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Simple and hyperhypersimple vector spaces1

Published online by Cambridge University Press:  12 March 2014

Allen Retzlaff
Allen Retzlaff*
Affiliation:
Cornell University, Ithaca, NY 14853 State University of New York, Purchase, NY 10577
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Abstract

Let V∞ be a fixed, fully effective, infinite dimensional vector space. Let be the lattice consisting of the recursively enumerable (r.e.) subspaces of V∞, under the operations of intersection and weak sum (see §1 for precise definitions). In this article we examine the algebraic properties of .

Early research on recursively enumerable algebraic structures was done by Rabin [14], Frölich 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 of r.e. sets to . We define hyperhypersimple vector spaces, discuss some of their properties and show if A, B ∈ , 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 ∈ 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.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 43 , Issue 2 , June 1978 , pp. 260 - 269
Copyright
Copyright © Association for Symbolic Logic 1978

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Footnotes

1

This research was partially supported by NSF grant number MPS 74-06378.

References

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