Embedding jump upper semilattices into the Turing degrees

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, ..., xn), of jpo with 0, which contains the formula x1 ≤ 0(m) & ... & xn ≤ 0(m) for some m, is realized in D, then every 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 ≤ 2א0 is embeddable in D. We show that for κ = 2א0 the answer is no, and that for κ = א1 it is independent of ZFC. (It is true if MA(κ) holds.) §