Journal of Symbolic Logic

Non-branching degrees in the Medvedev lattice of Π⁰₁ classes

Christopher P. Alfeld

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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.

Article information

Source
J. Symbolic Logic, Volume 72, Issue 1 (2007), 81-97.

Dates
First available in Project Euclid: 23 March 2007

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1174668385

Digital Object Identifier
doi:10.2178/jsl/1174668385

Mathematical Reviews number (MathSciNet)
MR2298472

Zentralblatt MATH identifier
1122.03043

Keywords
Π⁰₁ classes Medvedev lattice non-branching degree

Citation

Alfeld, Christopher P. Non-branching degrees in the Medvedev lattice of Π⁰₁ classes. J. Symbolic Logic 72 (2007), no. 1, 81--97. doi:10.2178/jsl/1174668385. https://projecteuclid.org/euclid.jsl/1174668385


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