Kimindependence in positive logic
Abstract
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 wellbehaved as in stable or simple theories, so it is replaced by another independence notion, called Kimindependence. We generalise Kimindependence over models in NSOP$_1$ 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 NSOP$_1$ and various other properties. We prove that, in a thick positive NSOP$_1$ theory, Kimindependence over existentially closed models has all the nice properties that it is known to have in a firstorder NSOP$_1$ theory. We also provide a KimPillay 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 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.
 Publication:

arXiv eprints
 Pub Date:
 May 2021
 DOI:
 10.48550/arXiv.2105.07788
 arXiv:
 arXiv:2105.07788
 Bibcode:
 2021arXiv210507788D
 Keywords:

 Mathematics  Logic;
 03C45;
 03C10
 EPrint:
 45 pages