Varieties of Languages in a Category

Eilenberg's variety theorem, a centerpiece of algebraic automata theory, establishes a bijective correspondence between varieties of languages and pseudovarieties of monoids. In the present paper this result is generalized to an abstract pair of algebraic categories: we introduce varieties of l...

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Bibliographic Details
Published in:2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science pp. 414 - 425
Main Authors: Adamek, Jiri, Myers, Robert S. R., Urbat, Henning, Milius, Stefan
Format: Conference Proceeding
Language:English
Published: IEEE 01.07.2015
Subjects:
algebra
Automata
Boolean algebra
coalgebra
duality
Eilenberg's theorem
Generators
Lattices
monoids
Structural rings
Tensile stress
varieties of languages
ISSN:1043-6871
Online Access:Get full text
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Summary:Eilenberg's variety theorem, a centerpiece of algebraic automata theory, establishes a bijective correspondence between varieties of languages and pseudovarieties of monoids. In the present paper this result is generalized to an abstract pair of algebraic categories: we introduce varieties of languages in a category C, and prove that they correspond to pseudovarieties of monoids in a closed monoidal category D, provided that C and D are dual on the level of finite objects. By suitable choices of these categories our result uniformly covers Eilenberg's theorem and three variants due to Pin, Polák and Reutenauer, respectively, and yields new Eilenberg-type correspondences.
ISSN:1043-6871
DOI:10.1109/LICS.2015.46