Generative complexity in semigroup varieties

We analyze the problem of generative complexity for varieties of semigroups. We focus our attention on varieties generated by a finite semigroup. A variety is said to have polynomial generative complexity if and only, if the number of k-generated semigroups it contains is bounded from above by a polynomial in k. We fully characterize the class of finite semigroups which generate varieties with polynomial complexity. It turns out that only semigroups of very special shape have this property. These are semigroups with zero multiplication or semigroups which are the product of a left-zero semigroup, a right-zero semigroup and an Abelian group. Moreover a characterization of finite semigroups generating varieties with linear complexity is given.