User:Jshflynn/Kleene Star is Linguistic Structure

Theorem

Let $\Sigma$ be an alphabet.

Let $\Sigma^{*}$ be the Kleene star of $\Sigma$ and $\circ$ denote concatenation.

Then $(\Sigma^{*}, \circ)$ is a linguistic structure.

Proof

As $\Sigma^{*} \subseteq \Sigma^{*}$ we have that $\Sigma^{*}$ is a formal language over $\Sigma$.

From Kleene Plus is Linguistic Structure we have that $\circ$ is closed on $\Sigma^{*}-\{\lambda \}$.

Including $\lambda$ in the underlying set we have from Empty Word is Two-sided Identity that $\Sigma^{*}$ is closed under $\circ$ and hence $(\Sigma^{*}, \circ)$ is a linguistic structure.

$\blacksquare$