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

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Aug 24th 2012, 02:32 PM
x3bnm

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 $x * e = x$ for $e$ ; if the equation cannot be solved there is no identity element. If it can be solved, it is
still necessary to check that $e * x = x * e = x$ for any $x \in \mathbb{R}$ . If it checks, then $e$ is an identity element."

Suppose there is an operation $*$ on $\mathbb{R}$ such that $x * y = \left | x + y \right|$ and we need to find the identity element.

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

$x * e = \left|x + e \right| = x$

so $x = x + e$ or $x = - (x + e)$

$e = 0$ or $e = 2x$

Now $x * e = x * 0 = \left| x + 0 \right| = \left| x \right| = x$

But $x * e = x * 2x = \left| x + 2x \right | = \left| 3x \right| \neq x$

So can I say that $e = 0$ 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 $e = 0$ is an identity element for this operation?
Aug 24th 2012, 02:41 PM
x3bnm

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

Sorry I got the solution.

You see, if $x = -1$ and $e = 0$ then $x * e = \left | -1 + 0 \right | = 1 \neq -1$

So no $e = 0$ is not an identity element for the operation $*$ on the set $\mathbb{R}$ in this case.
Aug 24th 2012, 02:47 PM
HallsofIvy

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.
Aug 24th 2012, 03:17 PM
x3bnm

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.