## Journal of Symbolic Logic

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

#### 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 in Project Euclid: 2 June 2009

http://projecteuclid.org/euclid.jsl/1243948330

Digital Object Identifier
doi:10.2178/jsl/1243948330

Mathematical Reviews number (MathSciNet)
MR2518814

Zentralblatt MATH identifier
05561769

#### Citation

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