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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Boolean algebra approximations


Authors: Kenneth Harris and Antonio Montalbán
Journal: Trans. Amer. Math. Soc. 366 (2014), 5223-5256
MSC (2010): Primary 03D45, 03G05
DOI: https://doi.org/10.1090/S0002-9947-2014-05950-3
Published electronically: June 3, 2014
MathSciNet review: 3240923
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Abstract | References | Similar Articles | Additional Information

Abstract: Knight and Stob proved that every low$ _4$ Boolean algebra is $ 0^{(6)}$-isomorphic to a computable one. Furthermore, for $ n=1,2,3,4$, every low$ _n$ Boolean algebra is $ 0^{(n+2)}$-isomorphic to a computable one. We show that this is not true for $ n=5$: there is a low$ _5$ Boolean algebra that is not $ 0^{(7)}$-isomorphic to any computable Boolean algebra.

It is worth remarking that, because of the machinery developed, the proof uses at most a $ 0^{\prime \prime }$-priority argument. The technique used to construct this Boolean algebra is new and might be useful in other applications, such as to solve the low$ _n$ Boolean algebra problem either positively or negatively.


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Additional Information

Kenneth Harris
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
Email: kaharri@umich.edu

Antonio Montalbán
Affiliation: Department of Mathematics, University of Chicago, Chicago, Illinois 60637
Email: antonio@math.uchicago.edu

DOI: https://doi.org/10.1090/S0002-9947-2014-05950-3
Keywords: Boolean algebra, back-and-forth, low, approximation
Received by editor(s): February 27, 2010
Received by editor(s) in revised form: September 5, 2012
Published electronically: June 3, 2014
Additional Notes: The second author was partially supported by NSF grant DMS-0901169, and by the AMS centennial fellowship
Article copyright: © Copyright 2014 American Mathematical Society

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