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Generalized Eilenberg Theorem: Varieties of Languages in a Category

Published: 20 December 2018 Publication History

Abstract

For finite automata as coalgebras in a category C, we study languages they accept and varieties of such languages. This generalizes Eilenberg’s concept of a variety of languages, which corresponds to choosing as C the category of Boolean algebras. Eilenberg established a bijective correspondence between pseudovarieties of monoids and varieties of regular languages. In our generalization, we work with a pair C/D of locally finite varieties of algebras that are predual, i.e., dualize on the level of finite algebras, and we prove that pseudovarieties of D-monoids bijectively correspond to varieties of regular languages in C. As one instance, Eilenberg’s result is recovered by choosing D = sets and C = Boolean algebras. Another instance, Pin’s result on pseudovarieties of ordered monoids, is covered by taking D = posets and C = distributive lattices. By choosing as C amp;equals; D the self-predual category of join-semilattices, we obtain Polák’s result on pseudovarieties of idempotent semirings. Similarly, using the self-preduality of vector spaces over a finite field K, our result covers that of Reutenauer on pseudovarieties of K-algebras. Several new variants of Eilenberg’s theorem arise by taking other predualities, e.g., between the categories of non-unital Boolean rings and of pointed sets. In each of these cases, we also prove a local variant of the bijection, where a fixed alphabet is assumed and one considers local varieties of regular languages over that alphabet in the category C.

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Cited By

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  • (2023)Eilenberg's variety theorem without Boolean operationsInformation and Computation10.1016/j.ic.2022.104916295(104916)Online publication date: Dec-2023
  • (2023)Minimisation in Logical FormSamson Abramsky on Logic and Structure in Computer Science and Beyond10.1007/978-3-031-24117-8_3(89-127)Online publication date: 2-Aug-2023
  • (2021)On Language Varieties Without Boolean OperationsLanguage and Automata Theory and Applications10.1007/978-3-030-68195-1_1(3-15)Online publication date: 1-Mar-2021

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    cover image ACM Transactions on Computational Logic
    ACM Transactions on Computational Logic  Volume 20, Issue 1
    January 2019
    234 pages
    ISSN:1529-3785
    EISSN:1557-945X
    DOI:10.1145/3301291
    • Editor:
    • Orna Kupferman
    Issue’s Table of Contents
    Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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    Publication History

    Published: 20 December 2018
    Accepted: 01 September 2018
    Revised: 01 August 2018
    Received: 01 June 2017
    Published in TOCL Volume 20, Issue 1

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    Author Tags

    1. Algebraic automata theory
    2. Eilenberg’s theorem
    3. bimonoids
    4. category theory
    5. coalgebra

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    Cited By

    View all
    • (2023)Eilenberg's variety theorem without Boolean operationsInformation and Computation10.1016/j.ic.2022.104916295(104916)Online publication date: Dec-2023
    • (2023)Minimisation in Logical FormSamson Abramsky on Logic and Structure in Computer Science and Beyond10.1007/978-3-031-24117-8_3(89-127)Online publication date: 2-Aug-2023
    • (2021)On Language Varieties Without Boolean OperationsLanguage and Automata Theory and Applications10.1007/978-3-030-68195-1_1(3-15)Online publication date: 1-Mar-2021

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