Formations of monoids, congruences, and formal languages

Abstract

The main goal in this paper is to use a dual equivalence in automata theory started in [25] and developed in [3] to prove a general version of the Eilenberg-type theorem presented in [4]. Our principal results conrm the existence of a bijective correspondence between three concepts; formations of monoids, formations of languages and formations of congruences. The result does not require niteness on monoids, nor regularity on languages nor nite index conditions on congruences. We relate our work to other results in the eld and we include applications to non-r-disjunctive languages, Reiterman's equational description of pseudovarieties and varieties of monoids.