The reconstruction of cycle-free partial orders from their automorphism groups

Is a cycle-free partial order recognisable from its abstract automorphism group? This
thesis resolves that question for two disjoint families: those cycle-free partial orders which
share an automorphism group with a tree; and those which satisfy certain transitivity
conditions, before giving a method for combining the two.
Chapter 1, the introduction, as well as introducing some notation and defining the cyclefree
partial orders (CFPOs), gives a list of the results that this thesis calls upon.
Chapter 2 gives a structure theorem for ℵ0-categorical trees, which is of particular
interest here as their reconstruction problem is completely solved, and for the ℵ0-
categorical CFPOs, which when combined with the results in Chapter 3, gives a complete
reconstruction result for ℵ0-categorical CFPOs.
Chapter 3 asks which CFPOs have an automorphism group isomorphic to one of a tree.
It gives conditions on the CFPO and the automorphism group that allow the invocation
of the work done by Rubin on the reconstruction of trees. In a brief epilogue the results
are also used to show that many of the model theoretic properties of the trees are also
properties of the CFPOs.
The second family is addressed in Chapter 4 using a method used by Shelah and Truss on
the symmetric groups of cardinals, which uses the alternating group on five elements.
In Chapter 5 I give a method of attaching structures of the first kind to structures of the
second, which admits a second order definition in the abstract automorphism group of the
automorphism groups of the components.
The last chapter is a discussion of how the work done here can be made more complete. I
have included an appendix, which lists the formulas used in Chapters 4 and 5, which the
reader can tear out and keep at hand to save flicking between pages.