Abstract.
Let w and M be the countable distributive lattices of Muchnik and Medvedev degrees of non-empty Π1 0 subsets of 2ω, under Muchnik and Medvedev reducibility, respectively. We show that all countable distributive lattices are lattice-embeddable below any non-zero element of w . We show that many countable distributive lattices are lattice-embeddable below any non-zero element of M .
Similar content being viewed by others
References
Balbes, R.: Projective and injective distributive lattices. Pacific J. Math. 21, 405–420 (1967)
Binns, S.E.: The Medvedev and Muchnik Lattices of Π0 1 Classes. PhD Thesis Pennsylvania State University, 2003, V+80 pages
Binns, S.E.: A splitting theorem for the Medvedev and Muchnik lattices. Math. Logic Quarterly 49, 327–335 (2003)
Cenzer, D., Hinman, P.G.: Density of the Medvedev lattice of Π0 1 classes. Archive for Math. Logic 43, 583–600 (2003)
COMP-THY e-mail list. http://listserv.nd.edu/archives/comp-thy. html, September 1995 to the present
Cooper, S.B., Slaman, T.A., Wainer, S.S., editors: Computability, Enumerability, Unsolvability: Directions in Recursion Theory. Number 224 in London Mathematical Society Lecture Notes. Cambridge University Press, 1996, VII+347 pages
FOM e-mail list. http://www.cs.nyu.edu/mailman/listinfo/fom/, 1997 to the present
Grätzer, G.A.: General Lattice Theory. Birkhäuser-Verlag, 2nd edition, 1998, XIX+663 pages
Jockusch, C.G., Jr., Soare, R.I.: Π0 1 classes and degrees of theories. Trans. Am. Math. Soc. 173, 35–56 (1972)
McKenzie, R., McNulty, G.F., Taylor, W.F.: Algebras, Lattices, Varieties. Vol. 1. Wadsworth and Brooks/Cole, 1987, XI+361 pages
Medvedev, Y.T.: Degrees of difficulty of mass problems. Doklady Akademii Nauk SSSR, n.s. 104, 501–504 (1955), In Russian
Muchnik, A.A.: On strong and weak reducibilities of algorithmic problems. Sibirskii Matematicheskii Zhurnal 4, 1328–1341 (1963), In Russian
Rogers, H., Jr.: Theory of Recursive Functions and Effective Computability. McGraw-Hill, 1967, XIX+482 pages
Sikorski, R.: Boolean Algebras. Springer-Verlag, 3rd edition, 1969, X+237 pages
Simpson, S.G., editor: Reverse Mathematics 2001. Lecture Notes in Logic. Association for Symbolic Logic, 2004, to appear
Simpson, S.G.: Π0 1 sets and models of WKL 0. In: [15]. Preprint, April 2000, 29 pages, to appear
Simpson, S.G.: FOM: natural r.e. degrees; Pi01 classes. FOM e-mail list [7], 1999
Simpson, S.G.: Medvedev degrees of nonempty Pi0 1 subsets of 2omega. COMP-THY e-mail list [5], 2000
Simpson, S.G.: Some Muchnik degrees of Π0 1 subsets of 2ω, 2001. Preprint, 7 pages, in preparation
Simpson, S.G., Slaman, T.A.: Medvedev degrees of Π0 1 subsets of 2ω, 2001. Preprint, 4 pages, in preparation
Soare, R.I.: Recursively Enumerable Sets and Degrees. Perspectives in Mathematical Logic. Springer-Verlag, 1987, XVIII+437 pages
Sorbi, A.: The Medvedev lattice of degrees of difficulty. In: [6], 1996, pp. 289–312
Author information
Authors and Affiliations
Corresponding author
Additional information
Simpson’s research was partially supported by NSF Grant DMS-0070718. We thank the anonymous referee for a careful reading of this paper and helpful comments.
Rights and permissions
About this article
Cite this article
Binns, S., Simpson, S. Embeddings into the Medvedev and Muchnik lattices of Π0 1 classes. Arch. Math. Logic 43, 399–414 (2004). https://doi.org/10.1007/s00153-003-0195-x
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00153-003-0195-x