## Journal of Symbolic Logic

### Non-branching degrees in the Medvedev lattice of Π⁰₁ classes

Christopher P. Alfeld

#### 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

https://projecteuclid.org/euclid.jsl/1174668385

Digital Object Identifier
doi:10.2178/jsl/1174668385

Mathematical Reviews number (MathSciNet)
MR2298472

Zentralblatt MATH identifier
1122.03043

#### 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