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

In the present article, we prove the following four assertions: (1) For every computable successor ordinal α, there exists a Δα0-categorical integral domain (commutative semigroup) which is not relatively Δα0-categorical (i.e., no formally Σα0 Scott family exists for such a structure). (2) For every computable successor ordinal α, there exists an intrinsically Σα0-relation on the universe of a computable integral domain (commutative semigroup) which is not a relatively intrinsically Σα0-relation. (3) For every computable successor ordinal α and finite n, there exists an integral domain (commutative semigroup) whose Δα0-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 Δα0(X) is not Δα0. In particular, for every finite n, there exists an integral domain (commutative semigroup) with presentations only in the degrees that are not n-low.