Journal of Symbolic Logic

Decidable Subspaces and Recursively Enumerable Subspaces

C. J. Ash and R. G. Downey

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A subspace $V$ of an infinite dimensional fully effective vector space $V_\infty$ 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 $V_\infty$ are natural analogues of recursive subsets of $\omega$. The set of r.e. subspaces forms a lattice $L(V_\infty)$ and the set of decidable subspaces forms a lower semilattice $S(V_\infty)$. We analyse $S(V_\infty)$ and its relationship with $L(V_\infty)$. We show: Proposition. Let $U, V, W \in L(V_\infty)$ 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, $\mathrm{Th}(S(V_\infty))$, is undecidable. This contrasts sharply with the result for recursive sets. Finally we examine various generalizations of our results. In particular we analyse $S^\ast(V_\infty)$, that is, $S(V_\infty)$ modulo finite dimensional subspaces. We show $S^\ast(V_\infty)$ is not a lattice.

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J. Symbolic Logic, Volume 49, Issue 4 (1984), 1137-1145.

First available in Project Euclid: 6 July 2007

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Ash, C. J.; Downey, R. G. Decidable Subspaces and Recursively Enumerable Subspaces. J. Symbolic Logic 49 (1984), no. 4, 1137--1145.

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