Geodesic
Isomorphisms, Definable Relations, and Scott Families for Integral Domains and Commutative Semigroups
Matematičeskie trudy, Tome 9 (2006) no. 2, pp. 172-190.

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In the present article, we prove the following four assertions: (1) For every computable successor ordinal $\alpha$, there exists a $\Delta^0_\alpha$-categorical integral domain (commutative semigroup) which is not relatively $\Delta^0_\alpha$-categorical (i. e., no formally $\Sigma^0_\alpha$ Scott family exists for such a structure). (2) For every computable successor ordinal $\alpha$, there exists an intrinsically $\Sigma^0_\alpha$-relation on the universe of a computable integral domain (commutative semigroup) which is not a relatively intrinsically $\Sigma^0_\alpha$-relation. (3) For every computable successor ordinal $\alpha$ and finite $n$, there exists an integral domain (commutative semigroup) whose $\Delta^0_\alpha$-dimension is equal to $n$. (4) For every computable successor ordinal $\alpha$, there exists an integral domain (commutative semigroup) with presentations only in the degrees of sets $X$ such that $\Delta^0_\alpha(X)$ is not $\Delta^0_\alpha$. In particular, for every finite $n$, there exists an integral domain (commutative semigroup) with presentations only in the degrees that are not $n$-low. 
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D. A. Tusupov. Isomorphisms, Definable Relations, and Scott Families for Integral Domains and Commutative Semigroups. Matematičeskie trudy, Tome 9 (2006) no. 2, pp. 172-190. http://geodesic.mathdoc.fr/item/MT_2006_9_2_a7/

[1] Bukur I., Delyanu A., Vvedenie v teoriyu kategorii i funktorov, Mir, M., 1972 | MR

[2] Goncharov S. S., “Problema chisla neavtoekvivalentnykh konstruktivizatsii”, Algebra i logika, 19:6 (1980), 621–639 | MR | Zbl

[3] Goncharov S. S., Ershov Yu. L., Konstruktivnye modeli, Nauchnaya kniga, Novosibirsk, 1999

[4] Maltsev A. I., “Konstruktivnye modeli. I”, Uspekhi mat. nauk, 16:3(99) (1961), 3–60 | MR

[5] Tusunov D. A., “Algebraicheskie struktury konechnoi $\Delta_\alpha^0$-razmernosti”, Algebra i logika, 2007 (to appear)

[6] Ash S. J. and Knight J. F., Computable Structures and the Hyper'arithmetical Hierarchy, Studies in Logic and the Foundations of Mathematics, 144, Elsevier, Amsterdam, 2000 | MR | Zbl

[7] Ash S. J., Knight J. F., Manasse M., and Slaman T., “Generic copies of countable structures”, Ann. Pure Appl. Logic, 42:3 (1989), 195–205 | DOI | MR | Zbl

[8] Chisholm J., “Effective model theory versus recursive model theory”, J. Symbolic Logic, 55:3 (1990), 1168–1191 | DOI | MR | Zbl

[9] McCoy S. F. D., “Finite computable dimension does not relativize”, Arch. Math. Logic, 41:4 (2002), 309–320 | DOI | MR | Zbl

[10] Goncharov S. S., Harizanov V. S., Knight J. F., McCoy C., Miller R., and Solomon R., “Enumerations in computable structure theory”, Ann. Pure Appl. Logic, 136:3 (2005), 219–246 | DOI | MR | Zbl

[11] Hirschfeldt D. R., Khoussainov V., Shore R. A., and Slinko A. M., “Degree spectra and computable dimensions in algebraic structures”, Ann. Pure and Appl. Logic, 115:1–3 (2002), 71–113 | DOI | MR | Zbl

[12] Kudinov O., An Integral Domain with Finite Dimension, Manuscript, Novosibirsk, 2006