I am surprised no one mentioned elliptic curves with its "usual" elliptic-curve-addition. You pick two points $a$ and $b$ on an elliptic curve. Then $a+b$ is defined as the negative of the third point of intersection between the elliptic curve and the straight line connecting $a$ and $b$. In this case, the additive identity is denoted $O$ which is the point at infinity. And this example is not just pathological. Elliptic curves are used widely with the most-real-world-example being cryptography.
↧