However, only assuming the existence of a left identity and a right inverse (or vice versa) is not sufficient to define a group. For example, consider the set G = { e , f } with the operator ⋅ ⋅ satisfying e ⋅ ⋅ e = f ⋅ ⋅ e = e and ⁠ e ⋅ ⋅ f = f ⋅ ⋅ f = f ⁠. This structure does have a left identity (namely, ⁠ e ⁠), and each element has a right inverse (which is e for both elements). Furthermore, this operation is associative (since the product of any number of elements is always equal to the rightmost element in that product, regardless of the order in which these operations are done). However, ( G , ⋅ ⋅ ) is not a group, since it lacks a right identity.