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

Theorem
Let $\Sigma$ be an alphabet.

Let $\map \PP {\Sigma^*}$ be the $P$-star of $\Sigma$

Let $\circ_L$ denote the language product operation.

Then $\struct {\map \PP {\Sigma^*}, \circ_L}$ is a monoid.

Proof
A monoid is an algebraic structure $\struct {\map \PP {\Sigma^*}, \circ_L}$, such that:



(This follows directly from Product of Languages is Language)



(This follows directly from Language Product is Associative)



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