Profinite Monads, Profinite Equations, and Reiterman’s Theorem

Chen, Liang-Ting; Adámek, Jiřı́; Milius, Stefan; Urbat, Henning

doi:10.1007/978-3-662-49630-5_31

Cited by 13 publications

(19 citation statements)

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“…An immediate next step to unleash the full power of our new variety theorem is to establish a Reiterman-type theorem for lattice bimodules leading to a description of pseudovarieties of lattice bimodules in terms of profinite equations. The recent categorical account of (profinite) equational theories [7,16] should provide inspiration in this direction. This may lead to new results on the decidability of basic varieties of regular languages, e.g.…”

Section: Discussionmentioning

confidence: 99%

On Language Varieties Without Boolean Operations

Birkmann¹,

Milius²,

Urbat³

2020

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Eilenberg's variety theorem marked a milestone in the algebraic theory of regular languages by establishing a formal correspondence between properties of regular languages and properties of finite monoids recognizing them. Motivated by classes of languages accepted by quantum finite automata, we introduce basic varieties of regular languages, a weakening of Eilenberg's original concept that does not require closure under any boolean operations, and prove a variety theorem for them. To do so, we investigate the algebraic recognition of languages by lattice bimodules, generalizing Klíma and Polák's lattice algebras, and we utilize the duality between algebraic completely distributive lattices and posets.

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Section: Discussionmentioning

confidence: 99%

On Language Varieties Without Boolean Operations

Birkmann¹,

Milius²,

Urbat³

2020

Preprint

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“…This line of extensions is being continued by several authors, e.g. [18,13,12], and represents one of leading research trends in the context of algebraic language theory.…”

Section: Introductionmentioning

confidence: 90%

Semi-galois Categories I: The Classical Eilenberg Variety Theory

Uramoto

2015

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This paper is an extended version of our proceedings paper [39] announced at LICS'16; in order to complement it, this version is written from a different viewpoint including topos-theoretic aspects of [39] that were not discussed there. Technically, this paper introduces and studies the class of semi-galois categories, which extend galois categories and are dual to profinite monoids in the same way as galois categories are dual to profinite groups; the study on this class of categories is aimed at providing an axiomatic reformulation of Eilenberg's theory of varieties of regular languagesa branch in formal language theory that has been developed since the mid 1960s and particularly concerns systematic classification of regular languages, finite monoids and deterministic finite automata. In this paper, detailed proofs of our central results announced at LICS'16 are presented, together with topos-theoretic considerations. The main results include (I) a proof of the duality theorem between profinite monoids and semi-galois categories, extending the duality theorem between profinite groups and galois categories; based on this results on semi-galois categories we then discuss (II) a reinterpretation of Eilenberg's theory from a viewpoint of duality theorem; in relation with this reinterpretation of the theory, (III) we also give a purely topos-theoretic characterization of classifying topoi BM of profinite monoids M among general coherent topoi, which is a topos-theoretic application of (I). This characterization states that a topos E is equivalent to the classifying topos BM of some profinite monoid M if and only if E is (i) coherent, (ii) noetherian, and (iii) has a surjective coherent point p : Sets → E . This topos-theoretic consideration is related to the logical and geometric problems concerning Eilenberg's theory that we addressed at LICS'16, which remain open in this paper.

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“…Related work. This paper is the full version of an extended abstract [8] presented at FoSSaCS 2016. Besides providing complete proofs of all results, the presentation is significantly more general than in op.…”

Section: Stonementioning

confidence: 99%

Reiterman's Theorem on Finite Algebras for a Monad

Adámek¹,

Chen²,

Milius³

et al. 2021

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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 T associated to the monad T, which gives a categorical view of the construction of the space of profinite terms.

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Profinite Monads, Profinite Equations, and Reiterman’s Theorem

Cited by 13 publications

References 19 publications

On Language Varieties Without Boolean Operations

On Language Varieties Without Boolean Operations

Semi-galois Categories I: The Classical Eilenberg Variety Theory

Reiterman's Theorem on Finite Algebras for a Monad

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