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\(\eta \)-representation of sets and degrees. (English) Zbl 1158.03025

The paper is devoted to \(\eta\)-representations of sets and degrees. It is shown that a set has an \(\eta\)-representation in a linear order if and only if it is the range of a \(0'\)-computable limitwise monotonic function. The author constructs also a \(\Delta_{3}\) Turing degree for which no set in that degree has a strong \(\eta\)-representation – this answers a question posed by R. Downey in 1998.

MSC:

03D28 Other Turing degree structures
03D30 Other degrees and reducibilities in computability and recursion theory
03D45 Theory of numerations, effectively presented structures
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References:

[1] Recursively enumerable sets and degrees (1987)
[2] DOI: 10.1007/BFb0090945 · doi:10.1007/BFb0090945
[3] DOI: 10.1305/ndjfl/1039724885 · Zbl 0891.03013 · doi:10.1305/ndjfl/1039724885
[4] Order 14 pp 107– (1998)
[5] DOI: 10.1305/ndjfl/1187031407 · Zbl 1146.03030 · doi:10.1305/ndjfl/1187031407
[6] Handbook of recursive mathematics 138 (1998)
[7] Bounding prime models 69 pp 1117– (2004)
[8] Handbook of recursive mathematics 138 pp 1177– (1998)
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