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

We investigate the intersection of two finitely generated submonoids of the free monoid on a finite alphabet A. 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 of A generated by prefix sets such that the product automaton associated to H∩K has a given special property then f rk(H ∩ K) ≤ f rk(H)f rk(K) where f rk(L) = max(0, rk(L) − 1) for any submonoid L of A.