Journal of Symbolic Logic

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

Serikzhan A. Badaev and Steffen Lempp

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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.

Article information

Source
J. Symbolic Logic Volume 74, Issue 2 (2009), 618-640.

Dates
First available: 2 June 2009

Permanent link to this document
http://projecteuclid.org/euclid.jsl/1243948330

Digital Object Identifier
doi:10.2178/jsl/1243948330

Zentralblatt MATH identifier
05561769

Mathematical Reviews number (MathSciNet)
MR2518814

Subjects
Primary: 03D45: Theory of numerations, effectively presented structures [See also 03C57; for intuitionistic and similar approaches see 03F55]

Keywords
d.c.e. sets Rogers semilattice Khutoretskii's Theorem

Citation

Badaev, Serikzhan A.; Lempp, Steffen. A decomposition of the Rogers semilattice of a family of d.c.e. sets. Journal of Symbolic Logic 74 (2009), no. 2, 618--640. doi:10.2178/jsl/1243948330. http://projecteuclid.org/euclid.jsl/1243948330.


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