Embedding jump upper semilattices into the Turing degrees

Journal of Symbolic Logic 68 (3):989-1014 (2003)
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 $\langle D, \leq_{T}, \vee, '\rangle$ 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, $p(x_{1},..., x_{n})$ , of jpo with 0, which contains the formula $x_{1} \leq 0{^(m)} \& ... \& x_{n} \leq 0^{(m)}$ 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 $\kappa \leq 2^{\aleph 0}$ is embeddable in D. We show that for $\kappa = 2^{\aleph 0}$ the answer is no, and that for κ = ℵ1 it is independent of ZFC. (It is true if $MA(\kappa)$ holds.)
Keywords No keywords specified (fix it)
Categories (categorize this paper)
DOI 10.2307/4147725
 Save to my reading list
Follow the author(s)
My bibliography
Export citation
Find it on Scholar
Edit this record
Mark as duplicate
Revision history Request removal from index
Download options
PhilPapers Archive

Upload a copy of this paper     Check publisher's policy on self-archival     Papers currently archived: 15,879
External links
Setup an account with your affiliations in order to access resources via your University's proxy server
Configure custom proxy (use this if your affiliation does not provide a proxy)
Through your library
References found in this work BETA

No references found.

Add more references

Citations of this work BETA

No citations found.

Add more citations

Similar books and articles

Monthly downloads

Sorry, there are not enough data points to plot this chart.

Added to index


Total downloads

1 ( #629,303 of 1,725,168 )

Recent downloads (6 months)

1 ( #349,161 of 1,725,168 )

How can I increase my downloads?

My notes
Sign in to use this feature

Start a new thread
There  are no threads in this forum
Nothing in this forum yet.