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March 2007 Non-branching degrees in the Medvedev lattice of Π⁰₁ classes
Christopher P. Alfeld
J. Symbolic Logic 72(1): 81-97 (March 2007). DOI: 10.2178/jsl/1174668385

Abstract

A $\Sigma^0_1$ class is the set of paths through a computable tree. Given classes $P$ and $Q$, $P$ is Medvedev reducible to $Q, P \leq_{M} Q$, if there is a computably continuous functional mapping $Q$ into $P$. We look at the lattice formed by $\Sigma^0_1$ subclasses of $2^\omega$ under this reduction. It is known that the degree of a splitting class of c.e. sets is non-branching. We further characterize non-branching degrees, providing two additional properties which guarantee non-branching: inseparable and hyperinseparable. Our main result is to show that non-branching iff inseparable if hyperinseparable if homogeneous and that all unstated implications do not hold. We also show that inseparable and not hyperinseparable degrees are downward dense.

Citation

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Christopher P. Alfeld. "Non-branching degrees in the Medvedev lattice of Π⁰₁ classes." J. Symbolic Logic 72 (1) 81 - 97, March 2007. https://doi.org/10.2178/jsl/1174668385

Information

Published: March 2007
First available in Project Euclid: 23 March 2007

zbMATH: 1122.03043
MathSciNet: MR2298472
Digital Object Identifier: 10.2178/jsl/1174668385

Rights: Copyright © 2007 Association for Symbolic Logic

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Vol.72 • No. 1 • March 2007
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