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Abstract
An important dividing line in the class of unstable theories is being NSOP1, which is more general than being simple. In NSOP1 theories forking independence may not be as wellbehaved as in stable or simple theories, so it is replaced by another independence notion, called Kimindependence. We generalise Kimindependence over models in NSOP1 theories to positive logic—a proper generalisation of firstorder 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 NSOP1 and various other properties. We prove that, in a thick positive NSOP1 theory, Kimindependence over existentially closed models has all the nice properties that it is known to have in a firstorder NSOP1 theory. We also provide a KimPillay style theorem, characterising which thick positive theories are NSOP1 by the existence of a certain independence relation. Furthermore, this independence relation must then be the same as Kimindependence. Thickness is the mild assumption that being an indiscernible sequence is typedefinable.
In firstorder logic Kimindependence 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 Lascarinvariant types, which do exist in thick positive theories. We also simplify certain tree constructions that were used in the study of Kimindependence in firstorder theories. In particular, we only work with trees of finite height.
In firstorder logic Kimindependence 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 Lascarinvariant types, which do exist in thick positive theories. We also simplify certain tree constructions that were used in the study of Kimindependence in firstorder theories. In particular, we only work with trees of finite height.
Original language  English 

Pages (fromto)  55113 
Number of pages  59 
Journal  Model Theory 
Volume  1 
Issue number  1 
DOIs  
Publication status  Published  24 Jun 2022 
Keywords
 Kimindependence
 Kimdividing
 positive logic
 NSOP1 theory
Projects
 1 Finished

Exponentially Algebraically closed fields.
Engineering and Physical Sciences Research Council
1/09/19 → 31/08/22
Project: Research