Weak Canonical Bases in Nsop Theories

Kim, Byunghan

doi:10.1017/jsl.2021.45

Cited by 2 publications

(1 citation statement)

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“…In the discussion following Definition 4.1 in [Kim21] it is stated that one may produce results for Kim-independence for the hyperimaginary extension M heq of a first-order structure M parallel with those for first-order structures, provided…”

Section: Nsopmentioning

confidence: 99%

Independence Relations in Abstract Elementary Categories

Mark

2022

Bull. symb. log

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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 $_1$ , 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 $_1$ -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 $_1$ 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 $_1$ 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:mark@markkamsma.nl.URL:https://markkamsma.nl/phd-thesis.

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Section: Nsopmentioning

confidence: 99%

Independence Relations in Abstract Elementary Categories

Mark

2022

Bull. symb. log

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Kim-independence in positive logic

Dobrowolski¹,

Mark

2022

Model Th.

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An important dividing line in the class of unstable theories is being NSOP 1 , which is more general than being simple. In NSOP 1 theories forking independence may not be as well behaved as in stable or simple theories, so it is replaced by another independence notion, called Kim-independence. We generalise Kim-independence over models in NSOP 1 theories to positive logic -a proper generalisation of full first-order logic where negation is not built in, but can be added as desired. For example, an important application is that we can add hyperimaginary sorts to a positive theory to get another positive theory, preserving NSOP 1 and various other properties. We prove that, in a thick positive NSOP 1 theory, Kim-independence over existentially closed models has all the nice properties that it is known to have in an NSOP 1 theory in full first-order logic. We also provide a Kim-Pillay style theorem, characterising which thick positive theories are NSOP 1 by the existence of a certain independence relation. Furthermore, this independence relation must then be the same as Kim-independence. Thickness is the mild assumption that being an indiscernible sequence is type-definable.In full first-order logic Kim-independence is defined in terms of Morley sequences in global invariant types. These may not exist in thick positive theories. We solve this by working with Morley sequences in global Lascar-invariant types, which do exist in thick positive theories. We also simplify certain tree constructions that were used in the study of Kim-independence in full first-order logic. In particular, we only work with trees of finite height. 65 4. Kim-dividing 70 5. EM-modelling and parallel-Morley sequences 74 6. Symmetry 81 7. Independence theorem 84 8. Transitivity 94 9. Kim-Pillay style theorem 96 10. Examples 101 Acknowledgements 111 References 111 MSC2020: 03C10, 03C45.

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Weak Canonical Bases in Nsop Theories

Cited by 2 publications

References 10 publications

Independence Relations in Abstract Elementary Categories

Independence Relations in Abstract Elementary Categories

Kim-independence in positive logic

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