Logical Methods pp 314-351

# 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  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  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  proved that any two effectively nowhere simple sets have effectively isomorphic lattices of supersets. Another such property is the outer splitting property .

## 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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