CHAPTER 1

Extended Introduction

1.1 Introduction

The present Memoir is devoted to a generalization of Droste's work on

homogeneous trees [5] to a richer class of homogeneous partially ordered

sets, which we call cycle-free partial orders, or CFPOs. In [30] Wielandt

proposed that an investigation should be carried out into the possible struc-

ture of infinite partial orders under suitable homogeneity assumptions, and

Droste's work certainly makes a big contribution to this programme. He

found that it was necessary to restrict the notions of homogeneity adopted

quite severely in the case of trees, but given these restrictions derived rather

striking characterizations and classifications. This approach is extended

here, though again it is necessary to make appropriate restrictions on ho-

mogeneity, explained below.

There is an extensive literature on the classification of finite graphs with

an unusually rich automorphism group, and in the last twenty-five years

considerable attention has also been paid to the classification of infinite

graphs, partial orders, digraphs, and similar structures, under strong ho-

mogeneity or transitivity assumptions on the automorphism group, notable

examples being the work of Cherlin on directed graphs [3], Droste on par-

tial orders [5, 6], Lachlan on tournaments [17], and Lachlan and Woodrow

[18] on graphs (stimulated by work of Gardiner [14] and Sheehan [25] in

the finite case).

A partially ordered set (M, ) is said to be k-transiiive if for every pair

A,B C M with A = B and |J4| = k, there is an automorphism of M carry-

ing A to B] and (M, ) is k-homogeneous if for every such pair A, i?, every

isomorphism of A with B is induced by an automorphism of M. Thus

Ar-transitivity says that if A, B are as above, then at least one isomorphism

extends to an automorphism, whereas ^-homogeneity says that every iso-

morphism from A to B extends. Unfortunately this terminology conflicts

with that customarily used for permutation groups (see [30] for instance),

but is the same as is used by Droste, and moreover, if we take homogeneous

to mean '^-homogeneous for all finite fc' then this accords with the general

usage in model theory. As an attempt at clarification, when we come to the

notions which turn out to be the relevant ones in this context, we therefore

preface them with the letters CS, to stand for 'connected set', so that k-

CS-transitive for instance will mean that the automorphism group of the

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