I have read that an operation can have at most one identity and fully understand the proof, however what if I define an operation $*$ on $\mathbb Q$ as follows?
$x*y=|x \times y|$, $\forall x,y \in \mathbb Q$
Surely both $1$ and $-1$ are identities? Or am I missing something obvious?
Note that neither $1$ nor $-1$ is an identity. We have that:
$$(-1)*(-1) = \;\mid 1 \mid\; = 1 \quad \quad (-1)*(1) = \;\mid -1 \mid\; = 1$$