A r-maximal vector space not contained in any maximal vector space

E-ISSN： 1943-5886|43|3|430-441

ISSN： 0022-4812

Source： The Journal of Symbolic Logic, Vol.43, Iss.3, 1978-09, pp. : 430-441

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Abstract

In [4], Metakides and Nerode define a recursively presented vector space V_∞ . over a (finite or infinite) recursive field F 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 any given n-tuple v ₀, …, v _n−1 from V _∞ is linearly dependent. Given a subspace W of V _∞, we write dim(W) for the dimension of W. Given subspaces V and W of V _∞, V + W will denote the weak sum of V and W and if V ∩ W = {0) (where 0 is the zero vector of V _∞), we write V ⊕ W instead of V + W. If W ⊇ V, we write W mod V for the quotient space. An independent set A ⊆ V _∞ is extendible if there is a r.e. independent set I ⊇ A such that I − A is infinite and A is nonextendible if it is not the case that A is extendible.