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TRANSITIVITY, LOWNESS, AND RANKS IN NSOP$_{1}$ THEORIES
Part of
- Model theory
ARTEM CHERNIKOV, BYUNGHAN KIM, NICHOLAS RAMSEY
Journal:

The Journal of Symbolic Logic / Volume 88 / Issue 3 / September 2023

Published online by Cambridge University Press:

09 June 2023, pp. 919-946

Print publication:

September 2023
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We develop the theory of Kim-independence in the context of NSOP$_{1}$ theories satisfying the existence axiom. We show that, in such theories, Kim-independence is transitive and that -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.