Varieties of algebras and algebraic varieties. Categories of algebraic varieties
type: Item
instance of: scholarly article
title: Varieties of algebras and algebraic varieties. Categories of algebraic varieties
zbMATH Open document ID: 0936.08007
published in: Siberian Advances in Mathematics
publication date: 2000-04-13T00:00:00Z
review text: The aim of the article is to study equations and their solutions in arbitrary algebras: not only in fields, as is done in classical algebraic geometry, but also in rings, groups, etc. For an algebra \(G\), the set of all solutions in \(G\) to a given system of equations is an algebraic variety. For a variety \(\Theta\) of algebras and an algebra \(G\) in \(\Theta\), consider the category \(K_{\Theta}(G)\) of algebraic varieties corresponding to these \(\Theta\) and \(G\); cosets of equivalent homomorphisms between free algebras over corresponding sets of variables are morphisms. The relations (geometrical equivalences) are given between algebras \(G_{1}\) and \(G_{2}\) which lead to isomorphy of the categories \(K_{\Theta}(G_{1})\) and \(K_{\Theta}(G_{2})\).
Mathematics Subject Classification ID: 08C05
Mathematics Subject Classification ID: 08B05
zbMATH DE Number: 1431911
zbMATH Keywords: algebraic variety
zbMATH Keywords: variety of algebras
zbMATH Keywords: category
zbMATH Keywords: geometrical equivalence of algebras
author: B. Plotkin
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1431911