Identity element

Let be a set  equipped with a binary operation ∗. Then an element  of  is called a if for all  in , and a if for all  in . If is both a left identity and a right identity, then it is called a ', or simply an '. An identity with respect to addition is called an Additive identity| (often denoted as 0) and an identity with respect to multiplication is called a '''' (often denoted as 1). This should not be confused with a unit in ring theory, which is any element having a multiplicative inverse. By its own definition, unity itself is necessarily a unit. The symbol was chosen because it is the first letter in Einheit'', the German for "unit" or "unity." ==Examples==

In the example S = {e,f} with the equalities given, S is a semigroup. It demonstrates the possibility for to have several left identities. In fact, every element can be a left identity. In a similar manner, there can be several right identities. But if there is both a right identity and a left identity, then they must be equal, resulting in a single two-sided identity. To see this, note that if is a left identity and is a right identity, then . In particular, there can never be more than one two-sided identity: if there were two, say and , then would have to be equal to both and . It is also quite possible for to have no identity element, such as the case of even integers under the multiplication operation. Another common example is the cross product of vectors, where 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. That is, it is not possible to obtain a non-zero vector in the same direction as the original. Yet another example of structure without identity element involves the additive semigroup of positive natural numbers. ==See also==