The paper introduces "synchronous algebras", an algebraic structure designed to recognize automatic (synchronous) relations. Key insights:
Synchronous algebras are typed and equipped with a dependency relation, which captures constraints between elements of different types. This is a novel feature compared to traditional algebraic structures like monoids.
The three pillars of algebraic language theory hold for synchronous algebras: (a) each relation has a unique canonical and minimal synchronous algebra recognizing it; (b) classes of synchronous relations with desirable closure properties (pseudovarieties) correspond to pseudovarieties of synchronous algebras; and (c) pseudovarieties of synchronous algebras are exactly the classes defined by profinite dependencies.
The paper shows how algebraic characterizations of classes of regular languages (pseudovarieties) can be "lifted" to characterize the corresponding classes of synchronous relations (V-relations). This is done both for ∗-pseudovarieties (corresponding to monoids) and +-pseudovarieties (corresponding to semigroups).
Key examples discussed include group relations and nilpotent relations, demonstrating the applicability of the synchronous algebra framework.
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