The translation theorem
type: Item
instance of: scholarly article
title: The translation theorem
zbMATH Open document ID: 0805.03029
DOI: 10.1007/BF01352931
published in: Archive for Mathematical Logic
publication date: 1994-07-07T00:00:00Z
review text: The paper brings new proofs for some known results concerning orbits in the lattice of recursively enumerable sets, in particular Soare's result on maximal sets and Maass's result on hyperhypersimple sets. The author claims that the new proofs should be more understandable. He shows that it suffices to construct uniformly \(0''\)-recursive enumerations rather than uniformly recursive enumerations; this is ensured by the Translation Theorem (showing translation of uniformly \(0''\)-recursive enumerations into uniformly recursive ones). The Translation Theorem is also used to show a modified version of Soare's Extension Theorem. In the proof of the Translation Theorem, the author follows the techniques used in his dissertation on automorphisms of the lattice of recursively enumerable sets; in particular in building a \(\Delta_ 3\)- branching tree. The author also mentions that Downey and Stob's results on hemimaximal sets and on Friedberg splittings of hyperhypersimple sets can also be recast in the format of the new proofs.
Mathematics Subject Classification ID: 03D25
zbMATH DE Number: 579320
zbMATH Keywords: orbits in the lattice of recursively enumerable sets
zbMATH Keywords: maximal sets
zbMATH Keywords: hyperhypersimple sets
zbMATH Keywords: Translation Theorem
zbMATH Keywords: Soare's Extension Theorem
author: Peter A. Cholak
reviewed by: Petr Jančar
MaRDI profile type: Publication
cites work: Automorphisms of the lattice of recursively enumerable sets
cites work: Automorphisms of the lattice of recursively enumerable sets: Orbits
cites work: Friedberg splittings of recursively enumerable sets
cites work: On the Lattice of Recursively Enumerable Sets
cites work: On the orbits of hyperhypersimple sets
full work available at URL: https://doi.org/10.1007/bf01352931
OpenAlex ID: W2084309100
description: scientific article; zbMATH DE number 579320