Should the identity of a binary operation be unique?

Question

I was trying to work out the identity of a binary operation and I found that there were infinitely many possible identities for a certain value. My confusion is whether this is a valid identity or should there be a unique identity element?

Answer 1

It's unique. If $e$ and $e'$ are two identities ($\therefore$ left and right identities), then:

• $e \star e' = e$ since $e'$ is a right identity.

• $e\star e' = e'$ since $e$ is a left identity.

So $e = e'$.