Schützenberger Products in a Category

The Schützenberger product of monoids is a key tool for the algebraic treatment of language concatenation. In this paper we generalize the Schützenberger product to the level of monoids in an algebraic category $\mathscr{D}$, leading to a uniform view of the corresponding constructions for monoids (Schützenberger), ordered monoids (Pin), idempotent semirings (Klíma and Polák) and algebras over a field (Reutenauer). In addition, assuming that $\mathscr{D}$ is part of a Stone-type duality, we derive a characterization of the languages recognized by Schützenberger products.

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