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Effectively and Noneffectively Nowhere Simple Subspaces

  • R. G. Downey
  • Jeffrey B. Remmel
Chapter
Part of the Progress in Computer Science and Applied Logic book series (PCS, volume 12)

Abstract

Let ωe denote the e th r.e. subset of the natural numbers N, i.e. ωe = {x: φe(x) converges} where φe is the partial recursive function computed by the e th Turing machine. Then an r.e. set A is called nowhere simple if A is coinfinite and for any r.e. set W e, there exists an r.e. set WeW e such that WeA = Ø and card(W eA) = ∞ if and only if card(We) = ∞. We say A is effectively nowhere simple if there is an effective procedure to find the index of such a We from e, i.e. if there is a recursive function f such that W f(e) = We. In this case, f is said to be a witness function for A. These classes of sets were introduced by Shore in [20] where he used them to investigate automorphism bases for ℒ(ω) (the lattice of r.e. sets) and ℒ*(ω) (ℒω) modulo finite sets). In particular he was able to give an easy proof of the main result of [19] that any nontrivial class of r.e. sets closed under recursive isomorphisms is an automorphism base for ℒ*(ω) using these sets. Subsequent to this paper, various investigations concerning automorphisms of ℒ(ω) and ℒ*(ω) have utilized properties of r.e. sets with rich lattices of supersets. For example, for effectively nowhere simple sets, Maass [11] proved that any two effectively nowhere simple sets have effectively isomorphic lattices of supersets. Another such property is the outer splitting property [11].

Keywords

Recursive Function Extendible Basis Independent Subset Recursion Theory Dependence Degree 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    Ash, C.J. and R.G. Downey [1984], Decidable subspaces and recursively enumerable subspaces. J. Symbolic Logic, 49, 1137–1145.MathSciNetzbMATHCrossRefGoogle Scholar
  2. [2]
    Downey, R.G. [1983], Nowhere simplicity in matroids. J. Austral. Math. Soc. (Series A), 28–45.Google Scholar
  3. [3]
    Downey, R.G. [1983], On a question of A. Retzlaff. Z. Math. Logik Grund. Math, 379–384.Google Scholar
  4. [4]
    Downey, R.G. [1984], Co-immune subspaces and complementation in V . J. Symbolic Logic, 49, 528–538.MathSciNetzbMATHCrossRefGoogle Scholar
  5. [5]
    Downey, R.G. [1984], A note on decompositions of recursively enumerable subspaces. Z. Math. Logik Grund. Math, 30, 465–470.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [6]
    Downey, R.G. [1987], Orbits of creative subspaces. Proc. Amer. Math. Soc, 99, 163–170.MathSciNetzbMATHCrossRefGoogle Scholar
  7. [7]
    Downey, R.G. and J.B. Remmel [1987], Automorphisms and r.e. structures. Z. Math. Logik Grund. Math, 33, 339–345.MathSciNetzbMATHCrossRefGoogle Scholar
  8. [8]
    Downey, R.G. and J.B. Remmel, Effective algebras and relational systems: Coding properties. In preparation.Google Scholar
  9. [9]
    Downey, R.G., J.B. Remmel and L.V. Welch [1987], Degrees of bases and splittings of recursively enumerable subspaces. Trans. Amer. Math. Soc, 302, 683–714.MathSciNetzbMATHGoogle Scholar
  10. [10]
    Guhl, R. [1973], Two Types of Recursively Enumerable Subspaces. Ph.D. Thesis, Rutgers University.Google Scholar
  11. [11]
    Maass, W. [1983], Characterizations of recursively enumerable sets with supersets effectively isomorphic to all recursively enumerable sets. Trans. Amer. Math. Soc, 279, 311–336.MathSciNetzbMATHCrossRefGoogle Scholar
  12. [12]
    Metakides, G. and A. Nerode [1977], Recursively enumerable vector spaces. Ann. Math. Logic, 11, 147–171.MathSciNetzbMATHCrossRefGoogle Scholar
  13. [13]
    Miller, D. and J.B. Remmel [1984], Effectively nowhere simple sets. J. Symbolic Logic, 49, 129–136.MathSciNetzbMATHCrossRefGoogle Scholar
  14. [14]
    Nerode, A. and J.B. Remmel [1982], Recursion theory on matroids. Patras Logic Symposion (ed. G. Metakides), North-Holland, Amsterdam, 129–136.Google Scholar
  15. [15]
    Nerode, A. and J.B. Remmel [1983], Recursion theory on Matroids, II. Southeast Asian Logic Conference (eds. C.T. Chong and M. Wicks), North-Holland, New York, 133–184.Google Scholar
  16. [16]
    Nerode, A. and J.B. Remmel [1985], A survey of the lattices of r.e. substructures. Recursion Theory (eds. A. Nerode and R.A. Shore), A.M.S. Publication, 323–376.Google Scholar
  17. [17]
    Remmel, J.B. [1980], On r.e. and co-r.e. vector spaces with non-extendible bases. J. Symbolic Logic, 45, 220–234.Google Scholar
  18. [18]
    Remmel, J.B. [1980], Recursion theory on algebraic structures with an independent set. Ann. Math. Logic, 18, 153–191.MathSciNetzbMATHCrossRefGoogle Scholar
  19. [19]
    Shore, R.A. [1977], Determining automorphisms of the recursively enumerable sets. Proc. A.M.S, 65, 318–326.MathSciNetzbMATHCrossRefGoogle Scholar
  20. [20]
    Shore, R.A. [1978], Nowhere simple sets and the lattice of recursively enumerable sets. J. Symbolic Logic, 43, 322–330.MathSciNetzbMATHCrossRefGoogle Scholar
  21. [21]
    Soare, R.I. [1987], Recursively Enumerable Sets and Degrees. Springer-Verlag (Berlin, Heidelberg, New York).Google Scholar
  22. [22]
    Stob, M. [1985], Major subsets and the lattice of recursively enumerable sets. Recursion Theory (eds. A. Nerode nad R.A. Shore), A.M.S. Publication, 107–116.Google Scholar

Copyright information

© Springer Science+Business Media New York 1993

Authors and Affiliations

  • R. G. Downey
    • 1
  • Jeffrey B. Remmel
    • 2
  1. 1.Department of MathematicsUniversity of VictoriaWellingtonNew Zealand
  2. 2.Department of MathematicsUniversity of California at San DiegoLa JollaUSA

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