Title: Taking model-complete cores
Abstract: A first-order theory T is a model-complete core theory if every first-order formula is equivalent moduloT 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. Whilst core companions may not exist in general, if they exist, they are unique. Moreover, ω-categorical theories always have a core companion, which is also ω-categorical.
In the first part of this talk, we show that many model-theoretic properties, such as stability, NIP, simplicity, and NSOP, are preserved when moving to the core companion of a complete theory.
In the second part of this talk, we study the notion of core interpretability, which arises by taking the core companions of structures interpretable in a given structure. We show that there are structures which are core interpretable but not interpretable in (
; =) or (
; <). We conjecture that the class of structures which are core interpretable in (; =) equals the class of ω-stable first-order reducts of finitely homogeneous relational structures, which was studied by Lachlan in the 80’s. We present some partial results in this direction, including the answer to a question of Walsberg.
; =) or (; <). We conjecture that the class of structures which are core interpretable in (; =) equals the class of ω-stable first-order reducts of finitely homogeneous relational structures, which was studied by Lachlan in the 80’s. We present some partial results in this direction, including the answer to a question of Walsberg.This is joint work with Manuel Bodirsky and Bertalan Bodor.