# Closure of Recognizable Languages under Kleene Star: Algebraic Proof?

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?

The reason is that recognizable languages in general are not closed under Kleene star. Or more specifically, in general we have rational sets that are not recognizable. See this thread for lots of examples. One particular simple example for Kleene star would be $$(1,1)^* \subseteq \mathbb N^2$$. This set is not recognizable in $$\mathbb N^2$$.