Syntactic Monoids in a Category
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 Polak (D = semilattices), and the syntactic associative algebras of Reutenauer (D = vector spaces). Assuming that D is an entropic 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 case the variety D is locally finite, we characterize the regular languages as precisely the languages with finite syntactic D-monoids.
Keywords
Cite
@article{arxiv.1504.02694,
title = {Syntactic Monoids in a Category},
author = {Jiri Adamek and Stefan Milius and Henning Urbat},
journal= {arXiv preprint arXiv:1504.02694},
year = {2015}
}