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Regular enumerations. (English) Zbl 1053.03024

The paper studies what the authors call, following I. N. Soskov [Arch. Math. Logic 39, 417–437 (2000; Zbl 0960.03037)], regular enumerations which are a kind of generic enumeration of families of sets. This paper deals with the transfinite case, which turns out to be a nontrivial extension of the finite case treated by Soskov (loc. cit.).
The results of this paper provide a unified treatment of several results. There are results on jump inversion, where the jump comes from enumeration reducibility. There are results characterizing the sets \(A\) such that
\[ (\forall X)[(\forall\gamma\leq \zeta)\,(B_\gamma\text{ is c.e. in }X^{(\gamma)}\text{ uniformly in }\gamma)\Rightarrow A\text{ is c.e. in }X^{(\alpha)}], \]
where \(\alpha\) is an arbitrary computable ordinal. There is a general result saying when the set
\[ {\mathcal S}_{\alpha,\beta}= \{X^{(\alpha)}:(\forall\gamma\leq \beta)\,(B_\gamma\text{ is c.e. in }X^\gamma\text{ uniformly in }\gamma)\} \]
has an element of least degree. This yields as corollaries some well-known results of C. J. Ash, C. G. Jockusch jun. and J. F. Knight [Trans. Am. Math. Soc. 319, 573–599 (1990; Zbl 0705.03022)] and of R. Downey and J. F. Knight [Proc. Am. Math. Soc. 114, 545–552 (1992; Zbl 0748.03027)], as well as some recent results of R. J. Coles, R. G. Downey and T. A. Slaman [J. Lond. Math. Soc., II. Ser. 62, 641–649 (2000; Zbl 1023.03036)] on existence and nonexistence of least jumps to which a given set is enumeration reducible, which have applications in computable algebra. The methods of the paper are interesting and demonstrate that enumeration reducibility plays a central role in a number of results in computable algebra. (To the reviewer’s knowledge the first such observation along these lines were in L. Richter’s thesis [Degrees of unsolvability of models, Ph.D. Thesis, University of Illinois at Urbana-Champaign (1977)]).

MSC:

03D25 Recursively (computably) enumerable sets and degrees
03D45 Theory of numerations, effectively presented structures
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References:

[1] Bulletin of the London Mathematical Society
[2] DOI: 10.1002/malq.19740201311 · Zbl 0304.02016 · doi:10.1002/malq.19740201311
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[5] DOI: 10.1007/s001530050156 · Zbl 0960.03037 · doi:10.1007/s001530050156
[6] DOI: 10.1002/malq.19710170139 · Zbl 0229.02037 · doi:10.1002/malq.19710170139
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[9] Theory of recursive functions and effective computability (1967) · Zbl 0183.01401
[10] Proceedings of the American Mathematical Society 114 pp 545– (1992)
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[12] Recursion Theory Week, Oberwolfach 1989 1432 pp 57– (1990)
[13] Higher Recursion Theory (1990) · Zbl 0716.03043
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