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A characterization of the δ20 hyperhyperimmune sets

Published online by Cambridge University Press:  12 March 2014

Roland Sh. Omanadze  and
Andrea Sorbi
Roland Sh. Omanadze
Affiliation:
Javakhishvili Tbilisi State University, Tbilisi 0186., Georgia, E-mail: r.omanadze@math.sci.tsu.ge
Andrea Sorbi
Affiliation:
Dipartimento di Scienze Matematiche ed Informatiche “R. Magari”, Università di Siena. 53100 Siena, Italy, E-mail: sorbi@unisi.it
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Abstract

Let A be an infinite set and let K be creative: we show that K ≤QA if and only if KA. (Here ≤Q denotes Q-reducibility, and is the subreducibility of ≤Q obtained by requesting that Q-reducibility be provided by a computable function f such that Wf(x) ∩ Wf(y) = ∅. if x ≠ y.) Using this result we prove that A is hyperhyperimmune if and only if no subset B of A is s-complete, i.e., there is no subset B of A such that ≤sB, where ≤s denotes s-reducibility, and denotes the complement of K.

Keywords

Q-reducibilitys-reducibilityhyperhyperimmune set
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 73 , Issue 4 , December 2008 , pp. 1407 - 1415
Copyright
Copyright © Association for Symbolic Logic 2008

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References

REFERENCES

[1]Cooper, S. B., Enumeration reducibility, nondeterministic computations and relative computability of partial functions, Recursion theory week, Oberwolfach 1989 (Ambos-Spies, K., Müller, G., and Sacks, G. E., editors), Lecture Notes in Mathematics, vol. 1432, Springer–Verlag, Heidelberg, 1990, pp. 57–110.Google Scholar
[2]Friedberg, R. M. and Rogers, H. Jr., Reducibility and completeness for sets of integers, Zeitschrift für Mathematische Logik and Grundlagen der Mathematik, vol. 5 (1959), pp. 117–125.CrossRefGoogle Scholar
[3]Gill, J. T. III and Morris, P. H., On subcreative sets and S-reducibility, this Journal, vol. 39 (1974), no. 4, pp. 669–677.Google Scholar
[4]Jockusch, C. G. Jr., The degrees of hyperhyperimmune sets, this Journal, vol. 34 (1969), no. 3, pp. 489–493.Google Scholar
[5]McEvoy, K., The structure of the enumeration degrees, Ph.D. thesis, School of Mathematics, University of Leeds, 1984.Google Scholar
[6]Morley, M. and Soare, R. I., Splitting theorems and δ20 sets, Fundamenta Mathematicae, vol. 90 (1975), pp. 45–52.CrossRefGoogle Scholar
[7]Omanauze, R. Sh. and Sorbi, A., Strong enumeration reducibilities, Archive for Mathematical Logic, vol. 45 (2006), no. 7. pp. 869–912.CrossRefGoogle Scholar
[8]Rogers, H. Jr., Theory of recursive functions and effective computability, McGraw-Hill, New York, 1967.Google Scholar
[9]Rozinas, M. G., Partial degrees of immune and hyperimmune sets, Siberian Mathematical Journal, vol. 19 (1978), pp. 613–616.CrossRefGoogle Scholar
[10]Soare, R. I., Recursively enumerable sets and degrees, Perspectives in Mathematical Logic, Omega Series, Springer–Verlag, Heidelberg, 1987.CrossRefGoogle Scholar
[11]Solov'ev, V. D., Q-reducibility and hyperhypersimple sets, Veroyatn. Metod. i Kibern., vol. 10-11 (1974), p. 1974, Russian.Google Scholar