Hostname: page-component-7c8c6479df-p566r Total loading time: 0 Render date: 2024-03-28T11:48:28.126Z Has data issue: false hasContentIssue false

EQUIVALENCE RELATIONS INVARIANT UNDER GROUP ACTIONS

Published online by Cambridge University Press:  01 August 2018

TOMASZ RZEPECKI*
Affiliation:
INSTYTUT MATEMATYCZNY UNIWERSYTET WROCŁAWSKI PL. GRUNWALDZKI 2/4 50-384 WROCŁAW, POLANDE-mail:tomasz.rzepecki@math.uni.wroc.pl

Abstract

We extend some recent results about bounded invariant equivalence relations and invariant subgroups of definable groups: we show that type-definability and smoothness are equivalent conditions in a wider class of relations than heretofore considered, which includes all the cases for which the equivalence was proved before.

As a by-product, we show some analogous results in purely topological context (without direct use of model theory).

Type
Articles
Copyright
Copyright © The Association for Symbolic Logic 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Becker, H. and Kechris, A. S., The Descriptive Set Theory of Polish Group Actions, London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 1996.CrossRefGoogle Scholar
Bourbaki, N., General Topology, Part 1, Elements of Mathematics, Hermann and Addison-Wesley, Paris and Reading, MA, 1966.Google Scholar
Casanovas, E., Lascar, D., Pillay, A., and Ziegler, M., Galois groups of first order theories. Journal of Mathematical Logic, vol. 1 (2001), no. 2, pp. 305319.CrossRefGoogle Scholar
Kaplan, I. and Miller, B., An embedding theorem of $\mathbb{E_0}$ with model theoretic applications. Journal of Mathematical Logic, vol. 14 (2014), no. 2, 1450010.CrossRefGoogle Scholar
Kaplan, I., Miller, B., and Simon, P., The Borel cardinality of Lascar strong types. Journal of the London Mathematical Society, vol. 90 (2014), no. 2, pp. 609630.CrossRefGoogle Scholar
Krupiński, K. and Pillay, A., Generalised Bohr compactification and model-theoretic connected components. Mathematical Proceedings of the Cambridge Philosophical Society, vol. 163 (2017), no. 2, pp. 219249.CrossRefGoogle Scholar
Krupiński, K., Pillay, A., and Rzepecki, T., Topological dynamics and the complexity of strong types. Israel Journal of Mathematics, 2018, to appear. Preprint available from arXiv:1510.00340.Google Scholar
Krupiński, K., Pillay, A., and Solecki, S., Borel equivalence relations and Lascar strong types. Journal of Mathematical Logic, vol. 13 (2013), no. 2, 1350008.CrossRefGoogle Scholar
Krupinski, K. and Rzepecki, T., Smoothness of bounded invariant equivalence relations, this Journal, vol. 81 (2016), no. 1, pp. 326356.Google Scholar
Miller, D. E., On the measurability of orbits in Borel actions. Proceedings of the American Mathematical Society, vol. 63 (1977), no. 1, pp. 165170.CrossRefGoogle Scholar
Newelski, L., The diameter of a lascar strong type. Fundamenta Mathematicae, vol. 176 (2003), no. 2, pp. 157170.CrossRefGoogle Scholar
Pillay, A., Type-definability, compact lie groups, and o-minimality. Journal of Mathematical Logic, vol. 4 (2004), no. 2, pp. 147162.CrossRefGoogle Scholar
Tent, K. and Ziegler, M., A Course in Model Theory, Lecture Notes in Logic, Cambridge University Press, Cambridge, 2012.CrossRefGoogle Scholar