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September 2003 Embedding jump upper semilattices into the Turing degrees
Antonio Montalbán
J. Symbolic Logic 68(3): 989-1014 (September 2003). DOI: 10.2178/jsl/1058448451

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 〈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.)

Citation

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Antonio Montalbán. "Embedding jump upper semilattices into the Turing degrees." J. Symbolic Logic 68 (3) 989 - 1014, September 2003. https://doi.org/10.2178/jsl/1058448451

Information

Published: September 2003
First available in Project Euclid: 17 July 2003

zbMATH: 1059.03038
MathSciNet: MR2004H:03089
Digital Object Identifier: 10.2178/jsl/1058448451

Rights: Copyright © 2003 Association for Symbolic Logic

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Vol.68 • No. 3 • September 2003
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