Embedding the diamond in the Σ2 enumeration degrees

Abstract

Lachlan [5] has shown that it is not possible to embed the diamond lattice in the r.e. Turing degrees while preserving least and greatest elements; that is, there do not exist incomparable r.e. Turing degrees a and b such that a ∧ b = 0 and a ∨ b = 0′. Cooper [3] has compared the r.e. Turing degrees to the enumeration degrees below 0e′ and has asked if the two structures are elementarily equivalent.In this paper we show that such an embedding is possible in the Σ2enumeration degrees, which implies a negative answer to Cooper's question.Theorem. There are low enumeration degreesaandbsuch thata ∧ b = 0eanda ∨ b = 0e′.Lower case italic letters denote elements of ω while upper case italic letters denote subsets of ω. D, E and F are reserved for finite sets, and K for ′. If D = {x0, x1, …, xn} then the canonical index of D is , and the canonical index of is ∅. Dx denotes the set with canonical index x. {Wi}i∈ω is any fixed standard listing of the r.e. sets, and <·, ·> is any fixed recursive bijection from ω × ω to ω.Intuitively, A is enumeration reducible to B if there is an effective algorithm for producing an enumeration of A from any enumeration of B. There is a natural one-to-one correspondence between all such algorithms and the r.e. sets.