The Reconstruction of Cycle-free Partial Orders from their Abstract Automorphism Groups I : Treelike CFPOs

In this triple of papers, we examine when two cycle-free partial orders can share an abstract automorphism group. This question was posed by M. Rubin in his memoir concerning the reconstruction of trees. In this first paper, we give a variety of conditions that guarantee when a CFPO shares an automorphism group with a tree. Some of these conditions are conditions on the abstract automorphism group, while some are one the CFPO itself. Some of the lemmas used have corollaries concerning the model theoretic properties of a CFPO.Comment: Part 1 of

Elementary Properties of Cycle-Free Partial Orders and Their Automorphism Groups

A classification was given in [1, 12, 13] of all the countable k-CS- transitive cycle-free partial orders for k 3. Here the elementary theories of these structures and their automorphism groups are examined, and it is shown that in many cases we can distinguish the structures or their groups by means of their first or second order properties. The small index property is established for weakly 2-transitive trees, and for several classes of cycle-free partial orders. 1 Introduction A natural generalization of the notion of the semilinearity of an ordering (linearity in just one direction, upwards or downwards) is provided by cycle-freeness, as was proposed by Rubin [9]. This notion was given a precise definition, and many of its basic properties were studied, in [13], and in that paper, and in [1] and [12], a wide class of sufficiently transitive `cycle-free partial orders' (CFPOs) was classified (principally, but not exclusively, in the countable case). In [11] the axiomatizabi..