On computable automorphisms of the rational numbers

Abstract The relationship between ideals I of Turing degrees and groups of I -recursive automorphisms of the ordering on rationals is studied. We discuss the differences between such groups and the group of all automorphisms, prove that the isomorphism type of such a group completely defines the ideal I , and outline a general correspondence between principal ideals of Turing degrees and the first-order properties of such groups.

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