eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2015-10-28
1
16
10.4230/LIPIcs.CALCO.2015.1
article
Syntactic Monoids in a Category
Adamek, Jiri
Milius, Stefan
Urbat, Henning
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol035-calco2015/LIPIcs.CALCO.2015.1/LIPIcs.CALCO.2015.1.pdf
Syntactic monoid
transition monoid
algebraic automata theory
duality
coalgebra
algebra
symmetric monoidal closed category
commutative variety