Primitivity of finitely presented monomial algebras

We study prime monomial algebras. Our main result is that a prime finitely presented monomial algebra is either primitive or it has GK dimension one and satisfies a polynomial identity. More generally, we show this result holds for the class of \emph{automaton algebras}; that is, monomial algebras that have a basis consisting of the set of words recognized by some finite state automaton. This proves a special case of a conjecture of the first author and Agata Smoktunowicz.