  In this paper, we look at an unambiguous version of Simon's forest
factorization theorem, a very deep result which has wide connections in
algebra, logic and automata. Given a morphism $\varphi$ from $\Sigma^+$ to a
finite semigroup $S$, we construct a universal, unambiguous automaton A which
is "good" for $\varphi$. The goodness of $\Aa$ gives a very easy proof for the
forest factorization theorem, providing a Ramsey split for any word in
$\Sigma^{\infty}$ such that the height of the Ramsey split is bounded by the
number of states of A. An important application of synthesizing good automata
from the morphim $\varphi$ is in the construction of regular transducer
expressions (RTE) corresponding to deterministic two way transducers.
