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Some remarks on the algebraic structure of the Medvedev Lattice

Published online by Cambridge University Press:  12 March 2014

Andrea Sorbi
Andrea Sorbi*
Affiliation:
Dipartimento di Matematica, Università di Siena, 53100 Siena, Italy
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Abstract

This paper investigates the algebraic structure of the Medvedev lattice . We prove that is not a Heyting algebra. We point out some relations between and the Dyment lattice and the Mučnik lattice. Some properties of the degrees of enumerability are considered. We give also a result on embedding countable distributive lattices in the Medvedev lattice.

Information

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 55 , Issue 2 , June 1990 , pp. 831 - 853
Copyright
Copyright © Association for Symbolic Logic 1990

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References

REFERENCES

[BD]Balbes, R. and Dwinger, P., Distributive lattices, University of Missouri Press, Columbia, Missouri, 1974.Google Scholar
[CO]Cooper, S. B., Partial degrees and the density problem, this Journal, vol. 47 (1982), pp. 854–859.Google Scholar
[DT]de Jongh, D. H. J. and Troelstra, A. S., On the connection of partially ordered sets with some pseudo-Boolean algebras, Indigationes Mathematicae, vol. 28 (1966), pp. 317–329.CrossRefGoogle Scholar
[DY1]Dyment, E. Z., Certain properties of the Medvedev lattice, Matematičeskiĭ Sbornik, vol. 101 (143) (1976), pp. 360–379; English translation, Mathematics of the USSR Sbornik, vol. 30 (1976), pp. 321–340.Google Scholar
[DY2]Dyment, E. Z., Exact bounds of denumerable collections of degrees of difficulty, Matematičeskie Zametki, vol. 28 (1980), pp. 899–910; English translation, Mathematical Notes, vol. 28 (1980), pp. 904–909.Google Scholar
[JP]Jockusch, C. G. Jr., and Posner, D., Automorphism bases for degrees of unsolvability, Israel Journal of Mathematics, vol. 40 (1981), pp. 150–164.CrossRefGoogle Scholar
[LE]Lerman, M., Degrees of unsolvability, Springer-Verlag, Berlin, 1983.CrossRefGoogle Scholar
[MA]Magari, R., Modal diagonalizable algebras, Bollettino dell'Unione Matematica Italiana B, ser. 5, vol. 15 (1978), pp. 303–320.Google Scholar
[ML]Lane, S. Mac, Categories for the working mathematician, Springer-Verlag, Berlin, 1971.CrossRefGoogle Scholar
[ME]Medvedev, Ju. T., Degrees of difficulty of the mass problems, Doklady Akademii Nauk SSSR, vol. 104 (1955), pp. 501–504. (Russian)Google Scholar
[MU]Mučnik, A. A., On strong and weak reducibility of algorithmic problems, Sibirskiĭ Matematičeskiĭ Žurnal, vol. 4 (1963), pp. 1328–1341. (Russian)Google Scholar
[PL]Platek, R. A., A note on the cardinality of the Medvedev lattice, Proceedings of the American Mathematical Society, vol. 25 (1970), p. 917.Google Scholar
[RO]Rogers, H. Jr., Theory of recursive functions and effective computability, McGraw-Hill, New York, 1967.Google Scholar
[RZ]Rozinas, M., The semilattice of e-degrees, Recursive functions (Poljakov, E. A., editor), Ivanovskiĭ Gosudarstvennyĭ Universitet, Ivanovo, 1978, pp. 71–84. (Russian) MR 82i:03057.Google Scholar
[SA]Sambin, G., An effective fixed point theorem in intuitionistic diagonalizable algebras, Studia Logica, vol. 35 (1976), pp. 345–361.CrossRefGoogle Scholar
[SO]Soare, R. I., Recursively enumerable sets and degrees, Springer-Verlag, Berlin, 1987.CrossRefGoogle Scholar
[SR1]Sorbi, A., On some filters and ideals of the Medvedev lattice (to appear).Google Scholar
[SR2]Sorbi, A., Embedding Brouwer algebras in the Medvedev lattice (to appear).Google Scholar
[SR3]Sorbi, A., The fine structure of the Medvedev lattice and the partial degrees, Ph.D. thesis, Graduate Center, City University of New York, New York, 1987.Google Scholar
[UR]Ursini, A., Intuitionistic diagonalizable algebras, Algebra Universalis, vol. 9 (1979), pp. 229–237.CrossRefGoogle Scholar