Transitivity, lowness, and ranks in NSOP${}_1$ theories

Abstract

We develop the theory of Kim-independence in the context of NSOP${}_1$ theories satsifying the existence axiom. We show that, in such theories, Kim-independence is transitive and that $\ind^{K}$-Morley sequences witness Kim-dividing. As applications, we show that, under the assumption of existence, in a low NSOP${}_1$ theory, Shelah strong types and Lascar strong types coincide and, additionally, we introduce a notion of rank for NSOP${}_1$ theories.