Co-immune subspaces and complementation in V∞

Journal of Symbolic Logic 49 (2):528 - 538 (1984)
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Abstract

We examine the multiplicity of complementation amongst subspaces of V ∞ . A subspace V is a complement of a subspace W if V ∩ W = {0} and (V ∪ W) * = V ∞ . A subspace is called fully co-r.e. if it is generated by a co-r.e. subset of a recursive basis of V ∞ . We observe that every r.e. subspace has a fully co-r.e. complement. Theorem. If S is any fully co-r.e. subspace then S has a decidable complement. We give an analysis of other types of complements S may have. For example, if S is fully co-r.e. and nonrecursive, then S has a (nonrecursive) r.e. nowhere simple complement. We impose the condition of immunity upon our subspaces. Theorem. Suppose V is fully co-r.e. Then V is immune iff there exist M 1 , M 2 ∈ L(V ∞ ), with M 1 supermaximal and M 2 k-thin, such that $M_1 \oplus V = M_2 \oplus V = V_\infty$ . Corollary. Suppose V is any r.e. subspace with a fully co-r.e. immune complement W (e.g., V is maximal or V is h-immune). Then there exist an r.e. supermaximal subspace M and a decidable subspace D such that $V \oplus W = M \oplus W = D \oplus W = V_\infty$ . We indicate how one may obtain many further results of this type. Finally we examine a generalization of the concepts of immunity and soundness. A subspace V of V ∞ is nowhere sound if (i) for all Q ∈ L(V ∞ ) if $Q \supset V$ then Q = V ∞ , (ii) V is immune and (iii) every complement of V is immune. We analyse the existence (and ramifications of the existence) of nowhere sound spaces

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Citations of this work

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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.
Maximal theories.R. G. Downey - 1987 - Annals of Pure and Applied Logic 33 (C):245-282.
Sound, totally sound, and unsound recursive equivalence types.R. G. Downey - 1986 - Annals of Pure and Applied Logic 31:1-20.
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References found in this work

The degrees of bi‐immune sets.Carl G. Jockusch - 1969 - Mathematical Logic Quarterly 15 (7‐12):135-140.
The degrees of bi-immune sets.Carl G. Jockusch - 1969 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 15 (7-12):135-140.
On a question of A. Retzlaff.Rod Downey - 1983 - Mathematical Logic Quarterly 29 (6):379-384.
Recursively enumerable vector spaces.G. Metakides - 1977 - Annals of Mathematical Logic 11 (2):147.

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