Proof of No Right Identity for Operation with Two Left Identities

[Punkyc7]

If an operation has two left identities, show that it has no right identity.
_{}
pf/
Let e_{1} and e_{2} be left identities such that e_{1}≠e_{2}. Assume there exist a right identity and call it r.

Then we have that
e_{1}x=x
e_{2}x=x and
xr=x.


From here I want to try and show that there can not be a right identity but I don't see where to go.

 

[gb7nash]

If an operation has two left identities, show that it has no right identity.
_{}
pf/
Let e_{1} and e_{2} be left identities such that e_{1}≠e_{2}. Assume there exist a right identity and call it r.

This is fine so far.

Try evaluating e₁r. What two pieces of information can you conclude? Similarly...

 

[Punkyc7]

wouldnt I get
e_{1}r=e_{1}=r

and

e_{2}r=e_{2}=r

So we get e_{2} and e_{1} are equal contradicting that they were distinct.
Is that right?

 

[gb7nash]

Correct.

 

[Punkyc7]

thanks, I was trying to figure it out with the x's and I couldn't come to any contradiction