Simple and hyperhypersimple vector spaces

Journal of Symbolic Logic 43 (2):260-269 (1978)
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Abstract

Let $V_\propto$ be a fixed, fully effective, infinite dimensional vector space. Let $\mathscr{L}(V_\propto)$ be the lattice consisting of the recursively enumerable (r.e.) subspaces of $V_\propto$ , under the operations of intersection and weak sum (see § 1 for precise definitions). In this article we examine the algebraic properties of $\mathscr{L}(V_\propto)$ . Early research on recursively enumerable algebraic structures was done by Rabin [14], Frolich and Shepherdson [5], Dekker [3], Hamilton [7], and Guhl [6]. Our results are based upon the more recent work concerning vector spaces of Metakides and Nerode [12], Crossley and Nerode [2], Remmel [15], [16], and Kalantari [8]. In the main theorem below, we extend a result of Lachlan from the lattice E of r.e. sets to $\mathscr{L}(V_\propto)$ . We define hyperhypersimple vector spaces, discuss some of their properties and show if $A, B \in \mathscr{L}(V_\propto)$ , and A is a hyperhypersimple subspace of B then there is a recursive space C such that A + C = B. It will be proven that if $V \in \mathscr{L}(V_\propto)$ and the lattice of superspaces of V is a complemented modular lattice then V is hyperhypersimple. The final section contains a summary of related results concerning maximality and simplicity

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Citations of this work

Direct Summands of Recursively Enumerable Vector Spaces.Allen Retzlaff - 1979 - Mathematical Logic Quarterly 25 (19‐24):363-372.
A note on r-maximal subspaces of V[infinity].David R. Guichard - 1984 - Annals of Pure and Applied Logic 26 (1):1.
Recursion theory on algebraic structures with independent sets.J. B. Remmel - 1980 - Annals of Mathematical Logic 18 (2):153.

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References found in this work

Theory of Recursive Functions and Effective Computability.Hartley Rogers - 1971 - Journal of Symbolic Logic 36 (1):141-146.
Recursively enumerable vector spaces.G. Metakides - 1977 - Annals of Mathematical Logic 11 (2):147.

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