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Convexity in Valued Fields and Semi-equations


This dissertation studies certain asymmetric (in the sense of not closed under complement) properties of families of sets, and how they relate to standard model-theoretic dividing lines and other combinatorial properties, particularly in the context of valued fields. In chapter 2, we investigate convex sets over valued fields, providing a classification result for them, and studying how the combinatorial properties satisfied by the family of convex sets over a valued field compares with the family of convex sets over $\mathbb{R}$. In chapter 3, we introduce and study two closely related concepts that we call semi-equationality, and weak semi-equationality, which are generalizations of equationality beyond the stable context, and also closely related to distality.

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