Just make some up!

For instance, let and * be such that

.

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

You should also notice that if 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...)