Growth sequences of finite semigroups

The growth sequence of a finite semigroup S is the sequence { d(S n ) }, where S n is the n th direct power of S and d stands for minimum generating number. When S has an identity, d(S n ) = d(T n ) + kn for all n , where T is the group of units and k is the minimum number of generators of S mod T . Thus d(S n ) is essentially known since d(T n ) is (see reference 4), and indeed d(S n ) is then eventually piecewise linear. On the other hand, if S has no identity, there exists a real number c > 1 such that d(S n ) ≥ c n for all n ≥ 2.