Families with one-element Rogers semilattice

1998 ◽  
Vol 37 (1) ◽  
pp. 21-34 ◽  
Author(s):  
S. S. Goncharov ◽  
S. A. Badaev
Keyword(s):  
Rogers Semilattice

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

2009 ◽  
Vol 74 (2) ◽  
pp. 618-640 ◽  
Author(s):  
Serikzhan A. Badaev ◽  
Steffen Lempp
Keyword(s):  
Principal Ideal ◽  
Inductive Step ◽  
Rogers Semilattice ◽  
Computable Numbering

AbstractKhutoretskii'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.

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