Isomorphisms,definable relations,and Scott families for integral domains and commutative semigroups |
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Authors: | J. A. Tussupov |
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Affiliation: | (1) Sobolev Institute of Mathematics, Novosibirsk, 630090, Russia |
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Abstract: | In the present article, we prove the following four assertions: (1) For every computable successor ordinal α, there exists a Δ α 0 -categorical integral domain (commutative semigroup) which is not relatively Δ α 0 -categorical (i.e., no formally Σ α 0 Scott family exists for such a structure). (2) For every computable successor ordinal α, there exists an intrinsically Σ α 0 -relation on the universe of a computable integral domain (commutative semigroup) which is not a relatively intrinsically Σ α 0 -relation. (3) For every computable successor ordinal α and finite n, there exists an integral domain (commutative semigroup) whose Δ α 0 -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 α 0 (X) is not Δ α 0 . 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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Keywords: | computable structure Scott family definable relation integral domain semigroup |
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