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Let $Rec(\Sigma)$ be the class of languages over $\Sigma^*$ recognizable by finite monoids.

To show that $Rec(\Sigma)$ is closed under Kleene star, one would usually refer to the equivalence of recognizable and regular languages and invoke Kleene's theorem. Can anyone think of a direct algebraic proof of this closure property on the level of monoids?

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For a pointer on concatenation-like operations (including Kleene star) and their impact on the algebraic structure, you can look to Schützenberger products (see here [1] for instance).

[1] Pin, Jean-Éric. "Theme and variations on the concatenation product." International Conference on Algebraic Informatics. Springer, Berlin, Heidelberg, 2011, https://www.irif.fr/~jep/PDF/ThemeVariations.pdf

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    $\begingroup$ Thanks for the reference, but I am afraid this paper does not answer the question, except in a very special case. $\endgroup$ – J.-E. Pin Jun 9 '18 at 18:29

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