0
$\begingroup$

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.

  • 2
    "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

1 Answers 1

7

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.

  • 5
    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.2012-02-26