Zoom link: https://huji.zoom.us/j/86286342430?pwd=vkoWBKRKwpEWEFUs4sSdrhHoViblQe.1
Meeting ID: 862 8634 2430
Passcode: 435280
Title: Taking model-complete cores
Abstract: A first-order theory T is a model-complete core theory if every first-order formula is equivalent modulo T to an existential positive formula; a core companion of a theory T is a model-complete core theory S such that every model of T maps homomorphically to a model of S and vice versa. If a core companion exists, it is unique up to equivalence of theories. Whilst core companions may not exist in general, they always exist for countably categorical theories, and in this case they are again countably categorical. We show that many model-theoretic properties, such as stability, NIP, simplicity, and NSOP, are preserved by moving to the core companion of a theory. On the other hand, we show that the classes of theories of structures interpretable over (N,=) and over (Q,<) are both *not* closed under taking core companions. The first class is contained in the class of theories of omega-stable first-order reducts of finitely homogeneous relational structures, which was studied by Lachlan in the 80's. We conjecture the two classes to be equal.
Joint work with Bertalan Bodor and Paolo Marimon, https://arxiv.org/abs/2512.21278