Does an identity exist for ALL functions?

Does an identity exist for all the functions? If not then what kinds of functions 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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The question is not clear. What do you mean by identity? –  Raskolnikov Feb 26 '12 at 17:28
@Raskolnikov I think he means functional equation. –  Pedro Tamaroff Feb 26 '12 at 17:29
@Peter I would assume so to. But the equation $f(x) = f(x)$ is a trivial functional equation satisfied by any function, and so the answer to the stated question would be trivially "yes". Perhaps the OP can enlighten us on what he/she actually has in mind. –  Willie Wong Feb 26 '12 at 17:34
I suspect it may be something like: for a function $f:S^2 \to S$ there is a value $i \in S$ such that for all $x \in S$ you have $f(x,i)=x$. Perhaps you also need $f(i,x)=x$. –  Henry Feb 26 '12 at 18:06
"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.