Corrigendum to "Kim-independence in positive logic"
Jan Dobrowolski, Mark Kamsma
针对正厚度NSOP₁理论中Kim-独立性的独立性定理原有证明存在漏洞的问题，本勘误在确认定理成立的前提下给出了新的证明方法。
2022年Dobrowolski与Kamsma提出的正厚度NSOP₁理论中Kim-独立性的独立性定理证明存在漏洞。该定理仍然成立，本勘误给出另一种证明方法。
The proof of the Independence Theorem for Kim-independence in positive thick NSOP$_1$ theories from (Dobrowolski and Kamsma, 2022) contains a gap. The theorem is still true, and in this corrigendum we give a different proof.
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We study NSOP$_{1}$ theories. We define Kim-independence, which generalizes non-forking independence in simple theories and corresponds to non-forking at a generic scale. We show that Kim-independence satisfies a version of Kim's lemma, local character, symmetry, and an independence theorem and that, moreover, these properties individually characterize NSOP$_{1}$ theories. We describe Kim-independence in several concrete theories and observe that it corresponds to previously studied notions of independence in Frobenius fields and vector spaces with a generic bilinear form.
We prove several results on the behavior of Kim-independence upon changing the base in NSOP$_{1}$ theories. As a consequence, we prove that Kim-independence satisfies transitivity and that this characterizes NSOP$_{1}$. Moreover, we characterize witnesses to Kim-dividing as exactly the $\ind^{K}$-Morley sequences. We give several applications, answering a number of open questions concerning transitivity, Morley sequences, and local character in NSOP$_{1}$ theories.
We study Kim-independence over arbitrary sets. Assuming that forking satisfies existence, we establish Kim's lemma for Kim-dividing over arbitrary sets in an NSOP$_{1}$ theory. We deduce symmetry of Kim-independence and the independence theorem for Lascar strong types.
We show that NSOP$_{1}$ theories are exactly the theories in which Kim-independence satisfies a form of local character. In particular, we show that if $T$ is NSOP$_{1}$, $M\models T$, and $p$ is a type over $M$, then the collection of elementary substructures of size $\left|T\right|$ over which $p$ does not Kim-fork is a club of $\left[M\right]^{\left|T\right|}$ and that this characterizes NSOP$_{1}$. We also present a new phenomenon we call dual local-character for Kim-independence in NSOP$_{1}$-theories.
An important dividing line in the class of unstable theories is being NSOP$_1$, which is more general than being simple. In NSOP$_1$ theories forking independence may not be as well-behaved as in stable or simple theories, so it is replaced by another independence notion, called Kim-independence. We generalise Kim-independence over models in NSOP$_1$ theories to positive logic -- a proper generalisation of first-order logic where negation is not built in, but can be added as desired. For example, an important application is that we can add hyperimaginary sorts to a positive theory to get anoth…
Kim-forking for hyperimaginaries in NSOP1 theories
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On some Fraisse limits with free amalgamation
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Transitivity, lowness, and ranks in NSOP$_1$ theories
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Abstract independence relations in neostability theory
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Non-forking independence in stable theories
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Conant-independence and generalized free amalgamation
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We introduce the framework of AECats (abstract elementary categories), generalising both the category of models of some first-order theory and the category of subsets of models. Any AEC and any compact abstract theory ("cat", as introduced by Ben-Yaacov) forms an AECat. In particular, we find applications in positive logic and continuous logic: the category of (subsets of) models of a positive or continuous theory is an AECat. The Kim-Pillay theorem for first-order logic characterises simple theories by the properties dividing independence has. We prove a version of the Kim-Pillay theorem for …
Some Remarks on Kim-dividing in NATP Theories
本文研究了NATP理论中Kim分割的性质，证明了Kim分割可由coheir Morley序列见证，并探讨了其与准分割的关系及在不同理论中的等价性。
NSOP$_1$-like independence in AECats
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Properties of independence in $\mathrm{NSOP}_3$ theories
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Simple Homogeneous Structures and Indiscernible Sequence Invariants
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We study expansions of NSOP$_1$ theories that preserve NSOP$_1$. We prove that if $T$ is a model complete NSOP$_1$ theory eliminating the quantifier $\exists^{\infty}$, then the generic expansion of $T$ by arbitrary constant, function, and relation symbols is still NSOP$_1$. We give a detailed analysis of the special case of the theory of the generic $L$-structure, the model companion of the empty theory in an arbitrary language $L$. Under the same hypotheses, we show that $T$ may be generically expanded to an NSOP$_1$ theory with built-in Skolem functions. In order to obtain these results, we…
Notions of rank and independence in countably categorical theories
本文在ω-分类理论中定义n-秩和n-独立性，证明其满足独立性关系的所有性质（可能除对称性外），并给出对称性成立的条件，进而得到理论是rosy的结论。
Dividing Lines between Positive Theories
本文在正逻辑中研究了正理论中OP、IP、TP、SOP等性质的关系及稳定性特征，证明了若干等价关系和新定理。
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本文为模型论学者提供正逻辑的自洽导论，讲解正模型论基础、可数范畴性、巨型模型、单纯与稳定理论等内容，介绍其在超虚元素、连续逻辑中的应用并附内容速览。
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