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, access via your institution.
Buy single article
Instant access to the full article PDF.
Price includes VAT (USA)
Tax calculation will be finalised during checkout.
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).
C. Ash, J. Knight, M. Manasse, and T. Slaman, “Generic copies of countable structures,” Ann. Pure Appl. Logic 42(3), 195–205 (1989).
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).
J. Chisholm, “Effective model theory versus recursive model theory,” J. Symbolic Logic 55(3), 1168–1191 (1990).
Yu. L. Ershov and S. S. Goncharov, Constructive Models (Plenum, New York, 2000).
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).
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)].
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).
O. Kudinov, An Integral Domain with Finite Dimension, Manuscript (Novosibirsk, 2006).
A. I. Mal’tsev, “Constructive algebras,” Uspekhi Mat. Nauk 16(3), 3–60 (1961) [Russian Math. Surveys (3), 77–129 (1961)].
C. F. D. McCoy, “Finite computable dimension does not relativize,” Arch. Math. Logic 41(4), 309–320 (2002).
J. A. Tussupov, “Computable graphs of finite Δ 0 α -dimension,” Vestn. Novosib. Gos. Univ., Ser. Mat. Mekh. Inform. (2007), to appear.
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
- computable structure
- Scott family
- definable relation
- integral domain