Languages with Finite String of Quantifiers on Uncountable Structures
Boris A. Romov
We establish a criterion for a structure M on an infinite set to have InvAut, the Galois closure on the set of relations that are invariant to all automorphisms of M, defined via infinitary predicate language with finite string of quantifiers, which extends Scott’s Definability Theorem to an uncountable domain. Based on this approach we present criteria for (weak) homogeneity, (strong) ω-homogeneity and a version of Scott’s Isomorphism Theorem for uncountable structures, as well as criteria for atomic and ω-categorical structures. Using the Back-and-forth Extendibility Lemma, we obtain the description of elimination sets with the property: the set of all their finite Boolean combinations is closed under infinitary intersection. This description is given for both countable and uncountable structures including finite elimination sets, as well as elimination sets for ω-categorical structures.
Keywords: (weakly) homogeneous structure, back-and-forth extendibility, language with finite string of quantifiers, elimination set, Galois closure