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Automorphisms of the truth-table degrees are fixed on a cone

Published online by Cambridge University Press:  12 March 2014

Bernard A. Anderson
Bernard A. Anderson*
Affiliation:
Department of Theoretical Computer, Science and Mathematical Logic, Faculty of Mathematics and Physics, Charles University, Malostranské Náměstí 25, 118 00 Prague 1, Czech Republic, E-mail: [email protected]
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Abstract

Let Dtt denote the set of truth-table degrees. A bijection π: Dtt → Dtt is an automorphism if for all truth-table degrees x and y we have x ≤tty ⇔ π(x) ≤ttπ(y). We say an automorphism π is fixed on a cone if there is a degree b such that for all x ≥ttb we have π(x) = x. We first prove that for every 2-generic real X we have X′ ≰ttX ⊕ 0′. We next prove that for every real X ≥tt 0′ there is a real Y such that Y ⊕ 0′ ≡ttY′ ≡ttX. Finally, we use this to demonstrate that every automorphism of the truth-table degrees is fixed on a cone.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 74 , Issue 2 , June 2009 , pp. 679 - 688
Copyright
Copyright © Association for Symbolic Logic 2009

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