Journal of Symbolic Logic

Generalized Nonsplitting in the Recursively Enumerable Degrees

Steven D. Leonhardi

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Abstract

We investigate the algebraic structure of the upper semi-lattice formed by the recursively enumerable Turing degrees. The following strong generalization of Lachlan's Nonsplitting Theorem is proved: Given $n \geq 1$, there exists an r.e. degree $\mathbf{d}$ such that the interval $\lbrack\mathbf{d, 0'}\rbrack \subset\mathbf{R}$ admits an embedding of the $n$-atom Boolean algebra $\mathscr{B}_n$ preserving (least and) greatest element, but also such that there is no $(n + 1)$-tuple of pairwise incomparable r.e. degrees above $\mathbf{d}$ which pairwise join to $\mathbf{0'}$ (and hence, the interval $\lbrack\mathbf{d, 0'}\rbrack \subset\mathbf{R}$ does not admit a greatest-element-preserving embedding of any lattice $\mathscr{L}$ which has $n + 1$ co-atoms, including $\mathscr{B}_{n + 1}$). This theorem is the dual of a theorem of Ambos-Spies and Soare, and yields an alternative proof of their result that the theory of $\mathbf{R}$ has infinitely many one-types.

Article information

Source
J. Symbolic Logic, Volume 62, Issue 2 (1997), 397-437.

Dates
First available in Project Euclid: 6 July 2007

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1183745235

Mathematical Reviews number (MathSciNet)
MR1464106

Zentralblatt MATH identifier
0884.03043

JSTOR
links.jstor.org

Citation

Leonhardi, Steven D. Generalized Nonsplitting in the Recursively Enumerable Degrees. J. Symbolic Logic 62 (1997), no. 2, 397--437. https://projecteuclid.org/euclid.jsl/1183745235


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