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 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.

We study computable embeddings for pairs of structures, i.e. for classes containing precisely two non-isomorphic structures. Surprisingly, even for some pairs of simple linear orders, computable embeddings induce a nontrivial degree structure. Our main result shows that {ωProposition (Greenberg and, independently, Kalimullin; see [13,14]). If K 0 ≤ c K 1 , then K 0 ≤ tc K 1 . The converse is not true.

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