# Thread: left identity of a set

1. ## left identity of a set

Hi,
I am looking for at least three examples of a set S and and operation * such that there is a left identity and no right identity.

2. Originally Posted by htani
Hi,
I am looking for at least three examples of a set S and and operation * such that there is a left identity and no right identity.
Just make some up!

For instance, let $S={a, b}$ and * be such that

$a*b = b$

$b*a = a$

$a^2 = a$

$b^2 = b$.

This is the right-zero semigroup of order 2. Note that every element is a left identity.

You should also notice that if $S$ has a right and a left identity then they must be the same element, a `common or garden' identity. So, each of your examples must have a left identity which is not a right identity, which is much easier to verify!

(This post would have been better suited to the algebra forum...)