User:Jshflynn/P-star is Monoid under Language Product

Theorem
Let $\Sigma$ be an alphabet.

Let $\mathcal{P}(\Sigma^{*})$ be the $P$-star of $\Sigma$ and $\circ_L$ denote the language product operation.

Then $(\mathcal{P}(\Sigma^{*}), \circ_L)$ is a monoid.

Proof
A monoid is an algebraic structure $(\mathcal{P}(\Sigma^{*}), \circ_L)$, such that:

(1) $(\mathcal{P}(\Sigma^{*})$ is closed under $\circ_L$.

(This follows directly from Product of Languages is Language)

(2) $\circ_L$ is associative on $\mathcal{P}(\Sigma^{*})$.

(This follows directly from Language Product is Associative)

(3) $\mathcal{P}(\Sigma^{*})$ has an identity under $\circ_L$.

(This follows directly from Null Language is Identity of Language Product)