Embedding jump upper semilattices into the Turing degrees
Montalbán, Antonio
J. Symbolic Logic, Tome 68 (2003) no. 1, p. 989-1014 / Harvested from Project Euclid
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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≤ 2ℵ0 is embeddable in 𝒟. We show that for κ=2ℵ0 the answer is no, and that for κ=ℵ1 it is independent of ZFC. (It is true if MA(κ) holds.)
Publié le : 2003-09-14
Classification:  
@article{1058448451,
     author = {Montalb\'an, Antonio},
     title = {Embedding jump upper semilattices into the Turing degrees},
     journal = {J. Symbolic Logic},
     volume = {68},
     number = {1},
     year = {2003},
     pages = { 989-1014},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1058448451}
}

Montalbán, Antonio. Embedding jump upper semilattices into the Turing degrees. J. Symbolic Logic, Tome 68 (2003) no. 1, pp.  989-1014. http://gdmltest.u-ga.fr/item/1058448451/