Embedding jump upper semilattices into the Turing degrees

Journal of Symbolic Logic 68 (3):989-1014 (2003)
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 $\langle D, \leq_{T}, \vee, '\rangle$ 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, $p(x_{1},..., x_{n})$ , of jpo with 0, which contains the formula $x_{1} \leq 0{^(m)} \& ... \& x_{n} \leq 0^{(m)}$ 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 $\kappa \leq 2^{\aleph 0}$ is embeddable in D. We show that for $\kappa = 2^{\aleph 0}$ the answer is no, and that for κ = ℵ1 it is independent of ZFC. (It is true if $MA(\kappa)$ holds.)
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