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

TL;DR

This paper investigates conditions under which cycle-free partial orders (CFPOs) can be reconstructed from their automorphism groups, focusing on when they are equivalent to trees, with implications for model theory.

Contribution

It provides new criteria linking CFPOs and trees via their automorphism groups, advancing the understanding of their structural and model-theoretic properties.

Findings

Conditions for CFPOs to share automorphism groups with trees

Corollaries on model-theoretic properties of CFPOs

Insights into reconstruction of trees from automorphism groups

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.