Decidable subspaces and recursively enumerable subspaces

Journal of Symbolic Logic 49 (4):1137-1145 (1984)
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 $V \oplus W = V_\infty$ . 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 and $U \oplus V = W$ . Then there exists a decidable subspace D such that U |oplus 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
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DOI 10.2307/2274266
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References found in this work BETA
Recursively Enumerable Vector Spaces.G. Metakides & A. Nerode - 1977 - Annals of Mathematical Logic 11 (2):147-171.
On a Question of A. Retzlaff.Rod Downey - 1983 - Mathematical Logic Quarterly 29 (6):379-384.
Direct Summands of Recursively Enumerable Vector Spaces.Allen Retzlaff - 1979 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 25 (19-24):363-372.
Direct Summands of Recursively Enumerable Vector Spaces.Allen Retzlaff - 1979 - Mathematical Logic Quarterly 25 (19‐24):363-372.

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Citations of this work BETA
Classifications of Degree Classes Associated with R.E. Subspaces.R. G. Downey & J. B. Remmel - 1989 - Annals of Pure and Applied Logic 42 (2):105-124.
Undecidability of L and Other Lattices of R.E. Substructures.R. G. Downey - 1986 - Annals of Pure and Applied Logic 32 (1):17-26.
On Speedable and Levelable Vector Spaces.Frank A. Bäuerle & Jeffrey B. Remmel - 1994 - Annals of Pure and Applied Logic 67 (1-3):61-112.
Sound, Totally Sound, and Unsound Recursive Equivalence Types.R. G. Downey - 1986 - Annals of Pure and Applied Logic 31 (1):1-20.

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