Reconstruction of homogeneous relational structures

Reconstruction results give conditions under which the abstract group structure of the automorphism group Aut(M) of an ω-categorical structure M determines the topology on Aut(M), and hence determines M up to biinterpretability, by [1]; they can also give conditions under which the abstract group Aut(M) determines the permutation group 〈Aut(M),M〉, so determines M up to bi-definability. One such condition has been identified by M. Rubin in [12], and it is related to the definability, in Aut(M), of point stabilisers. If the condition holds, the structure is said to have a weak ∀∃ interpretation, and Aut(M) determines M up to bi-interpretability or, in some cases, up to bi-definability. ∗This research was part of the first author’s PhD thesis at the University of Leeds, and was sponsored by the Istituto Nazionale di Alta Matematica