# Identity Element - Properties

Properties

As the last example shows, it is possible for (S, ∗) to have several left identities. In fact, every element can be a left identity. Similarly, there can be several right identities. But if there is both a right identity and a left identity, then they are equal and there is just a single two-sided identity. To see this, note that if l is a left identity and r is a right identity then l = lr = r. In particular, there can never be more than one two-sided identity. If there were two, e and f, then ef would have to be equal to both e and f.

It is also quite possible for (S, ∗) to have no identity element. The most common example of this is the cross product of vectors. The absence of an identity element is related to the fact that the direction of any nonzero cross product is always orthogonal to any element multiplied – so that it is not possible to obtain a non-zero vector in the same direction as the original. Another example would be the additive semigroup of positive natural numbers.