A set A ⊆ ω A\subseteq \omega is cototal if it is enumeration reducible to its complement,  A ¯ \overline {A} . The skip of  A A is the uniform upper bound of the complements of all sets enumeration reducible to  A A . These are closely connected:  A A has cototal degree if and only if it is enumeration reducible to its skip. We study cototality and related properties, using the skip operator as a tool in our investigation. We give many examples of classes of enumeration degrees that either guarantee or prohibit cototality. We also study the skip for its own sake, noting that it has many of the nice properties of the Turing jump, even though the skip of  A A is not always above  A A (i.e., not all degrees are cototal). In fact, there is a set that is its own double skip.

We say that A ≤LRB if every B-random number is A-random. Intuitively this means that if oracle A can identify some patterns on some real γ, oracle B can also find patterns on γ. In other words, B is at least as good as A for this purpose. We study the structure of the LR degrees globally and locally (i.e., restricted to the computably enumerable degrees) and their relationship with the Turing degrees. Among other results we show that whenever ∝ is not GL2 the LR degree of ∝ bounds degrees (so that, in particular, there exist LR degrees with uncountably many predecessors) and we give sample results which demonstrate how various techniques from the theory of the c.e. degrees can be used to prove results about the c.e. LR degrees.

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We show that if A and B form a nontrivial K-pair, then there is a semi-computable set C such that A ≤e C and B ≤e C. As a consequence, we obtain a definition of the total enumeration degrees: a nonzero enumeration degree is total if and only if it is the join of a nontrivial maximal K-pair. This answers a long-standing question of Hartley Rogers, Jr. We also obtain a definition of the "c.e. in" relation for total degrees in the enumeration degrees. 2010 Mathematics Subject Classification. 03D30. Key words and phrases. enumeration degrees, total enumeration degrees, automorphisms of degree structures.

a b s t r a c tThis paper continues the project, initiated in (Arslanov, Cooper and Kalimullin 2003) [3], of describing general conditions under which relative splittings are derivable in the local 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.

This paper continues the project, initiated in [ACK], of describing general conditions under which relative splittings are derivable in the local structure of the enumeration degrees.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, and a construction of a Π 0 1 e-degree unsplittable over a Δ 2 e-degree below it.In [ACK] it was shown that using semirecursive sets one can construct minimal pairs of e-degrees by both effective and uniform ways, following which new results concerning the local distribution of total e-degrees and of the degrees of semirecursive sets enabled one to proceed, via the natural embedding of the Turing degrees in the enumeration degrees, to results concerning embeddings of the diamond lattice in the e-degrees. A particularly striking application of these techniques was a relatively simple derivation of a strong generalisation of the Ahmad Diamond Theorem.This paper extends the known constraints on further progress in this direction, such as the result of Ahmad and Lachlan [AL98] showing the existence of a nonsplitting Δ 0 2 e-degree > 0 e , and the recent result of Soskova [Sos07] showing that 0 e is unsplittable in the Σ 0 2 e-degrees above some Σ 0 2 e-degree < 0 e . This work also relates to results (e.g. Cooper and Copestake [CC88]) limiting the local distribution of total e-degrees.For further background concerning enumeration reducibility and its degree structure, the reader is referred to Cooper [Co90], Sorbi [Sor97] or Cooper [Co04], chapter 11. Theorem 1. If a < h ≤ 0 , a is low and h is total and high then there is a low total e-degree b such that a ≤ b < h. Corollary 2. Let a < h ≤ 0 , h be a high total e-degree, a be a low e-degree. Then there are Δ 0 2 e-degrees b 0 < h and b 1 < h such that a = b 0 ∩ b 1 and h = b 0 ∪ b 1 .

Σ 0 2 e-degrees Non-splitting a b s t r a c tWe complete a study of the splitting/non-splitting properties of the enumeration degrees below 0 e by proving an analog of Harrington's non-splitting theorem for the Σ 0 2 enumeration degrees. We show how non-splitting techniques known from the study of the c.e. Turing degrees can be adapted to the enumeration degrees.

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