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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... – Rahul Feb 26 '12 at 18:15

Based on the clarifying edit, I think what you're really trying to ask here is whether all binary operations have an identity element. ("Function" is a broad term that includes functions that only take one input. "Functional identity" is not, as far as I know, a standard term in the sense in which you're using it in the comment.)

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.

However, we are often interested in groups and rings, for which the existence of an identity element is one of the axioms.

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Given a function, can we say if there exists an identity? – kunjan kshetri Feb 26 '12 at 19:47
You aren't listening, Kunjan. Functions don't have identities, any more than they have colors or bank accounts. It's binary operations that may, or may not, have identites. Ben has shown you that some binary operations have an identity element, and some don't. – Gerry Myerson Feb 26 '12 at 23:30

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