The Reconstruction of Cycle-free Partial Orders from their Abstract Automorphism Groups II : Cone Transitive CFPOs

TL;DR

This paper investigates the conditions under which cycle-free partial orders can be reconstructed from their automorphism groups, focusing on cone transitive CFPOs and adapting Shelah's methods within group theory.

Contribution

It introduces a novel method to define cone transitive CFPOs within their automorphism groups using group-theoretic language, advancing the reconstruction theory of partial orders.

Findings

Established a group-theoretic framework for cone transitive CFPOs

Extended Shelah's methods to the context of partial order automorphisms

Provided insights into the reconstruction of CFPOs 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 middle paper, we adapt a method used by Shelah in \cite{ShelahPermutation} and \cite{ShelahPermutationErrata}, and by Shelah and Truss in \cite{ShelahTrussQuotients} to define a cone transitive CFPO inside its automorphism group using the language of group theory.