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Independence Relations in Theories with the Tree Property


This thesis investigates theories with the tree property and, in particular, notions of independence in such theories. We discuss the example of the two-sorted theory of an infinite dimensional vector space over an algebraically closed field and with a bilinear form (which we refer to as T_∞), examined by Granger in his thesis. Granger notes that there is a well-behaved notion of independence, which he calls Γ-non-forking, in this theory, and that it can be viewed as the limit of the non-forking independence in the theories of its finite dimensional subspaces, which are ω-stable. He defines a notion of an ``approximating sequence'' of substructures, and shows that Γ-non-forking in a model of T_∞ corresponds to ``eventual" non-forking in an approximating sequence. We generalize his notion of approximation by substructures, and in the case of a theory whose large models can be approximated in this way, define a notion of independence which is the ``limit'' of the non-forking independence in the theories of the approximating substructures. We show that if the approximating substructures have simple theories, the limit independence relation satisfies invariance, monotonicity, base monotonicity, transitivity, normality, extension, finite character, and symmetry. Under certain additional assumptions, limit independence also satisfies anti-reflexivity and the Independence Theorem over algebraically closed sets.

We also consider the two-sorted theory of infinitely many cross-cutting equivalence relations, T^*_{feq}. We give a proof, explaining in detail the argument of Shelah and Usvyatsov for Theorem 2.1 in [Shelah-Usvyatsov'08], that T^*_{feq} does not have SOP_2 (equivalently, TP_1). The argument makes use of a theorem of Kim and Kim, from [Kim-Kim'11], along with several other lemmas involving tree indiscernibility.

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