The jump operator on the ω-enumeration degrees

Annals of Pure and Applied Logic 160 (3):289-301 (2009)
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Abstract

The jump operator on the ω-enumeration degrees was introduced in [I.N. Soskov, The ω-enumeration degrees, J. Logic Computat. 17 1193–1214]. In the present paper we prove a jump inversion theorem which allows us to show that the enumeration degrees are first order definable in the structure of the ω-enumeration degrees augmented by the jump operator. Further on we show that the groups of the automorphisms of and of the enumeration degrees are isomorphic. In the second part of the paper we study the jumps of the ω-enumeration degrees below . We define the ideal of the almost zero degrees and obtain a natural characterization of the class H of the ω-enumeration degrees below which are high n for some n and of the class L of the ω-enumeration degrees below which are low n for some n

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Citations of this work

The automorphism group of the enumeration degrees.Mariya I. Soskova - 2016 - Annals of Pure and Applied Logic 167 (10):982-999.
The ω-Turing degrees.Andrey C. Sariev & Hristo Ganchev - 2014 - Annals of Pure and Applied Logic 165 (9):1512-1532.

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References found in this work

Then-rea enumeration degrees are dense.Alistair H. Lachlan & Richard A. Shore - 1992 - Archive for Mathematical Logic 31 (4):277-285.
Jumps of quasi-minimal enumeration degrees.Kevin McEvoy - 1985 - Journal of Symbolic Logic 50 (3):839-848.
Definability of the jump operator in the enumeration degrees.I. Sh Kalimullin - 2003 - Journal of Mathematical Logic 3 (02):257-267.
Arithmetical Reducibilities I.Alan L. Selman - 1971 - Mathematical Logic Quarterly 17 (1):335-350.

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