Does there exist a system such that the additive identity is non-zero?

Solution 1:

Consider the tropical semiring $(\mathbb R\cup\{\infty\},\oplus,\otimes)$, where $x\oplus y=\min\{x,y\}$ and $x\otimes y=x+y$. The "additive" identity, meaning the identity element for the $\oplus$ operation, is $\infty$. That's pretty far away from $0$!

The number $0$ is present, but it is not the additive identity in this structure. For example, $0\oplus 79 = 0$. Instead, $0$ is the "multiplicative" identity: $0\otimes y=y$.

Solution 2:

In the integers under the operation $*$ defined by $n * m = n + m - 1$, $1$ acts as the additive identity.

Of course, this is identical to the ordinary additive group structure on the integers except for relabeling. But there are occasional situations where people find themselves using this operation, usually when there are historical off-by-one errors to contend with. For example, $*$ is the operation used to add musical intervals together.

Solution 3:

A lightswitch has two values, off and on. Assemble two lightswitches in parallel. You now have the following addition table:

$$\begin{array}{c|cc} + & \textrm{off} & \textrm{on} \\ \hline\\ \textrm{off} & \textrm{off} & \textrm{on} \\ \textrm{on} & \textrm{on} & \textrm{on} \end{array}$$

Obviously, $\textrm{off} + \textrm{on} = \textrm{on}$, and $\textrm{off} + \textrm{off} = \textrm{off}$, so the system satisfies the desired properties.

Solution 4:

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.