Skip to main content
SpringerLink
Log in

Isomorphisms, definable relations, and Scott families for integral domains and commutative semigroups

  • Published:
Siberian Advances in Mathematics Aims and scope Submit manuscript

Abstract

In the present article, we prove the following four assertions: (1) For every computable successor ordinal α, there exists a Δ 0 α -categorical integral domain (commutative semigroup) which is not relatively Δ 0 α -categorical (i.e., no formally Σ 0 α Scott family exists for such a structure). (2) For every computable successor ordinal α, there exists an intrinsically Σ 0 α -relation on the universe of a computable integral domain (commutative semigroup) which is not a relatively intrinsically Σ 0 α -relation. (3) For every computable successor ordinal α and finite n, there exists an integral domain (commutative semigroup) whose Δ 0 α -dimension is equal to n. (4) For every computable successor ordinal α, there exists an integral domain (commutative semigroup) with presentations only in the degrees of sets X such that Δ 0 α (X) is not Δ 0 α . In particular, for every finite n, there exists an integral domain (commutative semigroup) with presentations only in the degrees that are not n-low.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. C. Ash and J. Knight, Computable Structures and the Hyperarithmetical Hierarchy, vol. 144 of Studies in Logic and the Foundations of Mathematics (Elsevier, Amsterdam, 2000).

    Google Scholar 

  2. C. Ash, J. Knight, M. Manasse, and T. Slaman, “Generic copies of countable structures,” Ann. Pure Appl. Logic 42(3), 195–205 (1989).

    Article  MATH  MathSciNet  Google Scholar 

  3. I. Bucur and A. Deleanu, Introduction to the Theory of Categories and Functors, With the collaboration of Peter J. Hilton and Nicolae Popescu. Pure and Applied Mathematics, Vol. XIX (Interscience Publication John Wiley & Sons, Ltd., London-New York-Sydney, 1968).

    Google Scholar 

  4. J. Chisholm, “Effective model theory versus recursive model theory,” J. Symbolic Logic 55(3), 1168–1191 (1990).

    Article  MATH  MathSciNet  Google Scholar 

  5. Yu. L. Ershov and S. S. Goncharov, Constructive Models (Plenum, New York, 2000).

    MATH  Google Scholar 

  6. S. Goncharov, V. Harizanov, J. Knight, C. McCoy, R. Miller, and R. Solomon, “Enumerations in computable structure theory,” Ann. Pure Appl. Logic 136(3), 219–246 (2005).

    Article  MATH  MathSciNet  Google Scholar 

  7. S. S. Goncharov, “The problem of the number of nonautoequivalent constructivizations,” Algebra and Logika 19(6), 621–639 (1980) [Algebra and Logic 19 (6), 401–414 (1980)].

    MathSciNet  Google Scholar 

  8. D. R. Hirschfeldt, B. Khoussainov, R. A. Shore, and A. M. Slinko, “Degree spectra and computable dimensions in algebraic structures,” Ann. Pure Appl. Logic 115(1–3), 71–113 (2002).

    Article  MATH  MathSciNet  Google Scholar 

  9. O. Kudinov, An Integral Domain with Finite Dimension, Manuscript (Novosibirsk, 2006).

  10. A. I. Mal’tsev, “Constructive algebras,” Uspekhi Mat. Nauk 16(3), 3–60 (1961) [Russian Math. Surveys (3), 77–129 (1961)].

    Google Scholar 

  11. C. F. D. McCoy, “Finite computable dimension does not relativize,” Arch. Math. Logic 41(4), 309–320 (2002).

    Article  MATH  MathSciNet  Google Scholar 

  12. J. A. Tussupov, “Computable graphs of finite Δ 0 α -dimension,” Vestn. Novosib. Gos. Univ., Ser. Mat. Mekh. Inform. (2007), to appear.

Download references

Author information

Authors and Affiliations

  1. Sobolev Institute of Mathematics, Novosibirsk, 630090, Russia

    J. A. Tussupov

Authors
  1. J. A. Tussupov
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Original Russian Text © J. A. Tussupov, 2006, published in Matematicheskie Trudy, 2006, Vol. 9, No. 2, pp. 172–190.

About this article

Cite this article

Tussupov, J.A. Isomorphisms, definable relations, and Scott families for integral domains and commutative semigroups. Sib. Adv. Math. 17, 49–61 (2007). https://doi.org/10.3103/S1055134407010038

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.3103/S1055134407010038

Key words

Search

Navigation