Eilenberg Theorems for Free

TL;DR

This paper unifies and extends Eilenberg-type correspondences in algebraic language theory using monads and duality, covering finite and infinite languages and introducing new results.

Contribution

It provides a general variety theorem framework that unifies previous categorical approaches and extends local variety theorems to infinite words.

Findings

Unified Eilenberg-type correspondences via monads and duality.

Extended local variety theorem to infinite words.

Derived new results in algebraic language theory.

Abstract

Eilenberg-type correspondences, relating varieties of languages (e.g. of finite words, infinite words, or trees) to pseudovarieties of finite algebras, form the backbone of algebraic language theory. Numerous such correspondences are known in the literature. We demonstrate that they all arise from the same recipe: one models languages and the algebras recognizing them by monads on an algebraic category, and applies a Stone-type duality. Our main contribution is a variety theorem that covers e.g. Wilke's and Pin's work on $\infty$-languages, the variety theorem for cost functions of Daviaud, Kuperberg, and Pin, and unifies the two previous categorical approaches of Boja\'nczyk and of Ad\'amek et al. In addition we derive a number of new results, including an extension of the local variety theorem of Gehrke, Grigorieff, and Pin from finite to infinite words.