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Isomorphisms, definable relations, and Scott families for integral domains and commutative semigroups. (Russian, English) Zbl 1249.03065

Mat. Tr. 9, No. 2, 172-190 (2006); translation in Sib. Adv. Math. 17, No. 1, 49-61 (2007).
Summary: We prove the following four assertions:
(1) For every computable successor ordinal 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.

MSC:

03C57 Computable structure theory, computable model theory
13G99 Integral domains
20M14 Commutative semigroups
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