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Non-cupping, measure and computably enumerable splittings

Published online by Cambridge University Press:  01 February 2009

GEORGE BARMPALIAS
Affiliation:
School of Mathematics, University of Leeds, LS2 9JT, United Kingdom
ANTHONY MORPHETT
Affiliation:
School of Mathematics, University of Leeds, LS2 9JT, United Kingdom Email: awmorp@gmail.com

Abstract

We show that there is a computably enumerable function f (that is, computably approximable from below) that dominates almost all functions, and fW is incomplete for all incomplete computably enumerable sets W. Our main methodology is the LR equivalence relation on reals: ALRB if and only if the notions of A-randomness and B-randomness coincide. We also show that there are c.e. sets that cannot be split into two c.e. sets of the same LR degree. Moreover, a c.e. set is low for random if and only if it computes no c.e. set with this property.

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Copyright
Copyright © Cambridge University Press 2009

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