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Preprint Number 742

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742. Krzysztof Krupiński and Tomasz Rzepecki
Smoothness of bounded invariant equivalence relations
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Submission date: 29 June 2014.

Abstract:

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 get the conclusion.

Mathematics Subject Classification: 03C45; 03E15; 03C60

Keywords and phrases: bounded invariant equivalence relations, Borel cardinality, model-theoretic connected components

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