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SOME REMARKS ON KIM-DIVIDING IN NATP THEORIES

Part of: Model theory

Published online by Cambridge University Press:  21 October 2025

JOONHEE KIM Open the ORCID record for JOONHEE KIM [Opens in a new window]  and
HYOYOON LEE Open the ORCID record for HYOYOON LEE [Opens in a new window]
JOONHEE KIM*
Affiliation:
YONSEI UNIVERSITY REPUBLIC OF KOREA
HYOYOON LEE
Affiliation:
CENTER FOR NANO MATERIALS, G-LAMP SOGANG UNIVERSITY SEOUL REPUBLIC OF KOREA DEPARTMENT OF MATHEMATICS YONSEI UNIVERSITY SEOUL REPUBLIC OF KOREA E-mail: hyoyoonlee@sogang.ac.kr
*
E-mail: kimjoonhee@kias.re.kr
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Abstract

In this note, we prove that Kim-dividing over models is always witnessed by a coheir Morley sequence, whenever the theory is NATP.

Following the strategy of Chernikov and Kaplan [8], we obtain some corollaries which hold in NATP theories. Namely, (i) if a formula Kim-forks over a model, then it quasi-divides over the same model and (ii) for any tuple of parameters b and a model M, there exists a global coheir p containing $\text {tp}(b/M)$ such that for all $b'\models p|_{MB}$.

We also show that for coheirs in NATP theories, condition (ii) above is a necessary condition for being a witness of Kim-dividing, assuming that a witness of Kim-dividing exists (see Definition 4.1 in this note). That is, if we assume that a witness of Kim-dividing always exists over any given model, then a coheir $p\supseteq \text {tp}(a/M)$ must satisfy (ii) whenever it is a witness of Kim-dividing of a over a model M. We also give a sufficient condition for the existence of a witness of Kim-dividing in terms of pre-independence relations.

At the end of the article, we leave a short remark on Mutchnik’s recent work [17]. We point out that the class of N-$\omega $-DCTP$_2$ theories, a subclass of the class of NATP theories, contains all NTP$_2$ theories and NSOP$_1$ theories. We also note that Kim-forking and Kim-dividing are equivalent over models in N-$\omega $-NDCTP$_2$ theories, where Kim-dividing is defined with respect to invariant Morley sequences, instead of coheir Morley sequences as in [17].

Keywords

antichain tree propertyKim’s lemmaKim-dividing

MSC classification

Primary: 03C45: Classification theory, stability and related concepts

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Type
Article
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The Journal of Symbolic Logic , First View , pp. 1 - 24
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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