Author:
Badaev Serikzhan A.,Lempp Steffen
Abstract
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.
Publisher
Cambridge University Press (CUP)
Reference16 articles.
1. Recursively Enumerable Sets and Degrees
2. [LeLN] Lempp Steffen , Lecture notes on priority arguments, preprint available at http://www.math.wisc.edu/~lempp/papers/prio.pdf.
3. On the cardinality of the upper semilattice of computable enumerations
4. A unique positive enumeration;Goncharov;Siberian Advances in Mathematics,1994
5. Positive numerations of families with one-valued numerations
Cited by
19 articles.
订阅此论文施引文献
订阅此论文施引文献，注册后可以免费订阅5篇论文的施引文献，订阅后可以查看论文全部施引文献
1. On Concept Lattices for Numberings;Tsinghua Science and Technology;2024-12
2. A Family with a Single Minimal but Not Least Numbering;Siberian Mathematical Journal;2024-03
3. Two Theorems on Minimal Generalized Computable Numberings;Moscow University Mathematics Bulletin;2023-06
4. ON THE EXISTENCE OF UNIVERSAL NUMBERINGS;Herald of the Kazakh-British technical university;2023-04-01
5. Two theorems on minimal generally-computable numberings;Вестник Московского университета. Серия 1: Математика. Механика;2023