Affiliation:
1. Dept. Mathematics, Faculty of Electrical Engineering, Czech Technical University Prague, Prague, Czech Republic
2. Friedrich-Alexander-Universität Erlangen-Nürnberg, Erlangen, Germany
3. Technische Universität Braunschweig, Braunschweig, Germany
4. Universität Erlangen-Nürnberg, Erlangen, Germany
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
.
Funder
Deutsche Forschungsgemeinschaft
Publisher
Association for Computing Machinery (ACM)
Subject
Computational Mathematics,Logic,General Computer Science,Theoretical Computer Science
Cited by
3 articles.
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1. Minimisation in Logical Form;Samson Abramsky on Logic and Structure in Computer Science and Beyond;2023
2. Eilenberg's variety theorem without Boolean operations;Information and Computation;2022-05
3. On Language Varieties Without Boolean Operations;Language and Automata Theory and Applications;2021