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An automata-theoretic approach to the study of the intersectionof two submonoids of a free monoid
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- Journal:
- RAIRO - Theoretical Informatics and Applications / Volume 42 / Issue 3 / July 2008
- Published online by Cambridge University Press:
- 03 June 2008, pp. 503-524
- Print publication:
- July 2008
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- Article
- Export citation
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We investigate the intersection of two finitely generated submonoidsof the free monoid on a finite alphabet. To this purpose, weconsider automata that recognize such submonoids and we study theproduct automata recognizing their intersection. By using automatamethods we obtain a new proof of a result of Karhumäki on thecharacterization of the intersection of two submonoids ofrank two, in the case of prefix (or suffix) generators. In a moregeneral setting, for an arbitrary number of generators, we provethat if H and K are two finitely generated submonoids generatedby prefix sets such that the product automaton associated to $H \capK$
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.
