Embedding jump upper semilattices into the Turing degrees

Abstract

We prove that every countable jump upper semilattice can be embedded in 𝒟, where a jump upper semilattice (jusl) is an upper semilattice endowed with a strictly increasing and monotone unary operator that we call jump, and 𝒟 is the jusl of Turing degrees. As a corollary we get that the existential theory of 〈D,≤_T,∨,’〉 is decidable. We also prove that this result is not true about jusls with 0, by proving that not every quantifier free 1-type of jusl with 0 is realized in 𝒟. On the other hand, we show that every quantifier free 1-type of jump partial ordering (jpo) with 0 is realized in 𝒟. Moreover, we show that if every quantifier free type, p(x₁,…,x_n), of jpo with 0, which contains the formula x₁≤ 0^(m)∧…∧x_n≤ 0^(m) for some m, is realized in 𝒟, then every quantifier free type of jpo with 0 is realized in 𝒟.

We also study the question of whether every jusl with the c.p.p. and size κ≤ 2^ℵ₀ is embeddable in 𝒟. We show that for κ=2^ℵ₀ the answer is no, and that for κ=ℵ₁ it is independent of ZFC. (It is true if MA(κ) holds.)