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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