Alex Kruckman (Wesleyan University), Ideallic independence relations

Tue, 6 Dec 2022, 1:25 pm MST

Since the birth of modern model theory in the work of Morley and Shelah, notions of independence have been an important tool for studying complete first-order theories. Assumptions on the complexity of the theory often imply, and can in fact be characterized by, useful properties of these independence relations. For example, forking independence is a generalization of linear independence in vector spaces, which is defined in arbitrary theories; but a theory $T$ is stable if and only if forking independence is stationary, and $T$ is simple if and only if forking independence is symmetric. In this talk, I will describe a duality between certain notions of independence and certain ideals in the Boolean algebra of formulas, and I will show how constructions with ideals can be used to produce new notions of independence. As an application, I will prove a type amalgamation result that holds (surprisingly, at least to me) in arbitrary first-order theories.