Abstract Algebra - Monoids I'm trying to find the identity of a monoid but all the examples I can find are not as complex as this one.

(source: gyazo.com) 

It's straightforward to show that $\otimes$ is an associative binary operation, and as others have pointed out, the identity of the monoid is $(1, 0, 1)$. However, $(\mathbb{R}^3, \otimes)$ is not a group, since for example $(0, 0, 0)$ has no inverse element.
