When Is X Strongly Hyperimmune-free Relative to Y
نویسندگان
چکیده
The main topic of the present work is the relation that a set X is strongly hyperimmune-free (shif) relative to Y . Here X is shif-below Y if and only if for every partial X-recursive function p there is a partial Y -recursive function q such that every a in the domain of p is also in the domain of q and satisfies p(a) < p(q). It is shown that between degrees not above the halting problem this relation coincides with the Turing reducibility and that therefore only recursive sets have a strongly hyperimmune-free Turing degree while those sets Turing above the halting problem bound uncountably many sets with respect to the shif-relation. Furthermore, the shif-degree of the halting problem is definable with the shif-relation and every shifdegree coincides with the corresponding Turing degree.
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A reducibility related to being hyperimmune-free
The main topic of the present work is the relation that a set X is strongly hyperimmune-free relative to Y . Here X is strongly hyperimmune-free relative to Y if and only if for every partial X-recursive function p there is a partial Y -recursive function q such that every a in the domain of p is also in the domain of q and satisfies p(a) < p(a), that is, p is majorised by q. For X being hyperi...
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