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An automata-theoretic approach to the study of the intersection of two submonoids of a free monoid

Published online by Cambridge University Press:  03 June 2008

Laura Giambruno
Affiliation:
Dipartimento di Matematica e Applicazioni, Università di Palermo, via Archirafi 34, 90123 Palermo, Italy; lgiambr;restivo@math.unipa.it
Antonio Restivo
Affiliation:
Dipartimento di Matematica e Applicazioni, Università di Palermo, via Archirafi 34, 90123 Palermo, Italy; lgiambr;restivo@math.unipa.it
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Abstract

We investigate the intersection of two finitely generated submonoids of the free monoid on a finite alphabet. To this purpose, we consider automata that recognize such submonoids and we study the product automata recognizing their intersection. By using automata methods we obtain a new proof of a result of Karhumäki on the characterization of the intersection of two submonoids of rank two, in the case of prefix (or suffix) generators. In a more general setting, for an arbitrary number of generators, we prove that if H and K are two finitely generated submonoids generated by prefix sets such that the product automaton associated to $H \cap K$ has a given special property then $\widetilde{rk}(H \cap K) \leq \widetilde{rk}(H) \widetilde{rk}(K)$ where $\widetilde{rk}(L)=\max(0,rk(L)-1)$ for any submonoid L.

Type
Research Article
Copyright
© EDP Sciences, 2008

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