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  1. 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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  • Superstability, noetherian rings and pure-semisimple rings.Marcos Mazari-Armida - 2021 - Annals of Pure and Applied Logic 172 (3):102917.
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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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  • 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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  • Cellular Categories and Stable Independence.Michael Lieberman, Jiří Rosický & Sebastien Vasey - forthcoming - Journal of Symbolic Logic:1-24.
    We exhibit a bridge between the theory of cellular categories, used in algebraic topology and homological algebra, and the model-theoretic notion of stable independence. Roughly speaking, we show that the combinatorial cellular categories (those where, in a precise sense, the cellular morphisms are generated by a set) are exactly those that give rise to stable independence notions. We give two applications: on the one hand, we show that the abstract elementary classes of roots of Ext studied by Baldwin–Eklof–Trlifaj are stable (...)
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  • A topology for galois types in abstract elementary classes.Michael Lieberman - 2011 - Mathematical Logic Quarterly 57 (2):204-216.
    We present a way of topologizing sets of Galois types over structures in abstract elementary classes with amalgamation. In the elementary case, the topologies thus produced refine the syntactic topologies familiar from first order logic. We exhibit a number of natural correspondences between the model-theoretic properties of classes and their constituent models and the topological properties of the associated spaces. Tameness of Galois types, in particular, emerges as a topological separation principle. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.
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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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  • Interpreting groups and fields in simple, finitary AECs.Tapani Hyttinen & Meeri Kesälä - 2012 - Annals of Pure and Applied Logic 163 (9):1141-1162.
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  • Lascar Types and Lascar Automorphisms in Abstract Elementary Classes.Tapani Hyttinen & Meeri Kesälä - 2011 - Notre Dame Journal of Formal Logic 52 (1):39-54.
    We study Lascar strong types and Galois types and especially their relation to notions of type which have finite character. We define a notion of a strong type with finite character, the so-called Lascar type. We show that this notion is stronger than Galois type over countable sets in simple and superstable finitary AECs. Furthermore, we give an example where the Galois type itself does not have finite character in such a class.
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  • Reduction of database independence to dividing in atomless Boolean algebras.Tapani Hyttinen & Gianluca Paolini - 2016 - Archive for Mathematical Logic 55 (3-4):505-518.
    We prove that the form of conditional independence at play in database theory and independence logic is reducible to the first-order dividing calculus in the theory of atomless Boolean algebras. This establishes interesting connections between independence in database theory and stochastic independence. As indeed, in light of the aforementioned reduction and recent work of Ben-Yaacov :957–1012, 2013), the former case of independence can be seen as the discrete version of the latter.
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  • Cofinality Quantifiers in Abstract Elementary Classes and Beyond.Will Boney - forthcoming - Journal of Symbolic Logic:1-15.
    The cofinality quantifiers were introduced by Shelah as an example of a compact logic stronger than first-order logic. We show that the classes of models axiomatized by these quantifiers can be turned into an Abstract Elementary Class by restricting to positive and deliberate uses. Rather than using an ad hoc proof, we give a general framework of abstract Skolemizations. This method gives a uniform proof that a wide rang of classes are Abstract Elementary Classes.
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