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

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.

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

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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

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Key words

  • computable structure
  • Scott family
  • definable relation
  • integral domain
  • semigroup