Embedding jump upper semilattices into the Turing degrees
Antonio Montalbán · Journal of Symbolic Logic · 2003
Abstract We prove that every countable jump upper semilattice can be embedded in , where a jump upper semilattice (jusl) is an upper semilattice endowed with a strictly increasing and monotone unary operator that we call jump, and 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 . On the other hand, we show that every quantifier free 1-type of jump partial ordering (jpo) with 0 is realized in . Moreover, we show that if every quantifier free type,p(x1,…,xn), of jpo with 0, which contains the formulax1≤ 0(m)& … &xn≤ 0(m)for somem, is realized in , then every quantifier free type of jpo with 0 is realized in . We also study the question of whether every jusl with the c.p.p. and size is embeddable in . We show that for the answer is no, and that forκ= ℵ1it is independent of ZFC. (It is true if MA(κ) holds.)