Co-immune subspaces and complementation in V∞

Abstract

AbstractWe 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 M1, M2 ∈ L(V∞), with M1supermaximal and M2k-thin, such that M1, ⊕ V = M2 ⊕ V = V∞.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 ⊕ W = M ⊕ W = D ⊕ W = V∞.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 ⊃ 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.