Tusupov, D. A. 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. Cited in 1 Document MSC: 03C57 Computable structure theory, computable model theory 13G99 Integral domains 20M14 Commutative semigroups Keywords:computable structure; Scott family; definable relation; integral domain; semigroup PDFBibTeX XMLCite \textit{D. A. Tusupov}, Mat. Tr. 9, No. 2, 172--190 (2006; Zbl 1249.03065); translation in Sib. Adv. Math. 17, No. 1, 49--61 (2007) Full Text: DOI