Formations of monoids, congruences, and formal languages

The main goal in this paper is to use a dual equivalence in automata theory started in [RBBCL13] and developed in [BBCLR14] to prove a general version of the Eilenberg-type theorem presented in [BBPSE12]. Our principal results confirm the existence of a bijective correspondence between formations of (non-necessarily finite) monoids, that is, classes of monoids closed under taking epimorphic images and finite subdirect products, with formations of languages, which are classes of (non-necessarily regular) formal languages closed under coequational properties. Applications to non-r-disjunctive languages are given.