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

Abstract

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.

The $\aleph_{0}$-categorical Trees and Cycle-free Partial Orders

“…Additionally We say that M , a CFPO, is a CFPO n if M embeds Alt n or Alt * n , but not Alt n+1 or Alt * n+1 . We may therefore restrict our attention to the CFPO n s. However, it was shown in [1] that if M is a CFPO n then there is a coloured tree T (M ) such that Aut(M ) ∼ = Aut(T (M )), where this isomorphism is as permutations groups (i.e. the orbits of M and T (M ) are equal), so we conclude the following: Corollary 4.9 Let M be a CFPO n .…”

“…A formal definition is given in Definition 1. 1. A partial answer to the natural question 'Which trees are ℵ 0 -categorical?'…”