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Embedding finite lattices into the ideals of computably enumerable turing degrees

Published online by Cambridge University Press:  12 March 2014

William C. Calhoun  and
Manuel Lerman
William C. Calhoun
Affiliation:
Department of Mathematics, Computer Science and Statistics, Bloomsburg University, Bloomsburg, PA 17815-1301, USA, E-Mail: [email protected]
Manuel Lerman
Affiliation:
Department of Mathematics, University of Connecticut, Storrs, CT 06260-3009, USA, E-Mail: [email protected]
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Abstract.

We show that the lattice L20 is not embeddable into the lattice of ideals of computably enumerable Turing degrees (ℐ), We define a structure called a pseudolattice that generalizes the notion of a lattice, and show that there is a Π2 necessary and sufficient condition for embedding a finite pseudolattice into ℐ.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 66 , Issue 4 , December 2001 , pp. 1791 - 1802
Copyright
Copyright © Association for Symbolic Logic 2001

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References

REFERENCES

[1]Ambos-Spies, K. and Lerman, M., Lattice embeddings into the recursively enumerable degrees, this Journal, vol. 51 (1986), pp. 257–272.Google Scholar
[2]Calhoun, W. C., The lattice of ideals of recursively enumerable degrees, Ph.D. thesis, University of California, Berkeley, CA, 1990.Google Scholar
[3]Downey, R. G., Lattice non-embeddings and initial segments of the recursively enumerable degrees, Annals of Pure and Applied Logic, vol. 49 (1990), pp. 97–119.CrossRefGoogle Scholar
[4]Epstein, R. L. and Posner, D., Diagonalization in degree constructions, this Journal, vol. 43 (1978), pp. 280–283.Google Scholar
[5]Lachlan, A. H. and Soare, R. I., Not every finite lattice is embeddable in the recursively enumerable degrees, Advances in Mathematics, vol. 37 (1980), pp. 74–82.CrossRefGoogle Scholar
[6]Lempp, S. and Lerman, M., A finite lattice without critical triple that cannot be embedded into the enumerable Turing degrees, Annals of Pure and Applied Logic, vol. 87 (1997), pp. 167–185.CrossRefGoogle Scholar
[7]Lerman, M., A necessary and sufficient condition for embedding ranked finite partial lattices into the computably enumerable degrees, Annals of Pure and Applied Logic, vol. 94 (1998), pp. 143–180.CrossRefGoogle Scholar
[8]Lerman, M., A necessary and sufficient condition for embedding principally decomposable finite lattices into the computably enumerable degrees, Annals of Pure and Applied Logic, vol. 101 (2000), pp. 275–297.CrossRefGoogle Scholar
[9]Weber, F. P., Initial segments of the lattice of ideals of r. e. degrees, this Journal, vol. 59 (1994), pp. 1326–1350.Google Scholar
[10]Weinstein, B., On embeddings of the 1-3-1 lattice into the recursively enumerable degrees, Ph.D. thesis, University of California, Berkeley, CA, 1988.Google Scholar