Properties of factorization forests
It has been proved that every morphism f: A+ → S, with S a finite semigroup, admits a Ramseyan factorization forest of height at most 9|S|. In this paper we show that, up to a constant factor, this result is best possible. More precisely, we show that if S is a finite rectangular band and f(A) = S then every Ramseyan factorization forest admitted by f has height at least |S|.
In the second part, using induction on the height of vertices of factorization forests we obtain a new proof of a Theorem of T. C. Brown on locally finite semigroups. Our proof is constructive.
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