A Generalization of the Łoś-Tarski Preservation Theorem

We present new parameterized preservation properties that provide for each natural number k, semantic characterizations of the ∃∀ and ∀∃ prefix classes of first order logic sentences, over the class of all structures and for arbitrary finite vocabularies. These properties, that we call preservation under substructures modulo k-cruxes and preservation under k-ary covered extensions respectively, correspond exactly to the classical properties of preservation under substructures and preservation under extensions, when k equals 0. As a consequence, we get a parameterized generalization of the Loś-Tarski preservation theorem for sentences, in both its substructural and extensional forms. We call our characterizations collectively the generalized Loś-Tarski theorem for sentences. We generalize this theorem to theories, by showing that theories that are preserved under k-ary covered extensions are characterized by theories of ∀∃ sentences, and theories that are preserved under substructures modulo k-cruxes, are equivalent, under a well-motivated model-theoretic hypothesis, to theories of ∃∀ sentences. In contrast to existing preservation properties in the literature that characterize the Σ2 and Π 0 2 prefix classes of FO sentences, our preservation properties are combinatorial and finitary in nature, and stay non-trivial over finite structures as well.