Zur Modelltheorie von Kranzprodukten. (On the model theory of wreath products) (Q1100458)

Let the symbols \(\equiv\), \(\equiv^{\forall}\), \(\equiv^{pos}\), \(\equiv^{\forall -pos}\) denote coincidence of elementary, universal, positive, universal and positive theories of two groups, respectively. The main results are the following: Theorem 1.2. \(G_ i\equiv^{\forall-pos}H_ i\), \(i=1,2\Rightarrow\) \(G_ 1\wr G_ 2\equiv^{\forall-pos}H_ 1\wr H_ 2\). Theorem 1.3. \(G_ i\equiv^{\forall}H_ i\), \(i=1,2\Rightarrow\) \(G_ 1\wr (G_ 2)\equiv^{\forall}H_ 1\wr (H_ 2)\). Theorem 1.4. If the groups G, H are algebraically compact and A is an arbitrary group, then \(G\equiv^{pos}H \Rightarrow G\wr A\equiv^{pos}H\wr A\). Theorem 2.4. If the groups \(G_ i\), \(H_ i\), \(i=1,2\) are nontrivial, \(| G_ 2| >2\) or \(| H_ 2| >2\), and \(G_ 1\wr G_ 2\equiv H_ 1\wr H_ 2\), then \(G_ i\equiv H_ i.\) The author subjects to criticism the work of \textit{E. I. Timoshenko} [Algebra Logika, Sem. 7, No.4, 114-119 (1968; Zbl 0186.317)].