On r.e. and co-r.e. vector spaces with nonextendible bases

Extract

The concern of this paper is with recursively enumerable and co-recursively enumerable subspaces of a recursively presented vector spaceV_∞ over a (finite or infinite) recursive field F which is defined in [6] to consist of a recursive subset U of the natural numbers N and operations of vector addition and scalar multiplication which are partial recursive and under which V_∞ becomes a vector space. Throughout this paper, we will identify V_∞ with N, say via some fixed Gödel numbering, and assume V_∞ is infinite dimensional and has a dependence algorithm, i.e., there is a uniform effective procedure which determines whether or not any given n-tuple v₀, …, v_n−1 from V_∞ is linearly dependent. Various properties of V_∞ and its sub-spaces have been studied by Dekker [1], Guhl [3], Metakides and Nerode [6], Kalantari and Retzlaff [4], and the author [7].

Given a subspace W of V_∞, we say W is r.e. (co-r.e.) if W(V_∞ − W) is an r.e. subset of N and write dim(V) for the dimension of V. Given subspaces V, W of V_∞, V + W will denote the weak sum of V and W and if V ⋂ M = {0} (where 0 is the zero vector of V_∞), we write V ⊕ Winstead of V + W. If W ⊇ V, we write Wmod V for the quotient space. An independent set A ⊆ V_∞ is extendible if there is an r.e. independent set I ⊇ A such that I − A is infinite and A is nonextendible if it is not the case An is extendible. A r.e. subspace M ⊇ V_∞ is maximal if dim(V_∞ mod M) = ∞ and for any r.e. subspace W ⊇ Meither dim(W mod M) < ∞ or dim(V_∞ mod W) < ∞.