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Effectively nowhere simple sets

Published online by Cambridge University Press:  12 March 2014

D. Miller  and
J. B. Remmel
D. Miller
Affiliation:
Department of Computer Science, Depaul University, Chicago, Illinois 60614
J. B. Remmel
Affiliation:
University of California, San Diego, California 92093
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An r.e. set A is nowhere simple if for every r.e. set We such that We − A is infinite, there is an infinite r.e. set W such that W ⊆ We − A. The definition of nowhere simple sets is due to R. Shore in [4]. In [4], Shore studied various properties of nowhere simple sets and showed that they could be used to give an elegant and simple proof of the fact that every nontrivial class of r.e. sets C closed under recursive isomorphisms is an automorphism base for , the lattice of r.e. sets modulo finite sets, (that is, an automorphism α of is completely determined by its action on C; see Theorem 8 of [4]). Shore also defined the notion of effectively nowhere simple sets.

Definition. An r.e. set A is effectively nowhere simple if there is a recursive function f such that for every i, Wf(i) ⊆ Wi − A and Wf(i) is infinite iff Wi − A is infinite. f is called a witness function for A.

Other than to produce examples of effectively nowhere simple sets and nowhere simple sets that are not effectively nowhere simple, Shore did not concern himself with the properties of effectively nowhere simple sets since he felt that effectively nowhere simple sets were unlikely to be lattice invariant in either E, the lattice of r.e. sets, or in .

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 49 , Issue 1 , March 1984 , pp. 129 - 136
Copyright
Copyright © Association for Symbolic Logic 1984

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References

REFERENCES

[1]Lachlan, A. H., On the lattice of recursively enumerable sets, Transactions of the American Mathematical Society, vol. 130 (1968), pp. 1–37CrossRefGoogle Scholar
[2]Maass, W., Characterization of recursively emumerable sets with supersets effectively isomorphic to all recursively enumerable sets, Transactions of the American Mathematical Society, vol. 279 (1983), pp. 311–336.CrossRefGoogle Scholar
[3]Rogers, H. Jr., Theory of recursive functions and effective computability, McGraw-Hill, New York, 1967.Google Scholar
[4]Shore, R. A., Nowhere simple sets and the lattice of recursively enumerable sets, this Journal, vol. 43 (1978), pp. 322–330.Google Scholar