Categorical linearly ordered structures (Q2311213)

The authors consider isomorphisms computable relatively to hyperarithmetic hierarchies between computable structures from well-known algebraic classes and prove: Theorem 1. For every computable limit ordinal \(\alpha\) there exists a computable linear order \(A_\alpha\) such that (1) for every computable copy \(M\) there exists a \(\beta<\alpha\) such that \(M\cong_{\Delta_\beta} A_\alpha\); (2) for every \(\beta<\alpha\) there exists a computable copy \(B\cong A_\alpha \) such that \(M \ncong_{\Delta_\beta} A_\alpha\). Theorem 2. The properties in Theorem 1 can be witnessed by structures from the following classes: (i) ordered abelian groups, and (ii) real-closed fields of infinite transcendence degree.