Abstract. We show that every nonzero ∆ 0 2 e-degree bounds a minimal pair. On the other hand, there exist Σ 0 2 e-degrees which bound no minimal pair. 1.

We prove that there exists a noncappable enumeration degree strictly below 0 0 e . Two notions of relative computability, Turing and enumeration reducibility, are basic to any natural fine-structure theory for the classes of computable and incomputable objects. Of the theories for the corresponding degree structures (D D D and D D D e ), that for the Turing degrees is the better developed, mainly due to the depth of knowledge of specific local structure (see for example Lerman [Le83], Odifreddi [Od89] and Soare [So87]). Despite its importance (see [Co90]) in applications to nondeterministic computations, to relative computability involving partial information, in providing models of -calculus, and in setting 1991 Mathematics Subject Classification. 03D30. Key words and phrases. Enumeration operator, enumeration degree, \Sigma 2 -set. Research partially supported by British Council-MURST grant no. ROM/889 /92/81, S.E.R.C. Research Grant no. GR/H 02165 and EC Human Capital and Mobili...

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in fulfilment of the requirements for the degree of

We prove the following three theorems on the enumeration degrees of # 0 2 sets. Theorem A: There exists a nonzero noncuppable # 0 2 enumeration degree. Theorem B: Every nonzero # 0 2 enumeration degree is cuppable to 0 # e by an incomplete total enumeration degree. Theorem C: There exists a nonzero low # 0 2 enumeration degree with the anticupping property.

We construct an incomplete 3-c.e. enumeration degree which is maximal among the n-c.e. enumeration degrees for every n with 3 # n # #. Consequently the n-c.e. enumeration degrees are not dense for any such n. We show also that no low n-c.e. e-degree can be maximal among the n-c.e. e-degrees, for 2 # n # #.

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This paper continues the project, initiated in [ACK], of describing general conditions under which relative splittings are derivable in the lo-cal structure of the enumeration degrees, for which the Ershov hierarchy provides an informative setting. The main results below include a proof that any high total e-degree below 0 ′ e is splittable over any low e-degree below it, a non-cupping result in the high enumeration degrees which occurs at a low level of the Ershov hierarchy, and a ∅ ′′ ′-priority construction of a Π 0 1 e-degree unsplittable over a 3-c.e. e-degree below it. 1

We prove that there exists a noncappable enumeration degree strictly below 0 0 e .