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Initial segments of the lattice of ideals of r.e. degrees

Published online by Cambridge University Press:  12 March 2014

Frank P. Weber
Frank P. Weber*
Affiliation:
134 Zhong Shaw Road, Hsuei-Chia Chen, Tainan Shien, Taiwan ROC
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Extract

Embeddings into initial segments of (the lattice of ideals of r.e. degrees) are relevant for the investigation of decidability of (fragments of) the theory of (with constant-symbols denoting principal ideals). Interest in this investigation stems from open problems concerning decidability of fragments of the theory of R (the usl of r.e. degrees) and embeddings into R. Ambos-Spies and Lerman established comprehensive nonembeddability [AS/LR1] and embeddability [AS/LR2] conditions, but their complementarity is open. (For background on embeddings and decidability consult Lerman [LR1] and Lerman and Shore [LR/SH].) The first systematic investigation of embeddings into is in Calhoun [CL], where the existence of uniform bounds, in the case of embeddings into , on the number of “join traces” arising with nondistributive lattices is shown to remove “obstructions” to lattice-embeddings into R (such as typified in the nonembeddable lattice S8, see [LA/SR], [AS/LR1/). The analysis to follow extends this, by showing that it also removes the “obstruction” to lattice-embeddings into initial segments of R, arising with Lachlan's M5, which embeds into R, but not into every nontrivial initial segment R[0, a], by a result of Downey [DW]. The techniques are Lachlan's [LA], as adapted in Lerman's “pinball machine model” [LR1, LR2], combined with Fejer's “meet-trick” [FJ] as used by Downey [DW] in the proof that every distributive lattice is embeddable into every nontrivial initial segment of r.e. degrees R[0, a], a > 0. We present a sufficient condition, “allowing evasion” [Definition 4.2] for lattice embeddability into every [0, a], a > 0.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 59 , Issue 4 , December 1994 , pp. 1326 - 1350
Copyright
Copyright © Association for Symbolic Logic 1994

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References

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