Abstract - one sided identity question

[kathrynag]

Homework Statement

I'm trying to find what a a left and right identity element is.
Also, I want to see if a one sided element for * exists, if it is unique.

Homework Equations

The Attempt at a Solution

Ok, I just don't really know what a one sided element is.
I'm using e*s=s*e=s, but is e*s the left identity?
I think my problem with finding uniqueness is getting started and the fact that I don't really understand what a one sided identity element is?

 

[CompuChip]

e is a left identity, if e * s = s for all s.
It is a right identity if s * e = s for all s.
Usually, when speaking about "an identity element" we mean that it's both left and right-handed (e * s = s * e = e), although it can be shown that it suffices to demand only a left-handed identity (for example) in the group axioms, which will then automatically also be a right-handed identity element.

For the uniqueness: suppose you have two of them, e and e'.
Then e * s = s = e' * s.

 

[kathrynag]

ok, so for uniqueness, I use e*s=e'*s and show that e=e' for uniqueness?

 

[CompuChip]

Yep.
That's actually a very standard way of proving uniqueness (assume that there are two things with the defining property and show that they must be the same).

 

[kathrynag]

Thanks, that makes sense!