A Myhill-Nerode Theorem beyond Trees and Forests via Finite Syntactic Categories Internal to Monoids

The paper introduces recognizable languages as inverse images of sets of arrows from finite categories internal to monoids. The first result is the Myhill-Nerode Theorem as a conservative extension of the classic result for tree languages. The second result shows that a language of planar acyclic circuit diagrams whose gates have non-empty lists of input and output ports is recognizable if, and only if, it is accepted by an automaton in the sense of Bossut. The proof of the Myhill-Nerode Theorem hinges on a suitable counterpart of the syntactic monoid, dubbed syntactic category, which is obtained by endowing the syntactic congruence of a recognizable language with the structure of a finite category internal to monoids. This opens up a venue for future research on Eilenberg type correspondences for pseudovarieties of finite categories internal to monoids.