TITLE: Two identity element?
QUESTION [5 upvotes]: On $\Bbb N=\{0,1,2,...\}$ we define the operation $\otimes$ by $m\otimes n= |m-n|$. Are there any identity element?
I came up with two identity elements. $e= 0$ and $e=2m$ for an element $m$ in the set, but how is that possible? I thought on a operation $\otimes$ that it would always be one identity element?

REPLY [3 votes]: The definition of the identity element is $$\exists e, \forall x, e*x=x*e=x$$
and not
$$\forall x, \exists e, e*x=x*e=x$$