A variety of algebras of a fixed type (e.g. groups, rings, semigroups) that contains only finitely based subvarieties is called a Specht variety. Maximal Specht varieties of monoids exist, but maximal Specht varieties of semigroups do not.

A variety of algebras of a fixed type (e.g. groups, rings, Lie algebras, semigroups) that contains only finitely based subvarieties is called a Specht variety. It is known, as early as the 1980s, that maximal Specht varieties of semigroups do not exist. But some maximal Specht varieties of monoids were recently discovered; this is quite surprising given how close semigroups and monoids are.