Strong \(\Delta_1\)-definability of a model in an admissible set (Q1288078)

A model \(\langle M, P_1^{n_1},\ldots,P_m^{n_m}\rangle \) is called strongly \(\Delta_1\)-definable over an admissible set \(A\) provided there exist \(\Delta_1\)-formulas \(\varphi(x,\overline{y})\), \(\psi_1(x_1,\ldots,x_{n_1},\overline{y})\), \(\ldots\), \(\psi_m(x_1,\ldots,x_{n_m},\overline{y})\) and a tuple of elements \(\overline{a}\) in \(A\) such that \(M\) is isomorphic to the model \(\langle X, \overline{P}_1,\ldots,\overline{P}_m\rangle \), where \(X=\{x\in A\mid A\models\varphi(x,\overline{a})\}\) and \(\overline{P}_i= \{ \langle x_1,\ldots,x_{n_i}\rangle \mid A \models \psi_i (x_1,\ldots,x_{n_i},\overline{a}) \}\). The author calls a model \(M\) a \(B\)-model if, for every finite nonempty subset \(M_0\subseteq M\), there is a finite \(M^\ast_0\subseteq M\), \(M_0\subseteq M^\ast_0\) such that, for each finite subset \(M_1{\not\subseteq}M_0^\ast\) of \(M\), there exists an automorphism \(f\) of \(M\) such that \(f|{}M^\ast_0=\text{id}_{M_0^\ast}\) and \(fM_1\neq M_1\). The author proves that, if a rigid model \(C\) is definable over a Cartesian product \(M\times N\), where \(N\) is isomorphic to a recursive model and \(M\) is a \(B\)-model, then \(C\) is isomorphic to a recursive model. Then he uses this assertion for proving that every ordinal is \(\Delta_1\)-definable over at most countable Abelian groups and Boolean algebras if and only if it is recursive. This gives examples of Abelian groups and Boolean algebras whose respective Ulm types and ordinal types are not \(\Delta_1\)-definable in the superstructure of hereditarily finite sets over these algebras.