Varieties of algebras

Abstract

This chapter discusses the varieties of (non-)associative algebras, groups, semi-groups, lattices, and other classes of algebras. Many classes of algebras are defined by identities. According to G. Birkhoff's theorem, varieties are exactly the non-empty classes of algebras of a given type which are closed under subalgebras, homomorphic images, and direct products. A class of algebras means a class of algebras of a fixed type T. A class K is abstract if it is closed under isomorphisms of algebras. The theory of varieties of (non-)associative algebras is one of the most developed. It has some special methods and interesting results. The theory of associative algebra with a non-trivial identity (PI-algebras) is one of the most advanced. One of the most important identities in associative algebras is the standard identity. A theory of varieties of Lie algebras was influenced by the Burnside problem on local finiteness of groups with an identity x n = 1. The development of the theory of varieties of groups is greatly influenced by the Burnside problem on periodic groups. A survey of results on non-finitely based varieties of semi-groups is provided.