Hostname: page-component-cb9f654ff-rkzlw Total loading time: 0 Render date: 2025-08-30T13:10:46.420Z Has data issue: false hasContentIssue false
  • English
  • Français

Kleene index sets and functional m-degrees

Published online by Cambridge University Press:  12 March 2014

Jeanleah Mohrherr
Jeanleah Mohrherr*
Affiliation:
University of Illinoisat Chicago, Circle Chicago, Illinois 60680
*
Cleveland State University, Cleveland, Ohio 44115
Get access Rights & Permissions [Opens in a new window]

Abstract

A many-one degree is functional if it contains the index set of some class of partial recursive functions but does not contain an index set of a class of r.e. sets. We give a natural embedding of the r.e. m-degrees into the functional degrees of 0′. There are many functional degrees in 0′ in the sense that every partial-order can be embedded. By generalizing, we show there are many functional degrees in every complete Turing degree.

There is a natural tie between the studies of index sets and partial-many-one reducibility. Every partial-many-one degree contains one or two index sets.

Information

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 48 , Issue 3 , September 1983 , pp. 829 - 840
Copyright
Copyright © Association for Symbolic Logic 1983

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

BIBLIOGRAPHY

[1]Dekker, J.C., A theorem on hypersimple sets, Proceedings of the American Mathematical Society, vol. 5 (1954), pp. 791–796.CrossRefGoogle Scholar
[2]Ershov, Y., On a hierarchy of sets. I, Algebra i Logika, vol. 7 (1968), pp. 47–74 = Algebra and Logic, vol. 7 (1968), pp. 25–43.Google Scholar
[3]Ershov, Y., Completely enumerated sets, Sibirskiĭ Mathematičeskiĭ Žurnal, vol. 10 (1969), pp. 1048–1064 = Siberian Mathematics Journal, vol. 10 (1969), pp. 773–784.Google Scholar
[4]Ershov, Y., On a hierarchy of sets. III, Algebra i Logika, vol. 9 (1970), pp. 34–51 = Algebra and Logic, pp. 20–31.Google Scholar
[5]Ershov, Y., On inseparable pairs, Algebra i Logika, vol. 9 (1970), pp. 661–666 = Algebra and Logic, vol. 9 (1970), pp. 396–399.Google Scholar
[6]Ershov, Y., The theory of enumeration, Nauka, Moscow, 1977.Google Scholar
[7]Friedberg, Richard M., A criterion for completeness of degrees of degrees of unsolvability, this Journal, vol. 22 (1958), pp. 159–160.Google Scholar
[8]Hay, L., Isomorphism types of index sets of partial recursive functions, Proceedings of the American Mathematical Society, vol. 17 (1966), pp. 106–110.CrossRefGoogle Scholar
[9]Hay, L., A discrete chain of degrees of index sets, this Journal, vol. 37 (1972), pp. 139–149.Google Scholar
[10]Hay, L., Index sets in 0′, Algebra i Logika, vol. 12 (1973), pp. 713–729 = Algebra and Logic, vol. 12 (1973), pp. 408–416.Google Scholar
[11]Hay, L., A noninitial segment of index sets, this Journal, vol. 39 (1974), pp. 209–224.Google Scholar
[12]Lachlan, A.H., A note on universal sets, this Journal, vol. 31 (1966), pp. 573–574.Google Scholar
[13]Mal'cev, A.I., Algorithms and recursive functions, Wolters-Noordhoff, Groningen, 1970.Google Scholar
[14]Odifreddi, Piergiorgio, Strong reducibilities, Bulletin of the American Mathematical Society, vol. 4 (1981), pp. 37–85.CrossRefGoogle Scholar
[15]Rice, H.G., Classes of recursively enumerable sets and their decision problems, Transactions of the American Mathematical Society, vol. 74 (1953), pp. 358–366.CrossRefGoogle Scholar
[16]Rogers, H. Jr., Goedel numberings of partial recursive functions, this Journal, vol. 23 (1958), pp. 331–341.Google Scholar
[17]Rogers, H. Jr., Computing degrees of unsolvability, Mathematische Annalen, vol. 138 (1959), pp. 125–140.CrossRefGoogle Scholar
[18]Rogers, H. Jr., Theory of recursive functions and effective computability, McGraw-Hill, New York, 1967.Google Scholar
[19]Sacks, G.E., Degrees of solvability, Annals of Mathematic Studies, No. 55, Princeton University Press, Princeton, N.J., 1963.Google Scholar
[20]Selivanov, V.L., Structures of the degrees of unsolvability of index sets, Algebra i Logika, vol. 18 (1979), pp. 463–480 = Algebra and Logic, vol. 18 (1979), pp. 286–299.Google Scholar
[21]Yates, C.E.M., On the degrees of index sets, Transactions of the American Mathematical Society, vol. 121 (1966), pp. 309–328.CrossRefGoogle Scholar