Algebra Seminar talk
Jan Grebík (Akademie věd České republiky)
Reconstruction of $\omega$-categorical structures from their automorphism groups
By a standard result in model theory a set $A$ is $\emptyset$-definable in an $\omega$-categorical structure $M$ if and only if it is invariant under the action of Aut($M$). That suggests that a lot information about $M$ is coded in Aut($M$). We present two reconstruction theorems that are connected to this philosophy.
- Ahlbrandt and Ziegler's theorem says that $\omega$-categorical structures $M$, $N$ are bi-interpretable if and only if Aut($M$) is topologically isomorphic to Aut($N$).
- Rubin's theorem says that under an additional assumption on $M$ (weak $\forall\exists$-interpretation) it is enough to have Aut($M$) isomorphic to Aut($N$) as abstract groups.