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.