Hostname: page-component-7b7567c8c-tnzcf Total loading time: 0 Render date: 2026-11-15T01:55:24.803Z Has data issue: false hasContentIssue false
  • English
  • Français

Decidable subspaces and recursively enumerable subspaces

Published online by Cambridge University Press:  12 March 2014

C. J. Ash and
R. G. Downey
C. J. Ash
Affiliation:
Monash University, Clayton, Victoria, 3168, Australia
R. G. Downey
Affiliation:
National University of Singapore, Kent Ridge, 0511, Singapore
Get accessRights & Permissions [Opens in a new window]

Abstract

A subspace V of an infinite dimensional fully effective vector space V∞ is called decidable if V is r.e. and there exists an r.e. W such that V ⊕ W = V∞. These subspaces of V∞ are natural analogues of recursive subsets of ω. The set of r.e. subspaces forms a lattice L(V∞) and the set of decidable subspaces forms a lower semilattice S(V∞). We analyse S(V∞) and its relationship with L(V∞). We show:

Proposition. Let U, V, W ∈ L(V∞) where U is infinite dimensional andU ⊕ V = W. Then there exists a decidable subspace D such that U ⊕ D = W.

Corollary. Any r.e. subspace can be expressed as the direct sum of two decidable subspaces.

These results allow us to show:

Proposition. The first order theory of the lower semilattice of decidable subspaces, Th(S(V∞), is undecidable.

This contrasts sharply with the result for recursive sets.

Finally we examine various generalizations of our results. In particular we analyse S*(V∞), that is, S(V∞) modulo finite dimensional subspaces. We show S*(V∞) is not a lattice.

Information

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 49 , Issue 4 , December 1984, pp. 1137 - 1145
Copyright
Copyright © Association for Symbolic Logic 1984

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

REFERENCES

[1]Dekker, J. C. E., Countable vector spaces with recursive operations. Part I, this Journal, vol. 34 (1969), pp. 363–387.Google Scholar
[2]Baldwin, J. T., Recursion theory and abstract dependence, Patras Logic Symposion (Patras, 1980; Metakides, G., editor), North-Holland, Amsterdam, 1982, pp. 67–76.CrossRefGoogle Scholar
[3]Downey, R., On a question of A. Retzlaff, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 29 (1983), pp. 379–384.CrossRefGoogle Scholar
[4]Downey, R., Co-immune subspaces and complementation in V ∞, this Journal, vol. 49 (1984), pp. 528–538.Google Scholar
[5]Ladner, R. E., Mitotic recursively enumerable sets, this Journal, vol. 38 (1973), pp. 199–211.Google Scholar
[6]Lerman, M. and Remmel, J. B., The universal splitting property. I, Logic Colloquium '80 (van Dalen, D., editor), North-Holland, Amsterdam, 1982, pp. 181–207.Google Scholar
[7]Metakides, G. and Nerode, A., Recursively enumerable vector spaces, Annals of Mathematical Logic, vol. 11 (1977), pp. 147–171.CrossRefGoogle Scholar
[8]Metakides, G. and Nerode, A., Recursion theory on fields and abstract dependence, Journal of Algebra, vol. 65 (1980), pp. 36–59.CrossRefGoogle Scholar
[9]Nerode, A. and Smith, R., The undecidability of the lattice of recursively enumerable subspaces, Proceedings of the Third Brazilian Conference on Mathematical Logic (Arruda, A. I., da Costa, N. C. A. and Sette, A. M. A., editors), North-Holland, Amsterdam, 1982, pp. 245–252.Google Scholar
[10]Rogers, H. Jr., Theory of recursive functions and effective computability, McGraw-Hill, New York, 1967.Google Scholar
[11]Retzlaff, A., Direct summands of recursively enumerable vector spaces, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 25 (1979), pp. 363–372.CrossRefGoogle Scholar
[12]Soare, R. I., The Friedberg-Muchnik theorem re-examined, Canadian Journal of Mathematics, vol. 24 (1972), pp. 1070–1078.CrossRefGoogle Scholar
[13]Kalantari, I. and Retzlaff, A., Maximal vector spaces under automorphisms of the lattice of recursively enumerable vector spaces, this Journal, vol. 42 (1977), pp. 481–491.Google Scholar
[14]Downey, R., Nowhere simplicity in matroids, Journal of the Australian Mathematical Society, Series A, vol. 35 (1983), pp. 28–45.CrossRefGoogle Scholar
[15]Miller, D. and Remmel, J. B., Effectively nowhere simple sets, this Journal, vol. 49 (1984), pp. 129–136.Google Scholar
[16]Downey, R. G. and Remmel, J. B., The universal complementation property, this Journal, vol. 49 (1984), pp. 1125–1136.Google Scholar
[17]Remmel, J. B., On r.e. and co-r.e. vector spaces with nonextendible bases, this Journal, vol. 45 (1980), pp. 20–34.Google Scholar