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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
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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.

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

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