Abstract
One way to try to gain an understanding of the various degree-theoretic structures which recursion theorists study is to see what lattices can be embedded into them. Lattice embeddings have been used to show that such structures have an undecidable theory (via embeddings as initial segments) and to show that the theory of such structures is decidable up to a certain quantifier level. Many results in recursion theory can be stated as results about lattice embeddings even if they were not originally phrased that way.
This author was supported by a grant of the Stiftung Volkswagenwerk.
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References
Ambos-Spies, K. [1980], On the Structure of the Recursively Enumerable Degrees. PhD Thesis, University of Munich.
Ambos-Spies, K. [1984], Contiguous r.e. degrees. In Proceedings of Logic Colloquium’ 83 (Aachen), Lecture Notes in Mathematics, No. 1104, 1–37. Springer-Verlag, Berlin.
Ambos-Spies, K. and P.A. Fejer [1988], Degree theoretical splitting properties of recursively enumerable sets. J. Symbolic Logic, 53, 1110–1137.
Ambos-Spies, K., S. Lempp and M. Lerman, Lattice embeddings into the r.e. degrees preserving 0 and 1. To appear.
Ambos-Spies, K., S. Lempp and M. Lerman, Lattice embeddings into the r.e. degrees preserving 1. To appear.
Ambos-Spies, K. and M. Lerman [1986], Lattice embeddings into the recursively enumerable degrees. J. Symbolic Logic, 51, 257–272.
Ambos-Spies, K. and M. Lerman [1989], Lattice embeddings into the recursively enumerable degrees, II. J. Symbolic Logic, 54, 735–760.
Downey, R. [1990], Lattice nonembeddings and initial segments of the recursively enumerable degrees. Annals of Pure and Appl. Logic, 49, 97–119.
Downey, R. [1990], Notes on the 0″’-priority method with special attention to density results. In K. Ambos Spies, G.H. Müller and G.E. Sacks (eds.), Recursion Theory Week (Proc. of a Conference Held in Oberwolfach, FRG, March 19-25, 1989), Lecture Notes in Mathematics No. 1432, 114–140. Springer-Verlag, Berlin.
Fejer, P.A. [1982], Branching degrees above low degrees. Trans. of the Amer. Math. Soc, 273, 157–180.
Jockusch, C.G., Jr. and R.A. Shore [1983], Pseudo jump operators I: The r.e. case. Trans. of the Amer. Math. Soc, 275, 599–609.
Lachlan, A.H. [1966], Lower bounds for pairs of recursively enumerable degrees. Proc. of the London Math. Soc. (3), 16, 537–569.
Lachlan, A.H. [1972], Embedding nondistributive lattices in the recursively enumerable degrees. In W. Hodges (ed.), Conference in Mathematical Logic, London, 1970, Lecture Notes in Mathematics, No. 255, 149–177. Springer-Verlag, Berlin.
Lachlan, A.H. [1975], A recursively enumerable degree which will not split over all lesser ones. Annals of Math Logic, 9, 307–365.
Lachlan, A.H. [1979], Bounding minimal pairs. J. of Symbolic Logic, 44, 626–642.
Lachlan, A.H. [1980], Decomposition of recursively enumerable degrees. Proc. of the Amer. Math. Soc, 79, 629–634.
Lachlan, A.H. and R.I. Soare [1980], Not every finite lattice is embeddable in the recursively enumerable degrees. Advances in Mathematics, 37, 74–82.
Ladner, R.E. and L.P. Sasso, Jr. [1975], The weal; truth table degrees of recursively enumerable sets. Ann. of Math. Logic, 8, 429–448.
Shoenfield, J.R. and R.I. Soare [1978], The generalized diamond theorem. Recursive Function Theory Newsletter, 19. Abstract Number 219.
Slaman, T.A. [1991], The density of infima in the recursively enumerable degrees. Annals of Pure and Appl. Logic, 52, 155–179.
Soare, R.I. [1987], Recursively Enumerable Sets and Degrees: The Study of Computable Functions and Computably Generated Sets. Perspectives in Mathematical Logic, Ω Series. Springer Verlag, Berlin.
Thomason, S.K. [1971], Sublaltices of the recursively enumerable degrees. Zeitschrift f. Math. Logik u. Grundlagen d. Mathematik, 17, 273–280.
Yates, C.E.M. [1966], A minimal pair of recursively enumerable degrees. J. Symbolic Logic, 31, 159–168.
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Ambos-Spies, K., Decheng, D., Fejer, P.A. (1993). Embedding Distributive Lattices Preserving 1 below a Nonzero Recursively Enumerable Turing Degree. In: Crossley, J.N., Remmel, J.B., Shore, R.A., Sweedler, M.E. (eds) Logical Methods. Progress in Computer Science and Applied Logic, vol 12. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0325-4_2
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