5 research outputs found
A Categorical Approach to Syntactic Monoids
The syntactic monoid of a language is generalized to the level of a symmetric
monoidal closed category . This allows for a uniform treatment of
several notions of syntactic algebras known in the literature, including the
syntactic monoids of Rabin and Scott ( sets), the syntactic
ordered monoids of Pin ( posets), the syntactic semirings of
Pol\'ak ( semilattices), and the syntactic associative algebras of
Reutenauer ( = vector spaces). Assuming that is a
commutative variety of algebras or ordered algebras, we prove that the
syntactic -monoid of a language can be constructed as a
quotient of a free -monoid modulo the syntactic congruence of ,
and that it is isomorphic to the transition -monoid of the minimal
automaton for in . Furthermore, in the case where the variety
is locally finite, we characterize the regular languages as
precisely the languages with finite syntactic -monoids.Comment: arXiv admin note: substantial text overlap with arXiv:1504.0269