A Generalization of the {\L}o\'s-Tarski Preservation Theorem

Preservation theorems are amongst the earliest areas of study in classical model theory. One of the first preservation theorems to be proven is the Łoś-Tarski theorem that provides over arbitrary structures and for arbitrary finite vocabularies, semantic characterizations of the ∀ and ∃ prefix classes of first order logic (FO) sentences, via the properties of preservation under substructures and preservation under extensions respectively. In the classical model theory part of this thesis, we present new parameterized preservation properties that provide for each natural number k, semantic characterizations of the ∃∀ and ∀∃ prefix classes of FO 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 properties of preservation under substructures and preservation under extensions, when k equals 0. As a consequence, we get a parameterized generalization of the Łoś-Tarski theorem for sentences, in both its substructural and extensional forms. We call our characterizations collectively the generalized Łoś-Tarski theorem for sentences at level k, abbreviated GLT(k). To the best of our knowledge, GLT(k) is the first to relate counts of quantifiers appearing in the sentences of the Σ2 and Π 0 2 prefix classes of FO, to natural quantitative properties of models, and hence provides new semantic characterizations of these sentences. We generalize GLT(k) 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 modeltheoretic hypothesis, to theories of ∃∀ sentences. We also present natural variants of our preservation properties in which, instead of natural numbers k, we consider infinite cardinals λ, and show that these variants provide new semantic characterizations of Σ2 and Π 0 2 theories. In contrast to existing preservation properties in the literature that characterize Σ2 and Π 0 2 sentences, our preservation properties are combinatorial and finitary in nature, and stay non-trivial over finite structures as well. Hence, in the finite model theory part of the thesis, we investigate