Finite basis problem for varieties of algebraic systems

TL;DR

This survey explores the finite basis problem in algebraic systems, providing examples of both finitely based and non-finitely based varieties, with focus on semigroups, groups, and nonassociative algebras.

Contribution

It offers a comprehensive overview of known results and examples regarding the finite basis property across various algebraic varieties.

Findings

Numerous examples of non-finitely based varieties.

Examples of important finitely based varieties and their subvarieties.

Focus on varieties of semigroups, groups, and nonassociative algebras.

Abstract

This is a survey on the finite basis problem for varieties of algebraic systems. Our exposition is in two directions: (i) We give numerous examples of varieties which are not finitely based. (ii) We give examples of important varieties with the property that they and their subvarieties are finitely based. A special attention is paid on the varieties of semigroups, groups, associative, Lie and other nonassociative algebras.