Open Access
2017 Universal Structures
Saharon Shelah
Notre Dame J. Formal Logic 58(2): 159-177 (2017). DOI: 10.1215/00294527-3800985


We deal with the existence of universal members in a given cardinality for several classes. First, we deal with classes of abelian groups, specifically with the existence of universal members in cardinalities which are strong limit singular of countable cofinality or λ=λ0. We use versions of being reduced—replacing Q by a subring (defined by a sequence t¯)—and get quite accurate results for the existence of universals in a cardinal, for embeddings and for pure embeddings. Second, we deal with (variants of) the oak property (from a work of Džamonja and the author), a property of complete first-order theories sufficient for the nonexistence of universal models under suitable cardinal assumptions. Third, we prove that the oak property holds for the class of groups (naturally interpreted, so for quantifier-free formulas) and deals more with the existence of universals.


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Saharon Shelah. "Universal Structures." Notre Dame J. Formal Logic 58 (2) 159 - 177, 2017.


Received: 2 February 2012; Accepted: 1 September 2014; Published: 2017
First available in Project Euclid: 28 January 2017

zbMATH: 06751296
MathSciNet: MR3634974
Digital Object Identifier: 10.1215/00294527-3800985

Primary: 03C45
Secondary: 03C55

Keywords: Abelian groups , Classification theory , groups , model theory , the oak property , universal models

Rights: Copyright © 2017 University of Notre Dame

Vol.58 • No. 2 • 2017
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