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  1. Independence over arbitrary sets in NSOP1 theories.Jan Dobrowolski, Byunghan Kim & Nicholas Ramsey - 2022 - Annals of Pure and Applied Logic 173 (2):103058.
    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 NSOP1 theory. We deduce symmetry of Kim-independence and the independence theorem for Lascar strong types.
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  • Companionability characterization for the expansion of an o-minimal theory by a dense subgroup.Alexi Block Gorman - 2023 - Annals of Pure and Applied Logic 174 (10):103316.
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  • Independence in generic incidence structures.Gabriel Conant & Alex Kruckman - 2019 - Journal of Symbolic Logic 84 (2):750-780.
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  • Forking, imaginaries, and other features of.Christian D’elbée - 2021 - Journal of Symbolic Logic 86 (2):669-700.
    We study the generic theory of algebraically closed fields of fixed positive characteristic with a predicate for an additive subgroup, called $\mathrm {ACFG}$. This theory was introduced in [16] as a new example of $\mathrm {NSOP}_{1}$ nonsimple theory. In this paper we describe more features of $\mathrm {ACFG}$, such as imaginaries. We also study various independence relations in $\mathrm {ACFG}$, such as Kim-independence or forking independence, and describe interactions between them.
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  • Generic expansion of an abelian variety by a subgroup.Christian D'Elbée - 2021 - Mathematical Logic Quarterly 67 (4):402-408.
    Let A be an abelian variety in an algebraically closed field of characteristic 0. We prove that the expansion of A by a generic divisible subgroup of A with the same torsion exists provided A has few algebraic endomorphisms, namely. The resulting theory is NSOP1 and not simple. Note that there exist abelian varieties A with of any genus.
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  • Three Surprising Instances of Dividing.Gabriel Conant & Alex Kruckman - forthcoming - Journal of Symbolic Logic:1-20.
    We give three counterexamples to the folklore claim that in an arbitrary theory, if a complete type p over a set B does not divide over $C\subseteq B$, then no extension of p to a complete type over $\operatorname {acl}(B)$ divides over C. Two of our examples are also the first known theories where all sets are extension bases for nonforking, but forking and dividing differ for complete types (answering a question of Adler). One example is an $\mathrm {NSOP}_1$ theory (...)
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  • Recursive functions and existentially closed structures.Emil Jeřábek - 2019 - Journal of Mathematical Logic 20 (1):2050002.
    The purpose of this paper is to clarify the relationship between various conditions implying essential undecidability: our main result is that there exists a theory T in which all partially recursive functions are representable, yet T does not interpret Robinson’s theory R. To this end, we borrow tools from model theory — specifically, we investigate model-theoretic properties of the model completion of the empty theory in a language with function symbols. We obtain a certain characterization of ∃∀ theories interpretable in (...)
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  • Transitivity, Lowness, and Ranks in Nsop Theories.Artem Chernikov, K. I. M. Byunghan & Nicholas Ramsey - 2023 - Journal of Symbolic Logic 88 (3):919-946.
    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.
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  • Interpolative fusions.Alex Kruckman, Chieu-Minh Tran & Erik Walsberg - 2020 - Journal of Mathematical Logic 21 (2):2150010.
    We define the interpolative fusion T∪∗ of a family i∈I of first-order theories over a common reduct T∩, a notion that generalizes many examples of random or generic structures in the model-theo...
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  • Divide and Conquer: Dividing Lines and Universality.Saharon Shelah - 2021 - Theoria 87 (2):259-348.
    We discuss dividing lines (in model theory) and some test questions, mainly the universality spectrum. So there is much on conjectures, problems and old results, mainly of the author and also on some recent results.
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  • Pathological examples of structures with o‐minimal open core.Alexi Block Gorman, Erin Caulfield & Philipp Hieronymi - 2021 - Mathematical Logic Quarterly 67 (3):382-393.
    This paper answers several open questions around structures with o‐minimal open core. We construct an expansion of an o‐minimal structure by a unary predicate such that its open core is a proper o‐minimal expansion of. We give an example of a structure that has an o‐minimal open core and the exchange property, yet defines a function whose graph is dense. Finally, we produce an example of a structure that has an o‐minimal open core and definable Skolem functions, but is not (...)
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  • Generic Expansions of Geometric Theories.Somaye Jalili, Massoud Pourmahdian & Nazanin Roshandel Tavana - forthcoming - Journal of Symbolic Logic:1-22.
    As a continuation of ideas initiated in [19], we study bi-colored (generic) expansions of geometric theories in the style of the Fraïssé–Hrushovski construction method. Here we examine that the properties $NTP_{2}$, strongness, $NSOP_{1}$, and simplicity can be transferred to the expansions. As a consequence, while the corresponding bi-colored expansion of a red non-principal ultraproduct of p-adic fields is $NTP_{2}$, the expansion of algebraically closed fields with generic automorphism is a simple theory. Furthermore, these theories are strong with $\operatorname {\mathrm {bdn}}(\text (...)
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  • SOP1, SOP2, and antichain tree property.JinHoo Ahn & Joonhee Kim - 2024 - Annals of Pure and Applied Logic 175 (3):103402.
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  • Vector spaces with a dense-codense generic submodule.Alexander Berenstein, Christian D'Elbée & Evgueni Vassiliev - 2024 - Annals of Pure and Applied Logic 175 (7):103442.
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  • Generic multiplicative endomorphism of a field.Christian D'Elbée - 2025 - Annals of Pure and Applied Logic 176 (4):103554.
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  • Weak canonical bases in nsop theories.Byunghan Kim - 2021 - Journal of Symbolic Logic 86 (3):1259-1281.
    We study the notion of weak canonical bases in an NSOP $_{1}$ theory T with existence. Given $p=\operatorname {tp}$ where $B=\operatorname {acl}$ in ${\mathcal M}^{\operatorname {eq}}\models T^{\operatorname {eq}}$, the weak canonical base of p is the smallest algebraically closed subset of B over which p does not Kim-fork. With this aim we firstly show that the transitive closure $\approx $ of collinearity of an indiscernible sequence is type-definable. Secondly, we prove that given a total $\mathop {\smile \hskip -0.9em ^| \ (...)
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