Here is another part of the question:

For part(c), this is what I've done so far:

suppose both e and é are identities of $\displaystyle \mathcal{P}(M)$ and suppose $\displaystyle x \in \mathcal{P}(M)$. Therefore

xe=x .....(1)

xé=x .....(2)

Since $\displaystyle e,é \in \mathcal{P}(M)$, we can set x=é in equation (1) and set x=e in equation (2)

ée=é

eé=e

Thus e and é are equal and the identity is uniqe.

Is this correct? I think the problem here is that I've shown that the identity of $\displaystyle (\mathcal{P}(M), \star)$ is unique but I have not shown that it must have an identity. Can anyone help?

For part(d), I did this:

Let $\displaystyle a,b,c \in \mathcal{P}(M_n)$.

Suppose b and c are both inverses of a. Then $\displaystyle ab=e$ also $\displaystyle ac=e$. Hence $\displaystyle ab=bc \iff b=c$.

Again, I think my approach just shows the uniqueness, and not the existence of an inverse... I appreciate any help with this.