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Universal Structures

Notre Dame Journal of Formal Logic 58 (2):159-177 (2017)

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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.details 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 (...) Download Export citation Bookmark
Induced and higher-dimensional stable independence.Michael Lieberman, Jiří Rosický & Sebastien Vasey - 2022 - Annals of Pure and Applied Logic 173 (7):103124.details Download Export citation Bookmark 1 citation
Divide and Conquer: Dividing Lines and Universality.Saharon Shelah - 2021 - Theoria 87 (2):259-348.details 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. Download Export citation Bookmark 1 citation
Some Stable Non-Elementary Classes of Modules.Marcos Mazari-Armida - 2023 - Journal of Symbolic Logic 88 (1):93-117.details 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 (...) Download Export citation Bookmark 1 citation
On universal modules with pure embeddings.Thomas G. Kucera & Marcos Mazari-Armida - 2020 - Mathematical Logic Quarterly 66 (4):395-408.details 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 (...) Download Export citation Bookmark 6 citations
On properties of theories which preclude the existence of universal models.Mirna Džamonja & Saharon Shelah - 2006 - Annals of Pure and Applied Logic 139 (1):280-302.details We introduce the oak property of first order theories, which is a syntactical condition that we show to be sufficient for a theory not to have universal models in cardinality λ when certain cardinal arithmetic assumptions about λ implying the failure of GCH hold. We give two examples of theories that have the oak property and show that none of these examples satisfy SOP4, not even SOP3. This is related to the question of the connection of the property SOP4 to (...) Download Export citation Bookmark 2 citations

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