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Splitting properties and jump classes

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Abstract

We show that the promptly simple sets of Maass form a filter in the lattice ℰ of recursively enumerable sets. The degrees of the promptly simple sets form a filter in the upper semilattice of r.e. degrees. This filter nontrivially splits the high degrees (a is high ifa′=0″). The property of prompt simplicity is neither definable in ℰ nor invariant under automorphisms of ℰ. However, prompt simplicity is easily shown to imply a property of r.e. sets which is definable in ℰ and which we have called the splitting property. The splitting property is used to answer many questions about automorphisms of ℰ. In particular, we construct lowd-simple sets which are not automorphic, answering a question of Lerman and Soare. We produce classes invariant under automorphisms of ℰ which nontrivially split the high degrees as well as all of the other classes of r.e. degrees defined in terms of the jump operator. This refutes a conjecture of Soare and answers a question of H. Friedman.

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Author notes

  1. Wolfgang Maass

    Present address: Department of Mathematics, Cornell University, 14853, Ithaca, NY, USA

  2. Richard A. Shore

    Present address: Department of Mathematics, Calvin College, 49506, Grand Rapids, MI, USA

Authors and Affiliations

  1. Department of Mathematics, Massachusetts Institute of Technology, 02139, Cambridge, MA, USA

    Wolfgang Maass, Richard A. Shore & Michael Stob

  2. Department of Mathematics, University of Chicago, 60637, Chicago, IL, USA

    Wolfgang Maass, Richard A. Shore & Michael Stob

Authors
  1. Wolfgang Maass
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  2. Richard A. Shore
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  3. Michael Stob
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Additional information

During preparation of this paper, the first author was supported by the Heisenberg Programm der Deutschen Forschungsgemeinschaft, West Germany. The second author was partially supported by NSF Grant MSC 77-04013. The third author was partially supported by NSF Grant MSC 80-02937.

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Maass, W., Shore, R.A. & Stob, M. Splitting properties and jump classes. Israel J. Math. 39, 210–224 (1981). https://doi.org/10.1007/BF02760850

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  • DOI: https://doi.org/10.1007/BF02760850

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