Universal Cupping Degrees.

Cupping nonzero computably enumerable (c.e. for short) degrees to 0′ in various structures has been one of the most important topics in the development of classical computability theory. An incomplete c.e. degree a is cuppable if there is an incomplete c.e. degree b such that a∪b=0′, and noncuppable if there is no such degree b. Sacks splitting theorem shows the existence of cuppable degrees. However, Yates(unpublished) and Cooper [3] proved that there are noncomputable noncuppable degrees. After that, Harrington and Shelah were able to employ the cupping/noncupping properties to show that the theory of the c.e. degrees under relation ≤ is undecidable. Cuppable and noncuppable degrees were further studied later. See Harrington [7], Miller [10], Fejer and Soare [6], Ambos-Spies, Lachlan and Soare [1], etc.. [ABSTRACT FROM AUTHOR]

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