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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On the triple jump of the set of atoms of a Boolean algebra


Author: Antonio Montalbán
Journal: Proc. Amer. Math. Soc. 136 (2008), 2589-2595
MSC (2000): Primary 03D80
Published electronically: March 11, 2008
MathSciNet review: 2390531
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Abstract: We prove the following result concerning the degree spectrum of the atom relation on a computable Boolean algebra. Let $ \mathcal{C}$ be a computable Boolean algebra with infinitely many atoms and $ \mathbf{a}$ be the Turing degree of the atom relation of $ \mathcal{C}$. If $ \mathbf{d}$ is a c.e. degree such that $ \mathbf{a}^{\prime\prime\prime}\leq_T\mathbf{d}^{\prime\prime\prime}$, then there is a computable copy of $ \mathcal{C}$ where the atom relation has degree $ \mathbf{d}$. In particular, for every $ \mathrm{high}_3$ c.e. degree $ \mathbf{d}$, any computable Boolean algebra with infinitely many atoms has a computable copy where the atom relation has degree $ \mathbf{d}$.


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Additional Information

Antonio Montalbán
Affiliation: Department of Mathematics, University of Chicago, Chicago, Illinois 60637
Email: antonio@mcs.vuw.ac.nz

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09248-4
PII: S 0002-9939(08)09248-4
Keywords: Boolean algebra, atom, relation, degree spectrum
Received by editor(s): December 8, 2006
Received by editor(s) in revised form: April 12, 2007, April 22, 2007, and May 31, 2007
Published electronically: March 11, 2008
Additional Notes: This research was partially supported by NSF Grant DMS-0600824 and by the Marsden Foundation of New Zealand, via a postdoctoral fellowship.
Communicated by: Julia Knight
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.