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...)