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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

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