# Does every binary operation have an identity element?

Does every binary operation have an identity element? If not, then what kinds of operations do and do not have these identities?

For example, $1$ is a multiplicative identity for integers, real numbers, and complex numbers.

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"Most" binary functions do not have an identity. A couple of simple explicit examples are the $\min$ and $\max$ functions on the integers or the real numbers. However, $\max$ has an identity on the natural numbers... – Rahul Feb 26 '12 at 18:15

The answer to your question is no. For example, let the binary operation \$on the real numbers as be defined as$x\$y=|x|+|y|+1$. Then there is no left or right identity element for \$(because we always have$|x\$y|>|x|$ and $|x\$y|>|y|$). You can also have binary operations that have a left identity but not a right identity, or vice versa. For example, let$x\%y=|x|+y-7$. Then$7\%y=y$for any$y\$, but there is no number that plays the same role as 7 on the right-hand side.