Syntactic Monoids in a Category

Authors Jiri Adamek, Stefan Milius, Henning Urbat

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Jiri Adamek
Stefan Milius
Henning Urbat

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Jiri Adamek, Stefan Milius, and Henning Urbat. Syntactic Monoids in a Category. In 6th Conference on Algebra and Coalgebra in Computer Science (CALCO 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 35, pp. 1-16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)


The syntactic monoid of a language is generalized to the level of a symmetric monoidal closed category D. This allows for a uniform treatment of several notions of syntactic algebras known in the literature, including the syntactic monoids of Rabin and Scott (D = sets), the syntactic semirings of Polák (D = semilattices), and the syntactic associative algebras of Reutenauer (D = vector spaces). Assuming that D is a commutative variety of algebras, we prove that the syntactic D-monoid of a language L can be constructed as a quotient of a free D-monoid modulo the syntactic congruence of L, and that it is isomorphic to the transition D-monoid of the minimal automaton for L in D. Furthermore, in the case where the variety D is locally finite, we characterize the regular languages as precisely the languages with finite syntactic D-monoids.
  • Syntactic monoid
  • transition monoid
  • algebraic automata theory
  • duality
  • coalgebra
  • algebra
  • symmetric monoidal closed category
  • commutative variety


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