Syntactic Monoids in a Category

TL;DR

This paper generalizes the concept of syntactic monoids to symmetric monoidal closed categories, unifying various algebraic notions and providing a framework for analyzing regular languages through categorical algebra.

Contribution

It introduces a categorical framework for syntactic monoids in symmetric monoidal closed categories, extending classical notions and characterizing regular languages via finite syntactic D-monoids.

Findings

Syntactic D-monoids can be constructed as quotients of free D-monoids.

The syntactic D-monoid is isomorphic to the transition D-monoid of the minimal automaton.

Regular languages are characterized by having finite syntactic D-monoids when D is locally finite.

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.