On the triple jump of the set of atoms of a Boolean algebra
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- by Antonio Montalbán
- Proc. Amer. Math. Soc. 136 (2008), 2589-2595
- DOI: https://doi.org/10.1090/S0002-9939-08-09248-4
- Published electronically: March 11, 2008
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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 $\mathbf {d}$.References
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Bibliographic Information
- Antonio Montalbán
- Affiliation: Department of Mathematics, University of Chicago, Chicago, Illinois 60637
- Email: antonio@mcs.vuw.ac.nz
- 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
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 136 (2008), 2589-2595
- MSC (2000): Primary 03D80
- DOI: https://doi.org/10.1090/S0002-9939-08-09248-4
- MathSciNet review: 2390531