# Topological aspects of the Medvedev lattice

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## Abstract

We study the Medvedev degrees of mass problems with distinguished topological properties, such as denseness, closedness, or discreteness. We investigate the sublattices generated by these degrees; the prime ideal generated by the dense degrees and its complement, a prime filter; the filter generated by the nonzero closed degrees and the filter generated by the nonzero discrete degrees. We give a complete picture of the relationships of inclusion holding between these sublattices, these filters, and this ideal. We show that the sublattice of the closed Medvedev degrees is not a Brouwer algebra. We investigate the dense degrees of mass problems that are closed under Turing equivalence, and we prove that the dense degrees form an automorphism base for the Medvedev lattice. The results hold for both the Medvedev lattice on the Baire space and the Medvedev lattice on the Cantor space.

### Keywords

Medvedev reducibility Baire space Cantor space### Mathematics Subject Classification (2000)

03D30## Preview

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### References

- 1.Balbes R., Dwinger P.: Distributive Lattices. University of Missouri Press, Columbia (1974)MATHGoogle Scholar
- 2.Bianchini C., Sorbi A.: A note on closed degrees of difficulty of the Medvedev lattice. Math. Log. Q.
**42**, 127–133 (1996)CrossRefMATHMathSciNetGoogle Scholar - 3.Cooper S.B.: Computability Theory. Chapman & Hall/CRC Mathematics, Boca Raton, London, New York, Washington, DC (2003)Google Scholar
- 4.Dyment E.Z.: Certain properties of the Medvedev lattice. Mathematics of the USSR Sbornik,
**30**, 321–340 (1976) English TranslationCrossRefGoogle Scholar - 5.Ershov Y.L.: Note on a problem of Rogers. Algebra Log.
**8**, 285 (1969)CrossRefGoogle Scholar - 6.Kechris A.S.: Classical Descriptive Set Theory, volume 156 of Graduate Texts in Mathematics. Springer–Verlag, Heidelberg (1995)Google Scholar
- 7.Lerman M.: Degrees of Unsolvability. Perspectives in Mathematical Logic. Springer–Verlag, Heidelberg (1983)Google Scholar
- 8.Medevdev Y.T.: Degrees of difficulty of the mass problems. Dokl. Nauk. SSSR
**104**(4), 501–504 (1955)Google Scholar - 9.Muchnik A.A.: On strong and weak reducibility of algorithmic problems. Sibirskii Matematicheskii Zhurnal
**4**, 1328–1341 (1963) RussianMATHGoogle Scholar - 10.Platek R.A.: A note on the cardinality of the Medvedev lattice. Proc. Am. Math. Soc.
**25**, 917 (1970)MATHMathSciNetGoogle Scholar - 11.Rogers H. Jr: Theory of Recursive Functions and Effective Computability. McGraw-Hill, New York (1967)MATHGoogle Scholar
- 12.Rozinas, M.: The semilattice of
*e*-degrees. In: Recursive Functions, pp. 71–84. Ivanov. Gos. Univ., Ivanovo (1978). (Russian) MR 82i:03057Google Scholar - 13.Rozinas M.G.: Partial degrees of immune and hyperimmune sets. Siberian Math. J.
**19**, 613–616 (1978)CrossRefMathSciNetGoogle Scholar - 14.Shafer, P.: Characterizing the join-irreducible Medvedev degrees. To appear in Notre Dame J. Formal Log.Google Scholar
- 15.Simpson S.G.: Mass problems and intuitionism. Notre Dame J. Formal Log.
**49**(2), 127–136 (2008)CrossRefMATHMathSciNetGoogle Scholar - 16.Skvortsova E.Z.: Faithful interpretation of the intitionistic prpositional calculus by an initial segment of the Medvedev lattice. Sibirsk. Mat. Zh.
**29**(1), 171–178 (1988) RussianMATHMathSciNetGoogle Scholar - 17.Soare R.I.: Recursively Enumerable Sets and Degrees. Perspectives in Mathematical Logic, Omega Series. Springer–Verlag, Heidelberg (1987)Google Scholar
- 18.Sorbi A.: On some filters and ideals of the Medvedev lattice. Arch. Math. Log.
**30**, 29–48 (1990)CrossRefMATHMathSciNetGoogle Scholar - 19.Sorbi A.: Embedding Brouwer algebras in the Medvedev lattice. Notre Dame J. Formal Log.
**32**(2), 266–275 (1991)CrossRefMATHMathSciNetGoogle Scholar - 20.Sorbi A.: Some quotient lattices of the Medvedev lattice. Z. Math. Logik Grundlag. Math.
**37**, 167–182 (1991)CrossRefMATHMathSciNetGoogle Scholar - 21.Sorbi A., Terwijn S.: Intermediate logics and factors of the Medvedev lattice. Ann. Pure Appl. Log.
**155**(2), 69–86 (2008)CrossRefMATHMathSciNetGoogle Scholar - 22.Terwijn S.A.: The Medvedev lattice of computably closed sets. Arch. Math. Log.
**45**, 179–190 (2006)CrossRefMATHMathSciNetGoogle Scholar - 23.Terwijn S.A.: On the structure of the Medvedev lattice. J. Symbolic Log.
**73**(2), 543–558 (2008)CrossRefMATHMathSciNetGoogle Scholar