The Height of Factorization Forests

We show that for every homomorphism from A to a finite semigroup S there exists a factorization forest of height at most 3 |S|−1. Furthermore, we show that for every non-trivial group, this bound is tight. For aperiodic semigroups, we give an improved upper bound of 2 |S| and we show that for every n ≥ 2 there exists an aperiodic semigroup S with n elements which reaches this bound.