supp g = 0. Since f,g >_ e and for each weft, w(f A g) = min(wf,wg), it is clear that f A g = e. Thus (iii) =S> (i) . // From condition (ii) , we immediately obtain the following classifi cation of all po-sets ft such that /(ft) is a transitive ¿-group having the disjoint support property. COROLLARY 2.5. Let ft be a partially ordered set such that A(fl) is an ¿-group which has the disjoint support property. Then, if 4(Cl) is transitive on ft, Cl is a homogeneous chain. // 15 The following definition will be useful throughout the remainder of the dissertation. Let G be an ¿-group of automorphisms of the po-set ft and H be an ¿-group of automorphisms of the po-set E. Then the automorphism group (G,ft) is said to be ¿-isomorphic to (¿-embedded into) the automorphism group (H,Z) if and only if the following three conditions are satisfied: (i) There exists an ¿-isomorphism (¿-embedding) * from the ¿-group G to the ¿-group H. (ii) There exists a set order-isomorphism y : ft Z. (iii) For each a e ft and g e G, (ag)y - (oty) (g*) . The pair (*,y) will be called an ¿-isomorphism (¿-embedding) from the automorphism group (G,ft) to the automorphism group (H,Z). Now suppose that ft is a po-set such that >4(^) is an ¿-group which has the disjoint support property. Let {0^ : a e A) be the collection of orbits of A(ft) in ft. From condition (ii) of Theorem 2.4, we know that for each aeA, 0^ is a homogeneous chain and hence »4(0 ) is an ¿-group. Condition (iii) of Theorem 2.4 allows us to consider AW) as an ¿-subgroup of ff >4(0 ) as the aeA “ following theorem shows. COROLLARY 2.6. There exists an ¿-embedding * : AW) + "JT MO ) aeA “ given by: for f e A(ft)j f *==(... ,f , ...) e y]~ AW ), where “ aeA a f = f\r, , ttie function f restricted to the orbit 0 . ([A(ft)]*, ft) J a J 0 a 1 a is an automorphism group under the natural action, a'nd there is an ¿-isomorphism (*,y) from (A(ft), ft) to ([A (ft)]*, ft). Proof: It is clear that is a one-to-one group homomorphism 16 from /(ft) to [ | /(0 ). Condition (iii) of Theorem 2.4 implies that aeA * preserves the lattice operations. To show (/(ft),ft) is ¿-isomorphic to ([/(ft)]*,ft), we must verify that conditions (i), (ii), and (iii) of the definition are satisfied. We first define an action of the ¿-group [/(fi)]* on ft. Let weft and f* e [/(fi)]*- If w e 0a , define wf* = wf*(a^) = wfiQ . We now check the conditions of the definition, a 0 (i) has been demonstrated. For (ii), let x be the identity map. It is then clear that (iii) will be satisfied. Thus, (/(ft),ft) is ¿-isomorphic to ([/(ft)]*,ft). // For the remainder of this dissertation, whenever we consider ¿-groups /(ft) having the disjoint support property, we automatically will identify /(ft) with its image [/(ft)]* under the ¿-isomorphism described in Corollary 2.6, and we will identify the automorphism group (/(ft),ft) with the automorphism group ([/(ft)]*,ft). When ft is a chain, /(ft) enjoys another lattice-theoretic property which tends to distinguish /(ft) from ¿-groups in general, namely the property of being laterally complete [6]. An ¿-group is said to be laterally complete if whenever it has a subset (gg) such that for i £ j, gg a g. = e, then (gg) has a supremum. We now show that if ft is any po-set such that /(ft) is an ¿-group with the disjoint support property, then /(ft) is also laterally complete. 17 THEOREM 2.7. Let ft be a partially ordered set and suppose is an /-group under the induced order which enjoys the disjoint support property. Then A(Sl) is laterally complete. Proof: Let {g^ ) k e K) be a collection of pairwise disjoint positive elements of /(ft). For convenience, set K' = K U {“) and set e = g . Define a function g on ft by the following rule: ag, if there exists a keK such that a e supp g K Let a e ft, then ag ag^ if a i U SUPP Sv keK g is well-defined since algebraically disjoint elements of /(ft) have disjoint supports. Clearly g is one-to-one and onto. We check g is order preserving on ft. Let a,8 e ft with a £ B- We show ag < Bg- IP there exists a keK such that a,B e supp g^ or if ag = ag and Bg = Bgro, then clearly ag £ Bg. Therefore suppose i,j e K', i 0 j, ag = ag., and Bg = Bg-- Note that for each keK' and each me ft, m < mg. Then a £ agi = ag £ Bg± = B £ Bgj = Bg. Thus ag £ Bg and g preserves order on ft. Similarly g d preserves order on ft. Thus g e/(ft) . Clearly g = sup{g, }. Hence /(ft) is laterally complete. // keK k §4. Convex congruences and wreath products of permutation groups. Throughout §4, unless otherwise stated, ft will denote a totally ordered set. Suppose G is a subgroup of /(ft). A convex G-congruence on ft is an equivalence relation £ on ft such that (i) xCy implies xg£yg for all g e G, and (ii) if x £ y £ z 18 and x£z then x£y. A convex congruence is non-trivial if one of the equivalence classes contains more than one element and is not all of ft. AC ft is an o-block (with respect to G) if (i) A 0 0, (ii) A is a convex set, and (iii) for each g e G, either Ag = A or Ag Cl / = 0. The following theorem shows the correspondence between o-blocks and convex G-congruences. THEOREM 2.8 ([4], Theorem 1.8.6). Every o-block is a class of a convex G-congruence and, conversely, the classes of a convex G-congruence are o-blocks. If we assume G is transitive on£\~, every o-block is a class of a unique convex G-congruence. // If G is transitive on ft, the following theorem shows that the convex congruences on ft "line-up" in a nice fashion. THEOREM 2.9 ([4], Theorem 2.1.1). Suppose G is a transitive /-subgroup of -Af ft). The convex G-congruences form a chain. // We now generalize the above concepts to arbitrary partially ordered sets. Let Z be a partially ordered set and suppose G is a subgroup of /(£). Then, % is an o-congruence on Z (with respect to G) if and only if (i) A is an equivalence relation on Z, (ii) xXy implies xg/yg for all g e G and x,y e Z, and (iii) for every x,y,z,w e Z, if x)0y, z^fw, x is not related to z under A, and x < z, then y z and x A C Z is a block if A 0 0 and for each g e G, either Ag = A or Ag Pl A = 0. For any o-congruence A on Z, the set Z/30 can be partially ordered in a natural way. Also, we may construct a homo 19 morphism from the group G to the group /(E/X ) as the following theorem shows. THEOREM 2.10. If % is an o-congruence (with respect to the group G) on the po-set E, then the set l/)A can be partially ordered by letting xX <_y)( if and only if x y or xZCy. There is a homomorphism * of G into A () such that for g e G, g /—> g* s A(7-/ZC ) with (xX)g* = xgX . Proof: The relation defined above is well-defined by condition (iii) of the definition of o-congruence. The rest of the theorem follows by routine verification. // We again assume that ft is a totally ordered set and that G is a transitive ¿-subgroup of /-(ft). Convex G-congruences give rise to a convenient representation of G as an ¿-subgroup of the wreath product of two closely related ¿-groups. To describe this representa tion we need the following definitions. Let S and T be chains, and suppose K is a transitive ¿-subgroup of /(S) and H is a transitive ¿-subgroup of /(T). Set R = S x T and W = {( (‘J’gigg) ( Define an action of W on R by: Let (s,t) e R and ( /(R), and (W,R) is called the (large) wreath product of (K,S) by (H,T). (W,R) is denoted by (K,S)Wr(H,T). 20 Now suppose G is a convex G-congruence on ft. By using Theorem 2.10, we may define a total order on the set ft/6 . Let L(ft/£ ) = ig e G | xGg = xG for all G classes . G/L(ft/<5) is a transitive ¿-subgroup of AW/G) , and (G/L(ft/6) ,ft/6) is called the global action of G. If is an arbitrary G class, to G is totally ordered by setting < = < Set U 0 ~ft I |w00£ %fi = {g e G I to £}, and let denote the action of G- IWq'2/ restricted to Let L(tOg6) = {g e | yg - y for all y e (0^6), and finally let G(x^G) = gpL(tOg6)• Then G(tOg£) is a transitive ¿-subgroup of -4(u)g£), and (G (tOg£) ,tOg£) is called the local action of G. The following theorem relates (G,ft) to the global and local action of G relative to G . THEOREM 2.11 ([4], Theorem 4.1.1). There exists an ¿-embedding of the permutation group (G,ft) into the permutation group (W,R) = (GWqG),M0&)Wr(G/L(Sl/G)W/G). Moreover, W is the largest transitive ¿-subgroup of A(R) which has these local and global actions. // In [5], W. C. Holland generalized these notions to an arbitrary number of transitive permutation groups. Since the construction of these generalized wreath products will be used repeatedly in this dissertation, a brief summary of the methods involved will be given here. Suppose T is a totally ordered set. For each y e r, let ft^ be any homogeneous chain and any transitive ¿-subgroup of ^)-(ft^) . 21 We wish to construct Wr { (G , ft )). Let R = yp ft and choose any yeT Y Y yeT Y 0 e R (0 is a fixed,b ut arbitrary inference point). For r e R define supp r - (y e r | r(y) £ 0(y)}, and set R = {r e R | supp r is inversely well-ordered). For r,s e R, let T(r,s)= {y e T : r(y) £ s(y)}. We may define a total order on R as follows: Let r,s e R; r < s if and only if r(Y0) s(Yq) Y0 where y^ is the largest element of T(r,s) (Note that the existence of y^ is guaranteed since supp r and supp s are inversely well- ordered sets). We now construct the lattice-ordered group W. For each yeT, define a convex equivalence relation on R via: For r,s e R, r£^s if and only if r(y') = s(y’) for all y' 2 Y- Let W' = (g /(R) | for all yeT, r£^s if and only if rg£^sg}. Then, for each yeT, is a convex W'-congruence on R. For yeT, define RY to be the image of R under the natural ■projection R yp ft ,, denoted by r >—> rY. R is in one-to-one Y’>Y Y correspondence with the set of equivalence classes on R, and it inherits a total order in the natural way. For each g e W', yeT, and rY e RY, g induces an order preserving permutation of ft as follows: Let weft and choose s e R such that g Y Y Y,r sG r and s(y) = w; define wg = (sg)(y) e ft . Thus, each Y Y,rY Y V g e W' gives rise to a collection (g :-y e r and r' e R> Yi}. y,rY Finally, define W = (g e W' | g e G for all yeT, rY e RY), Y Y Y,r Then, W is a transitive ¿-subgroup of -4(R), and (W,R) is called 22 the wreath product of {(G^Sl^) | y e r). We denote (W,R) by Wr { (G ,ft . The verification of the facts presented above may be yeT T Y found in ([4], Chapter 4). Holland also derived an analogue to Theorem 2.11 for generalized wreath products. Before stating this theorem, we need to develop notions analogous to the global and local actions used in the case where we considered a single convex G-congruence £ . Let G be a transitive ¿-subgroup of /-(ft) and let £ and 9L be convex G-congruences such that £ Choose an arbitrary A'-class K, and set G. = {g e G j Kg = K). Let G(K) denote the action of (K) G(. K), restricted to K. £|1 K is a convex G(K) congruence on K, and, by Theorem 2.10, K/^ | can He totally ordered. Let L(K) = (h e G(K) [ ah = a for all a e K/£iv}. Then, G(K)/L(K) I K is a transitive ¿-subgroup of /(K/£jp, and (G(K)/L(K) ,K(£|K) is called the component of (G,9l) relative to Now suppose O( is a collection of convex G-congruences of ft and let G^> £ s&C. £* covers if (i) Ci & > and (ii) there exists no £ c 5t such that The pair (£^,£Y) is called a coverin pair in There exists e Of such that apibp but not and (ii) if o)pto2 c ft and e with not top“^, then there exists (£g,£ ) e Of such that to^ to2 and not to^a^. Let the elements of Of be indexed by the set T. For each y c T we may define components relative to (£p,£X), denoted by (G^,ft^), just as 23 above. By Theorem 2.9, the convex G-congruehces on ft form a chain under inclusion. We may therefore define a total order < of the Yi z- set T by setting < y^ if and only if £ Ç & THEOREM 2.12 [5]. Suppose G is a transitive ¿-subgroup of 4(Cl), Ot is a collection of convex G-congruences on ft, Ot is a plenary set of covering pairs in Ot indexed by T, and T is totally ordered by the rule given above. Let {(G^Sl ) | y e r} be the components of G which are associated with the plenary set of. Then, there is an immediate ¿-embedding of the permutation group (G, ft) into (W,R) = Wr {(G , Cl ) } (In this case the map from ft to R is yef V Y not necessarily onto). // We will need some notational conventions concerning wreath products Suppose now that ft is any partially ordered set such that /(ft) is an ¿-group where positive elements of the ¿-group are algebraically disjoint if and only if they have disjoint supports. For convenience of notation, suppose that /(ft) has precisely two orbits in ft, denoted by Og and 02> By Theorem 2.4, Og and O? are chains, and, in Corollary 2.6 we have identified /(ft) with an ¿-subgroup of /(Og) x/(02) by for f e/(ft), f^ (fjQ ,fjQ ) e /(Og) x^(02). Under this identification, we have defined an action on ft by for wf if we 0, 1 (f,g) e/(fi) and weft, w(f,g) = Let Pg denote Wg if W £ 0, the image of /(ft) under the projection map it onto the first coordinate of the elements of /(ft) and let P2 denote the image of 24 AW) under the projection map n onto the second coordinates. Then Pj and P2 are ¿-groups, Pj is an ¿-subgroup of >4(0p , and P2 is an ¿-subgroup of >4(O2). Then, C P x p?, the direct product of the ¿-group P^ with the ¿-group P2> P^ is an ¿-subgroup of -4(0p so we may consider convex Ppcongruences on 0^. Let denote a convex P^-congruence on Op By Theorem 2.11, there exists an ¿-embedding of the permutation group (P^,0^) into the permutation group (P^^ (s % ) ,s% pWr (P^/LfO^/Ai ,0^/A) p . Recall that P^ (s %2) is an ¿-subgroup of J?(sAp and P^/L(O^/^j ) is an ¿-subgroup of ^(O^/Ap). Thus, there is an ¿-embedding of the permutation group (p2’G]P into the permutation group (A(sA ) , sjV pWr (AW-^/Od p Wy/X. 2) • In this ¿-embedding, note that the mapping from 0^ is onto the set sp * 0 /# . We may make similar remarks for any convex Ppcongruence ?f2 on 02. Set (W1,R1) = M(sJf1),s^1)WrC4(01/jV 1),01/^'1) and set "(W2,R2) = (4(tjV2) ,tJY 2)WrC4(02/A/ 2) ,O2/jV 2) • Since there is an ¿-embedding of (P ,0p into (WpR ), there is an ¿-embedding of P into Wp Similarly, there is an ¿-embedding of P2 into W2. Quite often, when describing the structure of AW), we will wish to consider certain convex P^-congruences on 0^ and certain convex Ppcongruences on C>2. Since — P1 x P2’ P^ can be ¿-embedded in Wp and P2 can be ¿-embedded in W2, we may consider AW) as an ¿-subgroup of W x W2> To denote this fact we will often write AW) Q 1)Wr4(01/^' p] x [A(t%2)WT W2/A!2)]- The elements of W^ are permutations of the points of 0^ and the elements of W2 are permutations of the points of Op so we may consider W x W2 25 as a group of permutations of the set ft by defining an action as follows: Let weft and ( ( “(/gigg) if m e Og , («iy; g2)) Note that we are only ûûC<Î>2;S2^ if a) e O2 claiming that the elements of Wg x W2 are permutations of ft; these functions do not necessarily preserve the order on ft. There fore, we will often say that (/(ft) ,ft) C (Wg x W2,ft). Similar conventions should be made when /(ft) has an arbitrary number of orbits in ft. For further information on wreath products of permutation groups, the reader is referred to Holland and McCleary [8]. CHAPTER III /(ft) WITH TWO ORBITS IN ft Throughout this chapter we assume that ft is a partially ordered set such that /(ft) is an ¿-group with the disjoint support property. Corollary 2.5 provides a complete classification of the po-sets ft which satisfy our conditions and which have the property that /(ft) is transitive on ft. For ft a chain, the structure of the ¿-groups /(ft) has been carefully studied and will not be examined further here. The natural question to ask now is the following one: Suppose that /(ft) has precisely two orbits in ft. Which po-sets give rise to an ¿-group /(ft) satisfying these hypotheses and what is the structure of the ¿-groups which arise in this fashion? This question is answered in this chapter. For the general case where /(ft) has an arbitrary number of orbits in ft, we will be able to provide a classification by examining the orbits pair by pair. Thus, the main results necessary for the classification of the po-sets ft and the ¿-groups /(ft) which satisfy the disjoint support property are proved in the examination of the two orbit case. Therefore suppose ft is a po-set such that (i) /(ft) is an ¿-group having the disjoint support property, and (ii) /(ft) has precisely two orbits in ft, denoted by Og and 02■ By Theorem 2.4, is a total order of the set 0, and <, is a total order Hl 1 —£2 0, 0, -26- 27 of the set Op To describe the po-sets ft which satisfy our hypotheses, we must therefore examine the possible order relations which can exist between points of 0^ and points of 0^. By Corollary 2.6, the ¿-group >4(ft) will be an ¿-subgroup of j4(0p x ^(Op . As before, let denote the collection of elements of >4(0 ) obtained by projecting the elements of AW) onto their first coordinates and let P? denote the collection of elements of AW>2) obtained by projecting the elements of AW) onto their second coordinates. Then P is an ¿-subgroup of >4(0p, P2 is an ¿-subgroup of j4(0p , and AW) P^ x ^2- £o describe the form of AW, we therefore must specify P , specify P^, and specify which elements of P may be paired with which elements of P^ in the above direct product. In specifying the ¿-group P , we will need to consider several types of ¿-subgroups of >4(0^). Therefore, it is convenient at this time to make a brief study of several classes of ¿-subgroups of the group of order automorphisms of an arbitrary homogeneous chain. Once this study is made, we will introduce four classes of ¿-subgroups of the direct product j4(0^) x AW>2) ■ These four classes will be the only classes of ¿-groups which can occur as the full group of order automorphisms of a po-set which satisfies the hypotheses of this chapter. After these ¿-groups have been introduced, we will develop some notation and give a catalog of the po-sets and the corresponding ¿-groups AW) which satisfy our hypotheses. This catalog will be somewhat general, and the reader will find the explicit form of the po-sets ft and the ¿-groups A(ft) in the classification theorems which follow. 28 Therefore, suppose S is an arbitrary homogeneous chain, /(S) is the full group of order automorphisms of S, and S is the Dedekind completion of S. By Theorem 2.2, /(S) can be ¿-embedded in /(S) . Identify /(S) with its image in /(S) . Let T be any subset of an orbit of /(S) in S, and set /(S,T) = {f e /(S) | Tf = T). Then /(S,T) is clearly an ¿-subgroup of /(S). To illustrate this construction consider /(Q), the group of order automorphisms of the rational chain. K, the ordered set of irrationals, is a subset of an orbit of /((Q) in Q = R. In fact, /-((I}) has precisely two orbits in R (namely Q and H ) , and /-(Q, E ) = /-(Q) . However, if we let Q’ir denote the rational multiples of ir, /-(Q,Q’ir) /(Q) . In fact, /(Q,Q-Tr) is a proper transitive ¿-subgroup of /(Q). Next suppose z is any order preserving function from S into S such that for each s e S, (sz11 | n = + 1,+2,+3,... } is cofinal in S. Set & (S) = {f e/(S)[ for each s e S, s(zf) = s(fz)}. PROPOSITION 3.1. is an /-subgroup of A(S). Proof: Let s e S and f,g e (S) . Then s(zfg) = (sf)(zg) = s(fgz), so V (S) is closedunder products. Next, let s e S and f e & (S). We show sf ^z = szf \ s(f ^f)z = sz. Thus sf_1zf = sz. f_1 is a well defined permutation of S so (sf Tzf)f 1 = szf 1, i.e., sf z = szf 1. It now follows that since z preserves order, s(f v e)z = max(sfz,sz). sz(f v e) = szf v sz = max(szf,sz) = max(sfz,sz). Thus z(f v e) = (f v e)z, and f (S) is an ¿-subgroup of /(S). // ¿P (S) will be called the periodic ¿-subgroup of ACS) with period z. In [9], Stephen H. McCleary studied a more restricted notion of periodic ¿-subgroups of /(S). In addition to our hypotheses, he required that z be one-to-one and Sz should be dense in S, i.e., z could be extended to an order preserving permutation of S. These periodic ¿-groups played a fundamental role in the structure theory developed for transitive ¿-groups of order-preserving automorphisms of chains. The standard example of this type of periodic ¿-group is the collection of all order preserving permutations of the real line which commute with translation by 1. Now, generalizing the notation previously developed, for T a subset of an orbit of /(S) in S, set ¿P (S,T) = (f e /(S) | for each s e S, sfz = szf; Tf = T}. Again, it is easy to see that ¿P (S,T) is an ¿-subgroup of /(S) . The final class of ¿-subgroups of /(S) needed for the classifi cation is a class of ¿-subgroups of the ¿-groups If (S,T) discussed in the preceding paragraph. The restrictions necessary to define this class are technical, and are discussed in Theorem 3.24. We will not list the technicalities here, but will denote these further restrictions of zPz(S) by 7/z(S,T). We now list the four types of ¿-subgroups of /(Og) x /(02) which we claim are the only ¿-groups which can be the full group of 30 order automorphisms of a po-set ft satisfying the hypotheses of this chapter. Type £. /(Op x /(02) Type II. The assumptions needed for the description of this type are as follows: (a) S and S are homogeneous chains such that 1 3 1 1 3 Z °1 ’ Sl.l ' S1.2' (b) S2 i and S2 are homogeneous chains such that °2 = S2,l X S2,2’ (c) Tj 2 is a subset of an orbit of /(S^ p dri 2, and /(S, ?,T ) is transitive on S . 1 3 Z 1 3 Z 1 3 Z (d) T2 2 is a subset of an orbit of /(S2 2) in S2 2, and /(S2 2,T2 2) is transitive on S2 2- (e) There is an ¿-isomorphism * : /(S^ 2’Pl 2^ ^^^2 2’£2 2^' Then, the ¿-subgroup of /(Op x/(op we define is U(p;f),(p;f*)) E [(/(SbpWrASb2,Tb2)] x [/(S2, pwr/(s2 j2,t2j2)] }. Type III. The assumptions needed for the description of this type are as follows: (a) S , S and S, _ are homogeneous chains such that X 3 X X 3 Z X j D °1 = Sl,l X Sl,2 X Sl,3‘ (b) S9 , S , and S are homogeneous- chains such that / 3 X / 3 C~ C ) o °2 = S2,l X S2,2 X S2,3' (c) z, is fixed, T9 is a subset of an orbit of /(S X 1 3 Z X 3 Z in S T 2, and ¿P(S^ 2,T1 p is transitive on ST 2- 31 (d) z2 is fixed, T2 2 is a subset of an orbit of >4(S2 p in S2 2, and p^ (S2 pT2 is transitive on S2 p (e) There exists an order-isomorphism y : 3 G2 31 an ¿-isomorphism y : ,4(S2 p -> AW2 3) > and an ¿-isomorphism : ZPz1(S1,2’T1,2') Then, the ¿-subgroup of ,4(0p x >4(02) we define is {((‘I’pli'pg), W2W2;gy)) E PHSppWr^ (SppTx>2)Wr>i(S1 x W(S2;1)Wrp(S2>2,T2_2)Wr^S2_3)] | for every e S^, if ’PlC5! 3) = f then ^2^1 3X^ = Type IV. Repeat the description of type III replacing every occurrence of fi (S ,T by X (S _,T ) and replacing every Z J. , 4 1 j 4 Z -1,4 X , 4 occurrence of (S2 2,T2 p by Xz (s2 2’T2 2^‘ We now wish to give a catalog of the po-sets ft satisfying our hypotheses and the corresponding ¿-groups AW)- As noted previously, ft = 0^ 0 02 (the disjoint union of 0^ and Op is a total 0, order of the set Op and <: is a total order of the set 0, Therefore, to describe ft, we must examine the possible order relations which can exist between points of 0^ and points of Op To do this, we will make use of the following notation. For each Sq e Op let (Sq,+°°) = (s e 01 I sQ [sQ,+°°) = {s e 0J | SQ Also, for each tQ e Op define (tQ,+°°), [tp+oy), (-°°,t0), and 32 (-°°,t0]. For each sQ e Og, set Ug 2(sQ) = (t e 0? | sQ Lg 2(sQ) = e °2 J t’ <& sQ}, and for each tQ e 0^, set (tn) = {s e Og | tn segment of 02 and that Lg 2(s) is a (possibly empty) lower segment of 02- Similarly, for each t e 02> U2 g(t) and L2 g(t) are (possibly empty) upper and lower segments of Og. 1,2 Using this notation, we define relations £g 2 and £ on 2 1 Og and £2 g and £ ’ on 02- These relations are the crucial ones for purposes of the classification. For sa,s e saY s £n ~ s if and only if L o(s ) = Lq „(s ) (i.e., s and s_ a 1,2 0 7 1,2 a 1,28 a 8 1 2 have the same collection of lower bounds in 02), and say s^ £ ’ s^ if and only if U 7(s ) = U „ (s ) . Similarly define relations £4 1 3 Z Ct 1 3 Z p Z3I and £2,1 on 0Q. 7 2 PROPOSITION 3.2. £ J and £g g are convex P^-congruences on 2 1 the chain 0 . £ 3 and G. . are convex P .-congruences on the chain 1 2) 1 2 °2' 1 2 Proof: We check that £ ’ is a convex Pg-congruence on Og. The proofs for the other relations are similar. It is clear that 1 2 £ ’ is an equivalence relation on Og. Next, note that if ft is any po-set, a e ft, g e/(ft) , and U(ct) is the upper bounds of a in ft, then the upper bounds of ag are exactly U(a)g. To check the congruence property, suppose Sg £ ’ s2 and U(s^) is the collection of upper bounds of s^. Then U(Sg) A 02 = U(s2) A 02- Let f e Pg. Then there exists h in /(ft) such that hy = f. 33 Thus s^h = s^f and the upper bounds of s^f are U(sph. Now, since 02h = Op UCs^h A O2 = (U(sp A O^h = (U(s2) A O^h = U(s2)h A Op Hence s^f G 12’ s^f and G112 ’ is a congruence on Op 1 2 The fact that the classes of G ’ are convex follows from the 1 2 transitivity of the order relation on ft. Therefore G ’ is a convex P^-congruence on it . // We now define two other congruences on 0^ and on Op Let 26 denote the convex Ppcongruence on 0^ defined by s2^2 S2 £or every SpS2 e G2 (the universal congruence). Let denote the convex Ppcongruence on 0^ defined by: for SpS2 e si^l S2 if and only if s^ = s2 (the trivial congruence). Similarly define 2Z2 and £2 on Op By Theorem 2.9, the convex Ppcongruences on 0^ and the convex Ppcongruences on 02 form a totally ordered set when ordered by inclusion. In forming our catalog, we must therefore examine all possible containment relationships between the congruences 2, £1,2, and 2/^ on 0^ and £p <*2’1, and °n °2' These congruences are the natural ones to consider when making our catalog, since they complete describe the order relations which can exist between the points of 0^ and the points of Op The listing of all possible containment relationships between the congruences on 0^ and the congruences on 02 can be simplified by noting the following facts. Obviously, C 2/p C £1,2C 2/p <€2 — ^2 and $2 — £2,1 — &2’ Next’ either tfpCif’2 or ¿’2Ci12, and either Z^C^2’1 or 34 Also, the following proposition will simplify this procedure. PROPOSITION 3.3. (a) = %2 on 0^ if and only if 1 = ^2 on 0%. Cb) G g = ^2 on °2 and °n^y = ^2 °n °2' 1 2 Proof: IVe show £ ’ = implies tp j = ¿læ veTificati°n needed for the remainder of the proof is similar to this argument. If ip 2(s) ~ 0 for every s e 0^ then clearly L? (t) = 0 for every t e Cp. Then, for any t^,p e Cp, t^£2 1 £2' Thus S2,l’U2 °n °2‘ Therefore suppose ip 2(s) / 0 for each s e 02. Fix Sg e Cp and choose tg e U, Q(sn). We show that for each t e 07, 1,2^0 teip 2(sg)• Choose any t e 0^. Since /(ft) is transitive on Cp 1 2 there exists (f,g) e /(ft) such that tg(f,g) = t. £ ’ - so tg e ip 2(sQf (f’g) is order preserving on ft so sQf 1 < tQ implies Sg < tgg = t. Thus t e ip 2(sQ), i-e., ip 2(sQ) = °2 But ip 2(Sg) = ip 2(s) for.each s e Cp, i.e., each point of 02 is above each point of Cp. Then, obviously, 2(t) = Cp for each t e T. Hence (?2 = %2 on Cp’ If Since we are describing partially ordered sets in this classifica- :ion, it will be convenient to have a method of pictorially describing hese sets. The method will be described by a sequence of examples, he picture represents a partially ordered set which is the union of the two chains Cp and Cp. No order relations exist between 35 the chains Og and Cp. The picture indicates that points of Og have no upper bounds in 02 but only lower bounds, and points of only have upper bounds in Og and no lower bounds. 2,1 Moreover, £ = £ and £ The picture 1 3 Z 1 indicates that points of Og have both upper and lower bounds in 02, and points of 0^ have both upper and lower bounds in Og. Moreover, £ = £1’2 = £ and £ = £2’2 = £ . We also need a method for _1_ 3 / R, ) A. L showing when some of the congruences are not trivial but not universal. will indicate £g £1,2^2Yg £2 £ £2,1<7 ^2 resPectively- t will indicate £2 £ £2 i^-^2 and ¿1 ^1 2^^1 resPectively- We now make the catalog of results for the two orbit case. This is conveniently divided into six categories and we will use the four classes of ¿-subgroups of /(Og) x A(0p introduced previously. CATEGORY 1 : £g (pL £g 2 = C1 ’2 = ¿¿g on Og . By Proposition 3.3, this implies £2^ ^2 1 ’ = ^2 On G2‘ Here, /(ft) will be of type I. 36 1 2 CATEGORY 2: One of GI, f /7, and G ’ is the universal congruence on 0^ and one is not. By Proposition 3.3, this category divides into two cases: (a) and Ç G2 ’1 <= = (b) £2Ç ¿’2££1j2 and £2Ç £2a£ U2 - To examine this category, we only need consider case (a) since case (b) will be case (a) if we rename the chains 0^ and Op In both cases, AW) will be of type II. For (a) to hold, one can easily see that bl 2^ = $ £or ea<~h s e Op For illustration, assume £2 1 2 the universal congruence on 02> Here, we assume G 2 = £ ’ on Gi 2 1 and 1 = ’ on G2' We agaan have tw0 cases: (a) £1QG1 2 = GL,2^?li and for each s e Op inf U2 2(s) = sup L1 2(s) S2 — G2 2 = and for each t e Op inf U2 2(t) = sup L2 1(t). This will follow as a corollary to category 2 and AW) will be of type II. (b) Q. 2 = G1,2 2^ and for each s e Op 37 inf Ug 2(s) £ sup Lg 2(s). £2 C C2 g = %-2 and for each t e Cy, inf iy g(t) £ sup L2 g(t). Here /(ft) will be of type III, Zg will be an order preserving permutation of Sg 2, and z2 will be an order preserving permutation of $2 2- IT we assume £g = ^g 2 and ^2 = ^2 1’ we may draw ¿be -toi rel'b 4ft -Jo 1,2 CATEGORY 4: Both £ and £ are not the universal congruence. 1,2 2 1 Then, by Proposition 3.3, both £ ’ and £ are not the universal Z 3 1 congruence on 02- We assume here that on one chain the two congruences listed above are equal, whereas on the other, one of these refines the other. There are 4 cases • (a) 4 Ç S,2 £ c1’2 £ \ and 4 c ^2 4 9 1 c £2’^ (b) 4 C ^1,2 = and 4 4 9 c Z2’1^ (c) 4 Ç ^1,2^1 and 4 4, 1 Ç = £ 1 ’2$^ c 4 £ 2 (d) «1 %,2 and 4 2,1^ To examine this category, we need only consider (a) since (c) and (d) are dual to (a) and (b), and (b) becomes (a) if we rename the chains Og and 02> In all four cases, /(ft) is of type III. In (a), if we 38 draw the following picture Here, every point of a convex piece of 0^ has the same collection of upper bounds in Og, whereas each point of Og has a different collection of lower bounds in 0^. CATEGORY 5: £g g / 2/g, £1,2 £ Ip, g £ and £2,1 £ Zp. On both chains one of the two congruences refines the other, but the refinement is done so the chains are not symmetric. There are two cases: (a) yC £g,2£ ¿’2£#g and £2 C 1 £ zy, x . (b) £g£ ?’2£ £^2$^ and £2^ £2j1$ £2,1$ ^f2. To examine this category, we need only consider (a) since (b) becomes (a) if we rename the chains Og and 02> Here, /(ft) is of type III. 2 1 In (a), if we let <^g = ^g 2 and ~ ’ ¿ben we may draw 39 CATEGORY 6: ? £1,2 / T and z?2’1 / In both chains one of the two congruences refines the other, and the refinement is done so the chains are symmetric. There are two cases: (a) 2^£1,2^.^1 and GZ ^G2,1^^^. (b) and (b) is dual to (a), so we only examine (a). In both cases, AW) is of type IV. In (a), if we let = G± 2 and <*2 = 1’ we may draw tPe Thus, we need to prove six major theorems to perform the classifi cation. Before proving the first theorem, some technical notions must be discussed. We make the following conventions. Suppose for some s^ e 0^, U1 2 (S(P = °2* Then We Wil1 Say in£ U1 2^S0^ = ”°°' SuPPose taat for some s2 e C^, ^s^ = 0- Then we will saY in£ Uj 2*-sp = +°°- Next suppose that for some s2 e 0^, 2(s2) = G2‘ Then say sup 2^s2^ = +°°’ FinallY suppose that for some s$ e 0^, 40 bl 2^S3^ = H*611 let SUP lq 2^S3^ = ~m‘ Mabe similar conventions for the points of 0^. We order the set Cp U {-oo5+oo} (or Cp U {-“,+“}) in the obvious way: is less than each point of Cp and +°° is greater than each point of Cp. The ordered sets ______. rJ sj Cp U {-°°,+00} and 02 U {-°°}+oo} will be denoted by Cp and 0^ respectively. Keeping these conventions in mind, we now define maps between the chains Cp and 0^ which will prove useful in our classification. 12 ~ Define 2 and * ’ ^rom Ip to Cp via for each s e Cp, ’ 1 2 Sxl 2 = SUp L1 2^S^ and SX ’ = in£ U1 2^' Similarly define x2 j 2 1 ~ and x ’ from Cp to Cp via for each t e Cp, tXg j = SUP L2 j(t. and tx2,1 = inf dp .j/t). PROPOSITION 3.4. For e 0 , (f) si f-o S2 implies . 7 . 1,2 o 1,2 StX-j o S 9X-7 9J an Proof: Si <£ s2 implies 2(sp £ p 2(s2). This implies 5UP L1,2(SP ^O2 SUP L1,2(S2}’ i‘e” Slxl,2 ±O2 S2*l,2- Similar iroofs may be given for the other maps. // We may use these maps to give an alternative definition of the , A’2 PROPOSITION 3,5. For srs2 e °r (i) si^i 2 s2 only (ii) s~ çl’Z s2 if and only if tf s2^ij2 S2*l,2', and 41 7,2 2,2 yx s„x ' • Ws may make similar statements for each t ,t9 e 0 . Proof: We will verify (i). To show (i), we must show L1 2^S1^ = L1 2^S2^ if Snd °nly if SUP L1 2^S1^ = sup L1 • Tt is clear that Lg g(Sg) = Li 2^S2^ implies that sup Lg g(Sg) = sup Li 2^S2^ For the converse, suppose that Sg,s2 e and that sup L1>2(sp = sup L1>2(s2) We show Lg g(Sg) = 2^2)' If sup Lg 2(Sg) = sup Lg 2(s2) then Lg (Sg) L1,2^S2^ ~ If sup Lg 2(Sg) = sup Lg 2Cs2) = +c°, then Lx 2(s1) L1,2('S2^ = °2' Therefore suppose sup Lg 2(Sg) “ SUP g(Sg) ds a Poant oP G2- There are three possibilities for sup Lg g(Sg) (and sup Lg 2(Sg)): (a) sup Lg g(Sg) e °2 and S1 covers SUP 2(sj) dn (b) sup Lg g(Sg) £ G2 and SUp L1 2^S1^ dS tbe smallest point of Og which is not a lower bound of Sg in ft, (c) sup Lg 2(s|) e G2^G2 and S1 covers n0 points of 0? in ft. (a) holds for Sg if and only if (a) holds for s? since Sg can be mapped to s? by an order preserving permutation of ft. Similarly for (b) and (c) . Suppose Lg g(sj) / Lg g(Sg). Since Lg 2(Sg) and Lg g(Sg) are both proper lower segments of Og and sup Lg g(sj) = SUP Lg 2(s2), it must be the case that either (i) sup Lg g(Sg) £ Lg g(Sg) but sup Lg g(Sg) e Lg g(Sg), or (ii) sup Lg 2(Sg) e Lg g(Sg), but sup Lg g(Sg) / Lg 2(Sg). Neither of these can hold true by the above remarks. Thus Lg g(Sg) = Lg 2(s2)- // The form of the six classification theorems will be as follows. In all theorems we initially assume that ft is a po-set such that 42 (i) v4(ft) is an ¿-group with the disjoint support property, and (ii) ,4(ft) has precisely two orbits in ft, 0^ and C^. We also have some initial assumptions about the congruence structure on the chains 0^ and C^. From these assumptions we deduce properties of 12 2 1 the maps X2 2’ X ’ , x2 p and X ’ , and properties of the totally ordered sets 0^ and C^- We then explicitly describe the ¿-group >}(ft). To show that these results do indeed give a classification, we assume we have chains 0^ and 0^ and maps 12 2 1 y , y ’ , y. > and y ’ which have the properties previously 1 j 4 4,1 deduced. We then construct a partially ordered set ft such that (i) -4(ft) is an ¿-group with the disjoint support property, (ii) the orbits of j4(ft) in ft are precisely 0^ and O?, (iii) the congruence structures on the chains 0^ and 0^ are the same as those initially assumed, and (iv) the group AW) is of the same form as the one described in the converse. In the statement of each theorem, we make the following convention. Instead of repeating each of the hypotheses listed above, we will simply say, "Suppose ft is a po-set which satisfies our hypotheses." We now give our first classification theorem. THEOREM 3.6. Suppose ft is a po-set which satisfies oun hypotheses xnd suppose G2j2 = G^ 2 = On anG' = 2 I = ^2 °n G2’ hen one of the following three sets of conditions holds: 1 2 (i) y7 is a map from O to {-=), y 3 is a map from 0 2 j Z 1 2 to {■/-“}, 2 1 y is a map from 0 to and y 3 is a map from 43 02 to {A°°h 0^ and 0% are homogeneous chains such that 0^ is not order-isomorphic to 0%. 1 2 (ii) X2 2 ds a map from 0? to {+°°}j x 3 is a map from 0^ to {A“}, 2 1 X is a map from 0 to and x 3 is a map 1 c> from 0 2 to { -°°}. 0 and 09 are homogeneous chains and there exists no 1 partitions {A^B^} of 0^ and {A2,B2} of 02 such that A2 < Br A2 isomorphic to 0^. 1 2 ( iii) X7 9 is a map from 01 to x 3 is a map from 0^ J j Ci J- to 2 1 X2 2 ds a map from 02 to and x 3 is a map from C>2 to {A°°}. 0 and 09 are homogeneous chains and there exists no partitions of 0, and {A2,B2} of 02 such that .4 <„ B7, A < B9, Aq is order-isomorphic to J- Lu J Ci Lu di J- On, B is order-isomorphic to A9, and B is order- isomorphic to 02. In all three Cases, .4(0.) = -4(0^) *A(02). 1 2 Conversely, given chains 0, and 0%, maps x2 g and X — 2 1 ~ from 02 to 0%, and maps Xg i and X J from 02 to 0^ satisfying either (i), (ii), or (iii), we may construct a partially 44 ordered set ft satisfying our hypotheses such that (2) 61*2 = and C2’1 = £% i = 2b2, and (2) A (Cl) = A(0 ) xA(Og). Proof: By examining the proof of Proposition 3.3, we see that 1 2 £ ’ = 2Zg on 0^ implies that for each s c Og, Ug g(s) “ Og or Ug g(s) = 0. If Ug g(s) = Og for each s, then it must be the case that Lg g(s) = 0 for every s e Og. It also follows that for each t e Og, Ug g(t) = 0 and Lg g(t) = Og. Therefore assume that for each s e Og, Ug = 0, i.e., each s e Og has no upper bounds in the chain Og. In a proof similar to the one in Proposition 3.3, we show that for each s e Og, either L1 2('S) = °r L1 2^S) = °2‘ If botb L1 2^S-* = and U1 2^S^ = then for each t e Og, Lg g(t) = 0 and Ug g(t) = 0. If Lg g(s) = Og and Ug g(s) = 0, then for each t e Og, Lg g(t) = 0 and U (t) = 0r Thus far, we have shown that there are one of three possibilities 1 2 for the structure of the po-set ft. We now compute the maps X ’ , 2 1 Xg g’ X ’ , and Xg g for each case. First suppose that for each s e Og, Lg g(s) = 0 and bl 2^S-) = and tbat ¿or eacb F e Og, Lg g (t) = 0 and Ug g (t) = 0. 1 2 Then, by definition of the maps, X ’ maps Og to {+“}, Xg g maps 2 1 Og to {-“}, X ’ maps Og to {+“}, and Xg g maps Og to Next suppose that for each s e 0 , L (s) Og and 1 1 , z Ug 2^S-* = and tbat ¿or eacb t e Og, Lg g(t) 0 and 45 1 2 U„ (t) = 0. Then, by definition of the maps, X ’ and X map 9 ± 1 9 1 2 1 0^ to {-«>}, and X ’ and X2 2 map C>2 to {+«>}. The added conditions stated in each case of the theorem follow from the fact that 0^ and C>2 are orbits of AW) in ft. As noted previously,—is always- an-¿-subgroup of >4(0^4—x _4(02) A . straightforward case argument shows that each element (f,g) of >4(0p x j4(O2) is order preserving on the po-sets described above. Thus, AW) =^4(O2) x AW2') • 1 2 Conversely, suppose we have chains 0^ and C>2 and maps X ’ , 2 1 2, X ’ , and X2 satisfying the hypotheses of either (i), (ii), or (iii). We must construct a po-set ft satisfying our conditions. For each s e 0 , the proper upper bounds of s in ft are all 1 2 points in (s,+°=) C 0^ and all t e 02 such that sX ’ Gq The proper lower bounds of s in ft are all points in (-°°,s) C o and all t e 0 such that t of each t e 02 are described in a similar manner. By using this definition, the sets constructed are as follows. In (i), each point of 0 will be unrelated to each point of 02_ In (ii), each point of 0^ is above each point of C>2. In (iii), each point of Oj is below each point of 02> We must check that these sets have the desired conditions. The verification is for (i). The others are similar. Therefore suppose ft is a po-set such that ft = 0 O 02 0^ and C>2 are homogeneous chains, 0 is not order-isomorphic to C>2, and each point of 0^ is unrelated to each point of 02- We first check that 0^ and C>2 are blocks for j4(ft) in ft. Let f e >4(fi) 46 and suppose there exists an Sg e Cp such that Sgf e Cp. f is order preserving on ft, so points that are unrelated to Sg must have their images under f be unrelated to Sgf. Therefore, f maps each point of Cp to a point of Cp. Similarly, points that are related to Sq must have their images under f related to Sgf. Thus f maps each point of Cp to a point of Cp. The above argument shows each element of /(ft) either fixes the chains Cp and Cp, or exchanges the chains Cp and Cp. Hence Cp and Cp are blocks for /(ft) in ft. Note that Cp is a homogeneous chain and that each point of Cp has the same collection of upper and lower bounds in Cp. A similar remark also holds true for the chain Cp. Thus /(ft) is transitive on Cp and Cp. The functions in /(ft) cannot map the chain Cp to th e chain Cp since Cp is not order-isomorphic to Cp. Hence and Cp are the orbits of /(ft) in ft. Every orbit of /(ft) in ft is a chain. By Theorem 2.4, /(ft) is an ¿-group with the disjoint support property and /(ft) is an ¿-subgroup of /(0 ) x/(Cp). Define the convex -congruences 2 12 2 1 and G ’ on Cp and the convex P2-congruences and £ ’ on Cp. By the definition.of the partial order on ft, it is clear that. 2 = C1’2 = on (p and } = £2,1 & on Cp. Then, using the proof of the converse, /(ft) = /(Cp) x /-(Cp) // Before proceeding further, we introduce one definition which will simplify the statements of the remaining theorems of this chapter. In Theorem 2.4 we have characterized those partially ordered sets whose 47 groups of order automorphisms are ¿-groups enjoying the disjoint support property. In the work done at the beginning of this chapter, we have given further insight into the geometry of the sets ft by I 2 introducing the convex P^-congruences 2 and G 3 and the convex 2 1 P2-congruences G 2 2 and £ ’ • Primarily, the theorems in this chapter tell how to construct order relations between the points of the chains 0 and 0^ in order to obtain a particular congruence structure, and they also give the form of the ¿-group of order-auto morphisms of the resulting po-set ft. A technical problem in each category is guaranteeing that the orbits of AW in ft are 0^ and 02> so that A(ft) is indeed an ¿-group with the disjoint support property. To guarantee this, two problems must be overcome. First, we must prevent points of 0^ being mapped to points of 0^ and points of 02 being mapped to points of 0^ by elements of -4(ft). Second, we must check that the resulting ¿-group is transitive on the points of 0^ and transitive on the points of C>2. In Theorem 3.6, we dealt with the first problem by placing some rather awkward conditions on the chains 0^ and C^. Under different hypotheses, this problem can be handled in another way. For example, suppose x2 2 : °1 °2 ’ x2 1 : °2 °1 ’ and °1 and °2 are dense chains. To guarantee the transitivity of >4(ft) on 0 and C^, it must be the case that (i) °2X2 2 ~ °2 °r °lxl 2 = and (ii) O2x2 j = or °2X2 1 °1 = SuPPose that 0 Xi t = 0 and ^9X9 = 0 . We wish to specify the collection of lower bounds of each point of 0^ and each point of C>2. There is a choice. For each s^ e 0^, we may define 2 (sq) = {t e C>2 | t < s^Xj 2- 48 or Lg g(s^) = E °2 I ¿5 sp Since it is not the purpose of the theorems in this chapter to give equivalent conditions to those of Theorem 2.4, we make the following definition to avoid stating theorems which have an abundance of technical conditions. We will say that the homogeneous chains Og 12 2 1 and Og and the maps Xj 2’ * ’ ’ *21’ and * ’ are acceP'l'able if we can define a partial order <_ on the set ft = Og (J Og such that (i) qOl - y “d i|o2 ■ y i 1 2 (ii) For each sQ e Og, either Ug g(sQ) = (t c Og |sQx ’ Ll,2(s(P = {t’ e °2 I V ^Og S0xl,2} °r Ll,2(s(P = e °2 I "ôg S0Xl,2^’ 2 1 (iii) For each tQ e Og, either Ug g(tQ) = (s cO g | tQx ’ fy, s) or Ug g(tQ) = (s e Og ( t0x231 L2,l('t0') = {S' e °i ! s ' t0x2,l} °r L2,lfto’ ’ {S' E °1 I s' y '0*2,?• (iv) The choices in (ii) and (iii) can be made in such a way that 49 the full order automorphism group of the resulting partially ordered set has Cp and Cp as its collection of orbits. After the proof of each remaining theorem, we will give specific conditions to guarantee the choice in (iv) can be made in the cases which are usually encountered in constructing examples. We now present a sequence of lemmas which will be used to prove the remaining classification theorems. LEMMA 3.7. (i) G2’1 = implies that 0 x7 9 is a dense subset of . J J j z J 2 1,2 (U) G2,2 - G2 implies that CLx is a dense subset of 0. 1 2 ( iii) G 3 = G2 implies that 0 2*2 1 rs a dense subset of 0^. 2 1. — (iv) Cli2=f implies that Oa 3 is a dense subset of Cd. Proof: We prove (i). The remaining proofs are similar. First 2 1 note that £ ’ = £? implies that for each sQ e Cp, sQx1 2 +°° or s x, 9 i For if this was the case, by the transitivity of /(ft) on Cp we would have that for each s e Cp, L^ 2(s) ~ *p or L1 2*~S) ~ 0' Tben, for each t e Cp, either ip ^(t) = 0 or 2,1 U2 l(t) = °1’ i,e‘5 ’ & , a contradiction. Thus 2 : Cp Cp. For proof by contradiction, we now suppose that Glx(iv)l *2 * *d s* *n Ot dense in °2' Tben there exists t1,t2 e 02, t^ < t2, such that for no s e Cp is t^ <_ sup L^ 2(s) <_ t^. Since Cp is dense in 0^, we may choose points t^t^ e Cp such that t^ <_ t$ < t^ p t?. There exists no s e Cp such that t^ <_ sy^ 2 £ t2, so there also exists no s e Cp such that t^ £ sup L^ 2(s) £ t^. 50 t < t and £ is a transitive relation on ft imply that 2,1 i2 on 02 so U2jl(t3) / U2jl(t4). Thus Ug |(t4) £ U? ^(t^). Choose s g e U? such that sn i U. Ht.). Then t_ <_ sn but not t. < sn, i.e., t_ e L .(s.) 2,1 4 '3 ft 0 4 ft O’ 1,2 0J bui t4 i L1j2(s0) Thus t^ £ sup 2^S0^ — P4‘ This is a contra diction. Therefore 2 £s dense £n ^2' // LEMMA 3,8. (i) Gn = £implies that y 9 is a one-to-one map from 0 1,2 1,2 to 0 . 12 . 12 (ii) G 3 - $2 implies that y 3 is a one-to-one map from 0^ to Og. (iii) -, = £9 implies that y is a one-to-one map from 0c 2,1 2,1 to 0 . 2,1 2 1 ( iv) G Cg implies that y 3 is a one-to-one map from Oc to 0^. Proof: We prove (i) . Suppose s]ps2 e Oq and sq / s?. We show 1 1 slxl,2 S2X1,2- If S1X1,2 = S2X1,2’ then’ by Proposition 3.5, Sl^l 2S2’ But S1 S2 and 1 2 = ^1‘ This is a contradiction. is one-to-one. Thus Xl,2 // LEMMA 3.9. (i) If G2 g = and G2*1 = %23 then (a) y^ g can be extended to an order-isomorphism y^ g from , 0^ to Og, 2 1 . —2 1 (b) y 3 can be extended to an order-isomorphism y 3 from Og to 0^, and (a) (y-L^2) 1 = x2j2. 51 (ii) If C1,2, = p and £g i = £g, then (a) y2,2 can be —12 — — extended to an order-isomorphism x 3 from 0^ to Og, (b) can he extended to an order-isomorphism x_ x¿9j- ? J7. O) J from Og to Oj, and (c) (x2,2) 2 = x% Proof: We prove (i). Xg 2 ds order preserving by Proposition 3.4. By Lemma 3.8, £g g = £g imPlies ¿bat x 2 is a one-to-one map from Og to Og. Og is dense in Og. By Lemma 3.7, £ ’ = Qg implies that OgXg g is dense in Og. Thus Xj 2 is a one-to-one order preserving map from a dense subset of Og to a dense subset of 0 . x „ can be extended to an order-isomorphism x : 0 0 Z J. 3 Z J. 3 Z J. Z via: for each s e 0 , sx „ = sup{s Xl 9 | s £5 s}. Similarly, 2 1 —7 1 — — - ’ can be extended to an order-isomorphism x ’ • 027 Og. -1 •2,1 To complete the proof, we must show (Xj 2) X By way of — -1 —2 1 contradiction, suppose that (Xj 2) / X ’ • Then, since Og is dense in Og, there exists s® e Og and t^ e Og such that -2’1 t ?0. Set = t/’1. Since Oj is a S0xl,2 t0 but t0x chain, either Sg sQ or sQ Sg. First assume Sg sQ. If 1 1 sn does not cover Sg in 07, then there exists s e 07 such that 0 inf{s e 0 t is one-to-one and order preserving so bo "ft s- s Og sn0 and X 1,2 , i.e., sup{t e 0 j t < s} <- t . This means Sxl,2 S0xl,2 '3 8 ft Og 0 that tg s is not true, a contradiction. Thus s^ must cover Sg in Og. If this is the case, both s^ and Sg are elements of Og. For similar reasons to the ones above, t^ s^. Also, Xj 2 is 52 one-to-one and order preserving so 2 Thus, no such s^ and tQ exist, and we conclude that (xl,2)_1 = *2,1' 11 The following lemma is useful in constructing examples. Recall that for f e /(Cp) and g e /(Cp), f and g denote the extensions of f and g to the points of Cp and Cp respectively. LEMMA 3.10. (i) If C f , {g 1 g e A(O )} is transitive on the set 12 ¿j 1 Z °1*1, 2' (ii) If £ 3 / {g I g e ~4(O%)} is transitive on the set o,!2- (iii) If £ / ¿¿9, {f [ f sA(0 )} is transitive on the set Zj 7 Z 7 0 2*2,1 n -i _ (iv) If / 2L {f | f e d(O^)} is transitive on the set o 2^ 0 2* ' Proof: We prove (i) . p 2 implies that Cpxj 2Cp. Let e (Lx, 9- We must show that there exists age /(0?) 1)2 1 J. j / such that tpg = tp. ?1,t’2 e °lxl 2 S° there exist si’s2 E °1 such that ?1 = sup L} ^s^ and t2 = sup L1 2(s2). Cp is an orbit of /-(ft) in ft so there exists (f,g) e /(ft) such that s1(f,g) = s f = s?. (f,g) is order preserving on ft and s1(f,g) = s2 53 implies that SjXj 2(f,g) * S2X1 2’ i-e-’ ^pS = l2‘ g e so {g | g e /(CL)} is transitive on the set 0 x 1A1,2' // Frequently, if no ambiguities will arise, we will identify /(Op with its image in /(op under the ¿-embedding described in Theorem 2.2. If this identification is made, Lemma 3.10 (i) says that O^x^ 2 is a subset of an orbit of /(Cp) in 0^. The following lemma due to W. C. Holland will be used to compute the groups /(ft) . LEMMA 3.11 [7]. If x ■ S T is an order-isomorphism from the chain S onto the chain T, then * :^r(S) -+dr(T) is an /.-isomorphism, where for f c -A(S) and t e T, tf* - t(\ ^fx). // Suppose now that p : Cp -> Cp is order preserving. p induces a convex equivalence relation (called the kernel of p) denoted by Ker p via: for sps2 e Oj, Ker^ s2 if and only if s^p = s2p . OpKer p can be totally ordered by Theorem 2.10. Similarly, if it is any order preserving map from Cp to 0 , Ker n is a convex equivalence relation on 0^ and Cp/Ker n can be totally ordered. We will use the following notation. If ir is any order preserving map from Cp to Cp (i) define rp 2 : Cp/Ker Xj 2 [Cp/Ker ir] by: for s0 Ker Xj 2 e Cp/Ker Xj 2, (sQ Ker Xj 2)ip 2 = sup{t Ker ir e Cp/Ker ir | t s^ 2>, and 1,2 1,2 (ii) define n ’ : Cp/Ker x ’ [Cp/Ker it] by: for Sq Ker x1’•2 e OC2p//lK er x1,2 1,2 1,2 1,2 (sQ Ker x ’ )h inf{t Ker ir e Cp/Ker ir | s^x J t}. Similarly, 54 if p is any order preserving map from 0^ to 0^ (i) define n2 T : O2/Ker x2 2 -* [C^/Ker p] by: for tQ Ker x2 2 £ 02/Ker x? p (tQ Ker x2 1)n2 2 = sup(s Ker p e C^/Ker p | s tQx2 1>, and (ii) define p2’1 : 02/Ker x^’1 •* [O^/Ker p] by: for tQ Ker y2’1 £ O2/Ker y2’\ (tQ Ker x2’1)^2’1 = infis Ker p e O^Ker p | tgX2’1 5q s). LEMMA 3.12. (i) t\2 2 and t\2j2 are order preserving maps from Cf/Xer *1,2 1 2 and Oj/Ker x 3 respectively'to VOg/Ker mJ. 2 1 (ii) rig and p J one order preserving maps from 02/Ker X2J 1 and 0g/'Ker y2, 2 respectively to [0^/Ker p]. Proof: p 7 is well defined by definition of x^ 2 and it is 1,4 easy to check that p is order preserving. Similarly for the other ± 9 maps. // We are now ready to classify the po-sets ft satisfying our 1 2 hypotheses such that — 6^ 2^ C ’ = on Op and 9 1 Gt’ £2 = ^2 on Op We first prove the special case where 2 1 = G^ 2 and <^2 = G, ’ . We then make a classification for the general solution. LEMMA 3.13. Suppose ft is a partially ordered set satisfying 1,2 our hypotheses and suppose = G^ 2%. G = on C>2 and £ = (f‘> ^G =2jL on 09. Then, the following conditions are 4 Ci} 1 & 6 satisfied: 55 2 2 (i) X 3 : °2 an^ *2 1 ; °2 (ii) x2 2 : G1 G2 *S one~t°-one> order preserving, and 0^*2 2 is dense in Og. 2 1—. . 21 X 3 : 02 ^ 01 is one-to-one, order preserving, and OgX 3 is dense in 0^ (iii) X-, the extension of X7 9 to the points of 0 , is an 1, z 1 ) ¿J 1 — — —2 1 order-isomorphism from 02 to 0g. Similarly x 3 is an order-isomorphism from 09 to 0 . Moreover, ¿J 1 -1 -2,3 4/ X (iv) 0. and 0 „ and X- y2,2, X9 7, md X2,2 ar‘e acceptable ¿) 1 Set G2 2 = {f e A(02) | OgX?3 2f = O^3 7 ) and 2 2 \ — G 3 = {g e 4(0g) | O^Xg g£ = °f*l 2^' Then G2 g is a transitive ¿-subgroup of 4(0.) and G9 is a transitive ¿-subgroup of A(09). 1 ¿J) 1 ¿J Moreover, x7 9 induces an ¿-isomorphism *:G 9->G9 and 4 (Cl) = { (f, ft) I f £ G2 g}. Conversely, if we are given chains 02 and Og, maps x2 2 arbd 12 ~ 2 1 ~ X 3 from 02 to Og, and maps x2 j and x 3 from Og to 02 such that (i), (ii), (iii), (iv) are satisfied, then we may construct a po-set ft satisfying our conditions such that (i) £2 = ^2 2 = 2/^ on 02 and £g = G2j2^. 2 2 = 2 °n Og, and (ii) 4(Cl) = {(f,f*) ) f e G2 g} where * is the ¿-isomorphism from G2 g to Gg 2 induced by the mapping x2 2- We may geometrically represent the partially ordered sets which fall in this category by the following picture. 56 1 2 Proooff:: £ ’ = 26g. As in the proof of Theorem 3.6, we may show that for each s e Op either Ug ^(s) = 0 or Ug g(s) = Og. If for each s e Og Ug g(s) = Og, then each point of Og is above each point of Og. Thus Lg g(s) = 0 for each s e Og. This implies that £g g = 26g, a contradiction. Therefore each s e Og has no upper bounds in Og. If for each s e Og, Ug g(s) = 0, then it follows that for each t e Og, Lg (t) = 0. By definition of the maps y ’ and x9 , we now conclude that x ’ : 0 -* {+“} and Z 3 I 1 X2ji : 02 - Conditions (ii) and (iii) follow directly from Proposition 3.4 and Lt.pemmmqaqs 3.7,3.7. 3.8., and 3.9. Condition (f i vv)l is immedediatate from the hvynpothesees We now show that there exists an ¿-isomorphism * : G ? G 1 3 z Z 3 1 and that /(ft) = {(f,f*) | f e Gg g). If this set is indeed /(ft), then Pg = Gg g and ^2 = G2 1’ Since Pg and Pg are transitive ¿-subgroups of /(Og) and /(Og) respectively, the corresponding statements about Gg g and Gg g will hold true. We have noted previously that Xj 2 is an order-isomorphism — -1 —2 1 from Og to Og and tha_ (x „) = X ’ • By Lemma 3.11, we may 1 3 Z define an ¿-isomorphism ■ /(Og) /(Og) where for f e /(Og) and t e Og, tf* = ttXg^gfxg^g) Set 57 G1 2 = {f e /(Op J op = O1 and O^2’^ = Cpy2’1), and set G2 t = {g e/(Op | O2g = O2 and 0^ 2g = 0^ 2) • We claim that if * is restricted to G. * is an ¿-isomorphism from G 7 tc G2,l- Clearly G is an ¿-subgroup of /(0 ). We first show that 1 , Z 1 if f e G^ 2> Op* = G2 and Glxl 2£* ~ Glxl 2' By definition of it is clear that O^x^ 2£* ~ Glxl 2‘ Next, let t e 02. Then t£*= tx;>i,2 = ^2’W’V- tx2,1 e O2x2,1. f e G so tx2,1f e O2x2,1. Thus tx2’1f(x2jlb1 e 02. This shows Op* C cp. Similarly Opf*) 02. Thus Op* - 02- From this argument and the fact that * is an ¿-isomorphism from /(Op to /(Op we conclude that * is an ¿-embedding of G into G . The fact that * 1 3 Z Z 3 X is onto is proved similarly to the fact that * maps G^ 2 to G2 Thus G^ is ¿-isomorphic to G^ by *. Now let G and G be as in the theorem. Since 0^ is 1,2 2,1 dense in 0 , G is ¿-isomorphic to G ? by extension. Similarly, X J. 3 Z X 3 Z G~ is ¿-isomorphic to G_ by restriction. We may thus define an Z 3 X Z 3 X ¿-isomorphism * : G^ 2 -* G2 1 by first extending to G^ 2> mapping by k, and then restricting to G2 We show /(ft) = {(f,f*) | f e G^ 2). Fix (fQ,f0*) e {(f,f*) | f e G1 2). For each s e Cp, let s(fp,f0*) = sfn, and for each t e 0Q, define t(fn,fn*) = tfn*. It 0’ 0 is clear that (fQ,f *) is a permutation of the set ft. To show (fp,f *) e /(ft) , it remains to show (f^ip) is order preserving on ft. Let a,8 e ft with a <_ g. If either a,f$ e Cp, or a, 6 e Cp, 58 then clearly a(f ,f *) <_ £(fg,fg*). Since each point of 0^ has no upper bounds in 0^, the only remaining case is a e 0^, £ e 0 . By definition, By. . = sup L „(£). There are two possibilities for 1 5 Z 1 > Z Bx1,2’ Case d.:- -£ covers--By^£n and BXj 2 £ G2’’ a £-Lj 2W) - S° “Y/V fo*u(°2> 50 “? -6*l,2f0*- S*l,2f0* “ B*l,2*b2f0*l,2 = 6fob,2' By definition of q_2, this implies afQ* e Lj 2(£fQ). Case 2: £y 7 is not related to £ in ft. This case is 1 5 Z similar to the one above. Therefore, in all cases, (fg,f *) is order preserving on ft. Thus {(f,f*) | f £ Gj 2^ — ~4(W • Next let (g,g') e 4(ft) be arbitrary. Clearly g e G^ 2 and g' e G q. We must show g' = g*. By way of contradiction suppose 4,1 that g' / g*. Then GjXj 2 dense an G2 £mpl£es There exists SX1 2 £ °1X1,2 SUCh that SXl,2g' Sxl,2g*- s e °! and Sxl 2 = sup L1 2^' ^>S') is order preserving so (sXl 2)g’ = sup Lj 2(sg). But sup L1 2(sg) = (sg)x1;2 = a contradiction. Thus g' = g*. Therefore >4(ft) = {(f,f*) | f e G^ 2) Conversely, suppose we are given chains 0^ and C>2 and maps y ’ , X} 2’ X ’ , and x2 j such That (i), (ii), (iii), (iv) are satisfied. We must construct a po-set ft satisfying our conditions. By (iv), define a partial order of the set ft = 0 U C>2 using 12 2 1 the maps X-. X ’ , X9 , and X ’ such that the orbits of AW) 1 , Z Z y 1 59 in ft are precisely 0^ and 0^. Every orbit of ^4(ft) in ft is a chain. By Theorem 2.4, A(ft) is an ¿-group with the disjoint support 1 2 property. Define the convex P^-congruences 6^ 2 and 6 ’ on 0^ ' 2 1 and the convex Ppcongruences &2 and C ’ on C>2. By using the partial order given in the definition of "acceptability", clearly ^1 = S 2^ Cl’2 *= ^1 °n °1 Hnd ¿2 = G2 1 = ^2 °n °2" Then, as noted previously, -4(ft) = {(f,f*) | f e 2). // 12 2 1 In Lemma 3.13, if we use maps x ’ ’ x2 1’ and x ’ which satisfy (i), (ii), and (iii) to define a partial order of the set ft = 0 U 02, >4(ft) will map points of 0^ to points of 0^ and points of 02 to points of 0^ since points in 0^ have two chains of lower bounds, whereas points of 02 only have one chain of lower bounds. However, we must still guarantee transitivity on 0^ and Op By Lemma 3.10, 0 y must be a subset of an orbit of >4(0.) in 1 1 ; Z Z — 21 . — 02 and 02x ’ must be a subset of an orbit of _4(0p in Op In constructing examples, quite often we may guarantee transitivity by 2 1 letting 0^ 2 = °2 and °2X ’ = °1’ Note that in choosing the orbits needed in Lemma 3.10, we must choose so that (i) either °lxl 2 = °2 °T °1X1 2 °2 = 0’ Hnd either °2x2,1 = °1 or O2x2,1 0 0 = 0. We now classify the po-sets ft satisfying our hypotheses such that Q 2 61’2 = on 0}^ and j = °n Op Note that by Proposition 3.5, for spS2 e ^1’ Sl^l 2S2 d£ and 2 1 only if s1x1 2 = s2xi 2’ and that for tl’t2 £ °2’ if Hnd 2 1 2 1 only if tjX ’ - t2x ’ • Thus, we have that Ker Xj 2 = ^1 2 and 60 2 1 2 1 that Ker x ’ - £ 3 . In this case, we define the maps 0 /Ker x [0 /Ker x2*1] and 1,2 ’ 1 1,2 2,1 O2/Ker x 2,1 [Cp/Ker Xj 2] as follows: (i) for S0 KeT Xl,2 = S0^1,2 e °1/Ker Xl,2’ -2,1 ^S0Cl,2^nl,2 SUpit KeT X I 1 - THEOREM 3.14. Suppose ft is a po-set satisfying our hypotheses and suppose £2 = 2/^ on 0/ and G g £ J 2 £g 2 = ^2 1,2 on 09. Then, the following conditions are satisfied: Cl 2,2 (i) 0^ -> {L°} and x2 2 : °2 ^_co^ (ii) n2 2 ds one-t°-°ne order preserving, and [0^/Ker x2 glo^ g 2 2 is dense in [O^Ker x 3 ]• n'J is cne-to-one order 2,1, 2,1 preserving, and [C>2/Ker x 3 ]o bs dense in [0^/Ker x2 g] • (iii) n7 o, bbe extension of n7 9 bo the points of 1,2 1,2 [O^/Ter x2 gL bs an order-isomorphism from [C^/Ker x2 g] 2 2 to [CT/Ker x 3 ]• Similarly, T}2’1 : [Or/Xer x2j7] [0 /Ker x-, J an order-isomorphism. Cl 72 7 J C! ■2,1 Moreover, f n T n ij 2 1,2 (iv) The chains 0^ and C>2 and the maps X2 2J X ' , Xg 2> 2,1 and are acceptable. 61 We compute the group AW. Set G = {feA(O1/S1 2) | [(02/C23l)b2jl]f = (02/d3l)d31 and set 1,2 G2 1 = {g a A(O2/G2j2) I [(Og/£g 2)rig 2]g = (O2/£2 2)r\2 p. Then G^ 2 is a transitive ¿-subgroup of Afopip g) and Gg is a 2 1 transitive ¿^subgroup-of- AiOg/£ -23 *). For each-s—zO^, A(sp g)- 2 1 is a transitive ¿-group and for each t e Og, A(t£ 3 ) is a transitive ¿-group. Here, n7 9 induces an ¿-isomorphism 1,2 * : G1 2 G2 1 = {((pig), e \A(sG2 g)WrA(op£2 p] x [A(t£23l)WrA(0g/G231)] ( g g £ 2). Conversely, suppose 0^ and Og are totally ordered sets, Xg g 12 . ~ and x J are order preserving maps from 0^ to Og, and Xg g anG 2 1 x 3 are order preserving maps from Og to Og. Define ry g and 2 1 n 3 as before. Then, if (i), (ii), (iii), and (iv) are satisfied, there exists a po-set ft satisfying our hypotheses such that Sl~ Sl,2^^j2 = ^1 °n °1 ^2~C231^C2,1= ^2 °n °2' Furthermore, A(Sl) = { (( [Â(t£23l)WrA(0g/,^231,1 )] \g g £ }. 1,2 Proof: As in Lemma 3.13, £ 2/g on Og and £ g g = 2p on Og imply that (i) of the theorem is satisfied. (iv) is clear. £ / 2/_ and £2’^ ■/ U . The £ _ classes on 0 together with J. 3 Z J. Z J. 3 Z _L the £ 2 ’ 1 classes on Og form an o-congruence of ft with respect to the ¿-group /(ft). Call this o-congruence £ . Using the definition of Theorem 2.10, define a partial order on the set ft/£ ■ First note that 0g/£ = Og/£g 2’ ~ and ft/y = Og/£g 2 U Og 2,1 /£ ’ -< °l/<:i,2 is the natural total order of 62 the convex 6 ~ classes and i „ is the natural total order °2/i: ' 2 1 of the convex G ’ classes. Each point of 0/6 „ has lower bounds i z, z 2 1 2 1 in the chain O^/C ’ under A but it has no upper bounds in 0^/6 ’ 2 1 Similarly, each point of ’ has upper bounds in 0^/6"^ under 4 but ho lower bounds. Also, by definition of G , if 1, z s / s 6, ., then s & has a different collection of lower al,2 £1,2 a 1,2 2 1 bounds in 0./6 ’ than does s G Again, by definition of 6 2,1 Z p 1 f z ,2,1 different points of O2/6 ’ have different collections of upper bounds in the chain 0^/6 2‘ For the moment we replace the po-set ft be the po-set ft/6, the chain 0^ by the chain 0^/6^ 2’ and £be cha£n G2 by £he cba£n 2 1 0^/G ’ • As noted before the statement of the theorem, Ol^l 2 = °]/Ker Xj 2 and G2^2,1 = °2/Ker X2,1, As usua1’ define 2 1 maps ir1 2 : C^/Ker Xj 2 [02/Ker X ’ 1 and ir2’1 : 02/Ker y2’1 + [Oj/Ker Xj 2] as follows: (i) for s0 Ker *1,2 E °1/Ker *1,2’ 2 1 2 1 (sQ Ker X;L 2)tt1 2 = sup{t Ker y ’ | t Ker X ’ 4 sQ Ker xlj2}’ and (ii) for tg Ker y2’1 £ 02/Ker y2’1, (tQ Ker x2,1)tî2,1 = inf{s Ker X2 2 I to Ker X^ s Ker xl,2}' By 2 1 2 1 definition of the ordering on ft/6, H 2 = H 2 and 71 ’ = F ’ ’ 2 1 where p _ and p ’ are as defined before the statement of the X , z theorem. By the definitions, the remarks in the preceding paragraph, and the arguments used to prove Lemma 3.13, we conclude that the maps p^ 2 63 and n ’ have the properties claimed in (ii) and (iii). Also, by the same arguments as used in Lemma 3.13, it follows that G „ is 1 , z an ¿-subgroup of /(0^/£^ 2) > that G2 is an ¿-subgroup of 2 1 /(Cp/£ ’ ), and that n^ 2 induces an ¿-isomorphism * : G^ 2 -* G2 We return to the po-set ft and the chains Cp and Cp. /(ft) C /(0 ) x /(02) and the orbits of /(ft) in ft are Cp and 2 1 Cp. ^12 is a convex -congruence on Cp and £ ’ is a convex P2-congruence on Cp. As noted in Chapter 2, since /(ft) C P x P2, we also have that /(ft)C [/(s£^ 2)Wr/(O^/£^ 2)] x [/(t £2 ’ ^) Wr/(Cp/£2 ’ ^) ] Set G = {(((¡pig), (d>2;g*)) e [^(s^ 2) Wr^Cp/^ p] x 2 1 2 1 i [/(t£ ’ )Wr/(Cp/£ ’ )] I g e Gj . A computation similar to the one given in Lemma 3.13 shows each element of G is order preserving on ft. Theorem 2.10 implies that each element of /(ft) must induce an order automorphism of ft/£. To preserve order on tl/G,, it must there fore be the case that /(ft) = G. For the converse, suppose we are given chains Cp and Cp and 12 2 1 maps Xl 2’ * ’ ’ *2 1’ and x ’ SUCh that and (iv) are satisfied. We must construct a po-set ft = Cp U Cp satisfying our conditions. By (iv), define a partial order of ft using the maps 2> 12 2 1 y ’ , x9 > and x ’ such that the orbits of /(ft) in ft are precisely Cp and 02. Every orbit of /(ft) in ft is a chain. By Theorem 2.4, -/(ft) is an ¿-group with the disjoint support property. 1 2 Define the convex P^-congruences £p 2 and £ ’ on Cp and the 2 1 convex P2-congruences £p and £ ’ on Cp. As before, 64 £ O = Ker y, „, £1,2 - Ker x1’2, 1 " Ker x2 1’ and 1,2 *1,2’ .2,1 2,1 £ = Ker y ’ . By definition of the partial order on ft, clearly h- = on 02 and <$2 C £2 ’1 on Op Then, from the above, AW) = i ( > g) , ( [AKsGppWr^aySpp)] x [4 Ct^’bwr^^/i2’1)] I go G^}. // From the proof of Theorem 3.14, we may also classify the po-sets ft which satisfy the conditions of category 3a: (i) & 2 ~ and for each s e Op inf 2(s) = sup 2(s), and (ii) ^2— ^2 1 = £2’1g £¿2 and £or each 1 E °2’ inf U2 T(t) = sup L2 1(t). By using these assumptions and Proposition 3.5, we have that for S1’S2 E °1’ Sl^l 2S2 if and only if S1X1 2 = S2X1 2 if and °nly if 12 12 12 Sl^ ’ S2 i£ and °nly ££ six ’ = S2X ’ ' Similar relations hold for each tpt2 e Op These statements imply that C1 2 = Ker Xl 2 = ^’2 = Ker y1,2 and z?2 T = Ker x2p = 62’1 = Ker y2 ’1 We define 2 : O^/Ker Xj 2 [O2/Ker y2,1], n1’2 : OJ/Ker y1’2 + [O2/Ker y2 , p2’1 : O2/Ker y2’1 -> [Oj/Ker Xl 2], 1 2 and n2 ! : O2/Ker X2 l [Oj/Ker y ’ ] as follows: (i) for SgKer xJ ? e O^Ker xJ p 2 l . [sQ Ker Xp2]Pp2 = sup{t Ker y ’ | t SgXp2b (ii) for sQ Ker y1’2 e O^Ker y1’2, [s0 Ker yi^Jn1’2 = inf{t Ker y^ | sQ y1’2 t), 65 (iii) for tp Ker x2’^ e Cp Xer [tQ Ker x2’1]n2’1 = infis Ker xlj2 I bo*2’1 H3 s}’ and (iv) for tpKer x2 j e Cp/Ker x2jl> [f0 Ker X2jl]n2/1 = sup(s Ker y2’1 I s From the assumption that for each s e Cp inf ip 2(s) ~ SUP 2(s) 1,2 and the remarks above, it follows that 2 ’ n ’ Similarly, 2 1 2 1 n ’ = n9 Thus, we need only consider the maps n. ~ and n ’ Z 3 1 7 3 Z 2 1 and the sets Cp/Ker 2 and Cp/Ker x ’ in classifying the po-sets ft satisfying our hypotheses and having the stated congruence structure For this reason, we may directly apply Theorem 3.14 to these partially ordered sets. COROLLARY 3.15. Suppose ft is a po-set satisfying our hypotheses and suppose (a) 2b and for each s s Cd we have 7 7 j ¿j 7 7 inf ^(s) ~ sup g(s), and (b) G£0 7 = 2/ and for each t s 0~ we have inf Ur, ^(t) = sup L% 2(t). Then, the following -conditions are satisfied: ... 1,2 , 2,1 M X = *1 2 and *2,1 ~ * (ii) rp g ds one-to-one order preserving, and [O^/Ker 2^1 2 is dense in [O^/Ker x 3 ]• n 3 is one-to-cne order 2 12 1. preserving, and [Og/Ker x 3 ]n 3 is dense in [O2/Ker x2 2] ( iii) n7 9, the extension of n7 9 to the points of [O2/Ker x2 2] 1,2 1,2 is an order-isomorphism from [O2/Ker x2 to [O2/Ker x23h 66 Similarly, t}2,2 : [09/Ker y2j2] •* [0 /Ker y ] is an 1,T --7-27 order-isomorphism. Moreover, (t\ 9) = n J . 1, 2 7,2 (iv) The chains 0^ and Og and the maps y 7,23 X , *2, l3 and y2’2 are acceptable. The group A(Sl) is computed as follows: Set G2 2 = {f zA(O2/Ker Xg g) | [(Og/Ker y2jl)P3l]f = (Og/Ker y’2)T\2’2} and set G 2 2 = lg z A(Og/Ker y2,2) | {(O^Ker Xg 2\j2]g = (0.,/Ker y ) n 9}- Then G is a transitive ¿-subgroup of 1 1)21)2 1 ) 2 A(O./Ker x 9) and G is a transitive ¿-subgroup of 2 7 A( Og/Ker y 3 ). For each s e Og, A(s Ker Xg g) is a transitive 2 1. ¿-group, and for each t e 0^, A(t Ker y 3 ) is a transitive ¿-group. Here n 9 induces an ¿-isomorphism * : G.. 9 ■+ G9 7 and 1 ) 2i 1 ) 2 2 ) 1 AOi) = {((^2;g), ($2;g*)) e [Ats^ 2)WrA(O1/£1 2)} * [4(tC23l)WrA(02/^23, 1 \ g z G2 2~}. 1,2 Conversely, suppose 0^ and Og are chains, y and 7,2 2,1 are order preserving maps from 0^ to Og, and y2 g and are order preserving maps from Og to 0^. Define t\2 2 : O^/Ker X2 2 [Og/Ker y8,1] and t)2,2 : Og/Ker y^1 + [O2/Ker x7 9] 4,2J as in Theorem 3.14. Then, if (i)-(iv) are satisfied, there exists a po-set ft satisfying our hypotheses such that (a) C £g C and for every s e Og, inf Og 2(s) = sup L^ 2(s), and (b) £2^62 j = £23 2 U2 and for every t z Og, inf Ug pt) = sup Lg pt). Furthermore 67 AW) = (W2;g), W^g*)) e [A(sG2 2)Wr A(O2/G2 2)] * [Mte2sl)WrAW2/G2j1)] \ g a G2 2). // When constructing examples for this category, care must be taken so that points of 0^ are mapped to points of 0q and points of 0 1 1,2 are mapped to points of Op Suppose we want 2 = £ and ,2,1 = C. Then let 0 and 0. be dense chains, 2,1 1 2,1 xl,2 : °1 * °2’ and X : 0? -t- Op For each s e 0, define 1,2 Up2(s) = {t E 02 I sXp2 sx < t) and define 1,2 }. Then the points of 0 L1,2(s) = {t' E °2 I £' - sxl,2 SX cover points of Op but the points of 02 cover no points of ft. Another way to guarantee the desired behavior is to choose a Dedekind complete chain 0 which has two dense subsets 0^ and Op 0^ and 02 should be subsets of different orbits of >4(0) in 0. Set ft = 0 U 02 and let be the order on 0 restricted to the points of 0^ and Op Examples of both of these constructions will be given in Chapter 4. Note that both constructions result in ft being a totally ordered set. In each of the remaining categories (3b, 4, 5, 6) note that y^ 2 1,2 2,1 and y ’ are maps from 0^ to Op X2 j and are maps from 1,2 02 to Op for each s e 0^ Sxl,2 Sx and for each t e 0, 1,2 txl,2 #tX For the discussion of the four remaining cases, we define relations ¿8on 0^ and $2 on 02 in the following way: (i) For sj,s2 e sizG is2 ££ and °nly there exists nonnegative • >. j , , ,-1,2-2,l.m , integers m and n such that s2(y x 1 and 68 ,-1,2-2,1,11 , S2 Sl(x x ) ’ and (ii) for e °2’ £læ 2£2 d£ and only if there exists nonnegative integers m and n such that .-2,1-1,2m .-2,1-1,2,n -1,2 Ft If, Mf’Y’T and t9 <= t (x Here x^’“ and '1^2 2 '2 -O2 T -21 . 1221 X ’ denote the standard extensions of x ’ and x ’ to the points of 0^ and 02 respectively. Geometrically, two elements s and s2 of the chain 0^ are related by if and only if we can move s^ above s2 by "weaving back and forth" between the — — -12 -2 1 chains 0 and 0 by the maps x ’ and x ’ and we can also move s2 above s^ by this same weaving pattern. A similar geometric interpretation can be given for the relation LEMMA 3.16. dp is a convex P^-congruence on 0^. ¿£p is a convex P^-congruence on 0%. Proof: It is easy to see that ¿0^ is a convex equivalence relation on 0 . The fact that is a P^-congruence follows from -1 2-2 1 the fact that the map X ’ X ’ must commute with elements of /(ft)ig 1 Similar remarks hold for the relation 4p. // Recall that the convex P^-congruences on 0^ form a chain and that the convex P2~congruences on 02 form a chain. By definition of dp and £3 we see that for categories 3b, 4, 5, and 6, and Ggp on and *2-C2 l^'^ and ^2— inf U (t) 0 sup L (t). As before, we first assume that z 3 1 z 3 1 2 = £1,2 and £ = £2 g = £2,1, and then we give a classification for the general situation. For ease of statement we present the conditions necessary for a classification in a sequence of lemmas. LEMMA 3.17. Suppose ft is a po-set satisfying the hypotheses 1,2 given in the previous paragraph. Then, Xg 2 an(^ are one-to- one order preserving maps from Cd to a dense subset of 0 and for 1,2 2,1 each s e 0^ sXg g sy X2 2 and y are one-to-one order 2 preserving maps from 0. to a dense subset of Cd and for each 1,1 1 e °23 **2,1 <01 *3,2 x can be extended to order- -12 — — 2 1 isomorphisms x 9 and X J from Og to Og. y2 g and y 3 can 1) 2 -2,1 be extended to order-isomorphisms y2 g and yf3' from 02 to Og. -2,1 , - Moreover, X^ 2 X and (x ) = X 2,1' Proof: These conditions follow directly from the definitions and Proposition 3.4 and Lemmas 3.7, 3.8, and 3.9. // LEMMA 3.18. Suppose ft is a po-set satisfying the hypotheses of Lemma 3.17. Then there exist homogeneous chains and A2 and homogeneous chains and B2 such that there is an order-isomorphism Tg : 0^ A2 * Aan order-isomorphism t2 : 02 -+ B^ x B2, and an order-isomorphism t : A„ B which have the following properties: (i) For s ,s e Cd, there exists an a- e An such that a y 1 2 2 70 e A^ x {a^} if and only if there exist positive Z 2 2 m integers m and n such that s <- s (y 3 y J1) and a O2 y ,-l, 2-2,1 J s <- s (y y J . For t ,t e b% e B% such that t^T^tr^ £ B^x {b^} if and only if there exist positive integers m and n such that , „ , ,-2, 2-7, 2 ,7« , ,-2,1-1,2.n I O2 w * J \ <01 HF X > ■ ( ii) can be extended to an order-isomorphism t. : 0 -+ [A £ A.] 1 and t can be extended to an order-isomorphism t9 : 0 -+ [B £ B_]. (iii) For each s e 0^, if st2 e A^ x {a^} then sy^ and -1,2- sy ' Tg are elements of [B^ £ }]• Tor each t e 0%, if ti 9 £ B. x {b }, then ty and ty2,2i: are 1 2,1 1 elements of A- x {bri2}. 1 2 e> Proof: In Lemma 3.16 we have defined a convex P^-congruence ¿2^ on 0^ and a convex P2-congruence (8^ on Op Let s ¿8be an arbitrary ¿3^ class and ttB? be an arbitrary ¿22 class. Then the sets s$p O^/zS t<82’ and °2^ 2 Can be totally ordered in the natural way. Moreover, by Theorem 2.11, these four chains are homogeneous, and there exists order-isomorphisms : 0^ -> s ¿8 x 0^/23^ and T2 : G2 td^2 * G2/zB 2‘ We may define an order-isomorphism t : 0/20 -> 0 J Si 9 as follows: for s ¿8 e 0 /¿8 , ¿5 1 1 2 2 Uili j 2 _____ (SgfipTj = t0<£2 if and only if for every s e sQ28 , sy ’ e [t045’2] • Property (i) is satisfied by definition of ¿8^ and <®2. (ii) is 71 always satisfied since each chain is dense in its Dedekind completion. (iii) is clear by definition of . // By identifying x A2 with the chain Cp and B^ x with -12 the chain Cp we see that Xj 2 and * ’ are order_isomorpbisms ______- 2 1 from [A^ X A?] to [B^ £ B^] and x ’ and x2 j are order-isomorphisms from [B^ T ip] to [A^ J A21• From (iii) of Lemma 3.18 we may conclude even more. LEMMA 3.19. Suppose that ft is a po-set satisfying the hypotheses of Lerma 3.17 and t : A% -* B% is the order-isomorphism described in Lemma 3.18. Then, for each Î ^ag^ £ p * A%, the restrictions 1 2 °f Xi 9 and x 3 ore one-to-one order preserving maps from A^ x {a^} to a dense subset of [B^ f {ag^& }]• Similarly, for each B x {b„} e B x B , the restrictions of x 7 and x 3 ar‘e one-to- one order preserving maps from x {b^} to a dense subset of __ 2 [A^ x {b^T^ }]. For each A^ x (op) ep x A%, x^ g and X 3 -12 may be extended to order-isomorphisms x-^ g X 3 from [A^ x {Og}] bo [B? x ^agTi2 For‘ eac^ B1 * ^2^ E B1 * B2j X2 2 ond x 3 may Be extended to order-isomorphisms Xg 2 and X^l from [B^ x {b^}] to [A^ £ {b^pp}]. Moreover, ,- ,-2 -2,1 , .-2,2,-2 (*1,2 = * a™ * = *2,1' Proof: The lemma follows directly from Lemma 3.17 and Lemma 3.18. // 72 1,2 We now give the classification for £g 2 = £ ’ ~ °n G1 2,1 and £2a = £ £2 on Og. THEOREM 3.20. Suppose ft is a po-set satisfying our hypotheses such that (a) 2 = 2 = ^2 — ^7 °nd G°V eac^1 s e Gi3 inf ^(s) f sup 2^23 and (b) £2 = £^ g = &^2. Zi2 and for each t e 02, inf i/g g(t) f sup Lg jit). Then, the conditions stated in Lemmas 3.17, 3.18, and 3.19 are satisfied. Moreover, 12 2 1 (+) the chains Cf and 0% and the maps y 3 , ~X2 23 x * ■» and x9 are acceptable. 2,1 /(ft) is constructed in the following way. For each a% e A%, set 2(a2) = {f z A(A2 x {a2)J | f(y2j 2y2j 2) = (x232X2s2)f and ((B2 x {a^})y2 2)f = (B2 ? ta2T& ^*2 P3 ea°h 2 e B23 set G2 2(b2) = {g e /(Bg ? {2y}) | gty2,3 2y23 2) = (y23 2y23 2)g and (Mg x {b2Tig’Z};x'Zj 2)g = (A^x {b2^2})y23 2}. Then G2 2^a2^ '2s a transitive ¿-subgroup of A(A^ x {ay}) and G2 -jCb^ is a transitive ¿-subgroup of A(B^ x- {b2}). Here, Tg induces an ¿-isomorphism -1 2 : A(A2) r A(B2) and for each a2 z A2 and a^Q z B^ y 3 induces an ¿-isomorphism * : G^ 2^a22 G2 l^a21'A 2' Then A(n) = {(( Conversely, if we are given chains Cf and 02 and maps which 73 satisfy the conditions given in Lemmas 3.27, 3.18, and 3.19, and (t) above, then there exists a po-set ft satisfying the initial hypotheses of this theorem and whose group of order automorphisms is equal to the f-group A(9l) listed above. We may geometrically represent the partially ordered sets which fall in this category by the following picture. Proof: The assertions in Lemmas 3.17, 3.18, and 3.19 have already been proved under these hypotheses. (t) is obvious. We now compute /(ft). By Lemma 3.16, is a convex P^-congruence on 0 and ¿0^ ds a convex P^-congruence on 02- Thus, by the theorems and remarks in Chapter II, §4, P C/(szB 1)Wr/(O /43 T) and P2 Q/(t S3 p Wr / ((p/d? p . By the conditions in Lemma 3.18, O^/dl = A2, s43^ = A x {ap for any a2 e A2, 02/& 2 = B2’ and tz®2 = B1 * ^2^ £or any b2 E B2’ Thus Pj^/fA x {a2))Wr/(A2) and P2<^/(B1 x {bp)Wr/(Bp. Also by Lemma 3.18, Tq is an order-isomorphism from the chain A2 to the chain B2. By Lemma 3.11, we may define an ¿-isomorphism ~ : /(Ap +/(Bp where for g e/(Ap and b2 e B2, b2(gb = b2TzglgTiB ' maPs eacb class szB to the unique &2 class tZ8 2 such that [t£0 2] contains the images of the points 74 -12 in s ¿3 under the map y ’ . All elements of AW) must respect -1 2 X ’ . Therefore if an element of maps the ¿0. class s Æ , 1 a 1 to s &, it must map (s to (s ¿8 , , i.e.. Y 1 r a 1 œ) y 1 “ Ç {((^g), ((¡>2;g-)) e [4(A1 x {a2))Wr>4 (A2)] x [j4- (B x {t>2}) Wr >4(B2) ] | g e-4(A2)}. We now put restrictions on the functions >4(B1 t i2t& by £or £ e x Up), f* = (x1’2) ^X1’2- Set G^^) = <£ e >4(A1 t {a2}) | fix1’2?2’1) = (x1,2X2j1)£, (A1 x {a2))f = At x {a2), and x {a^ })x2 j)£ = * ia2Tz8 ^x2,p and G^Ca^) = {? e _4(B1 £ {a~^T) | ~g(y2,1y1,2) = W.2,1^’2)'^ (B1 x {a^T^ })g = BJ x {a2rs }, and (x {a2))y1,2)g = (A1 x {a^y1’2). We claim * may be restricted to an ¿-isomorphism from 2(a2) £o G2,1 ('a2T® } ‘ We first show * maps G^ 2(a2) to G2 (a^^ ). Let feG 2(a2). We check (N) V’2) = 6? ’ V ’2) (f^ • 3-.-2,1-1,2, .-1,2,-1?-1,2-2,1-1,2 .-1,2-1-1,2-2,13-1,2 f*(x X ) = (X ) £X X X = (X ) X X £X -2 1--1 ? -2 1-1 2 -12 -1--1 2 -2 1-1 2 x ’ fx ’ = (x ’ x ’ Hx ’ ) £x ’ - (y"’"x"’ H£*• Thus, elements of - . . , -2,1-1,2 G1 2 a2 * commute with y X The methods used to show functions -1,2 in G^ 2(a2J* maP elements of (A^ x {a2?)x ’ 1° elements of (Aj x {a })y1,2 and elements of B^ x {a^^ } to elements of B^ x {a2r(2 } are similar to the ones used in Lemma 3.13. Hence * 75 maps 2^a2'’ t0 G2 1 ^a,?T ¿-homomorphism, so is the restriction of *. To show * is onto, again we may use similar techniques to those used in Lemma 3.13. Therefore * is an ¿-isomorphism from G^ 2(a2) to G2 l^a2Tz3 ' By the composition of extension, *, and restriction, we now define an ¿-isomorphism * from G^ 2^a2^ t0 G2 1 ^a2TzB ’ where these sets are the ones defined in the statement of the theorem. Consider the set G = { (( [G2 jfbpWr/CBg)] ( for a2 e Ag if ^2^a2Tfl ) = By using the natural action of G on ft we wish to show each element of G is order preserving on ft. Choose ((p g) , (2; g'"')) e G and ay, ay e ft such that <_ay. We show ay ((<(y ;g), ( io ,ii) e 02 then the assertion is clearly true. Suppose next that U1 e G1 and W2 e G2' ¿here exists a2 e Ag and ty e ^2 such that ay e Aj x {a,?} and ay e B^ x {ty}. By definition of the action of G on ft, this implies that ay (( and ay ((<(> ; g) , ( the case that a2y 5g bg. If a2Tig < b2 then < bgg~, i.e., a2y y^gyg < tyg~, i.e., a^y? < tyg~. By the definition of Tg it is then clear that uy ( ( Therefore suppose that a2fQ> = bp Then, if 4y(a2) = ¿> 4>2 (b ) = ¿*• To complete the proof of this case, it now suffices to restrict the order relations on ft to the chains A^ x {a?} and B^ x {b2} and show the permutation (f,f*) preserves the order 76 relation between the points e x {a2) and w2 e B^ x {bp. -12 -12-- - ui f. u? so uiX ’ 5______“?• Thus to v ’ f* <_ to f*, i.e. , B1${b2} Z 1 2 -1 2-1 2-1--1 2 -- . --1 2 -- ’ x ’ fX ’ <_ io2f*, i-e. to-px ’ £m2f*. This implies to^f <_ top* and so the proof of this case is complete. The final case is that to^ e 02 and u2 e Op By using techniques similar to the ones above, we again consider chains A^ >< {ap and B^ i {bp such that o)2 e A2 x {ap, m1 c BJ x {bp, and a bp and a permutation (f, f*) of this set. <_ aig implies m^x- 2,1 r.< "m2. ' -2 1- - Therefore m^x ’ f f up. -2 1- -1 2 -1-1 2-2 1- -1,2-1=-1,2-2,1 3—2,1 _ “ix f = W1X ) X X f m x X fx X = u>x ,f*x • Thus __ 2 1 - ujb*X ’ 5. u2£ and S0 (b’f*) is order preserving. Therefore, in all cases we have shown that co C C;g), C2; g^)) £ w2 (((J^; g), ( Now, by using the methods of Lemma 3.13, it is easy to check that the set G defined is equal to__ 4(ft) .. . Conversely, suppose we are given chains 0^ and (p and maps 1,2 2,1 xi,2, X , x2jl, and X such that the conditions in Lemmas 3.17, 3.18, and 3.19 and (t) are satisfied. We must construct a po-set ft satisfying our conditions. Define the partial order on the set ft = 0^ U 02 to be the 1 2 natural partial order connected with the maps Xj 2, X ’ , x2 1’ and 2 1 . X ’ which is guaranteed to us by (+). The orbits of /(ft) in ft • are precisely 0^ and (p. Every orbit of /(ft) in ft is a chain. By Theorem 2.4, /(ft) is an ¿-group having the disjoint support property. Define the convex 77 1 2 -congruences 2 and £ ’ on Op and the convex Ppcongruences and £2’^ on q . T_h_e_n £ £ £1j2C S C U 2,1 2 ------V1 1,2 ^1-1’ ' &2 i = 62’#2 CZ &P for every s e Op inf ip 2(s) 0 sup L^ 2(s), and for every t e Op inf U2 (t) 0 sup (t) . By using the proof of the converse we see that /(ft) has the desired form. // Under these conditions, to construct examples so that (+) is true, apply the techniques mentioned when "acceptable" was first defined and also the techniques concerning transitivity mentioned after Lemma 3.13. We will do this in Chapter IV. 1 2 To generalize this theorem to the case when g C Ii- 1 2 1 and G2 t = £ ’ £ %.2, we apply the same methods as those used to generalize Lemma 3.13 to Theorem 3.14. As this generalization is completely straightforward, we only sketch the details. 1 2 Replace the chain 0^ by 0^/£^ 2 = 0]/^ ’ and ®2 by 2 1 O2/£2 1 = ^2^ ’ the statements °f the lemmas. Also, replace 12 12 the maps y ’ and y „ by n ’ and n ~ (which are defined in 1 3 C. 2 1 2 1 the usual way), and replace the maps y ’ and X2 2 by n ’ and n? To compute the group /(ft), we note the following facts. Z 3 1 12 2 1 G^ 2 = & ’ $ and 1 = i ^2' TzS again induces an ¿-isomorphism : /(0 /¿8 p -5->1 (02/^S 2) . Let sfflj be an arbitrary ¿8, class and t£87 be the image of s ¿8, under the map . 1 Consider 2 to be restricted to s and ((td32/£2 1)n2 1) f = (tzB 2/G2 1)n2 qb and 78 G2,lCtd32:) = {g o/(tz92/£2jl) I gCn2’1^1,2) = (n2’1^1’2)^ and g)n ’ )g = gJo^’ )• Then q1’2 induces an ¿-isomorphism * : G2 gtsfip + G2 j <8 2) • /(O^zGp, G2 gisfip, and /(s^ g) are transitive ¿-groups. Similarly, /(Og/fig), G2 l'-t^2"’’ and P are transitive ¿-groups. Then, /(ft) = {((’P1;1;g), (’¡,2’ (^(sfip = f then g C [s ¿B x] T^g ) = f*}. 1 2 We now examine category 4 (£^^ g $: £ ’ 2^ and 2 1 Pj-congruence ¿8^ on 0^ and the convex Pg-congruence & on Og. Then, ^1>2 $ on Oj and ^2 1= on Og. To state a classification theorem for this category, we assume = £^ g, ¿8^ = 2L^, £2 - £g p and fig = 2/g. The generaliza tion can be made by using the methods used to generalize Lemma 3.13 to Theorem 3.14 and by repeating three lemmas similar to Lemmas 3.17, 3.18, and 3.19 used to prove Theorem 3.20. Once the theorem has been proved under these extra hypotheses, we will state the general form of the ¿-group /(ft) for this category. THEOREM 3.21. Suppose ft is a po-set satisfying our hypotheses 1 2 such that (a) £i = Gi G 3 ^^2 ~ ^f on Gi3 an<^ (b) &2 ~ G2 1 = 2 = %2 °n G2‘ Then, the following four conditions are satisfied: (i) y., 7 is a one-to-one, order preserving map from 0 to a 1} 2 1 — 12 dense subset of 0%. y 3 is order preserving but not 79 1,2 one-to-one, and O^y is dense in <9 . Also, for each 7,2 2,7 s e 0-, Sy 1,2 0, SX is a one-to-one order preserving map from 0% to a dense subset of 0^. Kg 1 is one-to-one order preserving, but O^y^ is not dense 2,1 in 0 . Also, for each t e 0 , ty <- ty 2,7 02 2,7 (ii) x7,2 x can be extended to order-isomorphisms rWl . 77 . 77 x7 2 : °1 G2 and x J : °2 + °1’ Moreover, --7-27 72 ( of 02 and x2j 7 can be extended to the points of 0%. (iii) For each s e 0 , {s(y^J 2y2j 2 )n I n - +1, +2,...} is cofinal in 0^. For each t e 0%, {t ( y2* y~ * 2)m m +1, +2,... } is cofinal in 0%. 7,2 2,7 (iv) The chains 0^ and 0% and the maps y x7, 2J x J and x9 7 are acceptable. 2, 7 The group AW) is constructed as follows. Set ■1, 2-2,1. _ .-1,2-2,1 2,1 2,1. °1,2 = {f zA(°2) I fty^y’W = fr’TWf, and - (O2y’Wf = 02y’W and set C2 = {g zMOj | gty2’ 1y'3 2) = (x^y1* 2)g and W^i 2)g = G2^1 21* Then 2 a transitive ¿-subgroup of AW^) and GL 7 is a transitive ¿-subgroup of A(OA. Here, y . induces an ¿-isomorphism * : 2 G2 1J °:nG -4^) = {(f,f*) \ f e G^ 2~). Conversely, suppose we are given chains 0^ and 02 and maps 12 2 1 X J , X7 23 X J , and y2 2 such that (i), (ii), (iii) and (iv) are satisfied. Then we may construct a po-set ft which satisfies the initial hypotheses of this theorem, and AW) has the form described above. 80 We may geometrically represent the partially ordered sets of this category by the following picture. Proof: The proof of this theorem is almost identical to that of Theorem 3.20. Conditions (i) and (ii) follow from the definitions, Proposition 3.4, and Lemmas 3.7, 3.8, and 3.9. Condition (iii) follows from the fact that ZS and ¿8 2 = 2/^. Condition (iv) is obvious. We now verify only a few details for the computation of the group /(ft). From (ii), Xj 2 induces an ¿-isomorphism * : /(0^) -> /(C>2) . Set Gb2 = {f c/fop | f(x1,2X2,1) = (x1,2X2,1)f, Oj = 0p and (O2x2,1)f = O2x2,1} and set = {? e /(Op | giy2’1^’2) = (x^X1’2)!, 02g = 02, and (0^ 2)g = 0^ 2>- ¿hen can be restricted to an ¿-isomorphism from G ? to G . The only verification we will give — — -- -2 1-1 2 -2 1-1 2 -- for this statement is that for f e Gj 2, f*(x’x’)=(x’x’)f*- .3-, .-2,1-1,2, -I3- ,.-2,1-1,2, - -lj-1,2 - -lf.-1,2-2,1 - (f*)(x X ) = (xT 2fx1 2)(x X ) = Xlj2fx = xl,2fix X )X1,2 -2,1 .-1,2-2,1,5- -2,1-1,2 - -1-- X (x X )fXi,2 = (x X •)x1,-2£x1,2 = .-2,1-1,2,=- (X X )f*- After repeating the arguments in Theorem 3.20, we may define an ¿-isomorphism * : G -> G? . . Set G = {(f,f*) | f e G „}. To show X3ZZ3X J- 3 / 81 G = A(W > we will verify that each (f,f*) e G is order preserving on ft, i.e., GC’.A-(ft). The other containment is obvious. Let (f, f*) e G and with w w2- sh°w to (f,f*) £m2(f,f*). If oj1,aj2 e 0^ or topi^ e Op then clearly (0 (f,f*) <_w2(f,f*). Suppose next that m1 e 0 and m2 e Op 1 1 1 tw -b2 — -1 2-- Then to^ Wp f* e 4(02) S° œlX 5 £* — w2£*’ i.e. , 1,2- -lr 1,2-2,1,- < m7f*, i.e., toJ lXx xXql ,2 xl,2 V ±ft w2f*' i.e., to (f,f*) <^tü2(f,f*). The final case is that W1 E G2 and w2 E Gl ’ Then to^ <_ to2 implies to^ W2X1 2' -1 ?- f* e >4(O2) so to^f* <1^ 2f*, i.e., tüjf* <_ 2x T 2fxi,2’ i-e’’ “l£* - w2fxl,2' This implies w f* a^f, i.e., m^(f,f*) <_to2(f,f*). Thus, G = _4(ft) . The converse is almost identical to Theorem 3.20. Condition (iii) guarantees that ¿8^ = 2^ on 0^ and ¿8 2 = on Op // I,2 If we have chains 0^ and 02 and maps Xj p (2,1’ and x 2,’ 1 which satisfy (i), (ii) , and (iii) of Theorem 3.21, we may construct a po-set ft such that >4(ft) will map points of 0^ to points of 0^ and points of 02 to points of Op This follows 12' 2 1 from the fact that y ’ Is not one-to-one whereas x ’ is one-to-one, i.e., the order-relationships between the points of 0^ are not symmetric to the order-relationships between the points of Op However, we must still insure that AW will be transitive on 0^ and transitive on 0 82 Without giving the details, we now state the general form of the ¿-group /(ft) for the case where i on and £g £g 2 = £2’1^^2 — ^g On °2' proving a lemma similar to Lemma 3.18, we may construct an order-isomorphism r^ : 2 ■> Og/^g g. Tg induces an ¿-isomorphism - : /(O^fi 2) ^^(Og/iSg). Next, let s ¿81 be an arbitrary ¿8 class and let tfig be tbe ama§e o£ s & under the map . Consider £ to be restricted to sfi and 4,1 to be restricted to tCBg. 1 3 Z -L ffn1'2 -2 1 -1 2-2 1 - Set G2 g^s®p = if e 4) (s © 2^1 2^ 1 nJ = (n ’ n ’ )f and ((ti3 g/£g 1)n2’1)f = (tSg/^g^jn2,1} and G2,l(t®2) = {g ^4(t(32/C2>1) I i(n2’V’2) - (n2,1F1,2)g and We then have that n, 7 induces (sfi1/£1^g)n1 1 , Z an ¿-isomorphism * :: Gl,2(-S® P G2, 2 (t(3 g) • The three ¿-groups /(Oj/Æj), G^ g(s /(Og/i^g), Gg ^^(tZSg), and / (t£g ^ ) are transitive ¿-groups. Then, /-(ft) = { ((il^/pg), 0fj2; [/(t^’bwrGg ^tfigjWr/fOg/ffl 2)1 | ¿or j e ¿^/¿B p if We may obtain further information about the structure of the ¿-groups G. _(sé8 ) and G (t£8?) defined above. In category 3, I3Z 1 Z 3 1- Z the two corresponding ¿-groups were periodic in the sense of McCleary, -12-21 -21-12 i.e., n ’ h ’ and n ’ 1 ’ were order preserving permutations of 02 and Og respectively. Category 4 is the first category where the more general type of periodic ^¿-group arises. All of the categories 83 will be illustrated by example in Chapter 4. We present one example here to give further insight into the ¿-groups and G2,1(S<82)- EXAMPLE 3.22. We present an example of a po-set ft = 0 Û Og such that /(ft) is an ¿-group with the disjoint support property. Moreover, g Ç ¿A’ 2 Ç ¿8= 2/^ on 0^ and 4 ’ L.l “ C2,1ÇÆ2 ’ 4 °2‘ Let F = {0,1,2,...,°°) be an index set, and, for each y e F, let R be a copy of the real numbers. Each is a totally Ï Ï ordered set. For convenience, 0 will denote the vector in "]y R , yeF Y the direct product of the sets which has the zero element of appearing in each coordinate, and r = (r^,r ,....r^) will denote a typical element of yy R . Moreover, r will quite often denote yeT Y Y the y-th entry of the vector r. Set R = {r e ~]y R | r 0 for yeF y Y Y only finitely many yeT). The set F can be totally ordered in the obvious way. By noting this, we see that the set R will consist of those vectors in yy R which have inversely well-ordered support. yeT Y Define a total order of the set R in the following way: Let r,r' e R and let y^ be the largest element of r where r^ / r( . '0 Then say r <_ r' if and only if r By the methods used to construct generalized wreath products in Chapter II §4, R is a homogeneous chain and /(R) is ¿-isomorphic to Wr /(R ). To construct the example we let 0^ = Og = R. For yeT Y 84 notation, r = (r ,r^,r^,...,r ) will denote a typical element of 0 and s = [s ,s^,s2,•-•,s ] will denote a typical element of C^. Set ft = 0^ U 02, the disjoint union of 0^ and (J?, and define an order relation < = < on ft as follows: < — —ft —ft I for (rn,r1,r7,...,r_) e 0, and [sQ,s,s2,...,sj e 07, ^lo2 ’ ^2’ 0’ 1’ 2 the O-th coordinate of the vector (r ,r2>r ,...»r^+1), r2 appears in the 1-st coordinate,...,and rC O +1, which is the real number 1 added to the real number 2^,, appears in the °° coordinate) ; and for (rQ,r1, . . . ,r_) e 07 , and [sn,s7 , .. . ,sj e 07, 1 0’ I’ [Sq’5!’-- -,SJ ,r )x = [r^.r,,,...^] e 0- For Cr0’ri’r2’ 1,2 2,1 [s0,sl,s2, - • ,sj e 02, [sQ,s1, . . . ,sjy (Sq,s1,s2,. .. »sJ e and [so’Sl’S2’'’'’X2,1 - sup L2^([s^,ss2,...,s^]) sup {(r,s ,s ,...,s -1) e 0 }. Note that the vectors appearing in the reR0 last set are allowed to have any real number as their O-th coordinate, but they must have s^ as their first coordinate, s^ as their second coordiante’, . . . ’, s 00 -1 as their 00 coordinate. 85 1,2 We discuss some of the obvious properties of the maps x .2,1 , and x X-i 9 is a one-to-one order preserving map ll,2’ 2,1’ 1,4 from 0 onto 0. (namely the identity map). Since 0 is dense in 1 2,1 Ôp G1x1 2 as a dense subset of Op Similarly, x is a one-to- one order preserving map from 0? onto 0 (again the identity map) 2 1 and it th erefore follows that 02x ’ is a dense subset of Op Xj 2 2,1 and x can be extended to order-isomorphisms 2 ' G1 G2 and 7,2,1 - - - -1 -2 1 -121 : 02 Op and 2) = X ’ (Also note that Xj 2 = X ’ ) 1,2 X ' is not a one-to-one map because for any r e R^, the vector 1,2 (r,TpTp .. . ,rp has as its image under the map x the vector .1,2 1,2 [rpr2,...,rro+!]. is obviously order preserving and O^x °2 ’ again a dense subset of 0^. To discuss the map x2 j we make the following observations. For each r e Op r is contained in a countable tower of convex equivalence classes. These classes are classes of the convex equivalence relations which are defined as follows (1) Define X, on 0, by r-%, r' if andonly if r = r' for each 1 1 1 Y Y Y > 1, Y £ T; (2) Define 2 on 0 by ri-"^2 rT' ' if and only if r y = ry' for each y' —> 2,3 yt £ T s > (< °°z) Define X op on 0 byZ r XO O r ' if and only if rC O = rC 'O . Note that %1 Ç. )W4 Ç O Ç . . . X. 00 and each 9/ -class is the union of all the -classes, i = 1,2,..., °° 1 contained in it. These convex equivalence relations are the natural 12 A(R)-congruences on Op y ’ maps each (Y^-class to a unique point of Op and therefore x2 j maps each point of 0? to the supremum of its corresponding Pipdass. Thus, G2X2 1 £s not dense an but g2X2 2 is a subset of an orbit of -4(R) in Ôp X2 j is clearly 86 one-to-one and order preserving. We finally note that for each ?eOl and ?e°2’ ?X12 Pictorially, the po-set ft constructed is as follows: We now check to see that this is indeed an example of a po-set whose group of order-automorphisms is an ¿-group with the disjoint support property. First, /-(ft) must map points of 0^ to points of 0^ and points of ¿2 to points of 02- This is because each point of 0^ covers a unique point of ft (namely its image under Xj 2-^ ’ but P°ints °f 02 cover no points of ft. We may therefore consider /(ft) as a subgroup of -4(0p x /(op where the action of /-(ft) on ft is the natural one. We show /-(ft) is actually an ¿-subgroup. We have previously shown that on 0^ there is a tower of convex equivalence relations — ^2 — ••• — such that (i) 90 œ is the join of the 20 's, y e r\{0,°°}, (ii) 20æ is a proper equivalence relation on 0^, (iii) when ordered in the natural way, is order-isomorphic to R , and (iv) for y e r\{0,“}, 0-^/X is order-isomorphic to 0^. /(ft) (7 A(0^) x /(O^) . Let denote the projection of the elements of /(ft) onto their 87 first coordinates and let Pg denote the projection of elements of /(ft) onto their second coordinates. For any e ^1^1’ elements in r have the same image under y ’ . In fact, we may define via: r.V r’ if and only if ry ’ = r'y ’ . Elements of /(ft) must preserve order on ft, so it follows that % is a convex P^-congruence on 0^. Fix r = (r^,r^,Tg,•■•in 0^, and consider (s e Og | s = r^y^ g for some r^ e rX^}. This is a class of a convex equivalence relation on Og. Loosely speaking, this is the transfer of to Og by g, the identity map. Elements of ./(ft) must respect g, so *^1 as a P2_congruence on °2. is a Pg-congruence so (r^rg, . . . ,^+1) e Og is contained in the -class {(r,ro,r7,...,r~+l) e R | r e Rn). The collection of ’2 3 0' 1,2 all points of 0 whic are mapped to this class of y is {(r',r,r2,r3,...,rw) e R | r' e r e ^}, a class of %g. Again, 0’ 1 2 elements of /(ft) must respect y ’ , so 7(g is a convex P^-congruence on 0^. Transfer iVg to 0g by y^ g to obtain a Pg-congruence */”g. Continue this process obtaining identical towers of congruences on 0 and 0g: namely, % on 0^ and t/jCd^C . . .C^y < on 0g. Corresponding to each ^-class on 0^ there is a c/^-class on 0g and a point of 0g, corresponding to each #g-class of 0^ there is a /g-class and a t/^-class, etc. Also note that for (r0’r1’-"’rJ e °1’ CrQ,r1,...,roo)t/oo q (r1,r2,...,rOT+l)^oo = 0 on 0g. We now compute the group /(ft). As noted previously, /(fi)^/(0.) x/(0 ) and / (0 ) =/(0?) = Wr /(R ). We must restrict 1 yeF Y 88 the matrices of functions from the wreath products which appear in P 1 and P„. First, corresponding to the -class on 0^ which contains all vectors with last coordinate r is two t/ -classes, the one containing 00 00 vectors with last coordinate r (from the map v, „) and the one 00 ^1,2 1 2 containing vectors with last coordinate r^+1 (from x ’ )• Elements 1,2 2,1 of P must commute with x and elements of P? must commute 2 112 with y ’ X ’ • At this level these maps act in a one-to-one fashion and the composition corresponds to (+1), the element of v4(R ) which is translation by 1. By noting this and keeping in mind the correspondences mentioned earlier, we define = (g e Wr ^4(R ) = >4(0^) | Vy e T, y / and yeT 7 for all [SpSp..-,sœ+l] eOp [s3,...,s^+1] = gy+i,[-,sps2,. . ,3]’ and for y = °°, and any [s^,Sp...,sJ e Op g~ TS' S' S'l ( + 1) = ( + 1)gœ is' S' S'!1' Her6’ the ' 111 the ,‘-So,s1, • • • ,5^1 , lsq,s1, . . . ,sœj vector [-,s,,s„,...,s 1 means that any element of R. can be filled 1 4 œ U in where the is. Define an identical group Ip on Op Xj 2’ map, induces the identity ¿-isomorphism * : Hp From previous comments, it is clear that >4(ft) (Z {(f,f*) | f £ H }. A straightforward computation shows each element of this set is order preserving on ft. Thus J-(ft) = {(f,f*) [ f e H^}. P = H^ and P2 = Hp To complete the example we must show H^ is transitive on 0^ and H2 is transitive on 02. The proof is for Hp The proof for H2 is identical. 89 For each y e r, y 0 °°, the y-th component of is ¿-isomorphic to (_4(R),R), and, for y = °°, the “-component of is ¿-isomorphic to (G^,R) where G} = {g e 4(R) | g(+l) = (+l)g). Let (rf),r1 ,... ,rm) , (r^.r^,. .. ,r^) e 0^. We construct a permutation 0’ 1’ ‘ ‘ ’ ‘T’T f e H2 such that (r ,r ,...,rjf = (r^,r|,...,r^) G is a transitive ¿-subgroup of ^-(R). Choose any f e G, such that r f = r1. Set y 2 co oo % (r0,rj,r2 , .. . ,rj " Note that tr0’rl ■' ' '-L ■ (r0’S’ ’'' >r-’ E °1 implies each of these vectors have inversely well-ordered supports. For each yeF, if r 0 0 or r , f 0, choose any f e j4(R) which Y Y' Y maps r to r' If r = r' = 0, choose f to be the identity Y Y Y Y Y element of the group T(R). Fill in the matrix for the permutation f £ as follows: = f = f 2, (-,-,^,...,^-2) l,(-,r0,...,rro-l) 0,(rQ,r1,...,rJ 2,(-,r0,...,r_-l) fl,(rn,rq,...,rJ £0,(rq,rQ,...,rm+l) = f 0’ 1 1’ 2 £2,(r0,ri,r2,...,rJ ’ £l,(r1,r2,...,rQo+l) ^,(^,...,^+2) Outside of these entries, fill in the rest of the matrix for f with the identity element of _4-(R) . Note that after a certain point the matrix for f will be entirely set equal to the identity permutation. Then f e H1 and (rQ,.,rjf = (r^,r^,...,r^). Thus is transitive on 0^. It now follows that the orbits of j4-(ft) in ft are the chains 0 and 02- Thus .4 (ft) is an ¿-group with the disjoint support property. 90 12 2 1 Define ? and 6 ’ on Op and and € ’ on 0^. Then 2 " ^p <$1,2 = iYp and C - £2,1 = èThus ft has the properties desired. // We may generalize the results of the preceding example to the 1 2 case discussed in Theorem 3.21 (£ 2 2 C ’ Ç and 2 1 ¿2 = ^2 1 = * ^^2 = &?) ' I™ tlie general casej as an the example, 12 G1Y1 2 and G1X ’ are dense an ®2' using the denseness of these two sets in 0^ and the transferring argument described in Example 3.22, there is a countable tower of convex P^-congruences on 0^. In the example we noted that the join of the convex P^-congruences was a proper P2-congruence on 0^. We also noted that for (rQ,rp...,roo) e Op (r^Tp ., . fl (^,^,...,^+1)^ = 0 on Op These conditions were necessary to preserve the transitivity of P2 on Op These ideas also may be abstracted to the general case to preserve the transitivity of P2 on 0?. We may transfer 2 1 the same type of information to 0^ by the map y ’ • From the construction of generalized wreath products and the information above, we now see that the ¿-groups 2(s/B^) and G2 (s ¿3 2) may be represented as a wreath product of permutation groups with countably many factors and the countably infinite index set of the factors has a largest element. At the top level of these wreath products, the component groups will be periodic ¿-groups (in McCleary's sense). We now consider category 5 2 S <5 ’ on 0^ and ^2 — 1 $ ^2 ^2 on °2^ ' As in categ°rF 4> for 2 1 purposes of the theorem we assume that = 2’ &1 = ^1’ &2 ~ ’ 91 THEOREM 3.23. Suppose ft is a po-set satisfying our hypotheses such that (a) £2 = ¿T?3 ¿z <#-, = on 0^, and (b) £% = C?3'1' G2 2 $2 = %2 °n G 2‘ Th-en= following four conditions are satisfied: (i) X 9 is a one-to-one, order preserving map from 0 to a 1)2 1 1,2 dense subset of 0^. x is order preserving but not 1 2 one-to-one, and O^y 3 is not dense in 0^. Also, for 1,2 2,1 each s e 0^, sy? 2 <■& ax is a one-to-one order preserving map from 0% to a dense subset of 0^. X<> j £s order preserving, but not one-to-one, and O^y^ is not 2 1 dense in 0?. Also, for each t z 0^, ty^ <0 ^X 3 • 2,1 (ii) x2 2 and y can be extended to order-isomorphisms - - -2 1 - - X2 g : °i °2 x : °2 °1" Moreover, 2 _ -2,1 1,2 (^2,2 X Also, x can be extended to the points of Cf and xc , °an be extended to the points of 0 . 12)1 2 (/i•l•l•)I TF, or each, s z 0/I ^, {r s(/y- 1,2y- 2,2 ),n n +l,+2, .. . } is cofinal in 0^. For each t z 0%, {t(y2,2y^3 2)m m +l,+2, . . . } is cofinal in 02- 1,2 2,1 (iv) The chains O0^? aannd 02 and the maps y d,23 and xo 7 are acceptable. 2,1 The group A-(ft) is constructed as follows. Set -1,2-2,1. ,-1,2-2,! 2,1.^ . 2,1. = {f z A(02) I f(y" 3 '■ 3 ) = (y X )f, and (0 y )f = 0 y } 1,2 and set Gz 1 = z-&(02) | g(y2" 2y2j 2) = (y2j 2X2j 2)g and (°2^i 22g = °lxl 2^’ T2ien G1 2 a transitive ¿-subgroup of A(O^) and G2 2 a transitive ¿-subgroup of A(02). Here, y^ 2 induces 92 an ¿-isomorphism * : G- 9->G , and AW) = {(f,f*) ( f e G- Z Z Zj J -J j U Conversely, suppose we are given chains 0^ and 0% and maps y’2, x7 Q, y^, and x9 7 such that (i), (ii), (Hi) and (iv) are satisfied. Then we may construct a po-set ft which satisfies the initial hypotheses of this theorem, and A(ft) has the form described above. We may geometrically represent the partially ordered sets which fall in this category by the following picture. Proof: We may use the same proof as the proof of Theorem 3.21. // The comment about condition (iv) is the same as that following Theorem 3.21. Also, the generalization of this theorem is obtained in the same way as the generalization of Theorem 3.21. In the previous category, we showed that there must be a countable tower of congruences 1 2 on 0^ and Op This followed from the denseness of O^x ’ and 12 Ox, o in G.. In category 5, 0 v ’ is not dense in 0 and so 1 1, Z Z 1 a the argument sketched out previously is no longer valid. In fact, in Chapter 4 we give an example of a po-set satisfying the conditions of category 5 where 0^ and C>2 admit precisely one convex congruence. 93 1 2 The final category Is category 6 where — 2 ^1 ~ ^1 2 1 and <^2 — ^2 1^^ ’ $^2 — ^2' ASain’ we assume &2’ ¿0^=2/^, ^2 = ^2 1’ and &2 ~ %~2’ Snd tben we can make a generalization using the techniques previously developed. To examine category 6, we must make use of a number of methods which were not previously used. THEOREM 3.24. Suppose ft is a po-set satisfying our hypotheses such that (a) £ 1 - 2 = %1 °n °13 an^ (b) = Go 7 i G?32 $■ (3 = 2/. on 0Then, the following three ¿j ¿j j 7 ci ci u conditions cere satisfied: 12 . ■ (i) x 3 is not one-to-one but it is order preserving, and 12 - O^x 3 is a dense subset of 0%. y^ ds one-to-one, 12 order preserving, but OfXy g no^ dense in 0%. y ’ and x-, v may be extended to the points of 0 and for 12 2 1 each s e 0^, sx^ g but it is order preserving, and 0 y 2,1 is a dense subset of Cf. y9 7 is cne-to-one, order preserving, but 0%Xg 2 !' A2,l 2 1 is not dense in 0 X 3 and y may be extended to 2, 1 3,1 the points of 0% and for each t e 0%, ty^ 2 — 1 2—2 1 7i (ii) For each s e 0^, {s(x 3 x 3 ) \ n = +l,+2, ...} is cofinal -2 7-7 2 m in 0^. For each t e 0%, {t(x 3 X 3 ) m +1, +2, . . . } is co final in 0%. 1,2 2,1 (iii) The chains 0. and 0% and the maps y *1,2’ X and y9 7 acre acceptable. 1 The construction of the f-group A(&) is as follows. Let 94 1 2 H {/ e A(O^) | for all sctjSg e G23 sa ^‘er‘ X 3 sg an<^ onty s f Ker y"3 2sDf}, and let K = {g e .4(0 J | for all t ,t z 09, CX P z CX p Z 2 2 2 1 t& Ker x 3 t if and only if t&g Ker x 3 ^g} ■ ^or> eac^ f e #■> 2 2. 12 12 we may define a function ft^ z A(O^y 3 ) via: (sy 3 )f*j - s(fx 3 )• This permutation ft^ may be extended to ft^ zA(O^). For each 2 1 g z K, we may define a function g*2 e -4(0^ 3 ) via: 2 1 2 1 (ty 3 )g*2 = ¿^X 3 This permutation g*2 Tay he extended to a ___ _ _ 2 2-2 1 -1, 2-2, I,, permutation g*2 e Set G^ % = {f z H \ fy 3 y 3 X X A Ofy3 2f = 02y2i \ and 0^?^ = 0%} and set —2, 1-1, 2 _ -2,1-1,2- 1,29 17 ,29 G2,2 = E K ‘ = X~3/^3/, C>2x g = ^x and O.,g* = 0 }. Then, we may define an ¿-isomorphism *:G,.^-G by: 122 1 1^ 2 2j 1 for f z G2 2, f* = f’i1\0 and = \ f e G2 2^’ 2 Conversely, suppose we are given chains Oq and Op and maps '2 2 1,2 2f, 11 , X 3 X2 23 X c, j. satisfied. Then we may construct a po-set ft which satisfies the initial hypotheses of this theorem, and A(0.) has the form described above. We may geometrically represent the partially ordered sets which satisfy the hypotheses of this category by the following picture. 95 Proof: Condition (i) follows from the definitions, Proposition 3.4, and Lemmas 3.7, 3.8, and 3.9. Condition (ii) follows from the hypothesis that ¿8^ = and ^3^ = . Condition (iii) is obvious. We compute the group ~4(Q). First, we must verify that 1 2 ------f*l e -4-(OjX ’ ) and can be extended to f*^ e ^-(O^) ’ cbeck 1,2 1,2 f* is well-defined. Suppose s^y S1X IVe must show 1,2 l,2r 1,2. 1,2 , 1,2- , 1,2 sQy f*T = slX f*r soy f*T = sofx and SjX f*J = sJfx . 1,2 1,2 1,2 1,2 Since sQy S1X Sq Ker y ’ Sp f e H so s f Ker x ’ s^f, i.e., s^x1’2 = s1fx1’2. Thus s()x1’2f*1 = s1y1’2f*1, and f*J is well-defined. Next, O1y1’2f*1 °1X1’2- Let sy1,2 e C^y 1,2 Then 1,2 sfyl’2. f e xf-(O^) so sf e 0^. Thus, sfy"*"’2 e O^y"'"’2 sx f* 1 and the containment follows. 12 12 Third, f*i is one-to-one. Suppose s^y ’ / s^x ’ . We must show s0y1’2f*1 0 s1x1,2f*1. SoX1’2^*! = s0fy1,2 and 12 12 12 12 s y ’ = s fy ’ . Since s^x ’ / s^x ’ , it follows s^ is not 1 2 related to s^ under Ker y ’ . f e H so s^f is not related to 12 12 12 s^f under Ker y ’ , i.e., s^y1’2 0 Slfy ’ . Thus 12 12 SqX ’ f*i / s^y ’ and the one-to-one-ness follows. 12 12 Fourth, f*i is order preserving. Suppose s^y ’ s^y ’ . We show sQx1’2f*1 £ s1x1,2f*1. s0x1’2f*1 = s^y1’2 and siX1,2f*i = sifx1’2- 1,2 1,2 S0X - six so Sq < s^ in 0^ or 1,2 1,2 SOX S1X If Sq < Sp then, since f e-A(O^), s^f < s^f 12 12 12 and x ’ is order preserving so s^fx ’ s^fX ’ . If ,1,2 S0X = six.1,2 then So Ker .1,2 sl’ S° S0f Ker X.1,2 ’ Slf’ i-e-’ 96 , 1,2 - 1,2 12 12 sof* = Slf* • Thus, in either case, sgfX ’ £ s fy ’ . 12 12 Finally, f* is onto. Let s^y ’ e O^y ’ . We show there 1 l,: 12 12 12 exists s^y ’ e O^y ’ such that s^y ’ f*^ = s^y ’ . Since f e-4(Op there exists sq e 0q such that sqf = sn. Then 1 1 1 1,2. . 1,2 1,2 SjX f*j = Vx = SOX and onto follows. 12 12 - Therefore f*^ e >4(0^y ’ ) . Since O^y ’ is dense in Op f* can be extended to f* e >4(0 ) . Similar facts hold for the J. J. 4 function g*2- We now verify the correspondence * is an ¿-isomorphism from Set G {f I f e G, . } and G !,1 ig î g e G2,l' Gl,2 t0 G2,l 1,2 1,2- We show * defined from G to G? i by f* = f*i is an 1,2 ¿-isomorphism.- Then, by composing extension, * and restriction we get an ¿-isomorphism * ’ 2 G2 1' First we must show that * is well-defined and * maps G^ to G. .. . * is well-defined since the correspondence is well 4,1 J- defined. We next check that for each f e G^ f*y2’-'-y2’2 = y^’^y2’2?*. Since O^y1’2 Is dense in Op it suffices _ u „ u -I,2 n -I,2 -l,2---2,1-1,2 -1,2-2,1-1,2-- to show that for each sy e O^y , sy ±*x y = sy y y f* Note the following fact: From the fact that for each s e Op 12 12 - sy ’ f*^ = sfy ’ , and the fact that Cy is dense in Op we may ’ _ _ 2___ __ 12 -12 -12 infer that for each s e Op sy ’ f*j = sfy ’ . Choose sy ’ e O^y ’ . -1,2 -2,1-1,2--. -1,2 -2,1-1, 2-,—-, -1,2 -2,1--1,2 Then sy (y y f*) = sy (y y f*J sx Cx fx ) = --1 2-2 1-1 2 -!,2r> - 2,1-1,2. -1,2 .--2,1-1,2 s(fx X X ) sx Cf*xxv ’ X ) = sx ’ (f*x ’ X ’ ), so the desired property follows. It is obvious by definition of * that O,x1,2f* = °iX1,2 and O.f* = 0. . Next, for t ,t. e 0. we must 1A 1A 2 2 ’ a £ 2 21 . -- 2 1 — show t Ker x ’ t„ if and only if t f* Ker x ’ tnf*. Assume a p a p 97 2 1 2 1 2 1 t Ker x ’ t . Then t x ’ = toX ’ • We need to show CX p CX p t f*x2,1 = t f*x2’F. We claim that for each t e CL, tf*x2,1 - tx^’^f. ex p / -1,2 To verify this claim, since O^x is dense in Og, it suffices to _i 2 - -1 2-2 1- -1 2___2 1 show that for each sx ’ e Og, sx ’ x ’ f = sX ’ f*x ■ -1,2^--2,1 -1,2-=—-2,1 --1 2-2 1 -1 2-2 1- sX ’ f*X = sX ’ f*TX sfx ’ X ’ = sx ’ X ’ We have therefore -21 -21 established our claim. We have assumed t x ’ - t^X ’ ■ Then a p -2 1- -2 1- -- 2 1 t x ’ f = t,oX ’ f and by the claim this implies t f* Ker x ’ t f*. ex p ot p 2 1 Next assume t is not related to t under Ker x ’ • We must show a p 2,1 t f* is not related to t f* under Ker x i.e., we must show CX p taX2,1f t tgx2,1f- Not tQ Ker X2,1tg implies px■-2’1 * v2’1 -.2,1: -2,15 2,1. Then t x ’ £ / t y Therefore, for t ,t e CL, t Ker x CX p a’ 6 2,1. if and only if t f* Ker x ’ t„f*. Now, note that since CLf* = CL, a p z z 2 1 Og is dense in Og, and f* respects Ker x ’ , it makes sense to speak of the image of f* under *g. To finish proving that G C G? , we must show that 0 (f*)* = 0 . Note that 1 3 Z Z 3 J. -L ZJL -1 2-2 1 Ox ’ X ’ is dense in 0^. Therefore, to complete the proof of this step it suffices to show that for each f e G^ g and for each -1,2-2,1 . -1,2-2,1 -1,2-2,1? SX1’2X2’1[(£*)*2]• By using a SX X o 0lX X , SX X f -2 1-1 2 remark similar to the one used to show f* commutes with X ’ X ’ ; -1,2-2, ILL/; -i -1,2^--2,1 ?-l, 2-2,1 -1 2-2 1- sx X [(¿*)*9J = SX f*X = sfx X sx ’ x ’ f* Therefore, Opf*)*g = 0^ and it follows G^ g* C Gg Next, * preserves function composition. Let f,g e G^ g and 1,2 1,2 1,2 1,2, sx E 0lX We show (sx ’ )(f*)(g*) = (sx ’ )(fg) (sx1’2)(f*)(j*) = (sfX1,2)(g*) = sfgx1’2 = sx1,2(fg);. We now show * preserves the lattice operations. The verification is for v. The argument for a is similar. Let f,g e G^ g and 98 SX1’2 e OlX1,2. sx1’2^ V g)* = [s(f V g)]x1,2. Suppose s(f v g) = sf. (sx1’2)(f* v ¿i) = sx1’2?; v sx1,2g; = sfy1’2 V sgx1’2. Since s(f v g) = sf, we have sg <_ sf. By the order preserving 1,2 property of y sgx1’2 £ sfy1’2. Thus (sy1’2)(f* v g*) = 12 12 12 12 12 sfx ’ v sgy ’ = sfy ’ = [sf V sg]y ’ = [s(f V g)]y ’ = 12 (sy ’ )(f V g)*. A similar argument holds if s(f V g) - sg. We have therefore shown * is an ¿-homomorphism. Let e; denote the identity element of >4(0 ) and e^ denote °1 1 2 the identity element of >4(0^)• To show * is one-to-one, we suppose -1 2-2 1 f* = e^ and we wish to show f = e= . Since 0 y ’ X ’ is dense U2 U1 1 -1 2-2 1 -1 2-2 1 in Op it suffices to show that for each sy ’ y ’ e GjX ’ X ’ , -1 2-2 1- -1 2-2 1 SX ’ X ’ f = SX ’ X ’ • By using some of the remarks made previously -1 2-2 1- -1 2 ---- 2 1 -1 2-2 1 sy X ’ f = sy ’ (f*y ’ ) = sX ’ X , the last equality following from the fact that f* = e^ . Hence * is one-to-one. U2 To show * is onto, choose g e G? p restrict its action to the points of 02, take its image under *2, and extend this to a function on Op By using similar methods to the ones used to show G 7* G7 , we show this function is an element of G „ and its image under * is g. Thus * is an ¿-isomorphism from 2 to By the composition of extension, *, and restriction, we define an ¿-isomorphism * : G G . By similar arguments to the ones 1 , z z, 1 used in the previous categories, we see that AW {(f,f*) I f e G^ 2) To show .A(ft) is equal to this set, we must show each (f,f*) is order preserving on ft. Let upu2 E with w <_ i^. To show 99 m^(f,f*) <_ (jo2 (f, f*) • If either m2,m2 e or upu2 e ®2 tben tbe conclusion is obvious. Next, suppose w e 0^ and e 0^. Then w2. f* e AO.,) so w^x^’2f* <_m2f*, i.e., 1 2 w fX ’ <_w2f*. This implies o^f 5^ a^f*, i.e., m^(f,f*) <^m2(f,f*) -2 1 Finally, suppose e C>2 and m2 e 0^. Then o^x ’ w2 so _2 1- - __ 2 1 1 m^x ’ f £ w2£" Ph115 ’ S u2£ and dt f°H°ws that w^(f,f*) <(w2(f,f*). Therefore .£(&) is of the form described in the theorem. Conversely, suppose we are given chains 0^ and C>2 and maps 12 2 1 X ’ , Xj 2, X ’ , and x2 T such that W> (dl) > and Ciii) are satisfied. We must construct a po-set ft satisfying our conditions. By (iii) , define a partial order of the set ft = 0^ U C>2 using 1,2 .. 2,1 the maps Xj 2’ > X2jl’ and X such that the orbits of j4(ft) in jQ, are precisely 0^ and 02> Every orbit of W(ft) in ft is a chain. By Theorem 2.4, _4(ft) is an ¿-group with the disjoint 1 2 support property. Define 2’ » and on and & 2 1’ 2 1 £ ’ , and (3 on 0 . Then, by the properties of the maps 2,1. 2 £ on 02> Then, by the proof of the first part, j$(ft) = {(f,f*) I fe61 2). // Using the generalization techniques explicitly given for categories 2 and 3, we obtain the general form of ^4(ft) listed for category 6. The comment on condition (iii) should be similar to the one given for condition (t) in Theorem 3.20. For purposes of constructing examples, we have found it easiest to prevent mapping 0 to C>2 by making 0^ 100 not order-isomorphic to Og. This completes the verification of the catalog for the two orbit case. In Chapter IV we give examples of each of the categories introduced here, and in Chapter V we generalize these results to po-sets ft such that /(ft) is an ¿-group with the disjoint support property and /(ft) has an arbitrary number of orbits in ft. CHAPTER IV EXAMPLES In this chapter we present examples of each of the categories of partially ordered sets introduced in the previous chapter, and we explicitly compute the ¿-group of order automorphisms of each po-set. The examples given will demonstrate that the theorems proved in Chapter III are quite constructive. 1,2 EXAMPLE 4.1: Category 1 2 = on 0 and G = 62’1 = on 02). 2,1 Let Cy = Q and 02 =n. Then 0^ is not order-isomorphic to 02- Order the set ft = 0^ U 02 via: - <^; <_jQ = <_n, and each point of 0^ is unrelated to each point of 02, i.e., ft consists of two unrelated homogeneous chains. This is an example of the situation discussed in Theorem 3.6 (i). From this theorem it follows that AW = AW x A(P) ■ Now suppose we let 0^ = 2 and 02 = 2. We cannot order the set ft = 0^ 0 02 as above since we may map 0^ to 02 and 02 to 0^ and still preserve order. However, we may use the following construction: < £|n = —7’ and each point of 0 is an j '“'2 "*■ upper bound of each point of 02 in ft. Because of the discreteness of the two chains, we cannot map points of 0^ to points of Cy. -101- 102 This is the case discussed in Theorem 3.6 (ii). From this theorem, ^4(ft) =>4(Z) xj4(Z), which is ¿-isomorphic to Z * TL. If we let = C>2 = Q, we cannot construct an example for this category. The construction given in the first paragraph will not work because 0^ is order-isomorphic to 0^. The second construction also fails since a copy of the rational chain above a copy of the reational chain is order-isomorphic to the rational chain. Hence, the group of order automorphisms of this set would have only one orbit. // EXAMPLE 4.2: Category 2 ( ‘2 - e2,1^C2,l -^2 L' Let 0^ = Q and C>2 = H Then 0^ and 0? are dense subsets of the Dedekind complete set Let 2 ' ^1 ^2 be tbe adentaty 2 1 map and y ’ : 02 0^ be the identity map. Partially order the set ft = 0^ U 02 as follows: £|Q = <^, , for each s e 0p L 2(s) = {t e 02 | t < sx 2) and 2(s) = 0, and for 2,1 each t e 02 let L2 (t) =0 and U2 ^(t) = {s e 0^ | ty < s}. Pictorially, the set ft appears as follows: A Note that "X1.2 iS dense 9. 1 7 in C>2 and H y is J dense in °1- *1,2 and y 1 > 2 are one-to-one and beix+.cLx' ûÇ o order preserving. 103 X.. 7 can be extended to an order-isomorphism x, 7 (the 1 3 Z 1 3 Z identity map) from 6^ - R to Og = R. The map Xj 2 induces an ¿-isomorphism *, the identity ¿-isomorphism, from / (R) to >/(R). Note that G „ = {f e /(QJ) | H f = H } = /(QJ) and 1 3 Z Gg i = (g c /(H ) | Qg = Q) = /(H ). By composing extension, *, and restriction we define an ¿-isomorphism * : /(Q) -*/(H ). Then, by Lemma 3.13, /(ft) = {(f,f*) | f c/(Q)). The same construction as above may be repeated if we let 0^ = Q and Og - Q-tt\{0} (the rational multiples of ir with the number 0 eliminated). The difference is that G^ g = (f e /(Q) | Ogf = Og}C2/(Q) and Gg = (g e /(Og) | Qg = Q) Cf/(Og). However, we may still define the ¿-isomorphism * : G^ g -> Gg and /(ft) is again equal to {(f,f*) | f eGj g). // EXAMPLE 4.3: Category 2 (£ T = £ 1 g £ ’2 = on 0J and f2S C2’d Let 0^=2 and Og = R x 2. Define Xj 2 : + Og by ¿or 2 1 n0 e 2, ngX-L 2 = sup{(r,nQ) £ 1 * Z} and x ’ : Og + by for ’ rcR (r,n0) c R i z, (r,n0)x2,1 = n0- Xj 2 is one-to-one and order preserving, but 0 Xj 2 is not dense in Og. Moreover, O^x^ g is a - 2 1 subset of an orbit of /(Og) in Og. x ’ is not one-to-one, but it 2 1 - is order preserving, and OgX ’ is dense in 0-^ = Op Define a partial order on the set ft = 0^ U 0g by: i|o2 = W for each no E °1 set L1 2^0^ = 0 O2 1 (r,n) <0 rigXj p and ly 2(nQ) = and fOr 104 each (r,nQ) e °2 set L2 ]/(r’n(p) = $ and U2 pO’Og)) = in e 02 | nQ j_2 ob 2 1 The chain CL/Ker y ’ is order-isomorphic to 2, and the map 2,1 nl,2 : °1 " °2/Ker x is the identity map. Similarly, the map 2 1 2 1 ’ : O2/Ker y ’ ■* 0^ is the identity map. Since - Z is Dedekind complete, n induces the identity ¿-isomorphism 1,2 = n 1,2’ nl,2 * : .4(2) ->4?-(2). Then, by Theorem 3.14, ^(ft) = { (f ,(<}>; f*) ) e AW x [4(R) Wr>4(2) ]}. // 1 7 EXAMPLE 4.4: Category 3a (£^ = £ ? - & ’ 2^ and for each s e 01 inf Ux 2(s) = sup L^ 2(s), and = ^2 1 = ^2 and for each t e 02 inf U2 ^(t) = sup L2 ^(t)). Following the statement of Corollary 3.15, two methods of constructing examples for this category were suggested. We give examples of both methods here. First, let 0n - Q and 02 = Q*ir (the rational multiples of it) . 1 1,2 Define y^ 2 < 0^ -> 02 by for q e 0^, qx12 = q*1T and 2,1 2,1 define 02 -+ 0^ by for qir e 02, qiry qir/ir '2,1 y 2 is one-to-one, order preserving, and GjXj 2 = G2, a dense 2 1 subset of R. Similarly, y ’ is one-to-one, order preserving, and 2 1 02x ’ = Cy, a dense subset of R. Define a partial order of the set n - Oj 02 as follows: = <^, for each qQ e « set L^ 2(qQ) = {qir e C>2 | q*ir £ q^ir) and set U1 2^0^ = E °2 V < q7T^’ and £°r each £ SSt L2 1^0^ = {q E I q < VA} and set U2 T (qQTr) = (q e Q) | q^/ir £ qQ) 105 Then ft is a totally ordered set which is order-isomorphic to (ty. However, each rational point is replaced by a covering pair of points, the top member of the covering pair being an element of 0^ while the bottom member is an element of 0^. The group >?-(ft) is constructed in the same manner as those given in Example 4.2. To give an example of the second type of construction we make use of r, sets. We will state a number of facts about these sets .a without proof. For a thorough discussion of the ¿-group of order auto morphisms of an r) set, the reader is referred to ([4], page 289). Let a be any ordered number. A totally ordered set S is said to be an if whenever A,BCS, A < B, and |a|,|b| there exists an s e S such that A < s < B. Now, let s e S and consider {s e S I s < s) = T^S. Choose a subset T' of T of least cardinality such that sup T' = s. The cardinality of T' is called the initial character of s. The final character of s is defined dually. If s has initial character A’k and final character 4/ , we write c and call it the character of s. The extension of any order preserving permutation of S clearly must take points of the same character to points of the same character. Suppose that 0 is an r,^ set constructed by using a nontrivial ultrafilter on H and by using as structures the totally ordered sets R. 0 denotes the Dedekind completion of 0. Points in 0 have character c and points in 0\0 have characters conri, cTO’ °r C11‘ Set 0 = 0 U {s e OSO | character of s = c.^} and set 02 ={s e 0 [ character of s = c^^}. Let 0 = 0^ U 0^ and define the ordering on ft to be the natural total order on this set inherited from 106 6. 0^ and Og are dense subsets of 6 and are complete orbits of /(0) in 6. /(Op is ¿-isomorphic to /(0) by extension. The identity map * is an ¿-isomorphism from /(0) to /(0) . /(0) is ¿-isomorphic to /(Og) by restriction. Thus, there is an ¿-isomorphism, * : /(Op ->/(Og) defined via for f £ /(Op , f* = f*.n • Then, by Corollary 3.15, /(ft) = {(f,f* ) f e/(0 )}. |Og 1 Note that this same construction would not have worked if we let 0^ = Q and Og = H . For if we let the order on ft = Op Og be the natural total order on R, then /(ft) would have only one orbit in ft. In fact, /(ft) would be ¿-isomorphic to /(R) . // EXAMPLE 4.5: Category 3b ( 0 = Q = H . Define 9 ’ 0, % and Let and 0? y4, ,2 .1,2 .1,2 X : °l + °g via: for q0 £ °1’ q0Xl,2 = q0 and q0X - %-0 +l- 1,2 Then y and are one-to-one order preserving, and OjXj 2 1,2 and 0^y.1,2 ’ are dense subsets of 0g. Define X2 j : 0g -> 0^ and 2 1 - 2 1 X ’ : 0g 01 by: for iQ £ 0g, iQX2jl = iQ-l and iQx ’ = i O’ The properties of these maps are analogous to those of y^ 2 and 1 2 • X ’ . Partially order the set ft = Op O^ as follows: = 1^, o2 ’ > for each qo E °1 set h/V ’ fi e oJ2 I' i' Ht / and set Uj CqQ) = (i Og q + 1 < i), and for each i e 0 set 2 e U ih u / L2>i(i0) = {9 a 01 | q i0-H and set l^pp) = {q £ Oj iQ q). Pictorially, ft is as follows: 107 Define the convex equivalence relation ¿3^ on 0^ and (8 on G>2 in usual way. Then all points of 0^ are related under (8 and all parts of 02 are related under /8^. We claim that elements of A(ft) must map ^2 LU JJUJJlLb U1 U»2 • By noting that points that are unrelated must remain unrelated under the action of a function in AW and by noting that /S- 20^ and ¿82 ~ ^2’ we can sh°w that if we map any point of 0^ to a point of 02, we must map all points of 0^ to 02 and all points of 02 to Cy. This cannot be done, however, since 0^ is not order-isomorphic to 02- Hence, the claim is established. We may therefore consider >4(ft) as a subgroup of -A-(Op x^(02). We show it is an ¿-subgroup, y may be extended to an order-isomorphism y 7 (the identity map) from 0^ = R to 02 induces the identity ¿-isomorphism '1.2 - - -1 2-2 1 : A(0 ) A(O^) . Note that y ’ X ’ ~ + 1, the permutation which is 2,1-1,2 translation by 1, and y = +1 Set G = {f e J$-(Q) [ H f = H and f(+l) = ( + 1) f) = (f eA(Q) I f(+l) = (+l)f) and set G2 1 = (g e >4(11 ) | g(+l) = (+l)g) By using the arguments of Theorem 3.20, * an ¿-isomorphism * : G1 2 g2 i- Set G = Kf,f*) I f e Gj 2). Clearly ^(ft) Q G. A straightforwardc omputation shows each element of Gi s order 108 preserving on ft. Thus G - -4(^3 • G is transitive on 0^ and 0^ • Hence every orbit of j$(ft) in ft is a chain, and it follows that j4(ft) is an -¿-group with the disjoint support property. The congruence structure is clearly what was claimed. Note that in the above example = 2/^ and = U^. We may modify the example so that and Let 0 = Q) x 1 and 02 = H x Z. Define y^ 2 : 0^ -> 02 and y ’ : 0^ -> vaa: 1 2 for (q0,nQ) e 0^ (q0>n0)Xl ? = (q0,nQ) and (q0,nQ)y ’ = (q0+l,nQ). - 2 1 - Define Y2 j : °2 °1 and X ’ : °2 °1 by: £or e °2 ’ 2 1 (i0’m0-lx2 1 = (i0"1’m0-) and Ci0’m0)x ’ = 7116 Partial order on ft ~ 0^ U C>2 should be defined analogously to the one given above. Then, for each nQ e Z, QJ x {nQ} is a proper ¿8^ class and its corresponding ¿$2-class under the map defined in Lemma 3.18 is H x {n^}. By using the methods of Theorem 3.20, we define an -¿-isormophism ~ : A (Z) ->_4-(Z) and * : G^ 2 ^2 1 (where these groups are -¿-isomorphic to the ones defined in the first paragraph of Example 4.5). In this case, _4(fi) = {CC4>jig) , e [G1 2Wr^(Z)] x [G2 ^Vr^CZ)] | if (Q £ {nQ}) = f then 2(n * {nQ}) = An example for category 4 was given in Chapter III. We now give an example of category 5. EXAMPLE 4.6: Category 5 (£ = £ ? $ £ 1 ’ 2 $ on and ¿2 = £2,1£ C2,V? °n °2') ’ Let 0^ = I x J = 02> We denote the elements of 0^ by (s^,s ) and the elements of CL by (t ,t ). Define y „ : 0 0 and 2 a Y 1,2 1 2 109 1,2 :Oi*°2 V for (Vse)€°i- tsa’se)xi,2 ' H-V and 1,2 (sa’sg)x ’ = inf{(t,s +1) e O2 | t e R), X is one-to-one, 1 3 / 1,2 order preserving, and O^Xj 2 = ^2’ a ^ense subset °f Og■ is 1,2 not one-to-one, however it is order preserving, and O^x ’ is not 1,2 dense in 0, °1* is a subset of an orbit of /(Og) in Og (it consists of all lower endpoints of the natural convex equivalence 2,1 classes in Og) . Define X2 j : Og 0^ and : °2 + °! bx: for (t ,t ) e 0_, (t ,t )x- . = sup((s,t -1) e 0n I s e R) and cr y 2 cr y 2,1 y 1 ' 2 1 (t ,t )x ’ = (tpjt )• We may develop properties of these maps as 1 2 we did for Xi 9 and x ’ • Define a partial order of the set 1 3 z ft 0^ U Og analogously to the one given in the first paragraph of elements of 0g must be mapped to elements of 02 by members of /(ft) since points of 0^ cover a unique point of 02, whereas points of 02 cover no points of ft. Thus /(ft) is a subgroup of /(Op x /(0g) . We now show that it is an ¿-subgroup. - X^ 2 can be extended to an order-isomorphism Xj 2 ■ -> Og. X^ 2 induces the identity ¿-isomorphism * : /(Ôp /(Og) . Set no Gl,2 = {£ e>K°P I fix1’2/23) = Cx1,2X2,1)f, and (O^’bf = O?x2,1} = if e^COp | ffx1’2*2’1) = (x1,2x2,1)f} and set G2 1 = {g eZKOp | g(x2,1X1,2j = Cx2’ 1X1’2)g) • We may use the methods of Theorem 3.23 to define the identity ¿-isomorphism * : G ? G . To give a more familiar description of the ¿-groups G and G , 1,2 L,1 ~2 1-1 2 we examine the map X ’ X ’ more closely. Fix e Og' The -2 1-1 2 image of under x ’ X ’ is the lower endpoint of the natural convex equivalence class x (t +1}Ç.O„. In fact, any point p 2 with second coordinate t has this lower endpoint as its image under p |\je thus have a correspondence between the classes R £ {t } and the lower endpoints of the classes R £ {t^+l}. Elements of G2 respect this correspondence, i.e., these functions must commute with -2 1-1 2 X ’ X ’ • But, the action on a lower endpoint of a natural convex class corresponds to an action on the whole class. We therefore have a correspondence between the classes R x {t^l and the classes R x {t +1}, i.e., the global action of the elements of G on the p 2,1 natural classes of 02 must commute with translation by 1. As long as we respect the natural classes, we may apply any number of ,4(R) within a class and the resulting permutation will still commute with 22,1-1,2^ Set h = G = {(f,f*) | f e G ' }. Clearly A(ft) i G. By using the general form 1,2 of G. ? and G„ we see that each element of G preserves the order on ft. Thus G = ji(ft). G is transitive on the points of Cy and the points of C>2. Hence >4(ft) is an ¿-group with the disjoint Ill support property. The congruence structure on ft is clearly the one desired. // EXAMPLE 4.7: Category 6 (£ = £ 2 £ £ ’2 £ on Oj and ¿2 = ¿2,1^ ^2 °n °2K Example 4.7 is a variation of Example 3.22. As a result, not all the details of this example will be verified. Let T - {0,1,2,...,°=} and A = {0,1,2,...,°°} be two index sets. For each y e f, we define chains as follows: if y = 0 (mod 2) let S = Q; if y = 1 (mod 2) let S = II ; if y let Y Y S = Similarly, for each 6 e A, we define chains T as follows: if 6 = 0 (mod 2) let T. = H ; if 6=1 (mod 2) let T = Q; 6 o if 6 = °° let T. = S. Set S' = TT S and T’ = TT T.. 0_ will 6 y1s 'T Y 6r eAa 5 S denote the vector in S' which has the 0 element of Q appearing in each coordinate whose index is congruent to 0 (modulo 2), /2 appearing in each coordinate whose index is congruent to 1 (modulo 2), and the zero element of a appearing in the “-coordinate. Similarly, 0^ will denote the vector in T' which has the /2 appearing in each coordinate whose index is congruent to 0 (modulo 2), the zero element of Q appearing in each coordinate whose index is congruent to 1 (modulo 2), and the zero element of a appearing in the “-coordinate. Let s = (q ,i ,q2,i ,...»r^) denote a typical element of S' and t (iQ,q1,i2,q3,.. 1 denote a typical element of T' . Set S = {s E S' ! V °C for only finitely many SY y e F} and set T = {t E T' I t6 °T(5 for only finitely many 6 e A} By ordering F and A in the obvious way, we may define total orders 112 on the sets S and T analogously to the way the order was defined on R in Example 3.22. By the methods used to construct generalized wreath products in Chapter II §4, S and T are homogeneous chains, /(S) is ¿-isomorphic to Wr /(S ) , and /(T) is ¿-isomorphic to Wr /(Tj.) • yeF Y SeA To construct the example we let 0^=5, 0^ = T, and * 12 — ft = Oj U 0^. We define maps y ’ and Xj 2 ^rom p to and 2 1 maps x ’ and X2 j from 0^ to 0^ in the following way. For Cq0’1l’q2’13’'••’J“) e °1’ let (q0> ii’q2’13’ • ‘ • >rJx = ^il’q2’i3’ " ' ’rc°+1) e °2 and Cq0’il’q2’• ’ • ,r“')xl,2 = SUP{ fi’qo’il’q2’ • - ’ ,TJ e °2 I 1 e V- F°r (i0’ql’i2’q35'’‘’rcJ e °2’ let (i0’qi’i2’q3’---’r“}x2’1 = Cql’i23q3i • e °1 and (i0’t|l’i2’q3...... V*2,l = supUqU^qj.q,...,^-!) e (^ | q e SQ). By using these maps, define a partial order of the set ft as was done in Example 4.5. Then, by using the same argument as was given in Example 4.5, we show that elements of /(ft) must map points of 0^ to points of Oj and points of (p to points of 0^• Thus, we may consider /(ft) as a subgroup of /(Op x . We proceed to show that it is an ¿-subgroup of this ¿-group and that the po-set ft has the properties desired. . . 1,2 2,1 . . By examining the maps X , X1 2’ x ’ and x2 1 more cl°sely, we see that these maps have the properties stated in (i) and (ii) of Theorem 3.24. By using the properties of these maps, we may repeat the construction in Theorem 3.24 to obtain the general form of the group /(ft) . 113 Once this general form has been obtained, it is still not obvious that ,4(ft) is transitive on the points of 0^ and transitive on the points of 0^. To prove this, we give an analysis similar to the one given in Example 3.22. We have shown that j4(ft)C^4(O ) x ^4(CL) where j?(0 ) = Wr ^(S ) 1 yeT Y and = Wr j4(Ta). We must specify which members of these wreath <5eA ° products preserve the order on ft. There is a countable tower of natural proper convex equivalence relations — ^2 ~ • • • — on 0^. J is the join of the convex equivalence relations J. Similarly, there is a countable tower of natural proper convex equivalence relations on C^. These relations give rise to the components used to construct the generalized wreath products mentioned above. For the remainder of the proof it will simplify the notation to identify the index set T with the index set A. A typical element of j4(ft) will be denoted by (f,g). Choose s = (q„,1.,q„,1„,...,r j e u. . We recall that ÍVl’Lj’-'L e °1' 1,2 _ SX = (i1,q2,i3,. . . ,rro+l) and sXj 2 = sup{ (i , qQ, ^ , q2 , i3, . . . ,rj e 0, i e Tq}. Note that sy^ 2 is the supremum of a t/^-class. We fix For each s’ e and each y e T, y 1, =Sy’ For s’ e s xf and y = 0, s’ may be any element of = (5. It 1 2 now follows that each s’ e s, has the same image under y ’ . (In fact, Ker yI^’.~2 = Zp . Consider (s^1)x1 2 = {s’Xl 2 S’ E S This set consists of the supremum of all <3 ^-classes which contain elements of CL of the form (-,-,i,,q„,i7,...,r ) (The two -’s ¿ 1 ¿ j °° can be replaced by any element of T^ and any element of T ). The 114 action of a function on the supremum of a C^-class corresponds to an action of the class. By taking the union of the *=9^-classes mentioned, we obtain a convex e/2-class. Since elements (f,g) of 1 2 ^4 (ft) must respect y ’ and y , we see that it must be the case 1,2 that f, , . a = gn ,. . , t -i = 1 > (-, 1 j > ^2 ’ 23’ • • • ’rJ 0 > ’ ^2’ 13’' " ‘ ’ r<*> 1 The point (i^,q2,i^,...,r^+1) of is 2,( , ,i^,q2,i3>...»r^) contained in a -class. The collection of elements of 0, which 1 .1,2 are mapped to (i ,q ,i ,...,rro+l) by y ’ are all vectors of the form (-,-,-,i^,q^,...,r^). This subset of 0^ is an x/^-class. Again, (s By taking the union of these vJ^-classes we obtain all vectors of Cy of the form (-,-,-,-,i^,q^, ...,rj. It now follows that for (f,g) e AW, ¿■j >,1( - , - , - ,il 3,qn 4, • • • r ) _ g2 (- , - ,±i 3,qq 4, • • • , r +1) By continuing this procedure, we ’4,( , , , ,i5,q4, . . . ,rj conclude that for y e T, when y = 1 (mod 2), y , CqQ, ¿i ,q2, ■‘■3’ • • • >rJ Y-i, Ci j ,q2 ’i3’ • ■ • ’roo+^ gy+l,(-,-,i2,q2,i3>•••,rro)• The point (i , q2, i^, q4, . . . ,^+1) is also contained in a -class 12 All elements of si 2 get mapped to this W^-class by y ’ . We map this J^-class to 02 and take the union of the corresponding >=Z -classes. We thus obtain a «=/ -class. This procedure can again be 1 3 continued, and we may conclude that for y 2, yel, if y = 0 (mod 2) then f Y’^0’^l’^2’^3'‘' ' ’Y 1, (i|,q2,Ì3’• ■ • ,r0O+l^ ’y+l,(-,-,i1,q2,i3,...,roo) 115 We may now start with the vectors of 0? and use the maps x 2,1 and X9 -I • By similar arguments to the ones above, we see that for Z 3 J. (f,g) e /(ft) and y e r\{0,°°), f f . . = y i,,±2,43,±4,... ' ,roo-1^ ’ ^ip’cli’i2’Cl4’i4’ ' " ' ,T°^ ' As in Example 3.22, the above arguments set up a correspondence between the / CO -class which consists of vectors with “-coordinate ro and the /C O -class which consists of vectors with “-coordinate r co -1. There is also a correspondence between the -J^-class which contains vectors with “-coordinate r CO and the /c o -class which contains vectors with “-coordinate r +1. Let P denote the projection of elements of /(ft) onto their first coordinates and let P£ denote the projection of elements of /(ft) onto their second coordinates. By using the above anlaysis the components of (Pj’Op are as follows: (i) if ye r\{“} and y- = 0 (mod 2) then the y-th component of (P^,0^) is ¿-isomorphic to (/(Q) , Q) ; (ii) if y e r\{0,°°) and y = 1 (mod 2) then the y-th component of (P^,0^) is ¿-isomorphic to (/(H ),H ); and (iii) for y = “, the y-th component of (P ,0p is ¿-isomorphic to (H ,tt) where = {h e /(R) | h(-l) = (-l)h). We may similarly specify the components of (P2,O2). We may now modify the arguments in Example 3.22 to show that P is transitive on 0^ and P2 is transitive on 0^. We thus conclude that the orbits of / (ft) in ft are precisely 0^ and O2- Hence, >4 (ft) is an ¿-group with the disjoint support property. The desired congruence structure on 0^ and follows 12 2 1 from the properties of the maps X ’ , X-^ X ’ , and y^ j- // CHAPTER V THE GENERAL CLASSIFICATION We have previously considered po-sets ft such that AW is an ¿-group enjoying the disjoint support property and AW has either one or two orbits in ft. We now generalize the previous results to the case where AW has two or more orbits in ft. To do this, we need to develop some notation and to generalize some previous definitions. Suppose ft is a po-set such that A(ft) is an ¿-group which has the disjoint support property and further suppose that AW) has two or more orbits in ft. Let {0 I a e A) be the collection of orbits a 1 of u4(ft) in ft. Then, by Theorem 2.4, each 0^ is a homogeneous chain, and, by Corollary 2.6, AW is an ¿-subgroup of "(T ^4(0 ). aeA 01 Let P denote the projection of elements of AW onto their a-th a coordinates. Making the obvious generalizations from Chapter III, for each a,£ e A, a / 5, we define convex P -congruences g . and a a,£ ct 6 5 ,a C ’ on 0 , and we define convex P -congruences 0 and C, a 5 6,a on 0„. For each s e 0 5 a La,6W, and L congruence on 0^ and £ a Each element of AW between points of ft. In ; -116- 117 jQ(n) in ft and <_ is the partial order of ft, all functions in jHft) must respect the order relation on ft to the points of 0 and CL. We have defined relations r ™ g ,a,g convex P -congruences G o and 6 ’ on 0 and convex P -congruences a,g g,a G „ and G on 0 . By making similar conventions to those used 8,a g in Chapter III and by generalizing the notation in the obvious way, we a may define maps y D ar>d X from 0 to CL and maps yD and a ,p a p g ,a , ,a y ' from 0 to 0 . The congruence structure (when only considering g a G ’ , G o, Go , and G ’ ) must fit into one of the six categories a, g g ,a described in Chapter III, and we may derive properties of the four maps by using the methods of Chapter III. We wish to use to the derived properties of the maps to compute ^-(ft ). CX 3 p Let H = {(f,g) c P x P I there exists h e >4(ft) such that a, g a g h (a) = f and h(g) - g}. Clearly H >4-(ft D), and it follows a, g a, g that ^4(ft ) is transitive on the points of (L and transitive on a, g l the points of 0^. However, 0^ and 0^ may not be the orbits of ^■(ft ) in ft for, by only considering the order relations a, g a, g between 0 and 0^, we may still be able to map points of 0^ to points of Og and points of 0^ to points of 0^. When we consider the full partially ordered set ft, this undesired behavior does not occur. As an example of this phenomenon, consider the partially ordered set ft which is the disjoint union of three rational chains and whose partial order is described by the following picture: 0, °3 Here, the points of 0^ are not related to any points in 0^ or 0^, points of 0^ cover unique points of 0^, and points of 0^ have only upper bounds and no lower bounds in O2. If we restrict the order relation on ft to the chains 0^ and 0^, we may map the chain 0^ to the chain 0^ and the chain 0^ to the chain 0^ and still preserve order. However, each chain must be mapped back to itself by elements of Jf-(ft) . Therefore, instead of computing aifft , we will compute G „ {k ) 0 k = 0 and a,g a,6 a1 a a 0ok - 0Q}. Again, H^ Q£ G^ c so G~ Q is transitive on 0& and a, g a, g a , g 0 . Thus, the orbits of G „ inn ft „ are precisely 0 and 0„. g a,g a,g 1 7 a g g ,a We may now use the maps y , y a, 6 and directly CX > P and x apply the six category classification of Chapter III to derive the form of the group G . G will be one of the four types of a, g a, g .¿-groups introduced in the beginning of Chapter III. The action of G on the points of ft is the natural one. ex j p ex 5 p Before stating the general theorem, we make a few conventions. When we say that, "For each a,g e A, a / g, the chains 0^ and a,g g,a 0 and the maps y satisfy the conditions p ^a,g’ x *6,a1 a"d * given in one of the six classification theorems of Chapter III, "we are not necessarily referring to the precise conditions stated in the 119 theorems. In most of the theorems we assumed one of a number of possible congruence structures for a po-set of a particular category and proved a theorem under the added hypotheses. We then left it to the reader to make the obvious generalizations. The conditions we will refer to in the statement of Theorem 5.1 will be the generalized conditions. We also do not refer to the "acceptability condition” included in each theorem of Chapter III, since a broader notion of acceptability is needed for the general case. We redefine the term "acceptable" as follows: The homogeneous chains (0^ [ a e A} and the collection of maps {x“’e,x o>Y$’a,yo J « / g and a,g e a! are said to be a, p p,a J acceptable if we can define a partial order <_ on the set ft = ¿) 0 aeA such that (i) For each aeA, < (ii) For each pair HD a,g e A, a i g, for each s^ e 0^, and each tQ e 0 , either a,g a, g . -, U (s ) = {t e 0 V t} °r U«,6iSo’ ’ {t E °B V % t}- a, g ü g either La,e(so’ ’ {t e °6 I C V Va,? or L R(S ) = (t £ 0 1 <Ô Va,?’ either a, g 0 g p s) or U (t ) = {s e 0 g ,a ue,a(to’ = {s E °a 0 —0 g,a 0 a t0X "Ö S)- a a and either L (tn) = {s e 0 I s = (k e TT >4(0 ) I there exists (f,g) e G such that a, g ae1 A a 1 a,g k(a) = f and k(g) = g). Note that K is an ¿-subgroup of CX 3 p 120 jtrfO ) and, for each a, 6 e A, >4(ft) C K . For any collection Ct Ct 5 p oeA of homogeneous chains {0^ [aeA}, we may construct all possible ¿-groups K by referring to the classification in Chapter III. ct, P THEOREM 5.1 (The Generalization). Suppose ft is a po-set such that A(q) is an ¿-group which enjoys the disjoint support property. Further suppose A(tl) has at least two orbits in ft. Let {C I a e A} be the collection of orbits of A(Si) in ft, and for a 1 ct 6 each pair a, 6 e A, ct f &, define maps x X J j Xr j a'nd Otj p p.5 ct XPj as usual. Then, the following five conditions are satisfied: (i) For each a, £ e A, a f 5, the chains 0 and 0 and Ct p the maps x dj> Xd j and x’a satisfy the a, P Pj ct conditions given in one of the six classification theorems of Chapter III. (ii) The chains {0 [cteA} and the collection of maps a {yaj$sya gJX6jC\xg a [ a,B e A and a 6} are acceptable. (Hi) For each a, 5 e A where a / B and each s e 0 , a, 5 SXa, 5 SX • p (iv) For each a,5 e A where a f (Z and each s e 0^ and t s 0o, sxaj B t if and only if s <-, tv , and 6 ct t <-^ Sv if and only if tx 1 s. -0$ xa, 5 J s J a. (v) For each a,^>,y e A where a # £, 5 f y, and a f y and eaah s e 0^ sx“’Y sj’’T and Here, if for each a, 5 e A we define the ¿.-groups K as Ctj p was done previously, .A(si) = "^a £' a,5eA j or5 121 Conversely, if we are given chains (<2 | a £ A} and a collection of maps {y o,xa,^XD I a> 3 e A and a f £} such that (i)- otj p Pj a (v) are satisfied, we may construct a po-set ft such that A(ti) is an ¿-group which enjoys the disjoint support property, A(SV = K , and the collection of orbits of A(tt) in ft is a,f5e,4 a* c^3 precisely {0^ | a e A}. Proof: Condition (i) follows from the discussion preceding the theorem. Condition (ii) is clear. The remaining conditions are redun dant since they follow from the existence of the partial order asserted in (ii). We included these, however, to emphasize the compatibility needed between the maps to insure that an order relation does indeed exist. Modifications of the arguments in Chapter III show that QAK».e- a, peA To prove the converse, apply the same techniques used to prove the converse of the six classification theorems of Chapter III. // We conclude this dissertation by presenting one example and by listing some questions which would be interesting to study in view of the results presented here. EXAMPLE 5.2: (4(ft) with an infinite number of orbits in ft). Let T = {0,1,2,...} be an index set, and, for each y e T, let be a copy of the rational numbers. We wish to define a partial order of the set ft = (J (D . The most economical way to describe yeT ’ the relation is to draw a diagram and then give a brief explanation. fi) For m,n e T where m < n, y is the identity map. k J ’ m,n . n. m , , . r n, m . We wish x to be the inverse of xm s0 X 1S the identity map. (ii) Let m - 0 and n e r, n 1. For q e n n ' qx ’n = q + £ ir/21. Thus, for m = 0, n >_ 1, and q e Q , . i=l n qx o = <1 - "/21- 1=1 (iii) We now use (ii) to define xm’n and Xn m f°r m,n e r where m < n. The relation which should hold is -0,n -0,m-m,n . -m,n .-0,m.-l-0,n . X = X X , 1-e., X = (x ) X , i.e., c m . h f n .' -m,n - -0,n . -m,n r ,„i v ^/o1 X = X nX , i-e., X = - L V2 + I vll . m>u V i=i M i=i - -m,n -1 *n,m = U ) • Note that points of are covered by no points of ft, each point of is covered by one point of ft, each point of is covered by two points of ft, etc. Thus, elements of >4(^) must map each chain back to itself, and we may therefore consider -4(ft) as a 123 subgroup of “fi- »(Q ). Let G = {g e yp >i((D ) I ¿or each pair yeT Y ycT * y^y^ e T, g(y^) = g^) an^ there exists m/n e d} such that g(y) = (+ m/n)}. A straightforward computation shows each element of G is order preserving on ft so GC^(fi), Under the natural action. G is transitive on each of the chains 0 , so it follows Y that the collection of orbits of j4(ft) in ft is {0^ | y e T}. Thus, _A(ft) is an ¿-group with the disjoint support property. We proceed to analyze the po-set ft and the ¿-group A(A) further. By restricting the order relations on ft to Qg 0 Qp we obtain a partially ordered set belonging to category 3b of Chapter III. Call this restricted po-set ft 1. By Theorem 3.20, »(ftg p = { (f,f) £ ^-(Q) x -4(Q) I f(+ tt/2) = (+ ir/2)f}. If we project the elements of ^4(ft ■,) onto their first coordinates, we obtain an U > 1 ¿-subgroup P of ^i-(Q) which contains all permutations which are translations by a rational number as well as many other permutations. To see that P contains many more functions than rational translations, the reader is referred to ([4], p. 123). In fact, if we restrict the order relations of ft to any finite subcollection of the chains {(^ | y e r} to form a po-set E and we trace through the construction given in Theorem 5.1, ^4(E) will be ¿-isomorphic to an ¿-subgroup of the direct product of a finite number of the ¿-groups A(Q^.) . Each element of ^S-(E) will have the same permutation from >4(Q) appearing in each of its coordinates, and all the members of -4(Q) which may appear as the coordinate of some member of ~4-(E) must commute with a particular irrational number. The ¿-group formed by projecting the 124 elements of >4(2) onto one of their coordinates will again contain all rational translations as well as many more permutations. However, when we compute the ¿-group 4(£1) , we see that the ¿-group constructed by projecting the elements of ^4(Q) onto a fixed coordinate consists only of rational translations. This statement follows from the following facts. Let denote the projection of the elements of onto their y-th coordinate. Consider the additive subgroup H of R generated by n . { 1 ir/2 | n c I1}. H is a dense subgroup of R, and each element i=l of P must commute with each element of H to preserve order on Q. a r The only elements of >4(Q) which commute with translation by all numbers which are elements of a dense subset of R are the permutations which are rational translations. Thus, we have given an example of a po-set such that if the order relation on is restricted to a finite subcollection of the orbits of >4(fi) in fi, we obtain a po-set 2 such that >4(2) is an ¿-group which is "highly transitive" on each of its orbits in 2. But, if we consider all order relations on fi, _4(fi) is "uniquely transitive" on each of its orbits in Q. // Some open questions which are related to the topics studied in this dissertation are the following ones: (1) For any chain ft, _4(fi) is a completely distributive ¿-group ([4], p. 454). (An ¿-group G is said to be completely distributive if A V = V T A Sifm whenever id jeJ J fed id 1 J (g-j | i e I, j e J) (5s G and all the indicated suprema 125 and infima exist.) If we assume that Q is a po-set such that _4(ft) is an ¿-group enjoying the disjoint support property, is _4(C!) completely distributive? (2) Investigate the structure of periodic ¿-groups, where the definition of a periodic ¿-group is the one given in Chapter III. (3) It is known that every group is (group) isomorphic to _4(E) for some po-set E [3]. Is it true that every ¿-group is ¿-isomorphic to ^(E) for some po-set E? The groups JJ-(E) should be ordered under the induced order. BIBLIOGRAPHY 1. G. Birkhoff, Lattice Theory, Amer. Math. Soc., Providence, 1948. 2. P. F. Conrad, Lattice-ordered Groups, Lecture notes--Tulane University, New Orleans, 1970. 3. R. Frucht, "On the construction of partially ordered systems with a given group of automorphisms," Amer. J. Math., 72(1950), 195-199. 4. A. M. W. Glass, Ordered Permutation Groups, Bowling Green State University, Bowling Green, 1976. 5. W. C. Holland, "The characterization of generalized wreath products," J. Algebra, 13(1969), 152-172. 6. ____ , "The lattice-ordered group of automorphisms of an ordered set," Mich. Math. J., 10(1963), 399-408. 7. , "Transitive lattice-ordered permutation groups," Math. Zeit., 87(1965), 420-433. 8. ______and S. H. McCleary, "Wreath products of ordered permuta tion groups," Pacific J. Math., 31(1969), 703-716. 9. S. H. McCleary, "O-primitive ordered permutation groups," Pacific e J. Math. , 40(1972), 349-372. 10. E. B. Scrimger, Intransitive Lattice-ordered Groups of Order preserving Permutations of Chains, Ph.D. thesis, University of Wisconsin, Madison, 1970. -126-