User:Jshflynn/Kleene Star is Monoid

Theorem

Let $\Sigma$ be an alphabet.

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

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

Proof

Follows immediately from Kleene Star is Linguistic Structure, Concatenation is Associative and Empty Word is Two-sided Identity.

$\blacksquare$