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Uniform Martin's conjecture, locally


Author: Vittorio Bard
Journal: Proc. Amer. Math. Soc. 148 (2020), 5369-5380
MSC (2010): Primary 03D28; Secondary 03E15, 03E60
DOI: https://doi.org/10.1090/proc/15159
Published electronically: September 17, 2020
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Abstract: We show that part I of the uniform Martin's conjecture follows from a local phenomenon, namely that every non-constant uniformly Turing invariant $ f:[x]_{\equiv _T}\to [y]_{\equiv _T}$ satisfies $ x\le _T y$. Besides improving our knowledge about part I of the uniform Martin's conjecture (which turns out to be equivalent to Turing determinacy), the discovery of such local phenomenon also leads to new results that did not look strictly related to Martin's conjecture before. In particular, we get that computable reducibility $ \le _c$ on equivalence relations on $ \mathbb{N}$ has a very complicated structure, as $ \le _T$ is Borel reducible to it. We conclude by raising the question: Is part II of the uniform Martin's conjecture implied by local phenomena, too? and briefly indicating possible directions.


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Additional Information

Vittorio Bard
Affiliation: Dipartimento di Matematica “Giuseppe Peano”, Università di Torino, via Carlo Alberto 10, 10121 Torino, Italy
Email: vittorio.bard@unito.it

DOI: https://doi.org/10.1090/proc/15159
Keywords: Martin's conjecture, uniformly invariant functions, computable reducibility
Received by editor(s): July 24, 2019
Received by editor(s) in revised form: April 10, 2020
Published electronically: September 17, 2020
Communicated by: Heike Mildenberger
Article copyright: © Copyright 2020 American Mathematical Society