The Reconstruction of Cycle-free Partial Orders from their Abstract Automorphism Groups III : Decorated CFPOs

TL;DR

This paper introduces a method to construct decorated cycle-free partial orders (CFPOs) by attaching treelike CFPOs, and demonstrates that the automorphism groups of their components are definable within the overall automorphism group.

Contribution

It presents a novel construction of decorated CFPOs and proves the definability of component automorphism groups within the global automorphism group.

Findings

Automorphism groups of components are second order definable in the decorated CFPOs.

A new construction method for decorated CFPOs attaching treelike structures.

Enhanced understanding of automorphism group reconstruction for 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 final paper, we give describe a way of constructing `decorated' CFPOs by attaching treelike CFPOs to and between the elements of a cone transitive CFPO. We then show that the automorphism groups of the components of of a decorated CFPO are second order definable in the abstract automorphism group of the decorated CFPO.