Generalized Eilenberg Theorem I: Local Varieties of Languages

TL;DR

This paper generalizes Eilenberg's theorem by exploring the duality between algebraic and coalgebraic recognition of languages, establishing new correspondences across various algebraic structures.

Contribution

It introduces a unified approach to generalize the local Eilenberg theorem for different algebraic categories using duality theory.

Findings

Lattice of boolean algebras of regular languages is isomorphic to pseudovarieties of monoids.

Generalizations to distributive lattices, join-semilattices, and vector spaces over binary field.

New categorical frameworks unify recognition of languages across diverse algebraic structures.

Abstract

We investigate the duality between algebraic and coalgebraic recognition of languages to derive a generalization of the local version of Eilenberg's theorem. This theorem states that the lattice of all boolean algebras of regular languages over an alphabet {\Sigma} closed under derivatives is isomorphic to the lattice of all pseudovarieties of {\Sigma}-generated monoids. By applying our method to different categories, we obtain three related results: one, due to Gehrke, Grigorieff and Pin, weakens boolean algebras to distributive lattices, one weakens them to join-semilattices, and the last one considers vector spaces over the binary field.