[LX] Aliaume Lopez
Preservation theorems and locality of first order logic
In this talk, we will investigate the tight connection between the locality properties of first-order logic, and the relativisation (or lack thereof) of the Łós-Tarski Theorem to classes of structures. Starting with a gentle introduction to preservation theorems, we will then focus on two particular classes of sentences: existential formulas, and existential local formulas. We prove that existential local sentences are exactly those that can be rewritten in a positive variant of the Gaifman normal form, and can be characterized semantically as sentences that are preserved under local elementary embeddings. As expected, this semantic characterization fails in the finite. Quite surprisingly though, we can leverage this class of sentences to prove the following locality property for classes of finite structures: the Łós-Tarski Theorem relativises to a class C of finite structures if and only if the Łós-Tarski Theorem locally relativises to the class C, provided that the class C is hereditary and closed under disjoint unions. As a consequence of this result, we not only obtain new proofs of relativisation of Łós-Tarski Theorem, but also provide new classes where this relativisation holds.