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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The nonlow computably enumerable degrees are not invariant in $ \mathcal{E}$


Author: Rachel Epstein
Journal: Trans. Amer. Math. Soc. 365 (2013), 1305-1345
MSC (2010): Primary 03D25
Published electronically: July 18, 2012
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Abstract: We study the structure of the computably enumerable (c.e.) sets, which form a lattice $ \mathcal {E}$ under set inclusion. The upward closed jump classes $ \overline {\mathbf {L}}_n$ and $ \mathbf {H}_n$ have all been shown to be definable by a lattice-theoretic formula, except for $ \overline {\mathbf {L}}_{1}$, the nonlow degrees. We say a class of c.e. degrees is invariant if it is the set of degrees of a class of c.e. sets that is invariant under automorphisms of $ \mathcal E$. All definable classes of degrees are invariant. We show that $ \overline {\mathbf {L}}_{1}$ is not invariant, thus proving a 1996 conjecture of Harrington and Soare that the nonlow degrees are not definable, and completing the problem of determining the definability of each jump class. We prove this by constructing a nonlow c.e. set $ D$ such that for all c.e. $ A\leq _{\rm T} D$, there is a low set $ B$ such that $ A$ can be taken by an automorphism of $ \mathcal {E}$ to $ B$.


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Additional Information

Rachel Epstein
Affiliation: Department of Mathematics, Faculty of Arts and Sciences, Harvard University, 1 Oxford Street, Cambridge, Massachusetts 02138
Email: repstein@math.harvard.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-2012-05600-5
PII: S 0002-9947(2012)05600-5
Keywords: Computably enumerable, recursively enumerable, definability, automorphisms, invariance
Received by editor(s): January 28, 2011
Received by editor(s) in revised form: April 8, 2011
Published electronically: July 18, 2012
Additional Notes: The author would like to thank Bob Soare for many helpful comments and conversations.
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.