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Contiguity and distributivity in the enumerable Turing degrees

Published online by Cambridge University Press:  12 March 2014

Rodney G. Downey  and
Steffen Lempp
Rodney G. Downey
Affiliation:
Department of Mathematics, Victoria University, P. O. Box 600 Wellington, New Zealand E-mail: [email protected]
Steffen Lempp
Affiliation:
Department of Mathematics, University of Wisconsin, Madison WI 53706, USA E-mail: [email protected]
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Abstract

We prove that a (recursively) enumerable degree is contiguous iff it is locally distributive. This settles a twenty-year old question going back to Ladner and Sasso. We also prove that strong contiguity and contiguity coincide, settling a question of the first author, and prove that no m-topped degree is contiguous, settling a question of the first author and Carl Jockusch [11]. Finally, we prove some results concerning local distributivity and relativized weak truth table reducibility.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 62 , Issue 4 , December 1997 , pp. 1215 - 1240
Copyright
Copyright © Association for Symbolic Logic 1997

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References

REFERENCES

[1]Ambos-Spies, K., Contiguous r.e. degrees, Computation and proof theory (Börger, E., Oberschelp, W., Richter, M. M., Schinzel, B., and Thomas, W., editors), Springer Lecture Notes in Mathematics, no. 1104, Springer-Verlag, Berlin, 1984, pp. 1–37.Google Scholar
[2]Ambos-Spies, K., Antimitotic recursively enumerable sets, Z. Math. Logik Grundlagen Math., vol. 31 (1985), pp. 461–467.CrossRefGoogle Scholar
[3]Ambos-Spies, K., Ding, D., and Fejer, P., Embedding lattices preserving 1 below a nonzero recursive enumerable Turing degree, Logical methods (Crossley, J.et al., editors), Birkhäuser, Boston, 1994, pp. 92–129.Google Scholar
[4]Ambos-Spies, K. and Fejer, P., Degree theoretical splitting properties of recursively enumerable sets, this Journal, vol. 53 (1988), pp. 1110–1137.Google Scholar
[5]Ambos-Spies, K. and Soare, R.I., The recursively enumerable degrees have infinitely many one-types, Annals of Pure and Applied Logic, vol. 44 (1989), pp. 1–23.CrossRefGoogle Scholar
[6]Cholak, P. and Downey, R. G., Lattice nonembeddings and intervals in the recursively enumerable degrees, Annals Pure and Applied Logic, vol. 61 (1993), pp. 195–222.CrossRefGoogle Scholar
[7]Downey, R. G., degrees and transfer theorems, Illinois Journal Mathematics, vol. 31 (1987), pp. 419–427.CrossRefGoogle Scholar
[8]Downey, R. G., Localization of a theorem of Ambos-Spies and the strong antisplitting property, Archiv math. Logik Grundlag, vol. 26 (1987), pp. 127–136.CrossRefGoogle Scholar
[9]Downey, R. G., Subsets of hypersimplesets, Pacific Journal of Mathematics, vol. 127 (1987), pp. 299–319.CrossRefGoogle Scholar
[10]Downey, R. G., A contiguous nonbranching degree, Z. Math. Logik Grund. Math., vol. 35 (1989), pp. 375–383.CrossRefGoogle Scholar
[11]Downey, R. G. and Jockusch, C. G. Jr., T-degrees, jump classes and strong reducibilities, Transactions of the American Mathematical Society, vol. 301 (1987), pp. 103–136.CrossRefGoogle Scholar
[12]Downey, R. G. and Mourad, J., Superbranching degrees, Proceedings Oberwolfach 1989, Springer-Verlag Lecture Notes in Mathematics, 1990, pp. 175–186.Google Scholar
[13]Downey, R. G. and Remmel, J. B., Classification of degree classes associated with r.e. subspaces, Annals of Pure Applied Logic, vol. 42 (1989), pp. 105–125.CrossRefGoogle Scholar
[14]Downey, R. G. and Shore, R. A., Degree theoretical definitions of low2 recursively enumerable sets, this Journal, vol. 60 (1995), no. 3, pp. 727–756.Google Scholar
[15]Downey, R. G. and Slaman, T. A., Completely mitotic r.e. degrees, Annals of Pure and Applied Logic, vol. 41 (1989), pp. 119–152.CrossRefGoogle Scholar
[16]Downey, R. G. and Stob, M., Minimal pairs in initial segments of the recursively enumerable degrees, Israel Journal of Mathematics, to appear.Google Scholar
[17]Downey, R. G. and Stob, M., Structural interactions of the recursively enumerable W- and T-degrees, Annals of Pure and Applied Logic, vol. 31 (1986), pp. 205–236.CrossRefGoogle Scholar
[18]Downey, R. G. and Stob, M., Splitting theorems in recursion theory. Annals of Pure and Applied Logic, vol. 65 (1993), pp. 1–106.CrossRefGoogle Scholar
[19]Downey, R. G. and Welch, L. V., Splitting properties of r.e. sets and degrees, this Journal, vol. 32 (1986), pp. 137–151.Google Scholar
[20]Friedberg, R. and Rogers, H. Jr., Reducibilities and completeness for sets of integers, Z. Math. Logik Grundlagen Math., vol. 5 (1959), pp. 117–125.CrossRefGoogle Scholar
[21]Ingrassia, M., P-genericity for recursively enumerable sets, Thesis, University of Illinois at Urbana, 1981.Google Scholar
[22]Jockusch, C., Relationships between reducibilities, Transactions of the American Mathematical Society, vol. 162 (1969), pp. 229–237.CrossRefGoogle Scholar
[23]Jockusch, C. and Shore, R., Pseudo-jump operators I: The r. e. case, Transactions of the American Mathematical Society, vol. 275 (1983), pp. 599–609.Google Scholar
[24]Lachlan, A. H., Embedding nondistributive lattices in the recursively enumerable degrees, Conference in mathematical logic, London, 1970 (Hodges, W., editor), Springer-Verlag, New York, 1972, pp. 149–177.Google Scholar
[25]Lachlan, A. H., A recursively enumerable degree that will not split over all lesser ones, Annals of Mathematical Logic, vol. 9 (1975), pp. 307–365.CrossRefGoogle Scholar
[26]Ladner, R. E., A completely mitotic nonrecursive recursively enumerable degree, Transactions of the American Mathematical Society, vol. 184 (1973), pp. 479–507.CrossRefGoogle Scholar
[27]Ladner, R. E. and Sasso, L. P. Jr., The weak truth table degrees of recursively enumerable sets, Annals of Mathematical Logic, vol. 8 (1975), pp. 429–448.CrossRefGoogle Scholar
[28]Lerman, M. and Remmel, J. B., The universal splitting property, II, this Journal, vol. 49 (1984), pp. 137–150.Google Scholar
[29]Odifreddi, P., Strong reducibilities, Bulletin (New Series) of the American Mathematical Society, vol. 4 (1981), pp. 37–86.CrossRefGoogle Scholar
[30]Odifreddi, P., Classical recursion theory, North-Holland, Amsterdam, 1990.Google Scholar
[31]Sacks, G., The recursively enumerable degrees are dense, Annals of Mathematics, vol. 80 (1963), no. 2, pp. 211–231.CrossRefGoogle Scholar
[32]Slaman, T., The density of infima in the recursively enumerable degrees, Annals Pure and Applied Logic, vol. 52 (1991), pp. 155–179.CrossRefGoogle Scholar
[33]Soare, R. I., Recursively enumerable sets and degrees, Springer-Verlag, New York, 1987.CrossRefGoogle Scholar
[34]Stob, M., Wtt-degrees and T-degrees of recursively enumerable sets, this Journal, vol. 48 (1983), pp. 921–930.Google Scholar