An operation * has 2 possible identity element:which one's the real identity element?

The rule to find identity element is laid out on Pinter's Abstract Algebra book like this on page 24:

"First solve the equation for ; if the equation cannot be solved there is no identity element. If it can be solved, it is

still necessary to check that for any . If it checks, then is an identity element."

Suppose there is an operation on such that and we need to find the identity element.

So according to the above rule and by the definition of the operation :

so or

or

Now

But

So can I say that is an identity element for the operation because it fulfills the property of identity element?

Can anyone kindly tell me if I'm right or wrong to say that is an identity element for this operation?

Re: An operation * has 2 possible identity element:which one's the real identity elem

Sorry I got the solution.

You see, if and then

So no is not an identity element for the operation on the set in this case.

Re: An operation * has 2 possible identity element:which one's the real identity elem

Neither can "2x". An "identity element" for an algebraic stucture, set X with operation *, is member of X such that e+ x= x for **any** element of X. That is, an identity is a single member of x. There is not a different identity element for each element of X. That operation does not have **two** identities- it has **none**.

Re: An operation * has 2 possible identity element:which one's the real identity elem

Thanks HallsofIvy for clarifying the rule for being a identity element.