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 However, Yates(unpublished) and Cooper [3] proved that there are noncomputable noncuppable degrees.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.. In [13], Slaman and Steel proved that each nonzero degree below 0′ has a 1-generic complement. Thus, for any nonzero c.e. degree a, there is a 1-generic degree d cupping a to 0′. In [12], Seetapun and Slaman proved that for any given nonzero c.e. degree b, there is a minimal degree m cupping b to 0′. Cooper and Seetapun, and independently Li [9], announced that there is a degree below 0′ cupping every nonzero c.e. degree to 0′. Recently, Lewis [8] proved that there is a minimal degree cupping every nonzero c.e. degree to 0′. A natural generalization of computably enumerable sets is the n-c.e. sets. A set is n-c.e. if A has an effective approximation {As}s∈ω such that A0 = ∅ and for all x, |{s+ 1 | f(x, s) 6= f(x, s+ 1)}| ≤ n. Obviously, the 1-c.e. sets are just the c.e. sets and ∪n{A : A is n-c.e.} is just the Boolean algebra generated by the c.e. sets. Ershov [4] and [5] extended the hierarchy of n-c.e. sets to transfinite levels. In particular, the ω-c.e. sets are those having the following characterization: A ⊆ ω is ω-c.e. if and only if there are two computable functions f(x, s), g(x)