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Extension theorems, orbits, and automorphisms of the computably enumerable sets
 TRANS. AMER. MATH. SOC.
, 2008
"... We prove an algebraic extension theorem for the computably enumerable sets, E. Using this extension theorem and other work we then show if A and � A are automorphic via Ψ, then they are automorphic via Λ where Λ ↾ L ∗ (A) =ΨandΛ↾E ∗ (A) is∆0 3. We give an algebraic description of when an arbitrary ..."
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Cited by 4 (4 self)
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We prove an algebraic extension theorem for the computably enumerable sets, E. Using this extension theorem and other work we then show if A and � A are automorphic via Ψ, then they are automorphic via Λ where Λ ↾ L ∗ (A) =ΨandΛ↾E ∗ (A) is∆0 3. We give an algebraic description of when an arbitrary set �A is in the orbit of a computably enumerable set A. We construct the first example of a definable orbit which is not a ∆0 3 orbit. We conclude with some results which restrict the ways one can increase the complexity of orbits. For example, we show that if A is simple and �A is in the same orbit as A, then they are in the same ∆0 6orbit and, furthermore, we provide a classification of when two simple sets are in the same orbit.
Isomorphisms Of Splits Of Computably Enumerable Sets
 J. OF SYMBOLIC LOGIC
, 2002
"... We show that if A and A are automorphic via # then the structures SR (A) and SR ( 3 isomorphic via an isomorphism # induced by #. Then we use this result to classify completely the orbits of hhsimple sets. ..."
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Cited by 4 (4 self)
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We show that if A and A are automorphic via # then the structures SR (A) and SR ( 3 isomorphic via an isomorphism # induced by #. Then we use this result to classify completely the orbits of hhsimple sets.
The complexity of orbits of computably enumerable sets
 BULLETIN OF SYMBOLIC LOGIC
, 2008
"... The goal of this paper is to announce there is a single orbit of the c.e. sets with inclusion, E, such that the question of membership in this orbit is Σ1 1complete. This result and proof have a number of nice corollaries: the Scott rank of E is ωCK 1 + 1; not all orbits are elementarily definable; ..."
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Cited by 2 (0 self)
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The goal of this paper is to announce there is a single orbit of the c.e. sets with inclusion, E, such that the question of membership in this orbit is Σ1 1complete. This result and proof have a number of nice corollaries: the Scott rank of E is ωCK 1 + 1; not all orbits are elementarily definable; there is no arithmetic description of all orbits of E; for all finite α ≥ 9, there is a properly ∆0 α orbit (from the proof).