Definability patterns and their symmetries
Abstract
We identify a canonical structure J associated to any firstorder theory, the {\it space of definability patterns}. It generalizes the imaginary algebraic closure in a stable theory, and the hyperimaginary bounded closure in simple theories. J admits a compact topology, not necessarily Hausdorff, but the Hausdorff part can already be bigger than the KimPillay space. Using it, we obtain simple proofs of a number of results previously obtained using topological dynamics, but working one power set level lower. The Lascar neighbour relation is represented by a canonical relation on the compact Hausdorff part J; the general Lascar group can be read off this compact structure. This gives concrete form to results of Krupiński, Newelski, Pillay, Rzepecki and Simon, who used topological dynamics applied to large models to show the existence of compact groups mapping onto the Lascar group. In an appendix, we show that a construction analogous to the above but using infinitary patterns recovers the Ellis group of \cite{kns}, and use this to sharpen the cardinality bound for their Ellis group from $\beth_5$ to $\beth_3$, showing the latter is optimal. There is also a close connection to another school of topological dynamics, set theory and model theory, centered around the KechrisPestovTodor\v cević correspondence. We define the Ramsey property for a first order theory, and show  as a simple application of the construction applied to an auxiliary theory  that any theory admits a canonical minimal Ramsey expansion. This was envisaged and proved for certain Fraissé classes, first by KechrisPestovTodor\v cević for expansions by orderings, then by Melleray, Nguyen Van Thé, Tsankov and Zucker for more general expansions.
 Publication:

arXiv eprints
 Pub Date:
 November 2019
 DOI:
 10.48550/arXiv.1911.01129
 arXiv:
 arXiv:1911.01129
 Bibcode:
 2019arXiv191101129H
 Keywords:

 Mathematics  Logic
 EPrint:
 Various local edits, following lectures to the Oxford Logic Group and in Kazhdan's seminar. An example of a nonHausdorff pattern automorphism group has been added