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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...2012-02-26

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