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We generalise the main theorems from the paper “The Borel cardinality of Lascar strong types” by I. Kaplan, B. Miller and P. Simon to a wider class of bounded invariant equivalence relations. We apply them to describe relationships between fundamental properties of bounded invariant equivalence relations (such as smoothness or type-definability) which also requires finding a series of counterexamples. Finally, we apply the generalisation mentioned above to prove a conjecture from a paper by the first author and J. Gismatullin, showing that the key technical assumption of the main theorem (concerning connected components in definable group extensions) from that paper is not only sufficient but also necessary to obtain the conclusion.

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[1]Kaplan, Itay and Miller, Benjamin D., An embedding theorem of with model theoretic applications. Journal of Mathematical Logic, vol. 14 (2014), no. 2, 1450010.
[2]Kaplan, Itay, Miller, Benjamin, and Simon, Pierre, The Borel cardinality of Lascar strong types. Journal of London Mathematical Society (2), vol. 90 (2014), no. 2, pp. 609630.
[3]Gismatullin, Jakub and Krupiński, Krzysztof, On model-theoretic connected components in some group extensions, Version 2, 2013, arXiv: 1201.5221v2, submitted.
[4]Krupiński, Krzysztof, Pillay, Anand, and Solecki, Sławomir, Borel equivalence relations and Lascar strong types. Journal of Mathematical Logic, vol. 13 (2013), no. 2, 1350008.
[5]Conversano, Annalisa and Pillay, Anand, Connected components of definable groups and o-minimality I. Advances in Mathematics, vol. 231 (2012), no. 2, pp. 605623.
[6]Gismatullin, Jakub, Model theoretic connected components of groups. Israel Journal of Mathematics, vol. 184 (2011), no. 1, pp. 251274.
[7]Gismatullin, Jakub and Newelski, Ludomir, G-compactness and groups. Archive of Mathematical Logic, vol. 47 (2008), no. 5, pp. 479501.
[8]Kanovei, Vladimir, Borel Equivalence Relations, American Mathematical Society, Providence, RI, 2008.
[9]Pillay, Anand, Type-definability, compact lie groups, and o-minimality. Journal of Mathematical Logic, vol. 4 (2004), no. 2, pp. 147162.
[10]Newelski, Ludomir, The diameter of a lascar strong type. Fundamenta Mathematicae, vol. 176 (2003), no. 2, pp. 157170, doi. 10.4064/fm176-2-4.
[11]Rotman, Joseph J., Advanced Modern Algebra, American Mathematical Society, Providence, RI 2002.
[12]Casanovas, Enrique, Lascar, Daniel, Pillay, Anand, and Ziegler, Martin, Galois Groups of First Order Theories. Journal of Mathematical Logic, vol. 1 (2001), no. 2, pp. 305319.
[13]Becker, Howard and Kechris, A. S., The Descriptive Set Theory of Polish Group Actions, Cambridge University Press, Cambridge, 1996.
[14]Kechris, Alexander S., Classical Descriptive Set Theory, Springer, New York, 1995.
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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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