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Low5 Boolean Subalgebras and Computable Copies

Published online by Cambridge University Press:  12 March 2014

Russell Miller
Russell Miller*
Affiliation:
Department of Mathematics, Queens College–C.U.N.Y., 65-30 Kissena Blvd., Flushing, New York 11367, USA PH.D. Programs in Mathematics and Computer Science, C.U.N.Y. Graduate Center, 365 Fifth Avenue, New York, New York 10016, USA, E-mail: [email protected], URL: www.qc.edu/~rmiller
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Abstract

It is known that the spectrum of a Boolean algebra cannot contain a low4 degree unless it also contains the degree 0; it remains open whether the same holds for low5 degrees. We address the question differently, by considering Boolean subalgebras of the computable atomless Boolean algebra . For such subalgebras , we show that it is possible for the spectrum of the unary relation on to contain a low5 degree without containing 0.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 76 , Issue 3 , September 2011 , pp. 1061 - 1074
Copyright
Copyright © Association for Symbolic Logic 2011

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References

REFERENCES

[1]Csima, B., Harizanov, V., Miller, R., and Montalbán, A., Computability of Fraïssé limits, this Journal, vol. 76 (2011), no. 1, pp. 66–93.Google Scholar
[2]Downey, R. G. and Jockusch, C. G. Jr., Every low Boolean algebra is isomorphic to a recursive one, Proceedings of the American Mathematical Society, vol. 122 (1994), pp. 871–880.CrossRefGoogle Scholar
[3]Frolov, A., Harizanov, V., Kalimullin, I., Kudinov, O., and Miller, R., Degree spectra of highn and non-lown degrees, Journal of Logic and Computation, to appear.Google Scholar
[4]Goncharov, S. S. and Dzgoev, V. D., Autostability of models, Algebra and Logic, vol. 19 (1980), pp. 45–58 (Russian), 28–37 (English translation).Google Scholar
[5]Harizanov, V. S., Pure computable model theory, Handbook of recursive mathematics, vol. 1, Elsevier, Amsterdam, 1998, pp. 3–114.Google Scholar
[6]Harizanov, V. S. and Miller, R. G., Spectra of structures and relations, this Journal, vol. 72 (2007), no. 1, pp. 324–348.Google Scholar
[7]Harris, K. and Montalbán, A., On the n-back-and-forth types of Boolean algebras, Transactions of the American Mathematical Society, to appear.Google Scholar
[8]Harris, K. and Montalbán, A., Boolean algebra approximations, submitted for publication.Google Scholar
[9]Knight, J. F. and Stob, M., Computable Boolean algebras, this Journal, vol. 65 (2000), no. 4, pp. 1605–1623.Google Scholar
[10]Remmel, J. B., Recursive isomorphism types of recursive Boolean algebras, this Journal, vol. 46 (1981), pp. 572–594.Google Scholar
[11]Soare, R. I., Recursively enumerable sets and degrees, Springer-Verlag, New York, 1987.CrossRefGoogle Scholar
[12]Thurber, J. J., Every low2 Boolean algebra has a recursive copy, Proceedings of the American Mathematical Society, vol. 123 (1995), pp. 3859–3866.Google Scholar