Reiterman's Theorem on Finite Algebras for a Monad

TL;DR

This paper generalizes Reiterman's theorem to finite algebras for a monad, linking pseudovarieties with profinite equations using a new categorical construction called a profinite monad.

Contribution

It extends Reiterman's theorem to finite Eilenberg-Moore algebras for monads, introducing the concept of a profinite monad for a categorical perspective.

Findings

Proves the equivalence between pseudovarieties and profinite equations for T-algebras.

Introduces the concept of a profinite monad as a key technical tool.

Provides a categorical framework for the algebraic classification of formal languages.

Abstract

Profinite equations are an indispensable tool for the algebraic classification of formal languages. Reiterman's theorem states that they precisely specify pseudovarieties, i.e.~classes of finite algebras closed under finite products, subalgebras and quotients. In this paper, Reiterman's theorem is generalized to finite Eilenberg-Moore algebras for a monad T on a category D: we prove that a class of finite T-algebras is a pseudovariety iff it is presentable by profinite equations. As a key technical tool, we introduce the concept of a profinite monad associated to the monad T, which gives a categorical view of the construction of the space of profinite terms.