Journal of Symbolic Logic

Embedding jump upper semilattices into the Turing degrees

Antonio Montalbán

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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 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≤ 20 is embeddable in 𝒟. We show that for κ=20 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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J. Symbolic Logic, Volume 68, Issue 3 (2003), 989-1014.

First available in Project Euclid: 17 July 2003

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Montalbán, Antonio. Embedding jump upper semilattices into the Turing degrees. J. Symbolic Logic 68 (2003), no. 3, 989--1014. doi:10.2178/jsl/1058448451.

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