Extension theorems, orbits, and automorphisms of the computably enumerable sets
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- by Peter A. Cholak and Leo A. Harrington PDF
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Abstract:
We prove an algebraic extension theorem for the computably enumerable sets, $\mathcal {E}$. Using this extension theorem and other work we then show if $A$ and $\widehat {A}$ are automorphic via $\Psi$, then they are automorphic via $\Lambda$ where $\Lambda \restriction \mathcal {L}^*(A) = \Psi$ and $\Lambda \restriction \mathcal {E}^*(A)$ is $\Delta ^0_3$. We give an algebraic description of when an arbitrary set $\widehat {A}$ is in the orbit of a computably enumerable set $A$. We construct the first example of a definable orbit which is not a $\Delta ^0_3$ orbit. We conclude with some results which restrict the ways one can increase the complexity of orbits. For example, we show that if $A$ is simple and $\widehat {A}$ is in the same orbit as $A$, then they are in the same $\Delta ^0_6$-orbit and, furthermore, we provide a classification of when two simple sets are in the same orbit.References
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Additional Information
- Peter A. Cholak
- Affiliation: Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556-5683
- MR Author ID: 290865
- ORCID: 0000-0002-6547-5408
- Email: Peter.Cholak.1@nd.edu
- Leo A. Harrington
- Affiliation: Department of Mathematics, University of California, Berkeley, California 94720-3840
- Email: leo@math.berkeley.edu
- Received by editor(s): September 7, 2004
- Received by editor(s) in revised form: August 31, 2005
- Published electronically: October 29, 2007
- Additional Notes: This research was partially supported by NSF Grants DMS-96-34565, 99-88716, 02-45167 (first author), DMS-96-22290 and DMS-99-71137 (second author). We would like to thank Bob Soare and Mike Stob for their interest and helpful comments.
- © Copyright 2007 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 360 (2008), 1759-1791
- MSC (2000): Primary 03D25
- DOI: https://doi.org/10.1090/S0002-9947-07-04025-1
- MathSciNet review: 2366962