Preservation under Substructures modulo Bounded Cores
We investigate a model-theoretic property that generalizes the classical notion of preservation under substructures. We call this property preservation under substructures modulo bounded cores, and present a syntactic characterization via \(\Sigma_2^0\) sentences for properties of arbitrary structures definable by FO sentences. Towards a sharper characterization, we conjecture that the count of existential quantifiers in the \(\Sigma_2^0\) sentence equals the size of the smallest bounded core. We show that this conjecture holds for special fragments of FO and also over special classes of structures. We present a (not FO-definable) class of finite structures for which the conjecture fails, but for which the classical Łoś-Tarski preservation theorem holds. As a fallout of our studies, we obtain combinatorial proofs of the Łoś-Tarski theorem for some of the aforementioned cases.
KeywordsModel theory First Order logic preservation theorem
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