A decomposition of the Rogers semilattice of a family of d.c.e. sets
Serikzhan A. Badaev and Steffen Lempp
Source: J. Symbolic Logic Volume 74, Issue 2
(2009), 618-640.
Abstract
Khutoretskii's Theorem states that the Rogers semilattice of any family of c.e. sets has either at most one or infinitely many elements. A lemma in the inductive step of the proof shows that no Rogers semilattice can be partitioned into a principal ideal and a principal filter. We show that such a partitioning is possible for some family of d.c.e. sets. In fact, we construct a family of c.e. sets which, when viewed as a family of d.c.e. sets, has (up to equivalence) exactly two computable Friedberg numberings μ and ν, and μ reduces to any computable numbering not equivalent to ν. The question of whether the full statement of Khutoretskii's Theorem fails for families of d.c.e. sets remains open.
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Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.jsl/1243948330
Digital Object Identifier: doi:10.2178/jsl/1243948330
Zentralblatt MATH identifier: 05561769
Mathematical Reviews number (MathSciNet): MR2518814
References
Serikzhan A. Badaev and Sergey S. Goncharov, The theory of numberings: open problems, Computability theory and its applications. Current trends and open problems (Peter A. Cholak, Steffen Lempp, Manuel Lerman, and Richard A. Shore, editors), American Mathematical Society, Providence, RI, 2000, pp. 23--38.
Mathematical Reviews (MathSciNet): MR1770732
Serikzhan A. Badaev, Sergey S. Goncharov, Sergei Yu. Podzorov, and Andrea Sorbi, Algebraic properties of Rogers semilattices of arithmetical numberings, Computability and models. perspectives east and west (S. Barry Cooper and Sergey S. Goncharov, editors), Kluwer/Plenum, New York, 2003, pp. 45--77.
Mathematical Reviews (MathSciNet): MR2043593
Serikzhan A. Badaev, Sergey S. Goncharov, and Andrea Sorbi, Completeness and universality of arithmetical numberings, Computability and models. Perspectives east and west (S. Barry Cooper and Sergey S. Goncharov, editors), Kluwer/Plenum, New York, 2003, pp. 11--44.
Mathematical Reviews (MathSciNet): MR2043592
--------, Isomorphism types and theories of Rogers semilattices of arithmetical numberings, Computability and models. Perspectives east and west (S. Barry Cooper and Sergey S. Goncharov, editors), Kluwer/Plenum, New York, 2003, pp. 79--91.
Mathematical Reviews (MathSciNet): MR2043594
Yuri L. Ershov, Theory of numerations. Part I: General theory of numerations, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 19 (1973), pp. 289--388, German.
Mathematical Reviews (MathSciNet): MR422001
--------, Theory of numerations. Part II: Computable numerations of morphisms, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 21 (1975), pp. 473--584, German.
--------, Theory of numerations. Part III: Constructive models, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 23 (1977), pp. 289--371, German.
--------, Theory of numerations\phantom, Nauka, Moscow, 1977, Russian.
Mathematical Reviews (MathSciNet): MR506676
--------, Theory of numberings, Handbook of computability theory, North-Holland, Amsterdam, 1999, pp. 473--503.
Mathematical Reviews (MathSciNet): MR1720731
Digital Object Identifier: doi:10.1016/S0049-237X(99)80030-5
Zentralblatt MATH: 0948.03040
Sergey S. Goncharov, Computable single-valued numerations, Algebra i Logika, vol. 19 (1980), no. 5, pp. 507--551.
Mathematical Reviews (MathSciNet): MR623783
--------, Limit equivalent constructivizations, Trudy Inst. Matem. SO AN SSSR, vol. 2 (1982), pp. 4--12, Nauka, Novosibirsk, Russian.
Mathematical Reviews (MathSciNet): MR720197
--------, Positive numerations of families with one-valued numerations, Algebra and Logic, vol. 22 (1983), no. 5, pp. 345--350.
Zentralblatt MATH: 0572.03023
--------, A unique positive enumeration, Siberian Advances in Mathematics, vol. 4 (1994), no. 1, pp. 52--64.
Mathematical Reviews (MathSciNet): MR1259077
Aleksandr B. Khutoretskii, The power of the upper semilattice of computable numerations, Algebra i Logika, vol. 10 (1971), pp. 561--569.
Mathematical Reviews (MathSciNet): MR302433
Steffen Lempp, Lecture notes on priority arguments, preprint available at http://www.math.wisc.edu/~lempp/papers/prio.pdf.
Robert I. Soare, Recursively enumerable sets and degrees, Springer-Verlag, Berlin, New York, 1987.
Mathematical Reviews (MathSciNet): MR882921
