Cuppability of Simple and Hypersimple Sets

@article{Kummer2007CuppabilityOS,
  title={Cuppability of Simple and Hypersimple Sets},
  author={Martin Kummer and Marcus Sch{\"a}fer},
  journal={Notre Dame J. Formal Log.},
  year={2007},
  volume={48},
  pages={349-369}
}
An incomplete degree is cuppable if it can be joined by an incomplete degree to a complete degree. For sets fulfilling some type of simplicity property one can now ask whether these sets are cuppable with respect to a certain type of reducibilities. Several such results are known. In this paper we settle all the remaining cases for the standard notions of simplicity and all the main strong reducibilities. There are two sides to every question. —Protagoras, quoted in Diogenes Laertius, Lives of… 
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