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 $\alpha$, 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.