Abstract
Lachlan observed that the infimum of two r.e. degrees considered in the r.e. degres coincides with the one considered in the \(\Delta_2^0\) degrees. It is not true anymore for the d.r.e. degrees. Kaddah proved in [7] that there are d.r.e. degrees a, b, c and a 3-r.e. degree x such that a is the infimum of b, c in the d.r.e. degrees, but not in the 3-r.e. degrees, as a < x < b, c. In this paper, we extend Kaddah’s result by showing that such a infima difference occurs densely in the r.e. degrees.
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References
Arslanov, M.M., Lempp, S., Shore, R.A.: On isolating r.e. and isolated d.r.e. degrees. In: Cooper, S.B., Slaman, T.A., Wainer, S.S. (eds.) Computability, Enumerability, Unsolvability, pp. 61–80 (1996)
Cenzer, D., LaForte, F., Wu, G.: The nonisolating degrees are nowhere dense in the computably enumeralbe degrees (to appear)
Cooper, S.B., Harrington, L., Lachlan, A.H., Lempp, S., Soare, R.I.: The d.r.e. degrees are not dense. Ann. Pure Appl. Logic 55, 125–151 (1991)
Cooper, S.B., Lempp, S., Watson, P.: Weak density and cupping in the d-r.e. degrees. Israel J. Math. 67, 137–152 (1989)
Ding, D., Qian, L.: Isolated d.r.e. degrees are dense in r.e. degree structure. Arch. Math. Logic 36, 1–10 (1996)
Fejer, P.A.: The density of the nonbranching degrees. Ann. Pure Appl. Logic 24, 113–130 (1983)
Kaddah, D.: Infima in the d.r.e. degrees. Ann. Pure Appl. Logic 62, 207–263 (1993)
Kleene, S.C., Post, E.L.: The upper semi-lattice of degrees of recursive unsolvability. Ann. of Math. 59, 379–407 (1954)
Lachlan, A.H.: Lower bounds for pairs of recursively enumerable degrees. Proc. London Math. Soc. 16, 537–569 (1966)
LaForte, G.L.: The isolated d.r.e. degrees are dense in the r.e. degrees. Math. Logic Quart. 42, 83–103 (1996)
Soare, R.I.: Recursively Enumerable Sets and Degrees. Perspectives in Mathematical Logic. Springer, Heidelberg (1987)
Wu, G.: On the density of the pseudo-isolated degrees. In: Proceedings of London Mathematical Society, vol. 88, pp. 273–288 (2004)
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Liu, J., Wang, S., Wu, G. (2009). Infima of d.r.e. Degrees. In: Ambos-Spies, K., Löwe, B., Merkle, W. (eds) Mathematical Theory and Computational Practice. CiE 2009. Lecture Notes in Computer Science, vol 5635. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03073-4_34
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DOI: https://doi.org/10.1007/978-3-642-03073-4_34
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