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# An extension of the nondiamond theorem in classical and α-recursion theory

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Lachlan's nondiamond theorem [7, Theorem 5] asserts that there is no embedding of the four-element Boolean algebra (diamond) in the recursively enumerable degrees which preserves infima, suprema, and least and greatest elements. Lachlan observed that, essentially by relativization, the theorem can be extended to

Using the Sacks splitting theorem he concluded that there exists a pair of r.e. degrees which does not have an infimum, thus showing that the r.e. degrees do not form a lattice.

We will first prove the following extension of (1):

where an r.e. degree a is non-b-cappable if . From (2) we obtain more information about pairs of r.e. degrees without infima: For every nonzero low r.e. degree there exists an incomparable one such that the two degrees do not have an infimum and there is an r.e. degree which is not half of a pair of incomparable r.e. degrees which has an infimum in the low r.e. degrees. Probably the most interesting corollary of (2) is that the join of any cappable r.e. degree (i.e. half of a minimal pair) and any low r.e. degree is incomplete. Consequently there is an incomplete noncappable degree above every incomplete r.e. degree. Cooper's result [3] that ascending sequences of uniformly r.e. degrees can have minimal upper bounds in the set R of r.e. degrees is another corollary of (2).

References
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[1]Ambos-Spies, K., On the structure of the recursively enumerable degrees, Dissertation, Universität München, Munich, 1980.
[2]Ambos-Spies, K., On pairs of recursively enumerable degrees, Transactions of the American Mathematical Society (to appear).
[3]Cooper, S. B., Minimal upper bounds for sequences of recursively enumerable degrees, Journal of the London Mathematical Society, vol. 5 (1972), pp. 445450.
[4]Fejer, P. A., Structure of r.e. degrees (Abstract), Conference on Mathematical Logic, Storrs, Connecticut, 1979.
[5]Fejer, P. A., and Soare, R. I., The plus-cupping theorem for the recursively enumerable degrees, Logic Year 1979-80, Lecture Notes in Mathematics, vol. 859, Springer-Verlag, Berlin, 1981, pp. 4962.
[6]Friedberg, R. M., Two recursively enumerable sets of incomparable degrees of unsolvability, Proceedings of the National Academy of Sciences of the U.S.A., vol. 43 (1957), pp. 236238.
[7]Lachlan, A. H., Lower bounds for pairs of r.e. degrees, Proceedings of the London Mathematical Society, ser. 3, vol. 16 (1966), pp. 537569.
[8]Lachlan, A. H., Decomposition of recursively enumerable degrees, Proceedings of the American Mathematical Society, vol. 79 (1980), pp. 629634.
[9]Lerman, M., On suborderings of the α.-recursively enumerable α-degrees, Annals of Mathematical Logic, vol. 4 (1972), pp. 369392.
[10]Lerman, M., Least upper bounds for minimal pairs of a-r.e. x-degrees, this Journal, vol. 39 (1974), pp. 4956.
[11]Lerman, M. and Sacks, G. E., Some minimal pairs of x-recursively enumerable degrees, Annals of Mathematical Logic, vol. 4 (1972), pp. 415442.
[12]Miller, D. P., High recursively enumerable degrees and the anti-cupping property, Logic Year 1979-80, Lecture Notes in Mathematics, vol. 859, Springer-Verlag, Berlin, 1981, pp. 230245.
[13]Muchnik, A. A., Negative answer to the problem of reducibility of the theory of algorithms (Russian), Doklady Akadémii Nauk SSSR, vol. 108 (1956), pp. 194197.
[14]Robinson, R. W., Interpolation and embedding in the recursively enumerable degrees, Annals of Mathematics, vol. 93 (1971), pp. 285314.
[15]Sacks, G. E., Degrees of unsolvability, Annals of Mathematics Studies, vol. 55, Princeton University Press, Princeton, N.J., 1963.
[16]Shore, R. A., Splitting an α-recursively enumerable set, Transactions of the American Mathematical Society, vol. 204 (1975), pp. 6577.
[17]Shore, R. A., On the jump of an α-recursively enumerable set, Transactions of the American Mathematical Society, vol. 217 (1976), pp. 351363.
[18]Shore, R. A., >On the π∃-sentences of α-recursion theory, Generalized recursion theory. II (Fenstad, J. E., Gandy, R. O. and Sacks, G. E., editors), North-Holland, Amsterdam, 1978, pp. 331353.
[19]Shore, R. A., Some constructions in α-recursion theory, Recursion theory: its generalisations and applications, London Mathematical Society Lecture Notes Series, vol. 45, Cambridge University Press, Cambridge, 1980, pp. 158170.
[20]Simpson, S. G., Degree theory on admissible ordinals, Generalized recursion theory (Fenstad, J. E. and Hinman, P. G., editors), North-Holland, Amsterdam, 1974, pp. 165193.
[21]Soare, R. I., The Friedberg-Muchnik theorem re-examined, Canadian Journal of Mathematics, vol. 24 (1972), pp. 10701078.
[22]Soare, R. I., Computational complexity, speedable and levelable sets, this Journal, vol. 42 (1977), pp. 545563.
[23]Soare, R. I., Fundamental methods for constructing recursively enumerable degrees, Recursion theory: its generalisations and applications, London Mathematical Society Lecture Notes Series, vol. 45, Cambridge University Press, Cambridge, 1980, pp. 151.
[24]Yates, C. E. M., A minimal pair of r.e. degrees, this Journal, vol. 31 (1966), pp. 159168.
[25]Yates, C. E. M., On the degrees of index sets. II, Transactions of the American Mathematical Society, vol. 135 (1969), pp. 249266.
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