Prompt enumerations and relative randomness (Q2907062)

In the paper under review a notion of prompt non-low-for-randomness is introduced. It is analogous to the property of prompt simplicity and to the equivalent property of prompt permitting. The introduced property is a prompt form of non-low-for-random permitting, which is the natural notion of permitting in the context of relative randomness. It is shown that the class of Turing degrees of promptly non-low-for-random c.e. sets is non-trivial and a proper subclass of the non-low-for-random promptly simple degrees. The prompt non-low-for-randomness is also considered in the context of the LR-degrees and LR-reducibility, a weak reducibility arising naturally from relative randomness.