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.

Enumeration Degrees and Topology

“…For the latter, map A ⊆ N to {p ∈ N N | ∀n ∈ N n ∈ A ⇔ ∃k ∈ N p(k) = n + 1}. 1 In words, to embed an enumeration degree into the Medvedev degrees, we move from a set A ⊆ N to the set of all enumerations of A. If we start with a Turing degree, embed it as enumeration degree, and then embed that as a Medvedev degree, we obtain the same degree (although not the same set) as when we move from Turing degrees directly to Medvedev degrees.…”

“…For more on degrees and minimal subshifts, see [21]. The cototal enumeration degrees are further studied in [1,31].…”

Comparing the degrees of enumerability and the closed Medvedev degrees

“…Thus σ ⊇ f ↾ n, contradicting that f ↾ n / ∈ T . Thus A set A is called cototal if A ≤ e A, and an e-degree is called cototal if it contains a cototal set [1]. Every uniformly e-pointed tree w.r.t.…”

“…functions has cototal enumeration degree. This can be accomplished via the easy characterization of the cototal enumeration degrees in terms of the skip operator from [1].…”

“…A quasiminimal degree of enumerability that is compact. The existence of quasiminimal problems of enumerability that are equivalent to compact mass problems is a consequence of Theorem 17 and the fact that there are cototal quasiminimal e-degrees [1]. We think, however, that it is instructive to directly construct a quasiminimal uniformly e-pointed tree w.r.t.…”