In model theory, a branch of mathematical logic, we can classify mathematical structures based on their logical complexity. This yields the so-called stability hierarchy. Independence relations play an important role in this stability hierarchy. An independence relation tells us which subsets of a structure contain information about each other, for example, linear independence in vector spaces yields such a relation.
Some important classes in the stability hierarchy are stable, simple, and NSOP
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, each being contained in the next. For each of these classes there exists a so-called Kim-Pillay style theorem. Such a theorem describes the interaction between independence relations and the stability hierarchy. For example, simplicity is equivalent to admitting a certain independence relation, which must then be unique.
All of the above classically takes place in full first-order logic. Parts of it have already been generalised to other frameworks, such as continuous logic, positive logic, and even a very general category-theoretic framework. In this thesis we continue this work.
We introduce the framework of AECats, which are a specific kind of accessible category. We prove that there can be at most one stable, simple, or NSOP
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-like independence relation in an AECat. We thus recover (part of) the original stability hierarchy. For this we introduce the notions of long dividing, isi-dividing, and long Kim-dividing, which are based on the classical notions of dividing and Kim-dividing but are such that they work well without compactness.
Switching frameworks, we generalise Kim-dividing in NSOP
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theories to positive logic. We prove that Kim-dividing over existentially closed models has all the nice properties that it is known to have in full first-order logic. We also provide a full Kim-Pillay style theorem: a positive theory is NSOP
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if and only if there is a nice enough independence relation, which then must be given by Kim-dividing.
Abstract prepared by Mark Kamsma.
E-mail: [email protected].
URL: https://markkamsma.nl/phd-thesis.