On cototality and the skip operator in the enumeration degrees

Abstract

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.