License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.CALCO.2015.1
URN: urn:nbn:de:0030-drops-55235
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2015/5523/
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Adamek, Jiri ; Milius, Stefan ; Urbat, Henning

Syntactic Monoids in a Category

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Abstract

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.

BibTeX - Entry

@InProceedings{adamek_et_al:LIPIcs:2015:5523,
  author =	{Jiri Adamek and Stefan Milius and Henning Urbat},
  title =	{{Syntactic Monoids in a Category}},
  booktitle =	{6th Conference on Algebra and Coalgebra in Computer Science (CALCO 2015)},
  pages =	{1--16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-84-2},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{35},
  editor =	{Lawrence S. Moss and Pawel Sobocinski},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2015/5523},
  URN =		{urn:nbn:de:0030-drops-55235},
  doi =		{10.4230/LIPIcs.CALCO.2015.1},
  annote =	{Keywords: Syntactic monoid, transition monoid, algebraic automata theory, duality, coalgebra, algebra, symmetric monoidal closed category, commutative variety}
}

Keywords: Syntactic monoid, transition monoid, algebraic automata theory, duality, coalgebra, algebra, symmetric monoidal closed category, commutative variety
Collection: 6th Conference on Algebra and Coalgebra in Computer Science (CALCO 2015)
Issue Date: 2015
Date of publication: 28.10.2015

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