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Major subspaces of recursively enumerable vector spaces1

Published online by Cambridge University Press:  12 March 2014

Iraj Kalantari
Iraj Kalantari*
Affiliation:
University of California, Santa Barbara, CA 93106
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Extract

The main point of this paper is a further development of some aspects of the recent theory of recursively enumerable (r.e.) algebraic structures. Initial work in this area is due to Frölich and Shepherdson [4] and Rabin [10]. Here we are only concerned with vector space structure. The previous work on r.e. vector spaces is due to Dekker [2], [3], Metakides and Nerode [8], Remmel [11], Retzlaff [13], and the author [5].

Our object of study is V∞ a countably infinite dimensional fully effective vector space over a countable recursive field . By fully effective we mean that V∞. under a fixed Godel numbering has the following properties:

(i) Operations of vector addition and scalar multiplication on V∞ are presented by partial recursive functions on the Gödel numbers of elements of V∞.

(ii) V∞ has a dependence algorithm, i.e., there is a uniform effective procedure which applied to any n vectors of V∞ determines whether or not they are linearly independent.

We also study , the lattice of r.e. subspaces of V∞ (under the operations of intersection, ⋂ and (weak) sum, +). We note that if is not distributive and is merely modular (see [1]). This fact indicates the essential difference between the lattice of r.e. sets and .

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 43 , Issue 2 , June 1978 , pp. 293 - 303
Copyright
Copyright © Association for Symbolic Logic 1978

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Footnotes

1

These results form part of the author's dissertation.

References

BIBLIOGRAPHY

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