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Non-branching degrees in the Medvedev lattice of Π10 classes

Published online by Cambridge University Press:  12 March 2014

Christopher P. Alfeld
Christopher P. Alfeld*
Affiliation:
Department of Mathematics, University of Wisconsin, Madison, WI 53706, USA. E-mail: alfeld@math.wisc.edu
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Abstract

A class is the set of paths through a computable tree. Given classes P and Q, P is Medvedev reducible to Q, P ≤MQ, if there is a computably continuous functional mapping Q into P. We look at the lattice formed by subclasses of 2ω 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.

Keywords

classesMedvedev latticenon-branching degree

Information

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 72 , Issue 1 , March 2007 , pp. 81 - 97
Copyright
Copyright © Association for Symbolic Logic 2007

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References

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