Tuesday, December 17, 2013, 10.15
Seminar room 0.006, Mathematical Institute, University of Bonn
Speaker: Michael Pinsker (Vienna)
Title: Reconstructing omega-categorical structures from their clones
Abstract: Any countable omega-categorical structure Delta can be reconstructed up to first-order interdefinability from its automorphism group, by the theorem of Ryll-Nardzewski. Even more information about Delta is encoded into its polymorphism clone, i.e., the set of all homomorphisms from finite powers of Delta into itself: the polymorphism clone still “knows” Delta up to primitive positive interdefinability (Bodirsky+Nesetril).
If we consider the automorphism group of Delta as an abstract topological group, then Delta can still be recovered from this information up to first-order biinterpretability (Ahlbrandt+Ziegler). Recently, Bodirsky+Pinsker have shown that if we see the polymorphism clone of Delta as an abstract topological and algebraic structure, then we still know Delta up to primitive positive biinterpretability.
It turns out that for many structures, in particular for some of the most prominent homogeneous structures, the topological structure of their automorphism group is determined by its algebraic structure; consequently, those structures can be recovered from the abstract group structure of their automorphism group up to first-order biinterpretability. This situation was discussed by Rubin in his seminar talk two weeks ago.
We investigate when we can recover the topological structure of polymorphism clones from the algebraic laws which hold in them. We also outline the very recent proof that this is possible for the polymorphism clone of the random graph.