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arXiv · 1504.02694

Syntactic Monoids in a Category

Jiri Adamek·Stefan Milius·Henning Urbat
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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.

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Jiri Adamek, Stefan Milius, Henning Urbat. 2015-04-10. Syntactic Monoids in a Category. https://arxiv.org/abs/1504.02694

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