Equivalence Relations Invariant Under Group Actions

Rzepecki, Tomasz

doi:10.1017/jsl.2017.88

Cited by 2 publications

(3 citation statements)

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“…It is worth mentioning that after this paper was submitted, the third author made some further progress [Rze17]. In the current paper, the equivalence of smoothness and type-definability for bounded, invariant equivalence relations defined on the set of realizations of a single complete type over ∅ is proved.…”

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confidence: 82%

“…Example 4.4 of [KR16] shows that even under this assumption, in general, smoothness does not imply type-definability. In [Rze17], the third author introduced a new class of weakly orbital equivalence relations (which contains invariant relations defined on a single complete type over ∅ as well as orbital relations, such as E L , considered on the whole monster model), and proved that for such relations smoothness implies type-definability. This result generalizes Theorem 4.1, but one should emphasize that the proof of this generalization uses Theorem 4.1 and does not yield a new proof of Theorem 4.1.…”

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confidence: 99%

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Topological dynamics and the complexity of strong types

2018

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We develop topological dynamics for the group of automorphisms of a monster model of any given theory. In particular, we find strong relationships between objects from topological dynamics (such as the generalized Bohr compactification introduced by Glasner) and various Galois groups of the theory in question, obtaining essentially new information about them, e.g. we present the closure of the identity in the Lascar Galois group of the theory as the quotient of a compact, Hausdorff group by a dense subgroup.We apply this to describe the complexity of bounded, invariant equivalence relations, obtaining comprehensive results, subsuming and extending the existing results and answering some open questions from earlier papers. We show that, in a countable theory, any such relation restricted to the set of realizations of a complete type over ∅ is type-definable if and only if it is smooth. Then we show a counterpart of this result for theories in an arbitrary (not necessarily countable) language, obtaining also new information involving relative definability of the relation in question. As a final conclusion we get the following trichotomy. Let C be a monster model of a countable theory, p ∈ S(∅), and E be a bounded, (invariant) Borel (or, more generally, analytic) equivalence relation on p(C). Then, exactly one of the following holds:(1) E is relatively definable (on p(C)), smooth, and has finitely many classes, (2) E is not relatively definable, but it is type-definable, smooth, and has 2 ℵ0 classes, (3) E is not type definable and not smooth, and has 2 ℵ0 classes. All the results which we obtain for bounded, invariant equivalence relations carry over to the case of bounded index, invariant subgroups of definable groups. IntroductionGenerally speaking, this paper concerns applications of topological dynamics and the "descriptive set theory" of compact topological groups to model theory.2010 Mathematics Subject Classification. 03C45, 54H20, 03E15, 54H11.

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Topological dynamics and the complexity of strong types

2018

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“…(See also [Rze16,Corollary 4.10] for a generalization of Fact 1.1 to a certain class of strong types not necessarily defined on a single p(C). )…”

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Galois groups as quotients of Polish groups

Krupiński

Rzepecki

2020

J. Math. Log.

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We present the (Lascar) Galois group of any countable theory as a quotient of a compact Polish group by an F σ normal subgroup: in general, as a topological group, and under NIP, also in terms of Borel cardinality. This allows us to obtain similar results for arbitrary strong types defined on a single complete type over ∅. As an easy conclusion of our main theorem, we get the main result of [KPR15] which says that for any strong type defined on a single complete type over ∅, smoothness is equivalent to type-definability.We also explain how similar results are obtained in the case of bounded quotients of type-definable groups. This gives us a generalization of a former result from [KPR15] about bounded quotients of type-definable subgroups of definable groups.Date: November 9, 2018. 2010 Mathematics Subject Classification. 03C45, 54H20, 22C05, 03E15, 54H11.

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Equivalence Relations Invariant Under Group Actions

Cited by 2 publications

References 12 publications

Topological dynamics and the complexity of strong types

Topological dynamics and the complexity of strong types

Galois groups as quotients of Polish groups

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