论文标题
NATP理论中关于金·戴维的一些评论
Some Remarks on Kim-dividing in NATP Theories
论文作者
论文摘要
在本说明中，我们证明了Kim-Divident在NATP理论中的Coheir Morley序列总是见证了对模型的影响。 遵循切尔尼科夫和卡普兰[8]的策略，我们获得了一些在NATP理论中持有的推论。即，（i）如果kim fork在型号上是一个公式，那么它在同一型号上进行了准划分，（ii）对于任何参数元组$ b $和型号$ m $，都存在一个全球coheir $ p $，其中包含$ \ \ \ fext {tp}（tp}（b/m m）$ b/b \ ind $ b \ ind $ b b \ b b'$ al $ b'$ p | _ {mb} $。 我们还表明，对于NATP理论中的Coheirs，上面的条件（II）是作为Kim-Dividaning的见证的必要条件，假设存在Kim-Dividenting的见证人（请参阅本注释中的定义4.1）。也就是说，如果我们假设Kim-Dividing的见证总是在任何给定模型上存在，那么Coheir $ p \ supseteq \ text {tp}（a/m）$都必须满足（ii），每当它是$ a $ a $ a $ a $ a $ m $ $ m $的kim-divideing的见证人。我们还为在独立关系前的关系方面的存在提供了足够的条件。 在论文的结尾，我们就Mutchnik最近的作品留下了简短的评论[16]。我们指出，$ω$ -NDCTP $ _2 $理论的类是NATP理论类的子类，包含所有NTP $ _2 $ Theories和NSOP $ _1 $理论。我们还注意到，金·弗科（Kim-forking）和金（Kim-Dividing）等同于$ω$ -NDCTP $ _2 $理论中的模型，其中Kim-Dividing是根据不变的Morley序列定义的，而不是[16]中的Coheir Morley序列而不是Coheir Morley序列。
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].
