On the triple jump of the set of atoms of a Boolean algebra
Author:
Antonio Montalbán
Journal:
Proc. Amer. Math. Soc. 136 (2008), 25892595
MSC (2000):
Primary 03D80
Published electronically:
March 11, 2008
MathSciNet review:
2390531
Fulltext PDF Free Access
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Abstract: We prove the following result concerning the degree spectrum of the atom relation on a computable Boolean algebra. Let be a computable Boolean algebra with infinitely many atoms and be the Turing degree of the atom relation of . If is a c.e. degree such that , then there is a computable copy of where the atom relation has degree . In particular, for every c.e. degree , any computable Boolean algebra with infinitely many atoms has a computable copy where the atom relation has degree .
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 Rod Downey and Carl G. Jockusch.
Every low Boolean algebra is isomorphic to a recursive one. Proc. Amer. Math. Soc., 122(3):871880, 1994. MR 1203984 (95a:03044)
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 Rodney G. Downey and Michael F. Moses.
Recursive linear orders with incomplete successivities. Trans. Amer. Math. Soc., 326(2):653668, 1991. MR 1005933 (91k:03115)
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 Rod Downey.
Every recursive Boolean algebra is isomorphic to one with incomplete atoms. Ann. Pure Appl. Logic, 60(3):193206, 1993. MR 1216669 (94c:03059)
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Recursive isomorphism types of recursive Boolean algebras. J. Symbolic Logic, 46(3):572594, 1981. MR 627907 (83a:03042)
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 Jeffrey B. Remmel.
Recursive Boolean algebras with recursive atoms. J. Symbolic Logic, 46(3):595616, 1981. MR 627908 (82j:03055)
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 J. B. Remmel.
Recursive Boolean algebras. In Handbook of Boolean algebras, Vol. 3, pages 10971165. NorthHolland, Amsterdam, 1989. MR 991614
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Every Boolean algebra has a recursive copy. Proc. Amer. Math. Soc., 123(12):38593866, 1995. MR 1283564 (96b:03047)
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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/S0002993908092484
PII:
S 00029939(08)092484
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 DMS0600824 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.
