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June 2009 A decomposition of the Rogers semilattice of a family of d.c.e. sets
Serikzhan A. Badaev, Steffen Lempp
J. Symbolic Logic 74(2): 618-640 (June 2009). DOI: 10.2178/jsl/1243948330

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.

Citation

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Serikzhan A. Badaev. Steffen Lempp. "A decomposition of the Rogers semilattice of a family of d.c.e. sets." J. Symbolic Logic 74 (2) 618 - 640, June 2009. https://doi.org/10.2178/jsl/1243948330

Information

Published: June 2009
First available in Project Euclid: 2 June 2009

zbMATH: 1185.03071
MathSciNet: MR2518814
Digital Object Identifier: 10.2178/jsl/1243948330

Subjects:
Primary: 03D45

Rights: Copyright © 2009 Association for Symbolic Logic

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Vol.74 • No. 2 • June 2009
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