An Eilenberg Theorem for Arbitrary Languages

In algebraic language theory one investigates formal languages by relating them to finite algebras. The most important result along these lines is Eilenberg’s celebrated variety theorem [7]: varieties of languages (classes of regular finite-word languages closed under boolean operations, derivatives and preimages of monoid morphisms) correspond bijectively to pseudovarieties of monoids (classes of finite monoids closed under quotients, submonoids and finite products). Subsequently Reiterman [11] proved an equational characterization of pseudovarieties of monoids in the spirit of Birkhoff’s HSP theorem: they are precisely the classes of finite monoids presentable by profinite equations, and thus correspond to profinite equational theories. In [6, 12] we proved a strong generalization of both of these results: first, extending an idea of Bojánczyk [5], we replaced monoids by algebras for a monad T on an algebraic category D , and finite-word languages L ⊆ X∗ by T-languages L : TX → OD in D , where OD is some “object of outputs” in D . Second, we proved a Reiterman theorem for pseudovarieties of T-algebras, and derived from it an Eilenberg theorem for varieties of (finitely) recognizable T-languages, arising as a combination of our Reiterman theorem with a Stone-type duality: