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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.
