Sch\"utzenberger Products in a Category

TL;DR

This paper generalizes the Sch"utzenberger product to monoids within various algebraic categories, providing a unified framework and characterizing recognized languages under duality assumptions.

Contribution

It introduces a categorical generalization of the Sch"utzenberger product, unifying multiple algebraic structures and their language recognition properties.

Findings

Unified categorical framework for Sch"utzenberger products

Characterization of languages recognized by these products

Application to various algebraic structures like monoids, semirings, and algebras

Abstract

The Sch\"utzenberger product of monoids is a key tool for the algebraic treatment of language concatenation. In this paper we generalize the Sch\"utzenberger product to the level of monoids in an algebraic category $\mathscr{D}$, leading to a uniform view of the corresponding constructions for monoids (Sch\"utzenberger), ordered monoids (Pin), idempotent semirings (Kl\'ima and Pol\'ak) 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 Sch\"utzenberger products.