Varieties of algebras and algebraic varieties. Categories of algebraic varieties
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 \(K_{\Theta}(G_{2})\).
scientific article; zbMATH DE number 1431911