Abstract
We investigate differences in isomorphism types and elementary theories of Rogers semilattices of arithmetical numberings, depending on different levels of the arithmetical hierarchy. It is proved that new types of isomorphism appear as the arithmetical level increases. It is also proved the incompleteness of the theory of the class of all Rogers semilattices of any fixed level. Finally, no Rogers semilattice of any infinite family at arithmetical level n ≥ 2 is weakly distributive, whereas Rogers semilattices of finite families are always distributive.
Partial funding provided by grant INTAS Computability in Hierarchies and Topological Spaces no. 00–499; and by grant PICS-541-Kazakhstan.
Partial funding provided by grant INTAS Computability in Hierarchies and Topological Spaces no 00–499.
Partial funding provided by grant INTAS Computability in Hierarchies and Topological Spaces no 00–499; work done under the auspices of GNSAGA.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
S.A. Badaev, On minimal enumerations, Siberian Adv. Math, 1992, vol. 2, no. 1, pp. 1–30.
S.A. Badaev, S.S. Goncharov, A. Sorbi, Arithmetical numberings: completeness and universality. This volume.
S.A. Badaev, S.S. Goncharov, S.Yu. Podzorov, A. Sorbi, Arithmetical numberings: algebraic properties of Rogers semilattices. This volume.
V.D. Dzgoev, Constructive Enumerations of Boolean Lattices. Algebra i Logika, 1988, vol. 27, no. 6, pp. 641–648 (Russian); Algebra and Logic, 1988, vol. 27, no. 6, pp. 395–400 (English translation).
Yu.L. Ershov, Theory of Numberings. Nauka, Moscow, 1977 (Russian).
L. Feiner, Hierarchies of Boolean algebras. J. Symb. Logic. 1970, vol. 35, no. 3, pp. 365–374.
S.S. Goncharov Countable Boolean Algebras and Decidability., Plenum, Consultants Bureau, New York, 1997.
A.H. Lachlan, Standard classes of recursively enumerable sets. Zeit. Mat. Log. Grund. Math., 1964, vol. 10, pp. 23–42.
A.H. Lachlan, On the lattice of recursively enumerable sets. Trans. Amer. Math. Soc., 1968, vol. 130, pp. 1–37.
A.I. Mal’tsev, Algorithms and Recursive Functions. Nauka, Moscow, 1965 (Russian); Wolters-Noordoff Publishing, Groningen, 1970 (English translation).
S.S. Marchenkov, On computable enumerations of families of general recursive functions. Algebra i Logika, 1972, vol. 11, no. 5, pp. 588–607 (Russian); Algebra and Logic, 1972, vol. 11, no. 5, pp. 326–336 (English translation).
S.Yu. Podzorov, Initial segments in Rogers semilattices \( \sum\nolimits_n^0 {} \)-computable numberings, Algebra and Logic, to appear.
H. Rogers, Jr. Theory of Recursive Functions and Effective Computability. McGraw-Hill, New York, 1967.
R.I. Soare Recursively Enumerable Sets and Degrees. Springer-Verlag, Berlin Heidelberg, 1987.
V.V. V’jugin, On some examples of upper semilattices of computable enumerations. Algebra i Logika, 1973, vol. 12, no. 5, pp. 512–529 (Russian); Algebra and Logic, 1973, vol. 12, no. 5, pp. 277–286 (English translation).
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer Science+Business Media New York
About this chapter
Cite this chapter
Badaev, S., Goncharov, S., Sorbi, A. (2003). Isomorphism Types and Theories of Rogers Semilattices of Arithmetical Numberings. In: Computability and Models. The University Series in Mathematics. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-0755-0_4
Download citation
DOI: https://doi.org/10.1007/978-1-4615-0755-0_4
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4613-5225-9
Online ISBN: 978-1-4615-0755-0
eBook Packages: Springer Book Archive