On factorization forests of finite height

Abstract

Simon (Theoret. Comput. Sci. 72 (1990) 65–94) has proved that every morphism from a free semigroup to a finite semigroup S admits a Ramseyan factorization forest of height at most 9|S|. In this paper, we prove the same result of Simon with an improved bound of 7|S|. We provide a simple algorithm for constructing a factorization forest. In addition, we show that the algorithm cannot be improved significantly. We give examples of semigroup morphism such that any Ramseyan factorization forest for the morphism would require a height not less than |S|.