Algebraic geometry over groups. II: Logical foundations
The second in a series of three papers of the authors is devoted to the description of the logical foundations of the algebraic geometry over groups [for part I see \textit{G. Baumslag} and the authors, ibid. 219, No. 1, 16-79 (1999; Zbl 0938.20020)]. With a given \(G\)-group \(H\) one can associate several model-theoretic classes such as: \(\text{var }H\) (the variety of \(G\)-groups, generated by \(H\)), \(\text{qvar }H\) (the quasivariety of \(G\)-groups, generated by \(H\)), and so on, defined in the usual manner. The authors search for algebraic characterizations of model-theoretic classes as above. In particular, the class \(\text{pvar }H=SP(H)\), where \(S\) means the subgroup operator and \(P\) means the unrestricted Cartesian product operator, is considered. Some conditions allowing to give the classification needed in some important cases are derived. The paper under review contains many new definitions and a number of interesting results. There are the answers to two questions raised by Plotkin among these results.
scientific article; zbMATH DE number 1565860