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 ≤ 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.)

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this paper.

We prove that the two quantifier theory of the Turing degrees with order, join and jump is undecidable.

We show there is a non-recursive r.e. set A such that if W is any low r.e. set, then the join W # A is also low. That is, A is "almost deep". This answers a question of Jockusch. The almost deep degrees form an definable ideal in the r.e. degrees (with jump.) 1 Introduction Bickford and Mills in [1] defined an r.e. degree a to be deep in case, for all other r.e. degrees b, (b # a) # = b # . In other words, joining with a preserves the jump of every r.e. degree. They asked whether there are non-recursive deep degrees. Part of the motivation for asking this question is an interest in finding definable ideals in the r.e. degrees. Such ideals can help in understanding the global structure of the r.e. degrees; in particular they can provide ways of understanding definability properties and automorphisms. We know of few # Partially supported by NSF Grant DMS-96-3465. + Partially supported by NSF Grant DMS-97-96121. 1 non-trivial examples of such ideals. One example is the idea...