- 22 Downloads
Description is given of the isomorphism types of the principal ideals of the join semilattice of m-degrees which are generated by arithmetical sets. A result by Lachlan of 1972 on computably enumerable m-degrees is extended to the arbitrary levels of the arithmetical hierarchy. As a corollary, a characterization is given of the local isomorphism types of the Rogers semilattices of numberings of finite families, and the nontrivial Rogers semilattices of numberings which can be computed at the different levels of the arithmetical hierarchy are proved to be nonisomorphic provided that the difference between levels is more than 1.
Keywordsarithmetical hierarchy m-reducibility distributive join semilattice Lachlan semilattice numbering Rogers semilattice
Unable to display preview. Download preview PDF.
- 5.Badaev S. A., Goncharov S. S., and Sorbi A., “Isomorphism types and theories of Rogers semilattices of arithmetical numberings,” in: Computability and Models, Kluwer Plenum Publ., New York, 2003, pp. 79–91.Google Scholar
- 8.Grätzer G., General Lattice Theory, Birkhäuser, Basel (1978).Google Scholar
- 9.Ershov Yu. L., Theory of Numberings [in Russian], Nauka, Moscow (1977).Google Scholar
- 13.Badaev S., Goncharov S., and Sorbi A., “Completeness and universality of arithmetical numberings,” in: Computability and Models, Kluwer Plenum Publ., New York, 2003, pp. 11–44.Google Scholar
- 14.Badaev S. A., Goncharov S. S., Podzorov S. Yu., and Sorbi A., “Algebraic properties of Rogers semilattices of arithmetical numberings,” in: Computability and Models, 2003, Kluwer Plenum Publ., New York, pp. 45–78.Google Scholar
- 15.Goncharov S. S., Countable Boolean Algebras and Decidability [in Russian], Nauchnaya Kniga, Novosibirsk (1996).Google Scholar