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  1. On categoricity in successive cardinals.Sebastien Vasey - 2020 - Journal of Symbolic Logic:1-19.
    We investigate, in ZFC, the behavior of abstract elementary classes categorical in many successive small cardinals. We prove for example that a universal $\mathbb {L}_{\omega _1, \omega }$ sentence categorical on an end segment of cardinals below $\beth _\omega $ must be categorical also everywhere above $\beth _\omega $. This is done without any additional model-theoretic hypotheses and generalizes to the much broader framework of tame AECs with weak amalgamation and coherent sequences.
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  • On categoricity in successive cardinals.Sebastien Vasey - 2022 - Journal of Symbolic Logic 87 (2):545-563.
    We investigate, in ZFC, the behavior of abstract elementary classes categorical in many successive small cardinals. We prove for example that a universal $\mathbb {L}_{\omega _1, \omega }$ sentence categorical on an end segment of cardinals below $\beth _\omega $ must be categorical also everywhere above $\beth _\omega $. This is done without any additional model-theoretic hypotheses and generalizes to the much broader framework of tame AECs with weak amalgamation and coherent sequences.
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  • Abstract elementary classes stable in ℵ0.Saharon Shelah & Sebastien Vasey - 2018 - Annals of Pure and Applied Logic 169 (7):565-587.
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  • Some Stable Non-Elementary Classes of Modules.Marcos Mazari-Armida - 2023 - Journal of Symbolic Logic 88 (1):93-117.
    Fisher [10] and Baur [6] showed independently in the seventies that if T is a complete first-order theory extending the theory of modules, then the class of models of T with pure embeddings is stable. In [25, 2.12], it is asked if the same is true for any abstract elementary class $(K, \leq _p)$ such that K is a class of modules and $\leq _p$ is the pure submodule relation. In this paper we give some instances where this is true:Theorem.Assume (...)
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  • Superstability, noetherian rings and pure-semisimple rings.Marcos Mazari-Armida - 2021 - Annals of Pure and Applied Logic 172 (3):102917.
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  • A Note on Torsion Modules with Pure Embeddings.Marcos Mazari-Armida - 2023 - Notre Dame Journal of Formal Logic 64 (4):407-424.
    We study Martsinkovsky–Russell torsion modules with pure embeddings as an abstract elementary class. We give a model-theoretic characterization of the pure-injective and the Σ-pure-injective modules relative to the class of torsion modules assuming that the torsion submodule is a pure submodule. Our characterization of relative Σ-pure-injective modules extends the classical characterization of Gruson and Jenson as well as Zimmermann. We study the limit models of the class and determine when the class is superstable assuming that the torsion submodule is a (...)
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  • Algebraic description of limit models in classes of abelian groups.Marcos Mazari-Armida - 2020 - Annals of Pure and Applied Logic 171 (1):102723.
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  • Stability Results Assuming Tameness, Monster Model, and Continuity of Nonsplitting.Samson Leung - 2024 - Journal of Symbolic Logic 89 (1):383-425.
    Assuming the existence of a monster model, tameness, and continuity of nonsplitting in an abstract elementary class (AEC), we extend known superstability results: let $\mu>\operatorname {LS}(\mathbf {K})$ be a regular stability cardinal and let $\chi $ be the local character of $\mu $ -nonsplitting. The following holds: 1.When $\mu $ -nonforking is restricted to $(\mu,\geq \chi )$ -limit models ordered by universal extensions, it enjoys invariance, monotonicity, uniqueness, existence, extension, and continuity. It also has local character $\chi $. This generalizes (...)
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  • On universal modules with pure embeddings.Thomas G. Kucera & Marcos Mazari-Armida - 2020 - Mathematical Logic Quarterly 66 (4):395-408.
    We show that certain classes of modules have universal models with respect to pure embeddings: Let R be a ring, T a first‐order theory with an infinite model extending the theory of R‐modules and (where ⩽pp stands for “pure submodule”). Assume has the joint embedding and amalgamation properties. If or, then has a universal model of cardinality λ. As a special case, we get a recent result of Shelah [28, 1.2] concerning the existence of universal reduced torsion‐free abelian groups with (...)
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  • Simple-like independence relations in abstract elementary classes.Rami Grossberg & Marcos Mazari-Armida - 2021 - Annals of Pure and Applied Logic 172 (7):102971.
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  • Equivalent definitions of superstability in Tame abstract elementary classes.Rami Grossberg & Sebastien Vasey - 2017 - Journal of Symbolic Logic 82 (4):1387-1408.
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