Algebraic geometry over groups. II: Logical foundations
type: Item
instance of: scholarly article
title: Algebraic geometry over groups. II: Logical foundations
zbMATH Open document ID: 0970.20017
published in: Journal of Algebra
publication date: 2001-07-09T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 20E34
Mathematics Subject Classification ID: 20E05
Mathematics Subject Classification ID: 14A22
Mathematics Subject Classification ID: 20F10
Mathematics Subject Classification ID: 08B05
Mathematics Subject Classification ID: 08C15
Mathematics Subject Classification ID: 08A45
Mathematics Subject Classification ID: 20E06
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 20J15
Mathematics Subject Classification ID: 20F65
zbMATH DE Number: 1565860
zbMATH Keywords: algebraic geometry
zbMATH Keywords: free groups
zbMATH Keywords: varieties of groups
zbMATH Keywords: quasivarieties
zbMATH Keywords: equations
zbMATH Keywords: radicals
zbMATH Keywords: Cartesian products
zbMATH Keywords: Nullstellensatz
zbMATH Keywords: model-theoretic classes
author: Alexei Myasnikov
author: Vladimir Remeslennikov
reviewed by: Vitaliĭ Roman'kov
MaRDI profile type: Publication
OpenAlex ID: W2068665471
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DOI: 10.1006/JABR.2000.8414
description: scientific article; zbMATH DE number 1565860