A monoid S is said to be right coherent if every finitely generated subact of every finitely presented right S-act is finitely presented. Left coherency is defined dually and S is coherent if it is both right and left coherent. These notions are analogous to those for a ring R (where, of course, S-acts are replaced by R-modules). Choo et al. have shown that free rings are coherent. In this paper we prove that, correspondingly, any free monoid is coherent, thus answering a question posed by Gould in 1992.

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.