For every semigroup of finite exponent whose chains of idempotents are
uniformly bounded we construct an identity which holds on this semigroup but
does not hold on the variety of all idempotent semigroups. This shows that the
variety of all idempotent semigroups E is not contained in any finitely generated
variety of semigroups. Since E is locally finite and each proper subvariety of E is
finitely generated [1, 3, 4], the variety of all idempotent semigroups is a minimal
example of an inherently non-finitely generated variety.