Logical Methods pp 314-351 | Cite as

# Effectively and Noneffectively Nowhere Simple Subspaces

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## Abstract

Let ω_{e} denote the *e* ^{th} r.e. subset of the natural numbers *N*, i.e. ω_{e} = {*x*: φ_{e}(*x*) converges} where φ_{e} 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 *W*’_{e} ⊆ *W* _{e} such that *W*’_{e} ⋂ *A* = Ø and *card*(*W* _{e} − *A*) = ∞ if and only if *card*(*W*’_{e}) = ∞. We say *A* is **effectively nowhere simple** if there is an effective procedure to find the index of such a *W*’_{e} from *e*, i.e. if there is a recursive function *f* such that *W* _{f(e)} = *W*’_{e}. 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## Preview

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