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A decomposition of the Rogers semilattice of a family of d.c.e. sets

Published online by Cambridge University Press:  12 March 2014

Serikzhan A. Badaev  and
Steffen Lempp
Serikzhan A. Badaev
Affiliation:
Department of Mathematics, Al-Farabi Kazakh National University, Almaty 050038, Kazakhstan, E-mail: badaev@kazsu.kz
Steffen Lempp
Affiliation:
Department of Mathematics, University of Wisconsin, Madison, Wi 53706-1388, USA, E-mail: lempp@math.wisc.edu
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Abstract

Khutoretskii's Theorem states that the Rogers semilattice of any family of c.e. sets has either at most one or infinitely many elements. A lemma in the inductive step of the proof shows that no Rogers semilattice can be partitioned into a principal ideal and a principal filter. We show that such a partitioning is possible for some family of d.c.e. sets. In fact, we construct a family of c.e. sets which, when viewed as a family of d.c.e. sets, has (up to equivalence) exactly two computable Friedberg numberings μ and ν, and μ reduces to any computable numbering not equivalent to ν. The question of whether the full statement of Khutoretskii's Theorem fails for families of d.c.e. sets remains open.

Keywords

d.c.e. setsRogers semilatticeKhutoretskii's Theorem
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 74 , Issue 2 , June 2009 , pp. 618 - 640
Copyright
Copyright © Association for Symbolic Logic 2009

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References

REFERENCES

[BG00] Badaev, Serikzhan A. and Goncharov, Sergey S., The theory of numberings: open problems, Computability theory and its applications. Current trends and open problems (Cholak, Peter A., Lempp, Steffen, Lerman, Manuel, and Shore, Richard A., editors), American Mathematical Society, Providence, RI, 2000, pp. 23–38.CrossRefGoogle Scholar
[BGPS03] Badaev, Serikzhan A., Goncharov, Sergey S., Podzorov, Sergei Yu., and Sorbi, Andrea, Algebraic properties of Rogers semilattices of arithmetical numberings, Computability and models, perspectives east and west (Cooper, S. Barry and Goncharov, Sergey S., editors), Kluwer/Plenum, New York, 2003, pp. 45–77.CrossRefGoogle Scholar
[BGS03] Badaev, Serikzhan A., Goncharov, Sergey S., and Sorbi, Andrea, Completeness and universality of arithmetical numberings, Computability and models. Perspectives east and west (Cooper, S. Barry and Goncharov, Sergey S., editors), Kluwer/Plenum, New York, 2003, pp. 11–44.CrossRefGoogle Scholar
[BGS03a] Badaev, Serikzhan A., Goncharov, Sergey S., and Sorbi, Andrea, Isomorphism types and theories of Rogers semilattices of arithmetical numberings, Computability and models. Perspectives east and west (Cooper, S. Barry and Goncharov, Sergey S., editors), Kluwer/Plenum, New York, 2003, pp. 79–91.CrossRefGoogle Scholar
[Er73] Ershov, Yuri L., Theory of numerations. Part I: General theory of numerations, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 19 (1973), pp. 289–388, German.Google Scholar
[Er75] Ershov, Yuri L., Theory of numerations. Part IT. Computable numerations of morphisms, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 21 (1975), pp. 473–584, German.Google Scholar
[Er77] Ershov, Yuri L., Theory of numerations. Part III: Constructive models, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 23 (1977), pp. 289–371, German.Google Scholar
[Er77a] Ershov, Yuri L., Theory of numerations, Nauka, Moscow, 1977, Russian.Google Scholar
[Er99] Ershov, Yuri L., Theory of numberings, Handbook of computability theory, North-Holland, Amsterdam, 1999, pp. 473–503.CrossRefGoogle Scholar
[Gon80] Goncharov, Sergey S., Computable single-valued numerations. Algebra i Logika, vol. 19 (1980), no. 5, pp. 507–551.Google Scholar
[Gon82] Goncharov, Sergey S., Limit equivalent constructivizations, Trudy Inst. Matem. SO AN SSSR, vol. 2 (1982), pp. 4–12, Nauka, Novosibirsk, Russian.Google Scholar
[Gon83] Goncharov, Sergey S., Positive numerations of families with one-valued numerations. Algebra and Logic, vol. 22 (1983), no. 5, pp. 345–350.CrossRefGoogle Scholar
[Gon93] Goncharov, Sergey S., A unique positive enumeration, Siberian Advances in Mathematics, vol. 4 (1994), no. 1, pp. 52–64.Google Scholar
[Kh71] Khutoretskii, Aleksandr B., The power of the upper semilaltice of computable numerations, Algebra i Logika, vol. 10 (1971), pp. 561–569.Google Scholar
[LeLN] Lempp, Steffen, Lecture notes on priority arguments, preprint available at http://www.math.wisc.edu/~lempp/papers/prio.pdf.Google Scholar
[So87] Soare, Robert I., Recursively enumerable sets and degrees. Springer-Verlag, Berlin, New York, 1987.CrossRefGoogle Scholar