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
1
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