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Embedding jump upper semilattices into the Turing degrees

Published online by Cambridge University Press:  12 March 2014

Antonio Montalbán
Antonio Montalbán*
Affiliation:
Department of Mathematics, Cornell University, Ithaca, NY 14853, USA, E-mail: antonio@math.cornell.edu
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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 formula x1 ≤ 0(m) & … & xn ≤ 0(m) for some m, 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 for κ = ℵ1 it is independent of ZFC. (It is true if MA(κ) holds.)

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 68 , Issue 3 , September 2003 , pp. 989 - 1014
Copyright
Copyright © Association for Symbolic Logic 2003

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