Abstract
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.
This work was done while the author was visiting the Fachbereich Informatik of the Johann Wolfgang Goethe-Universität in Frankfurt am Main with partial support from FAPESP.
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References
T. C. Brown. An interesting combinatorial method in the theory of locally finite semigroups. Pacific J. Math., 36:285–289, 1971.
I. Simon. Factorization Forests of Finite Height. Technical Report 87-73, Laboratoire d'Informatique Théorique et Programmation, Paris, 1987.
H. Straubing. The Burnside problem for semigroups of matrices. In L. J. Cummings, editor, Combinatorics on Words, Progress and Perspectives, pages 279–295, Academic Press, New York, NY, 1983.
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© 1989 Springer-Verlag Berlin Heidelberg
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Simon, I. (1989). Properties of factorization forests. In: Pin, J.E. (eds) Formal Properties of Finite Automata and Applications. LITP 1988. Lecture Notes in Computer Science, vol 386. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0013112
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DOI: https://doi.org/10.1007/BFb0013112
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