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# How enumeration reductibility yields extended Harrington non-splitting

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§1. Introduction. Sacks [16] showed that every computably enumerable (c.e.) degree > 0 has a c.e. splitting. Hence, relativising, every c.e. degree has a Δ2 splitting above each proper predecessor (by ‘splitting’ we understand ‘nontrivial splitting’). Arslanov [1] showed that 0′ has a d.c.e. splitting above each c.e. a < 0′. On the other hand, Lachlan [11] proved the existence of a c.e. a < 0 which has no c.e. splitting above some proper c.e. predecessor, and Harrington [10] showed that one could take a = 0′. Splitting and nonsplitting techniques have had a number of consequences for definability and elementary equivalence in the degrees below 0′.

Heterogeneous splittings are best considered in the context of cupping and non-cupping. Posner and Robinson [15] showed that every nonzero Δ2 degree can be nontrivially cupped to 0′, and Arslanov [1] showed that every c.e. degree > 0 can be d.c.e. cupped to 0′ (and hence since every d.c.e., or even n-c.e., degree has a nonzero c.e. predecessor, every n-c.e. degree > 0 is d.c.e. cuppable). Cooper [4] and Yates (see Miller [13]) showed the existence of degrees noncuppable in the c.e. degrees. Moreover, the search for relative cupping results was drastically limited by Cooper [5], and Slaman and Steel [17] (see also Downey [9]), who showed that there is a nonzero c.e. degree a below which even Δ2 cupping of c.e. degrees fails.

We prove below what appears to be the strongest possible of such nonsplitting and noncupping results.

## References

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[1]Arslanov, M. M., Structural properties of the degrees below 0′, Doklady Akademii Nauk SSSR, vol. 283 (1985), pp. 270273.
[2]Arslanov, M. M., Cooper, S. B., and Kalimullin, I. Sh., Splitting properties of total enumeration degrees, Algebra and Logic, vol. 42 (2003), pp. 113.
[3]Arslanov, M. M. and Sorbi, A., Relative splittings of 0′e in the Δ2 enumeration degrees. Logic Colloquium'98 (Buss, S. and Pudlak, P., editors), Springer-Verlag, 1999.
[4]Cooper, S. B., On a theorem of C. E. M. Yates, 1974, handwritten notes.
[5]Cooper, S. B., The strong anti-cupping property for recursively enumerable degrees, this Journal, vol. 54 (1989), pp. 527539.
[6]Cooper, S. B., Enumeration reducibility, nondeterministic computations and relative computability of partial functions, Recursion Theory Week, Proceedings Oberwolfach 1989 (Ambos-Spies, K., Müller, G. H., and Sacks, G. E., editors), Lecture Notes in Mathematics, vol. 1432, Springer-Verlag, 1990. pp. 57110.
[7]Cooper, S. B., Computability Theory, Chapman & Hall/CRC Mathematics, 2004.
[8]Cooper, S. B., Sorbi, A., Li, A., and Yang, Y., Bounding and nonbounding minimal pairs in the enumeration degrees, this Journal, vol. 70 (2005), pp. 741766.
[9]Downey, R. G., degrees and transfer theorems, Illinois Journal of Mathematics, vol. 31 (1987), pp. 419427.
[10]Harrington, L., Understanding Lachlan s monster paper, 1980, handwritten notes.
[11]Lachlan, A. H., A recursively enumerable degree which will not split over all lesser ones. Annals of Mathematical Logic, vol. 9 (1975), pp. 307365.
[12]Lachlan, A. H. and Shore, R. A., The n-rea enumeration degrees are dense. Archive for Mathematical Logic, vol. 31 (1992), pp. 277285.
[13]Miller, D., High recursively enumerable degrees and the anti-cupping property. Logic Year 1979–80: University of Connecticut (Lerman, M., Schmerl, J. H., and Soare, R. I., editors), Lecture Notes in Mathematics, vol. 859, Springer-Verlag, 1981.
[14]Odifreddi, P. G., Classical Recursion Theory, Volume II, North-Holland/Elsevier, 1999.
[15]Posner, D. B. and Robinson, R. W., Degrees joining to 0′, this Journal, vol. 46 (1981), pp. 714722.
[16]Sacks, G. E., On the degrees less than 0′, Annals of Mathematics, vol. 77 (1963), no. 2, pp. 211231.
[17]Slaman, T. A. and Steel, J. R., Complementation in the Turing degrees, this Journal, vol. 54 (1989), pp. 160176.
[18]Soare, R. I., Recursively Enumerable Sets and Degrees, Springer-Verlag, 1987.

# How enumeration reductibility yields extended Harrington non-splitting

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