Relative primeness and Borel partition properties for equivalence relations
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- by John D. Clemens PDF
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Abstract:
We introduce a notion of relative primeness for equivalence relations, strengthening the notion of non-reducibility, and show for many standard benchmark equivalence relations that non-reducibility may be strengthened to relative primeness. We introduce several analogues of cardinal properties for Borel equivalence relations, including the notion of a prime equivalence relation and Borel partition properties on quotient spaces. In particular, we introduce a notion of Borel weak compactness, and characterize partition properties for the equivalence relations ${\mathbb F}_2$ and ${\mathbb E}_1$. We also discuss dichotomies related to primeness, and see that many natural questions related to Borel reducibility of equivalence relations may be viewed in the framework of relative primeness and Borel partition properties.References
- John D. Clemens and Samuel Coskey, New jump operators on equivalence relations, preprint, 2020.
- John D. Clemens, Dominique Lecomte, and Benjamin D. Miller, Dichotomy theorems for families of non-cofinal essential complexity, Adv. Math. 304 (2017), 285–299. MR 3558211, DOI 10.1016/j.aim.2016.08.044
- Clinton T. Conley, Canonizing relations on nonsmooth sets, J. Symbolic Logic 78 (2013), no. 1, 101–112. MR 3087064, DOI 10.2178/jsl.7801070
- Harvey Friedman and Lee Stanley, A Borel reducibility theory for classes of countable structures, J. Symbolic Logic 54 (1989), no. 3, 894–914. MR 1011177, DOI 10.2307/2274750
- Su Gao, Invariant descriptive set theory, Pure and Applied Mathematics (Boca Raton), vol. 293, CRC Press, Boca Raton, FL, 2009. MR 2455198
- L. A. Harrington, A. S. Kechris, and A. Louveau, A Glimm-Effros dichotomy for Borel equivalence relations, J. Amer. Math. Soc. 3 (1990), no. 4, 903–928. MR 1057041, DOI 10.1090/S0894-0347-1990-1057041-5
- G. Hjorth, Actions by the classical Banach spaces, J. Symbolic Logic 65 (2000), no. 1, 392–420. MR 1782128, DOI 10.2307/2586545
- Greg Hjorth, Classification and orbit equivalence relations, Mathematical Surveys and Monographs, vol. 75, American Mathematical Society, Providence, RI, 2000. MR 1725642, DOI 10.1090/surv/075
- Greg Hjorth and Alexander S. Kechris, New dichotomies for Borel equivalence relations, Bull. Symbolic Logic 3 (1997), no. 3, 329–346. MR 1476761, DOI 10.2307/421148
- Jared Holshouser, Partition Properties for Non-ordinal Sets Under the Axiom of Determinacy, ProQuest LLC, Ann Arbor, MI, 2017. Thesis (Ph.D.)–University of North Texas. MR 3797412
- Vladimir Kanovei, Borel equivalence relations, University Lecture Series, vol. 44, American Mathematical Society, Providence, RI, 2008. Structure and classification. MR 2441635, DOI 10.1090/ulect/044
- Vladimir Kanovei, Marcin Sabok, and Jindřich Zapletal, Canonical Ramsey theory on Polish spaces, Cambridge Tracts in Mathematics, vol. 202, Cambridge University Press, Cambridge, 2013. MR 3135065, DOI 10.1017/CBO9781139208666
- V. G. Kanoveĭ and M. Reeken, Some new results on the Borel irreducibility of equivalence relations, Izv. Ross. Akad. Nauk Ser. Mat. 67 (2003), no. 1, 59–82 (Russian, with Russian summary); English transl., Izv. Math. 67 (2003), no. 1, 55–76. MR 1957916, DOI 10.1070/IM2003v067n01ABEH000418
- Alexander S. Kechris, Classical descriptive set theory, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, New York, 1995. MR 1321597, DOI 10.1007/978-1-4612-4190-4
- Alexander S. Kechris and Alain Louveau, The classification of hypersmooth Borel equivalence relations, J. Amer. Math. Soc. 10 (1997), no. 1, 215–242. MR 1396895, DOI 10.1090/S0894-0347-97-00221-X
- Andrew Marks, Theodore A. Slaman, and John R. Steel, Martin’s conjecture, arithmetic equivalence, and countable Borel equivalence relations, Ordinal definability and recursion theory: The Cabal Seminar. Vol. III, Lect. Notes Log., vol. 43, Assoc. Symbol. Logic, Ithaca, NY, 2016, pp. 493–519. MR 3469180
- Christian Rosendal, Cofinal families of Borel equivalence relations and quasiorders, J. Symbolic Logic 70 (2005), no. 4, 1325–1340. MR 2194249, DOI 10.2178/jsl/1129642127
- Jack H. Silver, Counting the number of equivalence classes of Borel and coanalytic equivalence relations, Ann. Math. Logic 18 (1980), no. 1, 1–28. MR 568914, DOI 10.1016/0003-4843(80)90002-9
- Simon Thomas, Martin’s conjecture and strong ergodicity, Arch. Math. Logic 48 (2009), no. 8, 749–759. MR 2563815, DOI 10.1007/s00153-009-0148-0
- Jindřich Zapletal, Pinned equivalence relations, Math. Res. Lett. 18 (2011), no. 3, 559–564. MR 2802588, DOI 10.4310/MRL.2011.v18.n3.a15
- Jindřich Zapletal, Forcing borel reducibility invariants, preprint, 2013.
Additional Information
- John D. Clemens
- Affiliation: Boise State University, 1910 University Dr., Boise, Idaho 83725
- MR Author ID: 685049
- Email: johnclemens@boisestate.edu
- Received by editor(s): May 13, 2020
- Received by editor(s) in revised form: January 1, 2021
- Published electronically: November 5, 2021
- © Copyright 2021 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 375 (2022), 111-149
- MSC (2020): Primary 03E15; Secondary 03E02
- DOI: https://doi.org/10.1090/tran/8390
- MathSciNet review: 4358664