Some Remarks on Kim-dividing in NATP Theories
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arXiv
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2022
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| _version_ | 1866918367631245312 |
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| author | Kim, Joonhee Lee, Hyoyoon |
| author_facet | Kim, Joonhee Lee, Hyoyoon |
| contents | In this note, we prove that Kim-dividing over models is always witnessed by a coheir Morley sequence in NATP theories.
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, (ii) for any tuple of parameters $b$ and a model $M$, there exists a global coheir $p$ containing $\text{tp}(b/M)$ such that $B \ind^K_M b'$ 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 paper, we leave a short remark on Mutchnik's recent work [16]. We point out that the class of $ω$-NDCTP$_2$ 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 $ω$-NDCTP$_2$ theories, where Kim-dividing is defined with respect to invariant Morley sequences, instead of coheir Morley sequences as in [16]. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2211_04213 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Some Remarks on Kim-dividing in NATP Theories Kim, Joonhee Lee, Hyoyoon Logic 03C45 In this note, we prove that Kim-dividing over models is always witnessed by a coheir Morley sequence in NATP theories. 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, (ii) for any tuple of parameters $b$ and a model $M$, there exists a global coheir $p$ containing $\text{tp}(b/M)$ such that $B \ind^K_M b'$ 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 paper, we leave a short remark on Mutchnik's recent work [16]. We point out that the class of $ω$-NDCTP$_2$ 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 $ω$-NDCTP$_2$ theories, where Kim-dividing is defined with respect to invariant Morley sequences, instead of coheir Morley sequences as in [16]. |
| title | Some Remarks on Kim-dividing in NATP Theories |
| topic | Logic 03C45 |
| url | https://arxiv.org/abs/2211.04213 |