Syntactic Monoids in a Category

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. 1998 ACM Subject Classification F.4.3 Formal Languages Keywords and phrases Syntactic monoid, transition monoid, algebraic automata theory, dual...

• Type: Journal
• Year: 2015
• Where: CALCO
• Authors: Jirí Adámek, Stefan Milius, Henning Urbat
• Keywords: CALCO 2015, Mathematics