Abstract
In the present paper, we show the first-order definability of the jump operator in the upper semi-lattice of the \(\omega \)-enumeration degrees. As a consequence, we derive the isomorphicity of the automorphism groups of the enumeration and the \(\omega \)-enumeration degrees.
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Notes
Selman’s Theorem states that \(A\le _{e}B\iff (\forall X\subseteq \omega )[B\text { is c.e. in } X\Rightarrow A\text { is c.e. in } X]\).
A mapping \(\pi :{\mathcal {D}}_{\omega }\rightarrow {\mathcal {D}}_{\omega }\) is said to be jump preserving, if for each degree \(\mathbf {a}\in {\mathcal {D}}_{\omega }\), \(\pi (\mathbf {a}')=\pi (\mathbf {a})'\).
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The authors were partially supported by BNSF Bilateral Grant DNTS/Russia 01/8 from 23.06.2017 and Sofia University Science Fund project 80-10-128/16.04.2020.
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Ganchev, H., Sariev, A.C. The automorphism group and definability of the jump operator in the \(\omega \)-enumeration degrees. Arch. Math. Logic 60, 909–925 (2021). https://doi.org/10.1007/s00153-021-00766-7
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DOI: https://doi.org/10.1007/s00153-021-00766-7