<P> As the last example (a semigroup) 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 = l ∗ r = r . In particular, there can never be more than one two - sided identity . If there were two, e and f, then e ∗ f would have to be equal to both e and f . </P> <P> It is also quite possible for (S, ∗) to have no identity element . A common example of this is the cross product of vectors; in this case, 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 . </P>

0 is the identity element for addition with integers