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. (shrink)

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. (shrink)

We use some simple facts about the wtt-degrees of r.e. sets together with a construction to answer some questions concerning the join and meet operators in the r.e. degrees. The construction is that of an r.e. Turing degree a with just one wtt-degree in a such that a is the join of a minimal pair of r.e. degrees. We hope to illustrate the usefulness of studying the stronger reducibility orderings of r.e. sets for providing information about Turing reducibility.