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Some nondistributive lattices as initial segments of the degrees of unsolvability1

Published online by Cambridge University Press:  12 March 2014

Manuel Lerman
Manuel Lerman*
Affiliation:
Cornell University
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Extract

The question, “What do initial segments of the degrees of unsolvability look like?” has interested recursive function theorists for several years. Sacks [4] hypothesized that Sis a finite initial segment of degrees if and only if S is order-isomorphic to a finite initial segment of some upper semilattice with a least element. Lachlan [2] suggested the generalization, S is an initial segment of degrees if S is order-isomorphic to some countable upper semilattice with both least and greatest elements.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 34 , Issue 1 , 29 May 1969 , pp. 85 - 98
Copyright
Copyright © Association for Symbolic Logic 1969

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Footnotes

1

This work was partly supported by NSF grant GP 8732. We are grateful to A. Nerode for many helpful discussions on this problem.

References

[1]Birkhoff, G., Lattice theory, American Mathematical Society Colloquium Publications, vol. XXV, American Mathematical Society, Providence, R.I., 1948.Google Scholar
[2]Hugill, D. F. and Lachlan, A. H., Distributive initial segments of the degrees of unsolvability (to appear).Google Scholar
[3]Ryser, H. J., Combinatorial mathematics, The Carus Mathematical Monographs, number 14, Mathematical Association of America, Wiley, New York, 1963.Google Scholar
[4]Sacks, G. E., Degrees of unsolvability, Annals of mathematics studies, number 55, Princeton Univ. Press, Princeton, N.J., 1963.Google Scholar
[5]Spector, C., On degrees of recursive unsolvability, Annals of mathematics, vol. 64 (1956), pp. 581–592.CrossRefGoogle Scholar